1.1 Installing APL

APL is built and installed like this:

• download the tar file containing GNU APL, from ftp://ftp.gnu.org/gnu/apl or from one of its mirrors (see http://www.gnu.org/prep/ftp.html), and un-tar the file.
• read (and follow) the instructions given in file README in the unpacked directory.

For the experienced, but impatient reader: it is the well-known sequence

$ ./configure
$ make
$ sudo make install

On success, an executable file named ’apl’ will have been produced in the sub-directory ’src’.

File README contains further instructions about the installation GNU APL.

1.2 Starting APL

Last things first: before explaining how to start APL, it is important to remember how to stop (i.e. exit) it. Neither ^C nor ^D will stop APL - these serve other purposes. Instead, you leave APL with the command )OFF or )off (on a separate line) like this:

      )OFF

If APL is computing a function (and possibly caught in an endless loop), then you may have to press ^C (this is called ATTENTION in APL) to return to APL’s command mode so that the )OFF command can be entered. In some circumstances it may be necessary to press ^C twice within a short time interval (this is called INTERRUPT in APL).

Using the )OFF command will not automatically save your work, however, a history of your GNU APL commands will have been preserved. Having that said, APL is started like every other program - by entering its name and optional command line parameters, for example:

$ apl

or:

$ apl -id 1001

GNU APL is script-able; a text file whose first line looks like this (assuming the APL interpreter binary is called ’apl’ and is located in the current directory):

#! ./apl 

or (if the APL interpreter binary is not in the current directory but in /mypath/apl):

#! /mypath/apl

The path to the ’apl’ binary can be missing (like in the first example), relative, or absolute (second example). The exact details of how the first line of a GNU APL script shall look vary slightly among operating systems. Please consult the info (or man) pages for ’path_resolution’ and/or ’execve’ on your operating system. After the name of the binary, optional GNU APL command line options, usually –script, can be provided.

The text file must, of course, have execute permission, and should be ASCII or, more likely, UTF-8 encoded. The APL characters in the script shall be those defined in the Unicode character set (most of them in the U+2200 - U+23FF range).

1.3 Command Line Options

GNU APL understands the following command line options:

• -C new_root

  perform chroot("new_root") followed by chdir("/"). This restricts the access of the process running GNU APL to files in or below directory new_root, and it also changes the current directory (which could have resided above new_root before the chroot() was executed).

  The -C option is intended to be a security feature for GNU APL interpreters that are facing hostile environments such as the public Internet.

  NOTES:

  • For -C to work, new_root needs to contain a minimal set of binaries, in particular a shell, and possibly libraries needed by the shell. Consult ’info chroot invocation’ for issues to consider and common pitfalls.
  • GNU APL processes the -C option (i.e. it calls chroot("new_root")) before all other command line options. As a consequence, file names in other command line options of GNU APL are also affected by -C. That means that the file names in command line options are being interpreted relative to new_root and not relative to the current directory of the process that had started GNU APL.
  • GNU APL may automatically start other processes such as APserver and friends after processing the -C option. These processes (and the libraries that they depend on) should reside in the proper directory below new_root. For example, with the default configuration of GNU APL, GNU APL installs itself in /usr/local/bin and also expects APserver in the same directory. After -C new_root, however, /usr/local/bin is no longer accessible and one would need to copy /usr/local/bin/APserver to new_root/usr/local/bin/APserver
  • Depending on your platform, the process using -C may need root privileges.
• --cfg

  show ./configure options that were used to configure GNU APL, and exit.

• –[no]Color

  start with ]COLOR ON [OFF].

• –CPU_limit_secs sec

  set CPU time limit to sec seconds.

• -d

  run the APL interpreter (or APL script) in the background (i.e. as a daemon). For this to work you need to provide some input to the background process, e.g. via the -f option.

• --emacs

  run in (old) Emacs mode.

• --emacs_arg arg

  run in (new) Emacs mode with argument arg.

• --eval line

  evaluate one APL line and exit (-s without -f is implied). This option can be given several times; in that case several lines will be executed in command line order before GNU apl exits. Keep in mind that command line options are normally processed by the shell before being passed to apl. Therefore quoting the argument of –eval may often be necessary and it is common wisdom to always quote the –eval argument.

• -f file

  read input from file rather than from the keyboard. When the end of the file is reached, input is switched back to the keyboard. If you want to terminate the APL interpreter after executing the file, then use )OFF as last line in the file or include the –OFF option.

• --gpl

  show GNU APL license (GPL) and exit.

• -L wsname

  )LOAD wsname on start-up.

• --LX expr

  execute expr first. The workspace behaves as if ⎕LX (latent expression) were set to expr in the workspace. This can be used, for example, to start the same workspace with different start-up values.

• -h, --help

  print all command line options with a brief hint about what they do.

• --id proc

  use processor ID proc for this interpreter. If no ID is provided, then the first unused ID > 1000 is taken by this interpreter and the ID becomes used as long as the interpreter runs. Processor IDs are used by shared variables to identify share partners.

• -l num

  turn logging facility num ON (provided that dynamic logging was ./configure’d). The logging facility 37 (start-up messages) is of particular importance for troubleshooting and it works even if dynamic logging was not ./configure’d.

• --mem [memory-limit]

  tell the interpreter not to use more than memory-limit bytes of RAM. By using this option, the user is fully responsible for ensuring that the specified amount of memory will always be available. The following rules should be observed.

  The interpreter will exit at start-up if -mem is used and:

  • the platform on which the interpreter runs has no /proc/meminfo, or
  • the platform has no /proc/sys/vm/overcommit_memory, or
  • /proc/sys/vm/overcommit_memory is not 2 (aka. ’never overcommit’)

  On GNU/Linux systems these conditions are normally satisfied, but the root user has to set /proc/sys/vm/overcommit_memory to 2 which differs from the default value 0 (aka. overcommit allowed).

  If no memory-limit is given, then a memory-limit of 50% is used.

  If a memory-limit is provided then it must have a unit of %, kB, MB, or GB. If the unit is % then the limit is computed as that percentage (between 5% and 95%) of parameter ’MemFree:’ in /proc/meminfo. Otherwise the limit is the given amount in kB, MB, or GB.

  For example:

  • --mem (50% of MemFree: in /proc/meminfo are guaranteed)
  • --mem 80% (80% of MemFree: in /proc/meminfo are guaranteed)
  • --mem 5G (5 GB are guaranteed)

  WARNING: The memory-limit is checked against parameter ’MemFree’ in /proc/meminfo when GNU APL starts, but this does not protect against other processes consuming the available free memory sometime later.

  If that happens (and according to the rules above the user has the responsibility to prevent it), then ⎕WA becomes unreliable and the interpreter may crash badly (i.e. without a WS FULL error, and without any chance to )SAVE the workspace) when the available memory is exhausted.

• --echoCIN copy the input line (after editing) to stdout. For creating session logs.
• --noCIN

  do not echo stdin to stdout. Almost a must for scripting (unless you intend to troubleshoot a script).

• --to_COUT

  normally GNU APL writes its output to stderr (i.e. file descriptor 2) so that, when GNU APL is started in a script, the output of the script appears on stdout (i.e. file descriptor 1) while the output of GNU APL appears on stderr.

  This option redirects the stderr output of GNU APL to stdout. The same effect can be achieved with the option "OUTPUT-TO-COUT Yes" in a preferences file.

• --tcp_port PORT

  this option starts GNU APL as a server that listens on TCP port PORT. Every TCP connection accepted by the server forks a new GNU APL instance which has its stdin, stdout, and stderr redirected to the TCP connection.

  WARNING: This option is dangerous if PORT is directly exposed to the Internet!

• --noCONT

  do not load a SETUP or CONTINUE workspace on start-up.

• --OFF

  This option causes GNU APL to perform an automatic )OFF command after the last line of the last input file (as per -f option) was executed.

• --PW COLS

  set the initial value of ⎕PW to COLS (min. 30, max. 10000)

• --[no]SV

  do [not] start APserver (a shared variable server) on start-up. This disables communication with other workspaces or auxiliary processors through shared variables.

• -p N

  use profile number N in preferences files. A preference file may contain several sets of settings for different purposes; the profile number selects one of these sets.

• --par pproc

  use processor parent ID pproc (default: no parent ID).

• --rawCIN

  do not emit ESC sequences. Normally ESC sequences are emitted for colored output and during line editing. In scripts, however, ESC sequences usually are not wanted and can be turned off with this option.

• -s, --script

  this option is an abbreviation for: --silent --noCIN --noCONT -f - (which is a typical combination of options for APL scripts).

• -q, --silent

  suppress printing of the GNU APL welcome message. Useful for scripts.

• --safe

  disable shared variables and native functions

• --show_bin_dir

  display the binary directory where, according to ./configure, the programs apl, APserver, AP100, and AP210 are installed. Then exit.

• --show_doc_dir

  display the directory where, according to ./configure, documentation files for GNU APL are installed. Then exit.

• --show_etc_dir

  display the system configuration directory where, according to ./configure, the preferences file for GNU APL is installed. Then exit.

• --show_lib_dir

  display the library directory where, according to ./configure, shared library files and the workspaces shipped with GNU APL are installed. Then exit.

• --show_src_dir

  display the source directory where, according to ./configure, GNU APL was compiled. Then exit. This can be used, for example, by native functions that are built outside the GNU APL source tree to find GNU APL header files that are needed to compile the native function.

• --show_all_dirs

  display all the directories above. Then exit.

• -T testcases ... run testcases. Testcases are text files that contain both input to the APL interpreter and the expected response from the interpreter. The output from the interpreter is compared with the expected output in the testcase file(s) and differences are marked. In addition a summary file is created that tells whether or not each of the testcases was successful.
• --TM mode

  test mode. This option specifies how the interpreter shall behave when running a number of testcases (as specified with the -T option).

  --TM 0 (default) run all testcases and exit after the last testcase.

  --TM 1 like --TM 0 if no error was detected. However, if one of the testcases has failed, then the interpreter does not exit so that the user can investigate the state of APL (SI, variable values, etc.).

  --TM 2 like --TM 1, but stay in the interpreter even if all testcases have passed. This can be useful for quickly bringing the interpreter into a specific state and continue manual troubleshooting from that state.

  --TM 3 like --TM 1, but stop testcase execution after the first failed testcase (i.e. do not exit).

  --TM 4 like --TM 3, but exit after the first failed testcase. This is useful for automatic regression tests, where no errors are expected.

• --TR

  executes test case files in random order.

• --TS

  Normally, when the interpreter is run with the T option, an existing summary.log file is overwritten without notice. This option causes new test results to be appended to a possibly existing summary.log instead of overwriting it.

• -v, --version,

  show version information and exit.

• -u UID

  run as user with UID 0. This option can only be used by the root user (who then wants to run as a different user).

• -w milli

  wait milli milliseconds at start-up. Useful to give other programs that are started together with this interpreter time to initialize themselves.

• | +APPOPT|
• | ++APPOPT ARG1|
• |+++APPOPT ARG1 ARG2|
• ...

  Those command line options above that start with + are understood by the GNU APL binary and their arguments must follow the description given for them. In addition the interpreter also accepts command line options that start with +. However, these options are not checked by the interpreter in any way, but are merely copied to ⎕ARG (see below). In these options, APPOPT, ARG1, ARG2, ... are arbitrary strings that should not contain any whitespace characters. The purpose of these options is to control aspects of the APL application from the command line.

• -

  end of command line options for the interpreter. GNU APL provides the system variable ⎕ARG that returns all command line options with which the GNU APL interpreter was invoked (similar to variable argv in main(int argc, char * argv[]) in C/C++). Option - can be used to separate command line options for the APL interpreter from command line options understood by APL applications.

  Thus,

  (⎕ARG ⍳ ⊂'--') ↑ ⎕ARG returns the options for the APL interpreter, while

  (⎕ARG ⍳ ⊂'--') ↓ ⎕ARG returns the options for the APL application.

  All command line options after - are ignored by the interpreter (except for including them in ⎕ARG).

1.4 Configuration File for GNU APL

The default values for some of the command line options discussed in the previous section can be set in a configuration file for GNU APL. The name of the configuration file is ’preferences’ and it should live in one of the following directories:

• in the sub-directory gnu-apl.d of the system configuration directory, or
• in the sub-directory .config/gnu-apl of the user’s home directory (as per $HOME).

The system configuration directory is usually /etc or /usr/local/etc and is configurable via ./configure --sysconfdir. An empty (i.e. most settings commented out) preferences file is also installed in the system configuration directory when GNU APL is installed. You can edit it, use it as a template or read it to see which options can be controlled.

If file ’preferences’ exists in both directories, the settings in $HOME/.config/gnu-apl/preferences override settings in, for example, /etc/gnu-apl.d/preferences.

Command line options in turn override settings in ’preferences’ files.

1.5 File Names and Paths

The GNU APL interpreter is a binary file named apl. It is usually installed in directory /usr/bin/ or in /usr/local/bin/. The location where apl is installed can be changed via ./configure options (see file INSTALL).

GNU APL understands 4 file types:

1. APL workspaces that can be manipulated with the )LOAD, )SAVE, )COPY, and )DROP commands. APL workspaces are XML files and shall have a file extension of .xml to be accepted by GNU APL. APL workspaces can only be exchanged between machines that all run GNU APL.
2. APL exchange files can be manipulated with the )IN and )OUT commands. APL exchange files are text files in ⎕TF format defined by IBM (basically APL expressions that create variables or functions) and shall have a file extension of .atf to be accepted by GNU APL. APL exchange files can be exchanged between machines running APL interpreters from different vendors. The ⎕TF format can be easily emulated on machines that do not not provide compatible )IN and )OUT commands.
3. APL scripts consist of APL commands and APL expressions (including function definition via ∇) like they would be entered by the user. APL scripts should have a file extension of .apl but other extensions are also accepted by GNU APL. APL scripts are, for example, the files expected for the -f command line option. A workspace can be written in this format with the i)DUMP command and can become an active workspace using the )LOAD command.
4. APL testcase files are similar to APL scripts, but in addition to the APL commands and expressions they also contain the expected output from the commands. APL testcase files normally have a file extension of .tc for normal (functional) testcases and .pt for performance testcases. APL testcase files are, for example, the files expected for the -T command line option.

The following APL commands are related to file names:

• )LOAD [lib] name[.xml|.apl]
• )SAVE [lib] [name[.xml]]
• )COPY [lib] [name[.xml|.apl]]
• )PCOPY [lib] [name[.xml|.apl]]
• )DROP [lib] [name[.xml|.apl]]
• )IN [lib] name[.atf]
• )PIN [lib] name[.atf]
• )OUT [lib] name[.atf]
• )DUMP [lib] [name[.apl]]
• )LIB [lib]
• )LIBS [new-lib-root]
• )WSID [name]

The rules for how file names are constructed from the argument(s) of an APL command above are:

1. command arguments shown in brackets are optional.
2. lib is a number from 0 to 9. If lib is not present then 0 is taken as default.
3. if the name is optional and missing then the workspace ID (the name set with the )WSID command) is used.
4. if the file extension (i.e. .apl, .xml or .atf) is missing then it is appended automatically to name.

If the name starts with ’/’ then it is taken as an absolute path to the file (an absolute file name) and no further computation is done with the name.

Otherwise name is a relative path which is relative to some directory library-root and a sub-directory of library-root that is determined by the lib number. The library numbers 0-9 correspond to the following directories:

0: library-root/workspaces/

1: library-root/wslib1/

2: library-root/wslib2/

...

9: library-root/wslib9/

The command )LIBS without arguments shows the mapping between library numbers and paths. The command )LIBS with an argument sets a new lib-root.

The command )LIB [lib] shows the files in library (i.e. directory) lib.

The directory library-root is computed as follows when the interpreter starts:

If environment variable APL_LIB_ROOT is defined, then its value is used as the library-root. Otherwise the path from the current directory (".") up to the root directory ("/") is searched until a directory containing two sub-directories workspaces and wslib1 is found. Normally workspaces and wslib1 are directories, but for the computation of library-root files suffice.

If such a directory is found, then it is used as library-root; otherwise the current directory (i.e. ".") is used and converted to an absolute path.

For example, if library-root is "." then the command

)LOAD 2 test

will try to load the workspace file

./wslib2/test.xml

Using a library root implies that all 10 library directories are contained in the same directory. This is good enough for single-user environments but is often not adequate for multi-user environments where some directories are not writable by users and different users have different home directories.

For that reason the above library root scheme can be overridden by the GNU APL configuration files (named preferences). In these files you can un-comment any of the LIBREF-0 to LIBREF-9 settings (which correspond to library numbers 0 to 9) and provide your own paths. The library numbers that are NOT overridden in a preferences file still follow the library root scheme.

2.1 APL Scripting

As already mentioned, it is possible to write APL scripts. Similar to other nscript languages, an APL script is a text file whose first line is a so-called "shebang line", i.e. a line starting with #!, followed by the absolute path to the interpreter (in our case the GNU APL binary), followed by command line arguments that are passed on to the interpreter. In our case the shebang line could be, for example:

#! /usr/local/bin/apl --id 1010

There are essentially two ways to run an APL script: redirecting the script file to stdin of the interpreter or making the script executable and indicate apl as the script interpreter (followed by some command line arguments for apl).

• Redirect the script file to the stdin of the GNU APL interpreter
• Make the script file executable
• How command line arguments are handled
• Helpful Features for Scripting
• Double-quoted Strings
• Multi-Line Strings
• Multi-Line Literals
• Automatic )MORE
• Script Example

$ apl < SCRIPT.apl

$ apl -f SCRIPT.apl
$ apl -s SCRIPT.apl

    #! /usr/local/bin/apl --script

$ apl --silent -l 37 -- app1 app2
      ⊃⎕ARG
apl     
--silent
-l      
37      
--      
app1    
app2    

#! /usr/local/bin/apl --id 1010 --script

      ⊃⎕ARG     ⍝ show command line options
      )OFF      ⍝ leave the interpreter

$ ./SCRIPT.apl sarg1 sarg2

   ⊃⎕ARG
/usr/local/bin/apl
--script          
./SCRIPT.apl      
sarg1             
sarg2 

$ /usr/local/bin/apl
$ /usr/local/bin/apl --
$ /usr/local/bin/apl -s
$ /usr/local/bin/apl --script
$ /usr/local/bin/apl -s --
$ /usr/local/bin/apl --script --

    /usr/local/bin/apl --id 1010

      ⎕ENV ''

      BODY ← ⊂ 'First line'
      BODY ← BODY , ⊂ 'Second line'
      BODY ← BODY , ⊂ 'Third line'
   ...

      BODY←⎕INP 'END-OF-⎕INP'
First line
Second line
Third line
   ...
END-OF-⎕INP

      Z←'<?apl' '?>' ⎕INP 'END-OF-⎕INP'
First line
Time is now: <?apl ⍕⎕TS ?>
Third line
...
END-OF-⎕INP

      ⊃Z
First line                       
Time is now: 2022 8 4 15 15 2 177
Third line                       
...         

echo "Line1
   Line2"

Line1
   Line2

No string end found+

∇Z←FOO
 Z←"ABC
 DEF
 GHIJK"
∇

4 ⎕CR FOO
┏→━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━━┓ ┏→━━━━━┓┃
┃┃ABC┃ ┃ DEF┃ ┃ GHIJK┃┃
┃┗━━━┛ ┗━━━━┛ ┗━━━━━━┛┃
┗∊━━━━━━━━━━━━━━━━━━━━┛

∇Z←FOO
 Z←"
ABC
 DEF

  GHIJK"
∇

4 ⎕CR FOO
┏→━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏⊖┓ ┏→━━┓ ┏→━━━┓ ┏→━━━━━━┓┃
┃┃ ┃ ┃ABC┃ ┃ DEF┃ ┃  GHIJK┃┃
┃┗━┛ ┗━━━┛ ┗━━━━┛ ┗━━━━━━━┛┃
┗∊━━━━━━━━━━━━━━━━━━━━━━━━━┛

 4 ⎕CR """
ABC
 DEF

  GHIJK
       """
┏→━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━━┓ ┏⊖┓ ┏→━━━━━━┓┃
┃┃ABC┃ ┃ DEF┃ ┃ ┃ ┃  GHIJK┃┃
┃┗━━━┛ ┗━━━━┛ ┗━┛ ┗━━━━━━━┛┃
┗∊━━━━━━━━━━━━━━━━━━━━━━━━━┛

      «Hello world»
Hello world

      8 ⎕CR «««
→      Line 1
→        Line 2
→      »»»
┌→──────────────────┐
│┌→─────┐ ┌→───────┐│
││Line 1│ │  Line 2││
│└──────┘ └────────┘│
└ϵ──────────────────┘

      ⍝ provoke an error...
      ⍝
      8 ⎕CR «««
→      Line 1
→         Line 2
→            «««

*** WARNING: unexpected triplet ««« in mult-line string.
    Expected triplet: »»» (literal depth: 0).
    The triplet was added to the multi-line string.

→            »»»
┌→───────────────────────────────┐
│┌→─────┐ ┌→────────┐ ┌→────────┐│
││Line 1│ │   Line 2│ │      «««││
│└──────┘ └─────────┘ └─────────┘│
└ϵ───────────────────────────────┘

set matchpairs+=«:»

      ⊣««« ⍝⍝⍝
       Multi-line
       Comment
       ...
       »»» ⍝⍝⍝

                                         ┌─────────┐          
                       ┌─────────┐       │  MULTI  │    (text,
      (text,           │ LITERAL │       │  -LINE  │     multi line)
       single line)    └────┬────┘       │ LITERAL │
                            │            └────┬────┘
                            │                 │
                            │     input       │
                            ↓     rules       ↓
                            │                 │
                         ╔══╧══╗         ╔════╧════╗
      (vectors,          ║ APL ║         ║ GNU APL ║
       values)           ╚══╤══╝         ╚════╤════╝
                            │                 │
                            │     output      │
                            ↓     rules       ↓
                            │                 │
                         ┌──┴──┐           ┌──┴──┐
      (text,             │     │           │     │    (text,
       multi-line)       │ OUT │           │ OUT │     milti-line)
                         │     │           │     │
                         └─────┘           └─────┘

      ⍝ two-dimensional multi-line literal
      ⍝
      ⎕ ← <<<
          1 2 3
          4 5 6
          7 8 9
          >>>
1 2 3
4 5 6
7 8 9

      ⍝ three-dimensional multi-line literal
      ⍝
      ⎕ ← <<<
          1 2 3
          4 5 6
          7 8 9

          11 12 13
          14 15 16
          17 18 19
          >>>
 1  2  3
 4  5  6
 7  8  9

11 12 13
14 15 16
17 18 19

      ⍝ two-dimensional multi-line literal with comment
      ⍝
      ⎕ ← <<<   ⍝ same as 3 3⍴⍳9
          1 2 3
          4 5 6
          7 8 9
          >>>
1 2 3
4 5 6
7 8 9

      8 ⎕CR <<<   ⍝ two-dimansional with nested 'abc'
            1 2     3
            4 'abc' 6
            7 8     9
            >>>
┌→────────┐
↓1     2 3│
│4 ┌→──┐ 6│
│  │abc│  │
│  └───┘  │
│7     8 9│
└ϵ────────┘

      8 ⎕CR <<<   ⍝ two-dimansional with nested 10 11 12
            1 2          3
            4 (10 11 12) 6
            7 8          9
            >>>
┌→─────────────┐
↓1          2 3│
│4 ┌→───────┐ 6│
│  │10 11 12│  │
│  └────────┘  │
│7          8 9│
└ϵ─────────────┘

      8 ⎕CR <<<   ⍝ two-dimansional with sub-literal:
            1 2     3
            4 <<< ⍝ sub-literal
              10 11 12
              13 14 15
              16 17 18
              >>>   6
            7 8     9
            >>>
┌→─────────────┐
↓1          2 3│
│              │
│4 ┌→───────┐ 6│
│  ↓10 11 12│  │
│  │13 14 15│  │
│  │16 17 18│  │
│  └────────┘  │
│              │
│7          8 9│
└ϵ─────────────┘

      8 ⎕CR <<<   ⍝ two-dimansional with sub-string:
            1 2     3
            4 «««
              Hello,
              World
              »»»   6
            7 8     9
            >>>
┌→─────────────────────────────────────────────┐
↓1                                          2 3│
│4 ┌→───────────────────────────────────────┐ 6│
│  │┌→─────────────────┐ ┌→────────────────┐│  │
│  ││            Hello,│ │            World││  │
│  │└──────────────────┘ └─────────────────┘│  │
│  └ϵ───────────────────────────────────────┘  │
│7                                          8 9│
└ϵϵ────────────────────────────────────────────┘

      "ABC"[4]
INDEX ERROR+
      'ABC'[4]
      ^    ^
      )MORE
min index=⎕IO (=1), offending index=4, max index=⎕IO+2 (=3)
      ◊ ⍝ clears )MORE

      )MORE
NO )MORE ERROR INFO

      )MORE AUTO ON
Automatic )MORE is now: ON
      
      "ABC"[4]
min index=⎕IO (=1), offending index=4, max index=⎕IO+2 (=3)
INDEX ERROR+
      'ABC'[4]
      ^    ^

#! /usr/local/bin/apl --script

⊃⎕ARG   ⍝ show command line options
)OFF    ⍝ leave the interpreter

 /usr/local/bin/apl --silent <  ../workspaces/SCRIPT.apl

 or

 /usr/local/bin/apl --silent -f ../workspaces/SCRIPT.apl

      #! /usr/local/bin/apl --script
      
      ⊃⎕ARG     ⍝ show command line options
      )OFF      ⍝ leave the interpreter

../workspaces/SCRIPT.apl

 /usr/local/bin/apl --script ../workspaces/SCRIPT.apl 

../workspaces/SCRIPT.apl SCRIPTARG

 /usr/local/bin/apl --script ../workspaces/SCRIPT.apl SCRIPTARG 

      ∇Z←FOO B
[1] ∇
      
      ∇Z←FOO B
DEFN ERROR+
      ∇Z←FOO B
             ^
      )MORE
attempt to ∇-open existing function with new function header

2.2 Axis argument in defined functions

Defined functions and operators (including lambdas) accept an axis argument. For example:

∇Z←Average[X] B
 Z←(+/[X]B) ÷ (⍴B)[X]
∇

      Average[1] 5 5⍴⍳25
11 12 13 14 15

      Average[2] 5 5⍴⍳25
3 8 13 18 23

Syntactically, the axis is used in the same way as for primitive functions and operators.

There are no constraints on the axis such as being integers. Therefore you can use an axis as a third function argument. Keep in mind, however, that doing so will make your APL code incompatible with other APL interpreters. Use this feature carefully!

2.3 Function Groups

In the old APL1 every function of the interpreter was represented by a single APL character. More precisely, the character represented 1, 2, or 3 subfunctions and the subfunction was selected by the number of arguments provided:

• 1 for monadic functions,
• 2 or monadic functions with axis,
• 2 for dyadic functions (w/o axis), or
• 3 for dyadic functions with axis.

The APL character itself was designed in a way that made it easy to remember which functions it represented.

Unfortunately this 1:1 relationship between APL characters and interpreter functions did not scale very well. As the number of functions increased, a new and, at that time, better scalable concept was introduced: ⎕-functions (and variables). The APL standards define only a small number (less than 20) of them, but different APL vendors were eagerly adding more system functions to represent their proprietary extensions to th APL language. And so did GNU APL. With ⎕-functions the original visual appearance that helped to remember the many functions was replaced by abbreviations. That is, for example, ⎕AV stands for the Atomic Vector, ⎕TS for Time Stamp, and so forth.

As of this writing, the GNU APL interpreter has more than 100 non-standard functions for various purposes. For such a large number the ⎕-function approach is no longer scalable (try to remember 100 abbreviations and think about the readability of the code using them!). Rather than adding a new ⎕-function for every new function, GNU APL tries to keep the number of ⎕-functions as small as resonable and by grouping several interpreter functions into subfunction of a single top-level ⎕-function.

We call such a top-level ⎕-function a function group. The subfunction functions of a function group are somehow related by their purpose. The function groups of GNU APL are:

• ⌹: a collection of matrix factorizations (aka. decompositions)
• ⎕CR: a collection of data conversions (formerly known as monadic Charater Representations
• ⎕FFT: a collection of Fast Fourier Transformations
• ⎕FIO: a collection of File I/O functions
• ⎕MX: a collection of Matrix functions functions
• ⎕RVAL: generation of Random APL Values
• SQL: an interface to SQL databases

The syntax for function groups has evolved in 3 phases as elucidated in the following.

• Phase 1: Numerical Axis Argument
• Phase 2: String Axis Argument
• Phase 3: Member Notation
• Subfunction Lists

      4 ⎕CR B          ⍝ format B in a "boxed" fashion
      18 ⎕CR B         ⍝ convert APL string B to an UTF8-encoded byte vector

      ⎕FIO[3] B        ⍝ fopen file B for reading
      A ⎕FIO[3] B      ⍝ fopen file B with mode A

      ∇Zh ← As FIO∆fopen Bs
      ⍝⍝ fopen(Bs, As) filename Bs
       Zh ← As ⎕FIO[3] Bs
      ∇

      ∇Zh ← FIO∆fopen_ro Bs
      ⍝⍝ fopen(Bs, "r") filename Bs
       Zh ← ⎕FIO[3] Bs
      ∇

      'style_boxed' ⎕CR B        ⍝ format B in a "boxed" fashion
      'string_to_UTF8' ⎕CR B     ⍝ convert APL string B to UTF8

      ⎕FIO['fopen'] B            ⍝ fopen file B for reading
      A ⎕FIO['fopen'] B          ⍝ fopen file B with mode A

      ⎕CR.style_boxed B       ⍝ format B in a "boxed" fashion
      ⎕CR.string_to_UTF8 B    ⍝ convert APL string B to UTF8

      ⎕FIO.fopen B            ⍝ fopen file B for reading
      A ⎕FIO.fopen B          ⍝ fopen file B with mode A

2.4 Colored Output

The APL interpreter gets its input from the standard input (stdin), which is normally connected to the user’s keyboard, but can also be a file if APL scripting, the f option, or the T option is used.

The APL interpreter prints its results on either the standard output (stdout) for normal APL output, or to the error output (stderr) for additional trouble-shooting information.

You can print the 3 channels stdin, stdout, and stderr in different colors by means of the debug command ]XTERM. Command ]XTERM ON enables colored output while ]XTERM OFF disables it (for example to avoid annoying ANSI Escape sequences when forwarding stdout or stderr to a file).

Default colored output assumes a terminal (-emulation) that understands ANSI (or VT100) Escape sequences. The xterm that comes with most recent GNU/Linux distributions is a perfect choice supporting both colors and UTF-8 (Unicode) encoded character I/O.

Non-ANSI terminals, as well as other colors than the default ones, can be configured in the ’preferences’ file. The ’preferences’ file also contains a description of all possible color settings.

2.5 Comparison Rules

Both IBM APL2 and the ISO standard require that the arguments of <, ≤, ≥, and > (but not of = or ≠) are integer or real numbers. As a consequence, the argument(s) of ⍋ or ⍒ (which require comparison) must also be a vector of integer or real numbers.

In contrast, GNU APL also allows the comparison of characters and numbers or the comparison of complex numbers according to the following, more general, rules.

Let A and B be two APL values to be compared. The final result of comparing A and B is the first verdict (i.e. either A < B, or A > B, or A = B) obtained when following the rules below in the indicated order:

1. Comparison by rank: if (⍴⍴A) < (⍴⍴B) then A < B and vice versa.
2. Comparison by shape: if (⍴A) < (⍴B) then A < B and vice versa. The first differing shape item (from the left) decides.
3. Comparison of ravel elements: at this point (⍴A) ≡ (⍴B). If all corresponding ravel elements of A and B are equal (i.e. tolerantly equal within ⎕CT as defined in the ISO standard) then A = B.

   Otherwise let A1 and B1 be the first corresponding ravel elements of A and B with A1 ≠ B1. If A1 < B1 then A < B and vice versa. The comparison A1 < B1 is made according to the following rules 4 - 8 below.

4. Comparison by depth:
   • If A1 and B1 are both nested: the rules 1 - 3 above are (recursively) applied to corresponding ravel elements of ⊃A1 and ⊃B1 until a verdict is obtained.
   • if A1 is simple and B1 is nested then A < B and vice versa.
   • otherwise (i.e. A1 and B1 are both simple): A < B if A1 < B1 according to rules 5 - 8 below and vice versa.
5. Comparison by Unicode: if A1 and B1 are both character values and (⎕UCS A1) < (⎕UCS B1) then A < B and vice versa.
6. Comparison by type: If A1 is a character and B1 is numeric, then A < B and vice versa.
7. Comparison by numeric value: if A1 and B1 are both numeric values then:
   • Comparison by real part: if (9○A1) < (9○B1) then A < B and vice versa.
   • Comparison by imaginary part: otherwise if (11○A1) < (11○B1) then A < B and vice versa.
8. Otherwise: A = B.

Another way of describing the rules above is that the comparison of two values is comprised of sub-comparisons of certain properties of the values in the following order:

• the ranks of the values,
• the shapes of the values,
• the first differing ravel element (in row-major order) of the values,
• the depths of the differing ravel elements,
• the types (character vs. numeric) of the differing ravel elements,
• the Unicodes of the differing ravel elements (if applicable)
• the real parts of numeric values,
• the imaginary parts of numeric values,

Note: Rules 1 and 2 above are only relevant for comparisons made in the context of sorting (i.e. for ⍋ or ⍒). This is because for =, ≠, <, ≤, ≥, or > either a RANK ERROR or a LENGTH ERROR is raised if the ranks or shapes of A and B do not match:

      (9 8) < (1 2 3)
LENGTH ERROR
      9 8<1 2 3
      ^  ^

      ⍋(9 8) (1 2 3)
1 2

The reason for comparing complex numbers first by their real parts and then by their imaginary part and not, for example, first by their magnitude and then by their angle is that the chosen order gives more consistent results when comparing near-complex numbers or their true real companions. For example, a magnitude first comparison of complex numbers would make ¯2 < ¯1 < ¯2J1E¯20 for the near-complex number ¯2J1E¯20.

CAUTION: The comparison of two strings (i.e. nested character vectors) may give unexpected results because shorter strings come before longer strings. For example, ’Zoo’ comes before ’Adam’ even though one might expect the opposite.

      Z[⍋Z ← 'Adam' 'Zoo']
 Zoo Adam 

      Z[⍋Z ← 'Adam' 'Zora']
 Adam Zora

This pitfall can be avoided by enforcing the same length for all strings being compared or sorted. A simple way to achieve that is the use of ⊂[2]⊃ like this (assuming ⎕IO←1):

      Z[⍋Z ← ⊂[2]⊃ 'Adam' 'Zoo']
 Adam Zoo  

      Z[⍋Z ← ⊂[2]⊃ 'Adam' 'Zora']
 Adam Zora

2.6 Complex Numbers

Complex numbers are fully supported.

2.7 Debug Commands

In addition to the classical APL commands like )LOAD or )SAVE, GNU APL has a number of debug commands for debugging purposes. Regular APL commands start with ) and print their output on stdout. Debug commands start with ] and print their output on stderr. Normally you cannot easily distinguish between stdout and stderr, but another GNU APL feature, colored output, uses different colors for stdout and stderr.

Type )HELP or ]HELP in the interpreter for a list of all commands available.

2.8 Direct Functions (Lambdas)

GNU APL supports direct functions (aka. lambdas), but only in a rather limited form.

• Named Lambdas
• Unnamed Lambdas
• Limitations of GNU APL Lambdas

      FUN ← { body_statement } 

NAME ← { statement }

      FUN ← { body_expression }

      )FNS

      SUM ← { ⍺ + ⍵ }

      )FNS

      SUM ← { ⍺ + ⍵ }
      )FNS
SUM

      ∇SUM[⎕]∇
DEFN ERROR+
      ∇SUM[⎕]∇
             ^
      )MORE
function is a lambda

      ⎕CR 'SUM'
λ←⍺ λ1 ⍵
λ← ⍺ + ⍵ 

      SUM ← { ⍺ + ⍵ ;C;D }

      ⎕CR 'SUM'
λ←⍺ λ1 ⍵;C;D 
λ← ⍺ + ⍵     

      { ⍴ , ⍵ } ¨ 'a' 'ab' 'abc'
 1  2  3 

2.9 Commands )COPY_ONCE, )DUMP, and DUMP-HTML

In standard APL, workspaces are processed with the standard commands )LOAD, )COPY, and )SAVE. GNU APL provides additional commands to process workspaces.

• )DUMP Command
• )DUMP-HTML Command
• )COPY_ONCE Command

2.10 ]DOXY Command

A particularly useful debug command is ]DOXY. It dumps the current workspace in brows-able HTML format with listings of defined functions and hyperlinks between them.

]DOXY                   ⍝ write documentation to /tmp/WSNAME/*
]DOXY dest              ⍝ write documentation to dest/WSNAME/*

The starting point for browsing the documentation are the files:

/tmp/WSNAME/index.html         ⍝ for ]DOXY without arguments, or
dest/WSNAME/index.html         ⍝ for e.g. ]DOXY dest

The index.html files above usually correspond to the following URIs in your browser:

file:///tmp/WSNAME or
file:///absolute-path-to-dest/WSNAME respectively.

In the above examples WSNAME is the )WSID of the workspace in which the ]DOXY command was executed.

One can (and should make it a habit to) insert special comments into defined functions which are copied into proper places inside the documentation that is generated by the ]DOXY command. These "doxy" comments begin with ⍝⍝ (as opposed to "normal" APL comments that start with a single ⍝. Doxy comments are typically one-liners that briefly explain what a function is supposed to do.

For example:

∇Z←A SUM B
 ⍝⍝ Return the sum of A and B          ← ]DOXY comment:  (double ⍝)
 ⍝  A: numeric                         ← "normal" APL comments (single ⍝) ...
 ⍝  B: numeric
 Z←A + B
∇

But please note the following: in the original doxygen program (for C, C++, or other languages) source code lines may be a mix of code and comments (including doxygen comments). In the ]DOXY command of GNU APL a line of a defined function shall be either APL code or else a doxygen comment (i.e. starting with ⍝⍝). Doxygen comments indented with blanks are allowed (for the sake of readability).

Since a doxygen comment is also a valid APL comment, it is perfectly legal to put a doxygen comment at the end of a line that starts with APL code. However, such lines will be ignored by the ]DOXY command.

The parser generates no extra code from the doxygen comments, and therefore the use of many doxygen comments causes no performance problem (and neither do regular APL comments).

2.11 ]KEYB Command

In the early days of APL the user was typically faced with three problems: APL keyboards, APL screens, and APL printers.

The advent of Unicode has soved the last two problems rather eleqantly: the Uncode character set contains all APL characters, and screens (i.e CRT tubes in the old days and screen windows these days) and printers display them properly. What remains is the keyboard problem. The keyboard problem has two sub-problems: Keyboard caps that show APL symbols in addition to the normal ASCII characters, and generating proiper Unicodes when the keys are being pressed (typically with the ALT or CTRL keys held down).

GNU APL provides the debug command ]KEYB to help resolving these issues. There are two cases where ]KEYB may be useful:

• One can buy so called ’APL keyboards’ or APL stickers that can be glued on the key caps of a regular keyboard. As a matter of fact, APL keyboards are simply regular keyboards with differen key caps. The keycodes sent by APL keyboards do not differ from the keycodes of regular keyboards. One the other hand, many programmers type blindly; for them the keycaps are mostly irrelevant, and the substantial surcharge for APL keyboards may be unreasonable. A related problem is national keyboards: most APL keyboards have a US layout for their keycaps although a user may prefer a national layout. Most of this can be helped with a keuboard layout that is printed on the screen. A user that types bindly most of the time, but cannot remember the position of all APL symbols (in particular of rarely used ones) may find it useful to display the keyboard layout on the screen.
• The other use case for ]KEYB is troubleshooting the transformation of keys pressed on the keyboard to Unicodes sent to the APL interpreter. This transformation (which is explained in more detail below) is highly conficurable. The good news is that these days any keyboard can be configured to send any Unicode on any key as preferered by the user. The bad news is that this configuration differs a lot between different platforms and can be faily cumbersome. The ]KEYB command is intended to help figuring why pressing a partiicular key does not produce the APL character that it should. It maybe wothwhile to note that even an APL keyboard can (and most likely will) not produce the characters shown on its keypad if the transformation from key presses to Unicdes is misconfigured. In that respect APL keyboards are no better than keybiard stickers: Glueing a ⍴ sticker onto the R does not make the R send the Unicode for ⍴. Likewise, an APL keyboard with ⍴ printed in addition to R does not make that keyboard send the Unicode for ⍴. Both cases only work as expected if the transformation from keys to Unicdes is configured properly.

• The transformation of Keycodes to Unicodes
• xmodmap and friends
• The ]KEYB command

      ]KEYB [mode] [arguments]

      ]KEYB XMOD
US Keyboard Layout.    Source: xmodmap -pke

╔════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦═════════╗
║ ~  ║ !⌶ ║ @⍫ ║ #⍒ ║ $⍋ ║ %⌽ ║ ^⍉ ║ &⊖ ║ *⍟ ║ (⍱ ║ )⍲ ║ _! ║ +⌹ ║         ║
║ `◊ ║ 1¨ ║ 2¯ ║ 3< ║ 4≤ ║ 5= ║ 6≥ ║ 7> ║ 8≠ ║ 9∨ ║ 0∧ ║ -× ║ =÷ ║ BACKSPC ║
╠════╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦══════╣
║       ║ Q  ║ W⍹ ║ E⋸ ║ R  ║ T⍨ ║ Y¥ ║ U  ║ I⍸ ║ O⍥ ║ P⍣ ║ {⍞ ║ }⍬ ║ |⊣   ║
║  TAB  ║ q? ║ w⍵ ║ e∈ ║ r⍴ ║ t∼ ║ y↑ ║ u↓ ║ i⍳ ║ o○ ║ p⋆ ║ [← ║ ]→ ║ \⊢   ║
╠═══════╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩══════╣
║ (CAPS   ║ A⍶ ║ S  ║ D  ║ F  ║ G  ║ H⍙ ║ J⍤ ║ K  ║ L⌷ ║ :≡ ║ "≢ ║         ║
║ LOCK)   ║ a⍺ ║ s⌈ ║ d⌊ ║ f_ ║ g∇ ║ h∆ ║ j∘ ║ kλ ║ l⎕ ║ ;⍎ ║ '⍕ ║ RETURN  ║
╠═════════╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═════════╣
║             ║ Z  ║ Xχ ║ C¢ ║ V  ║ B£ ║ N  ║ Mμ ║ <⍪ ║ >⍙ ║ ?⍠ ║          ║
║  SHIFT      ║ z⊂ ║ x⊃ ║ c∩ ║ v∪ ║ b⊥ ║ n⊤ ║ m∣ ║ ,⍝ ║ .⍀ ║ /⌿ ║  SHIFT   ║
╚═════════════╩════╩════╩════╩════╩════╩════╩════╩════╩════╩════╩══════════╝

      ]KEYB XMOD KEYS
Physical Keyboard:      Source: GNU APL builtin

╔════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦═════════╗
║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc      ║
║ 49 ║ 10 ║ 11 ║ 12 ║ 13 ║ 14 ║ 15 ║ 16 ║ 17 ║ 18 ║ 19 ║ 20 ║ 21 ║ 22      ║
╠════╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦══════╣
║  Kc   ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║  Kc  ║
║  23   ║ 24 ║ 25 ║ 26 ║ 27 ║ 28 ║ 29 ║ 30 ║ 31 ║ 32 ║ 33 ║ 34 ║ 35 ║  51  ║
╠═══════╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩══════╣
║  Kc     ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc      ║
║  66     ║ 38 ║ 39 ║ 40 ║ 41 ║ 42 ║ 43 ║ 44 ║ 45 ║ 46 ║ 47 ║ 48 ║ 36      ║
╠═════════╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═════════╣
║    Kc       ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║   Kc     ║
║    50       ║ 52 ║ 53 ║ 54 ║ 55 ║ 56 ║ 57 ║ 58 ║ 59 ║ 60 ║ 61 ║   62     ║
╠════╦═════╦══╩══╦═╩════╩════╩════╩════╩════╩════╩══╦═╩═══╦╩════╬═════╦════╣
║ Kc ║ Kc  ║ Kc  ║                Kc                ║ Kc  ║ Kc  ║ Kc  ║ Kc ║
║ 37 ║115  ║ 64  ║                65                ║113  ║116  ║ 109 ║105 ║
╚════╩═════╩═════╩══════════════════════════════════╩═════╩═════╩═════╩════╝

Keyboard Layout.    Source: xmodmap -pke

╔════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦═════════╗
║ ~  ║ !⌶ ║ "  ║ #⍒ ║ $⍋ ║ %⌽ ║ ^⍉ ║ &⊖ ║ *⍟ ║ (⍱ ║ )⍲ ║ _! ║ +⌹ ║         ║
║ `◊ ║ 1¨ ║ 2⍫ ║ 3< ║ 4≤ ║ 5= ║ 6≥ ║ 7> ║ 8≠ ║ 9∨ ║ 0∧ ║ -× ║ =÷ ║ BACKSP  ║
╠════╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦══════╣
║       ║ Q  ║ W⍹ ║ E⋸ ║ R  ║ T⍨ ║ Y¥ ║ U  ║ I⍸ ║ O⍥ ║ P⍣ ║ {⍞ ║ }⍬ ║  |⊣  ║
║  TAB  ║ q? ║ w⍵ ║ e∈ ║ r⍴ ║ t∼ ║ y↑ ║ u↓ ║ i⍳ ║ o○ ║ p⋆ ║ [← ║ ]→ ║  \⊢  ║
╠═══════╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩══════╣
║  (CAPS  ║ A⍶ ║ S  ║ D  ║ F  ║ G  ║ H⍙ ║ J⍤ ║ K  ║ L⌷ ║ *≡ ║ "≢ ║         ║
║  LOCK)  ║ a⍺ ║ s⌈ ║ d⌊ ║ f_ ║ g∇ ║ h∆ ║ j∘ ║ kλ ║ l⎕ ║ +⍎ ║ '⍕ ║ RETURN  ║
╠═════════╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═════════╣
║             ║ Z  ║ Xχ ║ C¢ ║ V  ║ B£ ║ N  ║ Mμ ║ <⍪ ║ >⍙ ║ ?⍠ ║          ║
║    SHIFT    ║ z⊂ ║ x⊃ ║ c∩ ║ v∪ ║ b⊥ ║ n⊤ ║ m∣ ║ ,⍝ ║ .⍀ ║ /⌿ ║   SHIFT  ║
╠════╦═════╦══╩══╦═╩════╩════╩════╩════╩════╩════╩══╦═╩═══╦╩════╬═════╦════╣
║    ║     ║     ║                                  ║     ║     ║ !⌶  ║    ║
║CTRL║ Win ║ ALT ║              SPACE               ║ ALT ║ Win ║ 1¨9 ║CTRL║
╚════╩═════╩═════╩══════════════════════════════════╩═════╩═════╩═════╩════╝

      ]KEYB XMOD KEYS FUNK KPAD
US Keyboard Layout.    Source: GNU APL builtin

╔════╗    ╔════╦════╦════╦════╗    ╔════╦════╦════╦════╗    ╔════╦════╦════╦════╗    ╔════╦════╦════╗
║ Kc ║    ║ Kc ║ Kc ║ Kc ║ Kc ║    ║ Kc ║ Kc ║ Kc ║ Kc ║    ║ Kc ║ Kc ║ Kc ║ Kc ║    ║    ║ Kc ║ Kc ║
║ 09 ║    ║ 67 ║ 68 ║ 69 ║ 70 ║    ║ 71 ║ 72 ║ 73 ║ 74 ║    ║ 75 ║ 76 ║ 95 ║ 96 ║    ║    ║ 78 ║ 77 ║
╚════╝    ╚════╩════╩════╩════╝    ╚════╩════╩════╩════╝    ╚════╩════╩════╩════╝    ╚════╩════╩════╝

╔════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦═════════╗    ╔════╦════╦════╦════╗
║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc      ║    ║ Kc ║    ║ Kc ║ Kc ║
║ 49 ║ 10 ║ 11 ║ 12 ║ 13 ║ 14 ║ 15 ║ 16 ║ 17 ║ 18 ║ 19 ║ 20 ║ 21 ║ 22      ║    ║ 77 ║    ║ 82 ║ 82 ║
╠════╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦══════╣    ╠════╬════╬════╬════╣
║  Kc   ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc   ║    ║ Kc ║ Kc ║ Kc ║    ║
║  23   ║ 24 ║ 25 ║ 26 ║ 27 ║ 28 ║ 29 ║ 30 ║ 31 ║ 32 ║ 33 ║ 34 ║ 35 ║ 51   ║    ║ 79 ║ 80 ║ 81 ║ Kc ║
╠═══════╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩══════╣    ╠════╬════╬════╣ 86 ║
║ Kc      ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc      ║    ║ Kc ║ Kc ║ Kc ║    ║
║ 66      ║ 38 ║ 39 ║ 40 ║ 41 ║ 42 ║ 43 ║ 44 ║ 45 ║ 46 ║ 47 ║ 48 ║ 36      ║    ║ 83 ║ 84 ║ 85 ║    ║
╠═════════╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═════════╣    ╠════╬════╬════╬════╣
║  Kc         ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║ Kc ║  Kc      ║    ║ Kc ║ Kc ║ Kc ║    ║
║  50         ║ 52 ║ 53 ║ 54 ║ 55 ║ 56 ║ 57 ║ 58 ║ 59 ║ 60 ║ 61 ║  62      ║    ║ 87 ║ 88 ║ 89 ║ Kc ║
╠════╦═════╦══╩══╦═╩════╩════╩════╩════╩════╩════╩══╦═╩═══╦╩════╬═════╦════╣    ╠════╩════╬════╣108 ║
║ Kc ║ Kc  ║ Kc  ║              Kc                  ║ Kc  ║ Kc  ║ Kc  ║ Kc ║    ║ Kc      ║    ║    ║
║ 37 ║115  ║ 64  ║              65                  ║113  ║116  ║ 109 ║105 ║    ║ 90      ║    ║    ║
╚════╩═════╩═════╩══════════════════════════════════╩═════╩═════╩═════╩════╝    ╚═════════╩════╩════╝

      ⎕UCS "⍴"                    ⍝ Unicode for ⍴
9076
      ,5 ⎕CR 256 256 ⊤ ⎕UCS "⍴"   ⍝ Unicode for ⍴ in hex
2374

      ...
KeyPress event, serial 38, synthetic NO, window 0x3c00001,
    root 0x75c, subw 0x0, time 35629183, (-590,275), root:(284,737),
    state 0x0, keycode 27 (keysym 0x72, r), same_screen YES,
    XLookupString gives 1 bytes: (72) "r"
    XmbLookupString gives 1 bytes: (72) "r"
    XFilterEvent returns: False

keycode  27 =  r               R               U2374

#define NoSymbol         0L  /* special KeySym */
#define XK_Return    0xff0d  /* Return, enter */

2.12 ]NEXTFILE and ]PUSHFILE Commands

• ]NEXTFILE
• ]PUSHFILE

#!/usr/local/bin/apl

⍝ see documentation at the end of this file

<APL CODE...>

]NEXTFILE

This workspace does the following...

2.13 History and TAB completion

Until GNU APL 1.4 / SVN 465, GNU APL used libreadline for interactive user input. libreadline did provide two useful features: tab expansion (the tab key would expand file names) and history (the cursor up/down keys would recall previously entered lines).

Since SVN 465 libreadline was removed and the standard TAB expansion and history of libreadline were replaced by more context sensitive (i.e. APL aware) implementations:

1. Instead of simply recalling the last line entered by the user, there are now different histories for different input contexts:

1a. The input history in immediate execution recalls the last line entered in immediate execution (and not, for example, lines entered in function editing mode or ⍞ input.

1b. Likewise, ⍞ recalls the last line entered for ⍞-input

1c. ⎕ recalls the last line entered for ⎕-input

1d. In the ∇-editor, the other function lines of the function being edited can be recalled. This is far more handy than the ∇-editor commands for recalling function lines (which are not fully supported in GNU APL).

2. Instead of always TAB-completing file names, the tab character now understands different TAB-completion contexts:

2a. Input starting with . or / is completed as a filename like readline did.

2b. Input starting with ) or ] is completed as command-name name or, to some extent, as command arguments.

2c. Input starting with ⎕ is completed as a system function name or a system variable name.

2d. Input starting with letters, ∆, or ⍙ is completed as a user defined function or variable name.

2.14 Logging Facilities

The APL interpreter has over 30 logging facilities. Each logging facility can be ON (and then produces some logging output on stderr) or OFF. The decision which logging facility shall be ON and which shall be OFF can be made at compile time (of the APL interpreter) or at run-time.

If the decision is made at compile time - we call that static logging - then it cannot be changed later on. Otherwise - we call that dynamic logging - there is a debug command ]LOG that allows logging facilities to be turned ON or OFF.

• Static Logging
• Dynamic Logging

2.15 Rational Numbers

GNU APL has limited support for rational numbers. Instead of dividing integers (and possibly causing rounding errors), integer quotients are kept undivided internally until some function requires a conversion to a floating point (double) value.

Currently only +, -, ×, and ÷ preserve rational numbers where possible, but this list may grow in the future. Monadic + (a no-op for non-complex numbers) explicitly converts rational numbers to floating point numbers.

A quotient is internally stored as a 64-bit numerator and a 64-bit denominator. In some cases arithmetic with rational numbers is faster than with doubles, but in most cases it is slower.

For that reason support for rational numbers is disabled by default and must be enabled via ./configure (see README-2-configure).

2.16 Hex Numbers

GNU APL supports sedecimal numbers. They start with $ and can be uppercase or lowercase:

      $2a
42

      $2A
42

2.17 User-defined Commands

There is a simple mechanism to define additional APL commands. This mechanism is intended to introduce new commands by APL libraries. Like system commands, user-define commands can only be executed in immediate execution mode and not from user-defined functions or from ⍎. It is not intended to extend the functionality of user-defined commands beyond what is being described in the following.

A user-defined command ]NEW_COMMAND is created with the debug command ]USERCMD like this:

      ]USERCMD ]NEW_COMMAND APL_FUNCTION [mode]

APL_FUNCTION is an APL function that will be called when the command is entered in immediate execution mode. The entire line entered by the user, starting at ]NEW_COMMAND, is the right argument of APL_FUNCTION. If mode is missing (or 0) then APL_FUNCTION is called monadically. If mode is 1 then APL_FUNCTION is called dyadically; the left argument is a vector of strings that is the left argument broken down into individual argument strings.

The function APL_FUNCTION that implements a command need not exist when the command is created.

A single user-defined command ]UCMD, or all user-defined commands can be deleted like this:

      ]USERCMD REMOVE ]UCMD
      ]USERCMD REMOVE-ALL

2.18 Structured Variables and Associative Arrays

GNU APL has implemented two features that are closely related because, under the hood, they share the same implementation: structured variables and associative arrays.

• Structured Variables
• Associative Arrays

      PERSON.address.street

      PERSON.firstname ← 'Jane' ⍝ create variable PERSON with member 'firstname'
      PERSON.lastname  ← 'Doe'  ⍝ add a second member 'lastname' to PERSON

      PERSON.address.street ← '42 Main Street' ⍝ implicitly creates PERSON.address

      EMPTY ← 38 ⎕CR CAPACITY ← 32

      )ERASE PERSON.address      ⍝ OK, PERSON.address exists
      )ERASE PERSON.address      ⍝ error: PERSON.address does not exist anymore
NOT ERASED: PERSON.address

      PERSON.address.street ←  '42 Main Street'   ⍝ create member address.street
      PERSON.address.street   ⍝ reference member address.street of PERSON
42 Main Street

      PERSON.address.street ←  '44 Main Street'   ⍝ overwrite address.street
      PERSON.address.street
44 Main Street

      PERSON
.firstname: ┌→───┐
            │Jane│
            └────┘

.lastname: ┌→──┐
           │Doe│
           └───┘

.address: 
.address.street: ┌→─────────────┐
                 │44 Main Street│
                 └──────────────┘

      B.b.c←'leaf-Abc'        ⍝ OK, since B.b.c will be a leaf
      B.b←42                  ⍝ will fail since B.b is not a leaf
DOMAIN ERROR+
      B.b←42
       ^ ^
      )MORE
member access: cannot override non-leaf member A.b
)ERASE or ⎕EX that member first.

      )ERASE B.b
      B.b ← 'leaf-Ab'         ⍝ now OK, since B.b will now become a (new) leaf

      )SIC

      )ERASE A
      A.b.c ← 'leaf-Abc'      ⍝ variable A with leaf A.b.c
      C.d.e ← 'leaf-cde'      ⍝ variable C with leaf C.d.e
      A.b.c ← C               ⍝ overwrite leaf A.b.c of A
      A.b.c.d.e
leaf-cde

   A.key ← 42
   A.key
42
   A['key']
42
   A['key']←24
   A.key
24
   A['key' 'key']   ⍝ not allowed even though 'key' is a valid member
DOMAIN ERROR
      D['key' 'key']

      ASSOC ← 38 ⎕CR 8
      KEY←'key.dot'   ⍝ works, but avoid such keys
      ASSOC[KEY]←42
      ASSOC[KEY]
42
      ASSOC[KEY]←43
      ASSOC[KEY]
43
      ASSOC.key.dot   ⍝ won't work: 'key.dot' is a single key, but key.dot is 2 keys
VALUE ERROR+
      ASSOC.key.dot
               ^
      )MORE
member access: structure ASSOC has no member key

      ASSOC.key.dot←44   ⍝ works: 2 (nested) keys
      ASSOC.key.dot
44
      ASSOC['key.dot']   ⍝ works: one key containing '.'
43

      D.b.c←42
      D['b.c']   ⍝ will fail
INDEX ERROR+
      D['b.c']
      ^^
      )MORE
member access: member b.c was not found. The valid members are:
      b

      D['b']['c']   ⍝ will work
42

      ⍝ alternatively: use PICK
      ⍝
      'b' 'c' ⊃ D   ⍝ fails: 'b' 'c' is 'bc'
      "b" "c" ⊃ D   ⍝ works
42

      TMP ← 39 ⎕CR ASSOC_ARRAY   ⍝ save ASSOC_ARRAY as normal APL array
      ⊣ ⎕EX 'ASSOC_ARRAY'        ⍝ erase it so that it can be assigned
      ASSOC_ARRAY ← 38 ⎕CR TMP   ⍝ new associative array with ≥ twice the size

2.19 Monadic ⊢ and ⊣; dyadic ⊢ with Axis

Monadic ⊢ is the identity function. It returns its (committed or non-committed) right argument as a non-committed value.

Conversely, monadic ⊣ (called Hide in GNU APL) discards its (committed or non-committed) right argument and returns a committed integer scalar 0.

For the most part there is no difference between a committed value (= a value that was assigned to a variable, including ⎕ and ⍞) and a non-committed value. The point where it does make a difference is when the value is the final result of a statement (as opposed to an intermediate result inside a statement). In that situation (and only there) a non-committed value is printed but a committed value is not.

You can use ⊢ in a similar fashion as ⎕← at the left end of a statement, to print a value even though it was previously assigned to a variable.

The main motivation for ⊣ is that, at least in GNU APL, lambdas always return a value. However, if a lambda is used only for the sake of its side effects, say to print something, then the value returned by the lambda is often of no interest and only messes up the APL output. In that situation ⊣ can be used to suppress the printing of undesired return values from lambdas.

In earlier GNU APL versions, ⊣B and ⊢B would both return B; with ⊣ as committed value and with ⊢ as non-committed value. But since the only real-life purpose of ⊣ is to suppress the printing of B, the implementation of ⊣ was changed to returning a committed integer scalar 0 instead of committed B. That reduced the run-time of ⊣B from O(,B) to O(1). Also, ⊢B is marginally faster than ⎕←B.

Dyadic ⊢ with axis is a selection function that generalizes ⊣ and ⊢.

Let Z←A ⊢[X] B. Then:

• if X is a one-element vector and ↑X is 0 then Z ≡ A,
• if X is a one-element vector and ↑X is 1 then Z ≡ B,
• otherwise X selects items of A or B if the corresponding elements of X are 0 or 1 respectively. In that case, the shapes of A and B must match the shape of X, but one-element A or B are scalar extended to the shape of X.

Example:

      A←2 3⍴'abcdef'
      B←2 3⍴⍳6
      X←2 3⍴0 1 0 1 0 1
      A ⊢[X] B
a 2 c
4 e 6

   A ⊢[X] '*'
a*c
*e*

  '*' ⊢[X] B
* 2 *
4 * 6

2.20 Bit-wise Logical Functions ⊤∧, ⊤∨, ⊤⍲, ⊤⍱, ⊤≠, and ⊤=

The APL functions And (∧), Or (∨), Nand (⍲), and Nor (⍱) operate primarily on Boolean integers. Primarily means that the LCM variant for ∧ and the GCD variant for ∨ are not considered in this context. (The LCM and GCD variants are defined in the ISO standard and supported in GNU APL but not in IBM APL2).

However, probably more often than not one needs to compute Boolean functions between the bits of arbitrary (non-Boolean) integers and not between entire Boolean integers 0 or 1. Although that is possible to do in standard APL, the procedure is fairly awkward and, more importantly, inefficient:

• convert every integer argument X to a 64-item Boolean vector X←(64⍴2)⊤X,
• call the Boolean function ∧, ∨, ⍲, ⍱, =, or ≠ with the converted arguments, and
• convert the Boolean result vector R back to the integer result Z←2⊥R

Note: for Boolean arguments the APL functions ≠ and = can be used to compute the more customary Boolean functions XOR and XNOR respectively. In this context = and ≠ are treated as Boolean functions even though they accept non-Boolean arguments,

For example, using 5 ⎕CR (4⍴256)⊤X to display X in hex:

      5 ⎕CR (4⍴256)⊤   A←$ABBADEAD
ABBADEAD
      5 ⎕CR (4⍴256)⊤   B←$00FF00FF
00FF00FF

      5 ⎕CR (4⍴256)⊤   2⊥ ((64⍴2)⊤A) ∧ (64⍴2)⊤B
00BA00AD

With the bit-wise And (⊤∧) the same can be achieved in a simpler fashion and far more efficiently:

      ⍝ Traditional AND
      5 ⎕CR (4⍴256)⊤   A←$ABBADEAD
ABBADEAD
      5 ⎕CR (4⍴256)⊤   B←$00FF00FF
00FF00FF

      ⍝ bit-wise AND
      5 ⎕CR (4⍴256)⊤   A ⊤∧ B
00BA00AD

• Dyadic ⊤∧, ⊤∨, ⊤⍲, ⊤⍱, ⊤=, and ⊤≠
• Monadic ⊤∨ and ⊤⍱
• Monadic ⊤∧
• Character Arguments for Monadic ⊤⍱ and Dyadic ⊤∧, ⊤∨, ==, and ≠≠

      A ⊤∧ B    ←→    2⊥ ((64⍴2)⊤A) ∧ (64⍴2)⊤B      ⍝ aka. AND
      A ⊤∨ B    ←→    2⊥ ((64⍴2)⊤A) ∨ (64⍴2)⊤B      ⍝ aka. OR
      A ⊤⍲ B    ←→    2⊥ ((64⍴2)⊤A) ⍲ (64⍴2)⊤B      ⍝ aka. NAND
      A ⊤⍱ B    ←→    2⊥ ((64⍴2)⊤A) ⍱ (64⍴2)⊤B      ⍝ aka. NOR
      A ⊤≠ B    ←→    2⊥ ((64⍴2)⊤A) ≠ (64⍴2)⊤B      ⍝ aka. XOR
      A ⊤= B    ←→    2⊥ ((64⍴2)⊤A) = (64⍴2)⊤B      ⍝ aka. NXOR or XNOR

      ⊤∨ B    ←→    0 ⊤∨ B      ⍝ real B to nearby integer
      ⊤⍱ B    ←→    0 ⊤⍱ B      ⍝ bit-wise Not

      26 ⎕CR    1            ⍝ integer
16
      26 ⎕CR    1.1          ⍝ real
32
      26 ⎕CR    1÷1          ⍝ integer
16
      26 ⎕CR    1.1÷1.1      ⍝ real
32
      26 ⎕CR    ⊤∨ 1.1÷1.1   ⍝ integer
16

      26 ⎕CR    1.1J0÷1.1      ⍝ real
32
      26 ⎕CR    ⊤∨ 1.1J0÷1.1      ⍝ integer
16

      5⎕CR 'A'
41
      5⎕CR ⊤⍱'A'            ⍝ Note that 5⎕CR aka. ⎕CR.to_HEX ANDs with $FF
BE

      256 256 256 256⊤ ⎕UCS ⊤⍱ 'A'      ⍝ ⊤⍱ 'A' is FFFFFFBE
255 255 255 190
      256 256 256 256⊤$FFFFFFBE
255 255 255 190

2.21 ⊤ (Encode) with Axis

The dyadic primitive function A ⊥ B (Decode) computes the values of B in a number system with radices A. The most common case is a homogeneous radix, i.e. all items of A are the same. In that case the radix A can be specified as a scalar A which will be scalar-extended to the length of B (more precisely: to ↑⍴B). For example:

      2 2 2 2 ⊥ 1 1 0 1
13

      2⊥1 1 0 1              ⍝ scalar extension of 2 to length 4
13

      ((⍴B)⍴2) ⊤ B←1 1 0 1   ⍝ scalar extension of 2 to ↑⍴B
13

The dyadic primitive function A ⊤ B (Encode) computes the representation of B in a number system whose radices are A. The most common case is again a homogeneous radix A. In this sense, Encode is the inverse function of Decode. Unfortunately Encode can not scalar extend the radix A. This is because the shapes of its arguments do not provide sufficient information to determine the radix length that shall be used. As a consequence:

• the user needs to provide the radix A as a vector with identical items and with a length that is suitable for the items in B,
• if more radix items than needed are provided, then the result contains leading 0s (which is sort of OK because no information is lost), but
• if less radix items than needed are provided, then an oberflow occurs and the result is truncated. This is awkward because, for example

        2 2 2 2 2 2 2 2 ⊤ 1000     ⍝ radix too short for 1000
  1 1 1 0 1 0 0 0
  
        2 ⊥ 1 1 1 0 1 0 0 0        ⍝ therefore not 1000 !!!
  232
        232 + (2⋆8) + (2⋆9)        ⍝ add missing items
  1000
  

• The IBM APL2 language reference manual proposes to use the APL expression ⌊1∣+A⍟(∣B)+A=0 for the optimal length of the radix A.

We call a radix length ⍴A optimal if it is neither too long nor to short. In that case the result has no leading zeros and B ≡ A⊥A⊤N. For an optimal radix length are A⊤ and A⊥ the inverse functions of each other.

In order to improve on this, GNU APL offers the possibility to automatically (and without using A⍟B) compute the optimal radix length from A and B. The solution implemented in GNU APL is to provide the optimal radix length as an axis argument. There are two different cases for the axis argument:

• If the optimal radix length is known, say N > 0, then the radix length is given as axis argument N>0. In this case the left argument of A⊤B is slightly simplified:

        (N⍴↑A) ⊤ B   ←→   A ⊤[N] B   ⍝ optimal radix length N>0 is known
  

  This case frequently occurs when integer(s) B are encoded into hexadecimal 8-bit (i.e. radix length = 2 hex digits), 16-bit words (i.e. radix length = 4 hex digits), 32-bit longs (i.e. radix length = 8 hex digits), or 64-bit long longs (i.e. radix length = 16 hex digits).

• Otherwise the radix length is not known; this case is specified as axis argument 0. This is also the case where ⌊1∣+A⍟(∣B)+A=0 would be needed, and it simplifies A⊤B considerably:

        ((⌊1∣+A⍟(∣B)+A=0)⍴↑A) ⊤ B   ←→   A ⊤ [0]B   ⍝ unknown radix length
  

Please note the following:

• Obviously the case N>0 is faster than the case N=0 because the user has taken over radix length computation. However, A⊤[N]B is still faster than the classical (N⍴↑A)⊤B.
• If all items in B are non-negative, then the case N=0 uses the optimal radix length for unsigned numbers, i.e. without a sign. The result can be used in what is known as unsigned base-A arithmetic.
• If some item of B is negative, then the case N=0 uses a radix length that includes a sign (aka. two’s complement arithmetic if A=2). The sign is 0 for the non-negative items of B and A-1 for the negative items of B (= 1 for binary encodings).

Examples:

      2⊤[4] 13           ⍝ 4 digit radix, positive B
1 1 0 1

      2⊥1 1 0 1          ⍝ proof
13

      2⊤[5] ¯13          ⍝ 5 digit radix, negative B
1 0 0 1 1

      32 - 2⊥1 0 0 1 1   ⍝ proof
13

2.22 Generalized ⍳

• Generalized monadic ⍳
• Generalized dyadic ⍳

2.23 ⌹[X] - Miscellaneous Matrix and Vector Operations

• ⌹[1] and ⌹[2] - QR Factorizations
• The Impact of ⎕CT for ⌹

      ⊢B←3 3⍴ 1 1 3 2 4 2 4 8 7
1 1 3
2 4 2
4 8 7

      (Q R Ri)←⌹[⎕CT]B

      4⎕CR 4⍕Q
┏→━━━━━━━━━━━━━━━━━━━┓
↓ .2182  .9759  .0000┃
┃ .4364 ¯.0976 ¯.8944┃
┃ .8729 ¯.1952  .4472┃
┗━━━━━━━━━━━━━━━━━━━━┛

      4⎕CR 4⍕Q+.×⍉Q  ⍝ verify that Q is orthogonal
┏→━━━━━━━━━━━━━━━━━━━━┓
↓ 1.0000  .0000  .0000┃
┃  .0000 1.0000  .0000┃
┃  .0000  .0000 1.0000┃
┗━━━━━━━━━━━━━━━━━━━━━┛

      4⎕CR0 4⍕R       ⍝ verify that R is upper triangle
┏→━━━━━━━━━━━━━━━━━━━━┓
↓ 4.5826 8.9469 7.6376┃
┃  .0000 ¯.9759 1.3663┃
┃  .0000  .0000 1.3416┃
┗━━━━━━━━━━━━━━━━━━━━━┛

       Q+.×R         ⍝ verify that B is Q+.×R (i.e. B is (⍉Q)+.×R)
1 1 3
2 4 2
4 8 7

      4⎕CR0 4⍕Ri+.×R  ⍝ verify that Ri is the inverse of R
┏→━━━━━━━━━━━━━━━━━━━━┓
↓ 1.0000  .0000  .0000┃
┃  .0000 1.0000  .0000┃
┃  .0000  .0000 1.0000┃
┗━━━━━━━━━━━━━━━━━━━━━┛

      B←5 4⍴4J6 6J3 5J10 3J2 8J10 3J4 5J10 5J8 3J1 2J3 4J5 1J3 1J4 9J9 9J6 2J7 2J10 7J6 9J8 10J10

      (Q R Ri)←⌹[⎕CT]B

      4⎕CR 4⍕Q∘⍉Q  ⍝ verify that Q is orthogonal
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
↓ 1.0000J.0000  .0000J.0000  .0000J.0000  .0000J.0000  .0000J.0000┃
┃  .0000J.0000 1.0000J.0000  .0000J.0000  .0000J.0000  .0000J.0000┃
┃  .0000J.0000  .0000J.0000 1.0000J.0000  .0000J.0000  .0000J.0000┃
┃  .0000J.0000  .0000J.0000  .0000J.0000 1.0000J.0000  .0000J.0000┃
┃  .0000J.0000  .0000J.0000  .0000J.0000  .0000J.0000 1.0000J.0000┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      4⎕CR 4⍕R       ⍝ verify that R is upper triangle
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
↓ 8.5870J15.2557 10.1035J10.9309 13.5055J18.7452 10.0961J15.2530┃
┃  .0000J.0000    8.7353J5.5589   9.2429J.4120    2.5086J.7539  ┃
┃  .0000J.0000     .0000J.0000   ¯2.4869J4.4115  ¯5.9788J¯6.0801┃
┃  .0000J.0000     .0000J.0000     .0000J.0000    7.4038J¯3.9492┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

       Q∘5 4↑R         ⍝ verify that B is Q+.×R (i.e. B is (⍉Q)+.×R)
4J6  6J3 5J10  3J2
8J10 3J4 5J10  5J8
3J1  2J3 4J5   1J3
1J4  9J9 9J6   2J7
2J10 7J6 9J8  10J10

      B←?5 5⍴100   ⍝ a random matrix B
      (Q T Ti)←⌹[2]B
      (⌹B) = Ti∘⍉Q
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1

      ⍝ monadic, 1 indeterminant
      ⌹.poly_print ⎕←POLY←3 2 1
3 2 1
x² + 2x + 3

      ⍝ monadic, 2 indeterminants
      ⌹.poly_print ⎕←POLY←2 3⍴⍳6
1 2 3
4 5 6
6x²y + 5xy + 4y + 3x² + 2x + 1

      ⍝ monadic, 3 indeterminants
      ⌹.poly_print ⎕←POLY←3 3⍴⍳9
1 2 3
4 5 6
7 8 9
9x²y² + 8xy² + 7y² + 6x²y + 5xy + 4y + 3x² + 2x + 1

      ⍝ dyadic, two 1-character indeterminants X and Y
      'XY' ⌹.poly_print ⎕←POLY←2 3⍴⍳6
1 2 3
4 5 6
6X²Y + 5XY + 4Y + 3X² + 2X + 1

      ⍝ dyadic, two named indeterminants _Foo and _Bar
      '_Foo' '_Bar' ⌹.poly_print ⎕←POLY←2 3⍴⍳6
1 2 3
4 5 6
6_Foo²_Bar + 5_Foo_Bar + 4_Bar + 3_Foo² + 2_Foo + 1

      ⍝ single indeterminant x
      ⍝
      "A:"    (⌹.poly_print A ← 4 0 0 6 3 1)
 A: x⁵ + 3x⁴ + 6x³ + 4 

      "B:"    (⌹.poly_print B ← 5 0 2)
 B: 2x² + 5 

      "PROD:" (⌹.poly_print PROD ← A ⌹.poly_multiply B)
 PROD: 2x⁷ + 6x⁶ + 17x⁵ + 15x⁴ + 30x³ + 8x² + 20 

      ⍝ two indeterminants x and y
      ⍝
      "A:"    (⌹.poly_print A ← 2 2⍴0 1 1 0)
 A: x + y 

      "B:"    (⌹.poly_print B ← 2 2⍴0 ¯1 1 0)
 B: x - y 

      "PROD:" (⌹.poly_print PROD ← A ⌹.poly_multiply B)
 PROD: x² - y² 

      ⍝ show all monomials of indeterminants x, y, z
       ⍝ and max.powers 0, 1, and 2 (in lexicographic order)
      ⍝
      ⌹.poly_print 3 3 3⍴1
x²y²z² + x²y²z + x²y² + x²yz² + x²yz + x²y + x²z² + x²z + x² + xy²z² + xy²z + 
      xy² + xyz² + xyz + xy + xz² + xz + x + y²z² + y²z + y² + yz² + yz + y + 
      z² + z + 1

Let
      f = x2y + xy2 + y2,
      g₁ = xy − 1, and
      g₂ = y² − 1.

Dividing f by g₁ and then by g₂ gives:

      f = (x + y)(xy − 1)  +  (y² − 1)  +  (x + y + 1)
        = (x + y) g₁       +      g₂    +  (x + y + 1)

However, dividing f by g₂ and then by g₁ gives:

      f = x(xy − 1)  +  (x + 1)(y² − 1)  +  (2x + 1)
        = x g₁       +  ((x + 1) g₂      +  (2x + 1)

      Let LT(p) denote the largest term in polynomial p, and let       (1)

      q₁ = q₂ = ... = qₙ = 0   ⍝ quotient                              (1a)
      r  = 0                   ⍝ remainder                             (1b)
      p = f                    ⍝ dividend                              (1c)

      while p ≠ 0:                                                     (2)
            let i be the smallest i such that LT(gᵢ) divides LT(p).    (3)
            if such an i exists then:                                  (3a)
                  qᵢ = qᵢ + LT(p) ÷ LT(gᵢ)                             (4a)
                  p = p - gᵢ × LT(r) ÷ LT(gᵢ)                          (4b)
            otherwise:                                                 (3b)
                  r = r + LT(p)                                        (5a)
                  p = p - LT(gᵢ)                                       (5b)

      ⍝ construct the lexicographical order of all possible monom in P
      ⍝
      ORDER←-⎕IO-(⍴P)⍴⍳×/⍴P

      ⍝ append the desired monomial order O to a polynomial P.
      ⍝
      PO ← (2,⍴P)⍴(,P),,O

      ⍝ create the dividend
      ⍝
      "A:" A←⌹.poly_scan "x³ - x² - xy² - 2x + y² + 4"
 A:    4 0  1 
     ¯2 0 ¯1 
     ¯1 0  0 
      1 0  0 

      ⍝ create the divisor
      ⍝
      "B:" B←⌹.poly_scan "x - y"
 B:   0 ¯1 
      1  0 

      ⍝ create the monomial orders for A and B
      ⍝
      ORDER_A←-⎕IO-(⍴A)⍴⍳×/⍴A
      ORDER_B←-⎕IO-(⍴B)⍴⍳×/⍴B

      ⍝ append the orders to their polynomial
      ⍝
      ARG_A ← (2,⍴A)⍴(,A),,ORDER_A
      ARG_B ← (2,⍴B)⍴(,B),,ORDER_B

      ⍝ call ⌹.poly_divideN and poly_divideNO
      ⍝
      (Q1 R1) ←     A ⌹.poly_divideN B
      (Q2 R2) ← ARG_A ⌹.poly_divideNO ARG_B

      ⍝ the result should be the same, because
      ⍝ ORDER_A is what ⌹.poly_divideN uses.
      ⍝ 
      (Q1 R1) ≡ (Q2 R2)
1

      ⊢Z←⌹.poly_scan "1 + 2xy + 3y²"
1 0 3
0 2 0

      ⌹.poly_print Z
2xy + 3y² + 1

      ⍝ monadic (same as A=1)
      ⍝
      ⌹.integral "sqrt(x³)"
2*x*sqrt(x³)/5

      ⍝ A=1 : vector result
      ⍝
      1 ⌹.integral "sqrt(x³)"
2*x*sqrt(x³)/5

      ⍝ A=2 : matrix result
      ⍝
      2 ⌹.integral "sqrt(x³)"
      _____
2⋅x⋅╲╱  x³ 
───────────
     5     

2.24 ⎕CC - Character Classes

Some older APL interpreters (e.g. APL68000) had system variables for character classes, such as ⎕D for digits 0..9 or ⎕A for letters A..Z. GNU APL provides a single system function ⎕CC for that purpose.

• monadic ⎕CC B: Return character class
• dyadic A ⎕CC B: Test character class

      ⍝ character class 1: digits (old ⎕D)
      ⍝
      ⎕CC 1
0123456789

      ⍝ character class 2: uppercase letters A..Z (old ⎕A)
ABCDEFGHIJKLMNOPQRSTUVWXYZ

      ⍝ character class 3: lowercase letters a..z
abcdefghijklmnopqrstuvwxyz

      ...

      ⍝ Line drawing characters
      ⎕CC 7
┌─┬─┐╒═╤═╕
├─┼─┤╞═╪═╡
└─┴─┘╘═╧═╛
╓─╥─╖╔═╦═╗
╟─╫─╢╠═╬═╣
╙─╨─╜╚═╩═╝

      ⍝ Some mathematical symbols
      ⎕CC 8
⎲⎛⎞⎡⎤⎧⎫
⎳⎜⎟⎢⎥⎨⎬
↔⎝⎠⎣⎦⎮⎮
ℕℤℚℝℂ⎩⎭

      ⎕CC ⍬
      ⎕CC   1:  digits 0-9 (same as ⎕CC 10)
      ⎕CC   2:  uppercase characters A-Z (same as ⎕CC 26)
      ⎕CC   3:  lowercase characters a-z (same as ⎕CC ¯26)
      ⎕CC   4:  ASCII characters (same as ⎕CC 127)
      ⎕CC   5:  superscripts ( ²³¹ʲᵏᵐᵗ⁰ⁱ⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾ⁿ )
      ⎕CC   6:  subscripts ( ᵢ₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎ₖₘₙⱼ )
      ⎕CC   7:  line drawing chatacters
      ⎕CC   8:  octal digits 0-7
      ⎕CC   9:  some mathematical symbols
      ⎕CC  10:  decimal digits 0-9
      ⎕CC  16:  UPPERCASE HEXADECIMAL DIGITS 0-9 A-F
      ⎕CC ¯16:  lowercase hexadecimal digits 0-9 A-F a-f
      ⎕CC  17:  Hexadecimal Digits 0-9 A-F a-f
      ⎕CC  26:  UPPERCASE CHARACTERS A-Z
      ⎕CC ¯26:  lowercase characters A-Z
      ⎕CC  33:  base32 encoding (RFC 4648)
      ⎕CC  48:  greek characters A-Ω α-ω
      ⎕CC  52:  characters A-Z a-z
      ⎕CC  65:  base64 encoding (RFC 4648)
      ⎕CC  95:  printable characters 0x20 ... 0x7E
      ⎕CC 128:  ASCII characters

      'a45Y' ⎕CC 1
0 1 1 0

      'a45Y' ⎕CC 1 2
0 1 1 1

       (⎕CC 7) ⎕CC 7
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2.25 Dyadic ⎕CR

The ⎕CR function has now an optional left argument that selects one of several formatting styles and conversion functions in addition to the well-known monadic form.

Calling ⎕CR monadically without an axis and with an empty right argument shows a list of all functions provided by ⎕CR:

      ⎕CR ''

Definition: A byte vector is an integer (!) vector with numbers having a (signed or unsigned) 8-bit value (i.e. a value from -128 to 255 inclusive). Such byte vectors are frequently used arguments and results of ⎕FIO functions.

Let Z←A ⎕CR B.

Then the left argument A of ⎕CR selects one of several subfunctions of ⎕CR:

• A∈0 1 2 3 4 7 8 9 29: various formatting styles (boxed, APL input/output, etc.). Just try them out.
• A∈5 6: convert byte vector B to a string of uppercase or lowercase i hex digits respectively. Every byte in B becomes 2 characters in Z.
• A=10: convert variable named in B to an APL expression producing it.
• A=11: convert value B to byte vector Z in CDR ("Common Data Representation", an IBM standard) format (similar to 3 ⎕TF).
• A=12: convert byte vector B in CDR format to value Z.
• A=13: convert hex string B to byte vector Z.

  If a conversion has an inverse conversion (like 12 being the inverse of 11) then the inverse conversion can be expressed as the negative of the conversion number. For example, 12 ⎕CR B is the same as ¯11 ⎕CR B.

• A=14: conversion 11 followed by conversion 13 (Value to hex string in CDR format)
• A=15: conversion 13 followed by conversion 12 (hex string in CDR format to Value)
• A=16: encode byte vector B into Z (base64 encoding, RFC 4648)
• A=17: decode base64 vector B into byte vector Z (base64 encoding, RFC 4648)
• A=18: convert text vector B into byte vector Z (UTF8 encoding, RFC 3629)
• A=19: convert byte vector B into text vector Z (UTF8 encoding, RFC 3629)
• A=20-25: like 3,4,i or 7-9 but using a formatting similar to NARS APL ⎕FMT (showing the axis lengths as numbers instead of → and ↓)
• A=26: Z is the cell types of the ravel elements of B, i.e.
  • 2: character,
  • 16: integer,
  • 32: real number,
  • 64: complex number.
• A=27: Z[I] is the primary data representation (for example the real part of a complex number, or the numerator of a rational number) of B[I].
• A=28: Z[I] is the additional data representation (for example the imaginary part of a complex number, or the denominator of a rational number) of B[I].
• A=30: Z is B with all top-level elements conformed to a common rank and shape (as required by the ⍤ operator). This conversion is primarily used internally by the GNU APL interpreter.
• A∈31 32: These conversions are used internally by ⎕INP.
• A=33: convert tagged byte vector to a TLV (Tag/Length/Value) buffer. The TLV buffer can be sent over a byte stream (socket) and easily decoded at the other end. Say B = B[1], B[2], ..., B[n] such that B1 is an Integer (the tag) and B[j] is a character in the range 0-255 for j > 1. Let Z←33 ⎕CR B with Z = Z[1], Z[2], ... Z[m]. Then Z[1 2 3 4] is the 4 byte tag, Z[5 6 7 8] is the 4 byte data length (n-1) = (m - 8), and 1↓B == 8 ↓ Z. In other words, the first 4 bytes of Z are the tag in big endian byte order, the next 4 bytes are the length of B except the tag, and the rest of Z is B except the tag.

  Example:

        Tag←55   ⍝ hex 37
        5 ⎕CR 33 ⎕CR Tag,'Value'
  000000370000000556616C7565
  

• A=34: this is the inverse of 33 ⎕CR. The intended use for 33 ⎕CR and 34 ⎕CR is the transmission of a tagged byte vector over e.g a TCP socket:

    Sender                                              Receiver
    ──────                                              ────────
    T,Data →→→ 33 ⎕CR T,Data   →→→ TCP connection →→→   34 ⎕CR T,Data →→→ T,Data
  

  Example:

        34 ⎕CR ¯5 ⎕CR '000000370000000556616C7565'
  55 Value
  

  The TLVs constructed by 33 ⎕CR by the sender and decoded with 34 ⎕CR by the receiver can be sent back-to-back over a TCP connection or similar in such a way that the receiver knows exactly after which byte a TLV ends (and possibly the next TLV, if any, starts). This is perfect for connections over which data is sent sporadically. 33 ⎕CR and 34 ⎕CR are particularly useful for encoding and decoding TLV byte buffers exchanged between GNU APL and processes that were forked by GNU APL with ⎕FIO[57] (aka. fork() and execve()).

• A=35: convert a string of lines (containing LF characters as line delimiters) to a nested vector of lines.
• A=36: the inverse of ⎕CR[35]. Convert a nested vector of lines to a string with lines terminated with LF characxters.
• A=37: Same as ⎕CR B but without removing any indentation.
• A=38: Return an empty structured variable with (integer) capacity Bi, or convert an N×2 matrix B to a structured variable.
• A=39: convert a strucured variable B to an N×2 variable.
• A=40: pack a boolean variable Bi (experimental).
• A=41: unpack a boolean variable Bi (experimental).
• A=42: tokenize the APL statement Bs.
• A=43: parse the APL statement Bs.
• A=44: decode the (integer) token tag B or the (nested) token B.

Most dyadic ⎕CR variants whose argument B is expected to be a byte vector throw:

• RANK ERROR if 1≠⍴⍴B
• DOMAIN ERROR if one of the B[j] is not a proper byte value

A proper byte value is either an integer in the range -128...255 inclusive, or a (Unicode) character with a code point between U+FF80 and U+FFFF (inclusive, corresponding to a negative signed char in C/C++) or between U+0000...U+00FF (inclusive, corresponding to an unsigned char or to a signed positive char in C/C++). Real, Complex, or rational numbers are never proper byte values even if their value is close to an integer. Nor are nested APL values or values being assigned.

⎕CR is a function group, therefore its subfunctions can be selected with a name. For example: monadic ⎕CR.hex_to_bytes is the same as monadic ⎕CR[13] or dyadic 13 ⎕CR.

2.26 Dyadic ⎕FX (Native Functions)

A Native Function is a function that can be called in APL like a normal user defined APL function, but is implemented in C++.

A native function is created with A ⎕FX B. A is a string that is the path of a shared library and B is the name of the function in APL.

The GNU APL package contains a shared library file_io.so that contains the implementation of a native function for reading and writing files (fopen(), fclose(), ...), For example:

      ⍝ fix native function in lib_file_io.so as FILE_IO
      ⍝
      'lib_file_io.so' ⎕FX 'FILE_IO'
FILE_IO

      ⍝ show overview of subfunctions in FILE_IO
      ⍝
      FILE_IO ''
   Functions provided by this library.
   Assumes 'lib_file_io.so'  ⎕FX  'FUN'

   Legend: e - error code
           i - integer
           h - file handle (integer)
           s - string
           A1, A2, ...  nested vector with elements A1, A2, ...

           FUN     ''    print this text on stderr
        '' FUN     ''    print this text on stdout
           FUN[ 0] ''    print this text on stderr
        '' FUN[ 0] ''    print this text on stdout

   Zi ←    FUN[ 1] ''    errno (of last call)
   Zs ←    FUN[ 2] Be    strerror(Be)
   Zh ← As FUN[ 3] Bs    fopen(Bs, As) filename Bs mode As
   Zh ←    FUN[ 3] Bs    fopen(Bs, "r") filename Bs
      ...

Recent versions of GNU APL have replaced the native FILE_IO function above by the system function ⎕FIO. ⎕FIO need not be ⎕FX’ed and is otherwise backward compatible to the native function. New function numbers are, however, only added to ⎕FIO and not to the old native function FILE_IO. The parameters of the functions are described in the man pages for, e.g. strerror, fopen, ... and are fairly obvious.

Many functions in FILE_IO have byte vectors as arguments or return byte vectors. A byte vector is an integer vector whose numbers fit into a byte (so they are integers between -128 and 255). Often ⎕UCS and the functions in dyadic ⎕CR are used to convert such byte vectors to/from, for example, Unicode strings.

The GNU APL package also contains other shared libraries as templates for your own native functions. Copy one of the files src/native/template_F0.cc (for niladic native functions), src/native/template_F12.cc (for nomadic native functions), src/native/template_OP1.cc (for monadic native operators), or src/native/template_OP2/cc (for dyadic native operators) to your own .cc file and adjust src/native/Makefile.am accordingly.

Note: The )IN and )OUT commands of GNU APL support native functions, but to do so they have to use dyadic ⎕FX. This renders the workspace interchange file (.atf files) written by )OUT incompatible with all other APL interpreters if the workspace contains native functions. The )OUT command prints a warning when it is used with a workspace that contains native functions.

Note: As of GNU APL 1.6, the native function FILE_IO has been turned into the system function ⎕FIO. The syntax of ⎕FIO is the same as for FILE_IO. The )CLEAR workspace command will close all open files.

2.27 ⎕ARG - Interpreter command line arguments

⎕ARG contains the command line arguments with which GNU APL was invoked. See APL Scripting.

2.28 ⎕DLX - Knuth’s Dancing Links Algorithm

⎕DLX is an implementation of Donald Knuth’s Dancing Links Algorithm (called DLX by Knuth himself, but is sometimes also referred to as Knuth’s Algorithm X). ⎕DLX is a generic backtracking machine that can be used to dramatically simplify problems like the 8 queens problem on a chess board or sudokus.

The monadic form of ⎕DLX, i.e. ⎕DLX B, is a shortcut for 0 ⎕DLX B. It computes the first solution for the constraint matrix B.

The dyadic form of ⎕DLX, i.e. A ⎕DLX B, has an integer scalar A as left argument which determines the details of the computation as follows:

A > 0: The algorithm tries to find all solutions, but stops when A solutions have been found. This is handy while debugging code using ⎕DLX. The result is a nested vector with one vector item per solution.

A = 0: The algorithm stops when the first solution was found. In this case the solution is a simple (non-nested) numeric vector.

A = ¯1: like A > 0 but finding all solutions

A = ¯2: like A = 0 but instead of returning the first solution, the number of solutions (i.e. 0 or 1), the number of backtracks, and the number of link dances is returned as a 3-element numeric vector.

A = ¯3: like A = ¯1 but instead of returning all solutions, the number of solutions, the number of backtracks, and the number of link dances is returned as a 3-element numeric vector.

A = ¯4: A number of single steps in Knuth’s Algorithm are performed. Let e.g. A←¯4 r1 r2 r3. Then Z←A ⎕DLX B is the matrix B after 3 steps r1, r2, and r3 have been performed. r1, r2, and r3 are valid (as per ⎕IO) row numbers of B, and a step with a given row changes B as follows:

• the given row is set to 0 (this row becomes part of the final result),
• all other rows that have a 1 or a 2 in the same column as B are set to 0 (these 1s or 2s in the other rows conflict with the 1 or 2 in the given row and are therefore removed from B), and
• all columns that have no 1s or 2s left are removed from B (these columns have a 1 or 2 in B and the constraint is therefore satisfied).

The purpose of ¯4 ⎕DLX is:

• To demonstrate how Knuth’s algorithm works, and/or
• To preset some initial values in B. For example, one can first initialize the constraints for an empty Sudoku (which is the same matrix for all Sudokus) and then enter the initial digits of a particular Sudoku using 4 ⎕DLX. If there are, say, 20 initial digits in the Sudoku then A has 21 elements (¯4 plus the 20 rows representing the 20 initial digits. The result of ¯4 ⎕DLX B has the same number of rows as B (with some rows now cleared), but fewer columns than B.

The right argument B of ⎕DLX B or A ⎕DLX B is a constraints matrix whose columns consist of either 0s and 1s (called a primary column) or 0s and 2s (called secondary columns). The 0s, 1s and 2s can be the integers 0, 1, or 2, characters ’0’, ’1’, or ’2’ respectively, or ’ ’ meaning ’0’. The character representation is useful when B becomes large and shall be printed because the spaces in the numerical variant will not be printed. In the following, B is assumed to be numeric.

Let Z←A ⎕DLX B, and let R be a solution in Z, that is, Z itself (A = 0) or R is Z[k] for some k if A ≠ 0. And let S←+⌿B[R].

Then S = s1 s2 s3, ... sN where N is the number of columns in B and sj=1 if j is a primary column of B and sj∈0 1 if j is a secondary column of B.

In other words, ⎕DLX B computes a subset of the rows of B in such a way that for every column j of B exactly one (for primary column j) or at most one (for a secondary column) of the rows in a solution has its j’th element set to 1 and all other words set their j’th element set to 0.

In yet other words, for every solution returned by ⎕DLX B, a 1 in one row prevents all other rows that also have a 1 in that column, and all rows together have exactly one 1 in every primary column and at most one 1 in every secondary column. In the absence of secondary columns, the problem solved by ⎕DLX is also known as the "exact cover problem"

If all that sounds weird and useless, consider the following APL program for finding all solutions of the 8 Queens problem on a chess board (which probably every programmer has programmed at some point in time):

      RC←8↑'1' ◊ D←15↑¯8↑'2'   ⍝ helpers for constructing Q8
      ⍝     rows   cols   diag1    diag2
      Q8←⊃{(R⌽RC),(C⌽RC),((C-R)⌽D),((¯7-R+C)⌽D)⊣(R C)←-8 8⊤⍵-⎕IO} ¨ ⍳64
      Z←¯1 ⎕DLX Q8
      {⎕UCS (65+⌊⍵÷8)(49+8∣⍵←⍵-⎕IO)} ¨ ⊃Z[1 2 3 92] ⍝ solutions 1, 2, 3, and 92
 A1 B5 C8 D6 G2 E3 F7 H4 
 A1 B6 C8 D3 E7 F4 G2 H5 
 A1 B7 C4 E8 D6 G5 F2 H3 
 A8 B4 C1 D3 G7 E6 F2 H5 

      8 8⍴("+" "Q")[⎕IO+(⍳64)∈⊃Z[1]]   ⍝ visualize solution Z[1]
Q + + + + + + + 
 + + + + Q + + + 
 + + + + + + + Q 
 + + + + + Q + + 
 + + Q + + + + + 
 + + + + + + Q + 
 + Q + + + + + + 
 + + + Q + + + + 

Obviously Z contains solutions for the 8 Queens problem; the total number of solutions is well known to be 92 and we showed only the first three and the last solution above.

The constraint matrix Q8 is the key to success. The matrix has 64 rows - one row for every field of the chess board. And it has 8 + 8 + 15 + 15 columns. The first 8 columns of Q8 are constraints that prevent more than one Queen from being placed in the same row of the chess board (the argument ⍵ is the field number counting from left to right and from bottom to top). The next 8 columns of Q8 are constraints that prevent more than one Queen from being placed in the same column of the chess board.

If we would call ⎕DLX with only these constraints, i.e. ⎕DLX T8←64 16↑Q8, then we would get the solutions of the 8 tower problem. However, we continue and add 15 more constraints for each of the two diagonals. The resulting constraint matrix Q8 is this:

      Q8

1       1              2                     2
1        1              2                   2 
1         1              2                 2  
1          1              2               2   
1           1              2             2    
1            1              2           2     
1             1              2         2      
1              1              2       2       
 1      1             2                     2 
 1       1             2                   2  
 1        1             2                 2   
 1         1             2               2    
 1          1             2             2     
 1           1             2           2      
 1            1             2         2       
 1             1             2       2        
  1     1            2                     2  
  1      1            2                   2   
  1       1            2                 2    
  1        1            2               2     
  1         1            2             2      
  1          1            2           2       
  1           1            2         2        
  1            1            2       2         
   1    1           2                     2   
   1     1           2                   2    
   1      1           2                 2     
   1       1           2               2      
   1        1           2             2       
   1         1           2           2        
   1          1           2         2         
   1           1           2       2          
    1   1          2                     2    
    1    1          2                   2     
    1     1          2                 2      
    1      1          2               2       
    1       1          2             2        
    1        1          2           2         
    1         1          2         2          
    1          1          2       2           
     1  1         2                     2     
     1   1         2                   2      
     1    1         2                 2       
     1     1         2               2        
     1      1         2             2         
     1       1         2           2          
     1        1         2         2           
     1         1         2       2            
      1 1        2                     2      
      1  1        2                   2       
      1   1        2                 2        
      1    1        2               2         
      1     1        2             2          
      1      1        2           2           
      1       1        2         2            
      1        1        2       2             
       11       2                     2       
       1 1       2                   2        
       1  1       2                 2         
       1   1       2               2          
       1    1       2             2           
       1     1       2           2            
       1      1       2         2             
       1       1       2       2              

To see what, for example, the first solution looks like and how it relates to the constraints matrix Q8:

      ⍝ the rows in Q8 of the first solution
      ⍝
      Z[1]
 1 14 24 27 39 44 50 61 

      ⍝ the first solution translated back into the problem domain
      ⍝
      {⎕UCS (65+⌊⍵÷8)(49+8∣⍵←⍵-⎕IO)} ¨ ⊃Z[1]
 A1 B6 C8 D3 E7 F4 G2 H5 

      ⍝ the constraints of the rows of the first solution
      ⍝
      Q8[⊃Z[1];]
1       1              2                     2
 1           1             2           2      
  1            1            2       2         
   1      1           2                 2     
    1         1          2         2          
     1     1         2               2        
      1  1        2                   2       
       1    1       2             2           

      ⍝ all primary constraints met?
      ⍝
      +⌿ ' '≠ Q8[⊃Z[1];] 
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 
      1 0 0 0 0 1

And that’s it: call ⎕DLX to get the solution(s). In general all problems that can be solved by ⎕DLX contain 3 steps:

• translate the problem into a constraints matrix B,
• Z←⎕DLX B, and
• translate the result Z back into the problem domain.

Another application of ⎕DLX is solving sudokus. The constraints matrix is a little more complicated, but the principle is the same. GNU APL is shipped with two workspaces: sudoku.apl (which solves sudokus without using ⎕DLX) and sudoku_DLX.apl (which solves sudokus using ⎕DLX).

2.29 ⎕ENV - Environment Variables

⎕ENV contains the environment variables of the process that is running GNU APL. See APL Scripting.

2.30 ⎕FIO - File I/O Functions

As of GNU APL 1.6, the native function FILE_IO has been replaced by the system function ⎕FIO. ⎕FIO normally takes a numeric axis argument which selects one of many different functions, most of which are contained in the standard C library. The arguments for these functions are usually the same as the corresponding C functions and the man page for each function describes the meaning of the arguments.

For example, ⎕FIO[3] corresponds to fopen() and ’man fopen’ explains what fopen does (opening a file).

Calling ⎕FIO monadically without an axis and with an empty right argument shows a list of all functions provided by ⎕FIO:

      ⎕FIO ''
   Functions provided by ⎕FIO...

   Legend: a - address family, IPv4 address, port (or errno)
           d - table of dirent structs
           e - error code (integer as per errno.h)
           h - file handle (integer)
           i - integer
           n - names (nested vector of strings)
           s - string
           u - time divisor: 1       - second
                             1000    - milli second
                             1000000 - micro second
           y4 - seconds, wday, yday, dst (
           y67- year, mon, day, hour, minute, second, [dst]
           y9 - year, mon, day, hour, minute, second, wday, yday, dst
           A1, A2, ...  nested vector with elements A1, A2, ...

           ⎕FIO     ''    print this text on stderr
        '' ⎕FIO     ''    print this text on stdout
           ⎕FIO[ 0] ''    print this text on stderr
        '' ⎕FIO[ 0] ''    print this text on stdout

   Zi ←    ⎕FIO[ 1] ''    errno (of last call)
   Zs ←    ⎕FIO[ 2] Be    strerror(Be)
   Zh ← As ⎕FIO[ 3] Bs    fopen(Bs, As) filename Bs mode As
   Zh ←    ⎕FIO[ 3] Bs    fopen(Bs, "r") filename Bs

File I/O functions:

   Ze ←    ⎕FIO[ 4] Bh    fclose(Bh)
   Ze ←    ⎕FIO[ 5] Bh    errno (of last call on Bh)
   Zi ←    ⎕FIO[ 6] Bh    fread(Zi, 1, 5000, Bh) 1 byte per Zi
   Zi ← Ai ⎕FIO[ 6] Bh    fread(Zi, 1, Ai, Bh) 1 byte per Zi
   Zi ← Ai ⎕FIO[ 7] Bh    fwrite(Ai, 1, ⍴Ai, Bh) 1 byte per Ai
   Zi ←    ⎕FIO[ 8] Bh    fgets(Zi, 5000, Bh) 1 byte per Zi
   Zi ← Ai ⎕FIO[ 8] Bh    fgets(Zi, Ai, Bh) 1 byte per Zi
   Zi ←    ⎕FIO[ 9] Bh    fgetc(Zi, Bh) 1 byte
   Zi ←    ⎕FIO[10] Bh    feof(Bh)
   Zi ←    ⎕FIO[11] Bh    ferror(Bh)
   Zi ←    ⎕FIO[12] Bh    ftell(Bh)
   Zi ← Ai ⎕FIO[13] Bh    fseek(Bh, Ai, SEEK_SET)
   Zi ← Ai ⎕FIO[14] Bh    fseek(Bh, Ai, SEEK_CUR)
   Zi ← Ai ⎕FIO[15] Bh    fseek(Bh, Ai, SEEK_END)
   Zi ←    ⎕FIO[16] Bh    fflush(Bh)
   Zi ←    ⎕FIO[17] Bh    fsync(Bh)
   Zi ←    ⎕FIO[18] Bh    fstat(Bh)
   Zi ←    ⎕FIO[19] Bh    unlink(Bc)
   Zi ←    ⎕FIO[20] Bh    mkdir(Bc, 0777)
   Zi ← Ai ⎕FIO[20] Bh    mkdir(Bc, AI)
   Zi ←    ⎕FIO[21] Bh    rmdir(Bc)
   Zi ← A  ⎕FIO[22] 1     printf(         A1, A2...) format A1
   Zi ← A  ⎕FIO[22] 2     fprintf(stderr, A1, A2...) format A1
   Zi ← A  ⎕FIO[22] Bh    fprintf(Bh,     A1, A2...) format A1
   Zi ← Ac ⎕FIO[23] Bh    fwrite(Ac, 1, ⍴Ac, Bh) 1 Unicode per Ac, Output UTF8
   Zh ← As ⎕FIO[24] Bs    popen(Bs, As) command Bs mode As
   Zh ←    ⎕FIO[24] Bs    popen(Bs, "r") command Bs
   Ze ←    ⎕FIO[25] Bh    pclose(Bh)
   Zs ←    ⎕FIO[26] Bs    return entire file Bs as byte vector
   Zs ← As ⎕FIO[27] Bs    rename file As to Bs
   Zd ←    ⎕FIO[28] Bs    return content of directory Bs
   Zn ←    ⎕FIO[29] Bs    return file names in directory Bs
   Zs ←    ⎕FIO 30        getcwd()
   Zn ← As ⎕FIO[31] Bs    access(As, Bs) As ∈ 'RWXF'
   Zh ←    ⎕FIO[32] Bi    socket(Bi=AF_INET, SOCK_STREAM, 0)
   Ze ← Aa ⎕FIO[33] Bh    bind(Bh, Aa)
   Ze ←    ⎕FIO[34] Bh    listen(Bh, 10)
   Ze ← Ai ⎕FIO[34] Bh    listen(Bh, Ai)
   Za ←    ⎕FIO[35] Bh    accept(Bh)
   Ze ← Aa ⎕FIO[36] Bh    connect(Bh, Aa)
   Zi ←    ⎕FIO[37] Bh    recv(Bh, Zi, 5000, 0) 1 byte per Zi
   Zi ← Ai ⎕FIO[37] Bh    recv(Bh, Zi, Ai, 0) 1 byte per Zi
   Zi ← Ai ⎕FIO[38] Bh    send(Bh, Ai, ⍴Ai, 0) 1 byte per Ai
   Zi ← Ac ⎕FIO[39] Bh    send(Bh, Ac, ⍴Ac, 0) 1 Unicode per Ac, Output UTF8
   Zi ←    ⎕FIO[40] B     select(B_read, B_write, B_exception, B_timeout)
   Zi ←    ⎕FIO[41] Bh    read(Bh, Zi, 5000) 1 byte per Zi
   Zi ← Ai ⎕FIO[41] Bh    read(Bh, Zi, Ai) 1 byte per Zi
   Zi ← Ai ⎕FIO[42] Bh    write(Bh, Ai, ⍴Ai) 1 byte per Ai
   Zi ← Ac ⎕FIO[43] Bh    write(Bh, Ac, ⍴Ac) 1 Unicode per Ac, Output UTF8
   Za ←    ⎕FIO[44] Bh    getsockname(Bh)
   Za ←    ⎕FIO[45] Bh    getpeername(Bh)
   Zi ← Ai ⎕FIO[46] Bh    getsockopt(Bh, A_level, A_optname, Zi)
   Ze ← Ai ⎕FIO[47] Bh    setsockopt(Bh, A_level, A_optname, A_optval)
   Ze ← As ⎕FIO[48] Bh    fscanf(Bh, As)
   Zs ←    ⎕FIO[49] Bs    return entire file Bs as nested lines
   Zs ← LO ⎕FIO[49] Bs    ⎕FIO[49] Bs and pipe each line through LO.
   Zi ←    ⎕FIO[50] Bu    gettimeofday()
   Zy4←    ⎕FIO[51] By67  mktime(By67)  Note: Jan 2, 2017 is: 2017 1 2 ...
   Zy9←    ⎕FIO[52] Bi    localtime(Bi) Note: Jan 2, 2017 is: 2017 1 2 ...
   Zy9←    ⎕FIO[53] Bi    gmtime(Bi)    Note: Jan 2, 2017 is: 2017 1 2 ...
   Zi ←    ⎕FIO[54] Bs    chdir(Bs)
   Ze ← As ⎕FIO[55] Bh    sscanf(Bs, As) As is the format string
   Zs ← As ⎕FIO[56] Bs    write nested lines As to file named Bs

Benchmarking functions:

           ⎕FIO[200] Bi    clear statistics with ID Bi
   Zn ←    ⎕FIO[201] Bi    get statistics with ID Bi
           ⎕FIO[202] Bs    get monadic parallel threshold for primitive Bs
        Ai ⎕FIO[202] Bs    set monadic parallel threshold for primitive Bs
           ⎕FIO[203] Bs    get dyadic parallel threshold for primitive Bs
        Ai ⎕FIO[203] Bs    set dyadic parallel threshold for primitive Bs

A new feature of ⎕FIO (which is not available with the native function FILE_IO) is ⎕FIO[49]. ⎕FIO[49] is a monadic operator which takes a monadic conversion function as function argument. For example:

Z←F ⎕FIO[49] 'filename'

reads the file named filename line by line. For every line read, the conversion function F is called and the result returned by F is enclosed and stored in Z. In other words,

Z←F ⎕FIO[49] 'filename'

does:

Z F¨Z←⎕FIO[49] 'filename'

2.31 ⎕FFT - Fast Fourier Transform

For those interested in signal processing and the like, GNU APL provides ⎕FFT:

The monadic form ⎕FFT B is a shortcut for the dyadic form 0 ⎕FFT B. It computes the FFT of complex or real B without applying a window function.

The dyadic form A ⎕FFT B provides more control over what ⎕FFT computes. A is an integer scalar which falls into one of three ranges.

The first range from ¯15 to ¯10 does not compute an FFT, but returns the result of multiplying B with one of several window functions, that are frequently used in the context of FFTs. The result has same shape as B and can be used for analyzing or troubleshooting FFTs:

• A=¯10: no FFT, return the Hann window applied to B
• A=¯11: no FFT, return the Hamming window applied to B
• A=¯12: no FFT, return the Blackman window applied to B
• A=¯13: no FFT, return the Blackman-Harris window applied to B
• A=¯14: no FFT, return the Blackman-Nuttal window applied to B
• A=¯15: no FFT, return the Flat-Top window applied to B

The second range around 0 contains the computation of the forward and inverse FFTs:

• 0 ⎕FFT B returns the "normal" (aka. forward) FFT of the numeric array B.
• ¯1 ⎕FFT B returns the inverse FFT of the numeric array B.

The third range from 10 to 15 corresponds to the first range and first multiplies B with a window function and then computes the FFT:

• A=10: FFT(B × Hann window)
• A=11: FFT(B × Hamming window)
• A=12: FFT(B × Blackman window)
• A=13: FFT(B × Blackman-Harris window)
• A=14: FFT(B × Blackman-Nuttal window)
• A=15: FFT(B × Flat-Top window)

⍴⍴B can be 1 (one-dimensional FFT, the most common case) or more. The implementation of ⎕FFT uses libfftw3, aka. "The fastest Fourier Transform in the West." GNU APL checks the presence of libfftw3 when it is ./configure’d. If libfftw3 is present then ⎕FFT will hopefully return the expected result; if not then a SYNTAX ERROR will be raised when ⎕FFT is used.

⎕FFT honors the presence of /etc/fftw/wisdom (see man fftw-wisdom) to speed up the computations performed by ⎕FFT. Creating /etc/fftw/wisdom will take a few hours, though, so that creating it will not pay off for most mortals.

2.32 ⎕GTK - GTK Interface

GTK (Gimp ToolKit) is a rather powerful library for creating graphical user interfaces (GUIs). ⎕GTK makes a subset of the almost 10000 functions in the different GTK libraries available to GNU APL programs. With ⎕GTK a GNU APL program can, for example, replace the somewhat crude ⎕ or ⍞ input methods of standard APL with a more intuitive GUI.

The details of using ⎕GTK would go far beyond the scope of this info manual and has therefore been put into a separate document. See: HOWTOs/Quad-GTK.html.

2.33 ⎕JSON - JSON Parsing

GNU APL provides ⎕JSON for decoding and encoding JSON strings and files. A (valid) JSON string is mapped to an APL value as follows:

• JSON value ←→ structured or non-structured APL value
• JSON number ←→ APL number
• JSON string ←→ APL string
• JSON literal ←→ enclosed APL string (one of either ⊂’null’, ⊂’true’, or ⊂’false’ )
• JSON array ←→ APL vector
• JSON object ←→ structured APL value (associative array)

• Monadic ⎕JSON
• Dyadic ⎕JSON

2.34 ⎕MAP - Map Value

⎕MAP changes the ravel items of its right argument according to a mapping table provided as its left argument. Let Z←A ⎕MAP B.

The left argument A of Z←A ⎕MAP B shall be a N×2 matrix. Each 2-element row of A, say A[J;], specifies a separate mapping A[J;1] → A[J;2]. The result Z has the same shape as the right argument B. The items Z[...] of their result are constructed from their corresponding items B[...] in B as follows:

• if B[...] ≡ A[J;1] for some J then Z[...] is A[J;2].
• otherwise B[...] is different from all A[1;J] and then Z[...] is B[...]

In other words, Z is B, but with items of B found in A[;1] replaced by their mapped item A[;2].

In practice the left argument A is frequently created from a literal APL value such as 5 2⍴’eEwWaAzZ92’ in the example below. To simplify these cases, A ⎕MAP B also accepts a vector instead of a N×2 matrix:

A ⎕MAP B ←→ ((N 2)⍴A) ⎕MAP B if (2×N) ←→ ⍴A

Examples:

      ⍝ the map A
      ⊢A←5 2⍴'eEwWaAzZ92'
eE    ⍝ map 'e' → 'E'
wW    ⍝ map 'w' → 'W'
aA    ⍝ map 'a' → 'A'
zZ    ⍝ map 'z' → 'Z'
92    ⍝ map '9' → '2'

      ⍝ the value B being mapped
      ⊢B←'Halloween'
Halloween

      ⍝ the result of A ⎕MAP B
      A ⎕MAP B
HAlloWEEn

      A←'eEwWaAzZ92'   ⍝ vector A instead of N×2 matrix
      A ⎕MAP B
HAlloWEEn

NOTES:

• The keys of the mapping A (i.e. the elements in column A[;1]) must be unique. If they are not then the mapping is ambiguous and a DOMAIN ERROR is raised.
• Nested items in B: if an item of B is nested, then it is either equal to some (also nested) key, say A[N;1], (in that case it is being mapped to A[N;2] in the result), or it is different from all keys in A[;1] (and in that case the corresponding item in the result is the item in B.
• Nested items in A: if a key, say A[N;1], is nested and matches an equal (hence also nested) item in B, then the corresponding item in the result will be A[N;2] (which may or may not be nested). If a (nested or not nested) key A[N;1] is equal to an item in B, then the corresponding item in the result will be nested if A[N;2] is nested. If, for some N, the depths of A[N;1] and A[N;2] differ, then depths of B and A ⎕MAP B may differ as well. That is, ⎕MAP conserves the shape of B, but not necessarily the depth of B.
• By default the mapping of B is non-recursive, i.e. the top-level items of B are compared with the top-level keys A[;1], and nested keys and items of B are handled as described above.

  Sometimes, however, it is desirable to recursively descend into the nested sub-values of B (though never of A). This can be achieved by enclosing A as shown in the examples below.

      A←5 2⍴'eEwWaAzZ92'
      B←'Hal' 'low' 'een'   ⍝ nested B
      4 ⎕CR A ⎕MAP B        ⍝ non-recursive (none of the keys in A[;1] matches)
┏→━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━┓ ┏→━━┓┃
┃┃Hal┃ ┃low┃ ┃een┃┃
┃┗━━━┛ ┗━━━┛ ┗━━━┛┃
┗∊━━━━━━━━━━━━━━━━┛

      4 ⎕CR (⊂A) ⎕MAP B   ⍝ recursive (some simple keys in A[;1] match)
┏→━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━┓ ┏→━━┓┃
┃┃HAl┃ ┃loW┃ ┃EEn┃┃
┃┗━━━┛ ┗━━━┛ ┗━━━┛┃
┗∊━━━━━━━━━━━━━━━━┛

2.35 ⎕MX - Matrix/Statistics

⎕MX was kindly contributed by Henrik Moller

⎕MX[x] is a system function that provides a collection of matrix and statistical operations such as covariance, eigensystems, and so on, along with random number generation in several statistical distributions. Particular operations are selected through the use of the axis specifier x:

            ⎕MX[1] B       (monadic) Determinant
        (A) ⎕MX[2] B       (nomadic) Cross product
          A ⎕MX[3] B       (dyadic)  Vector angle
            ⎕MX[4] B       (monadic) Eigenvector
            ⎕MX[5] B       (monadic) Eigenvalue
            ⎕MX[6] B       (monadic) Ident
            ⎕MX[7] B       (monadic) Rotation matrix
          A ⎕MX[8] B       (dyadic)  Homogeneous transform
            ⎕MX[9] B       (monadic) Norm
        (A) ⎕MX[10] B      (nomadic) Randoms
        (A) ⎕MX[11] B      (nomadic) Covariance
          A ⎕MX[12] B      (dyadic)  Histogram
          A ⎕MX[13] B      (dyadic)  Print
            ⎕MX[14] B      (monadic) Set RNG seed

(These may by shown by entering ⎕MX ” or ⎕MX[0]0 on the APL command line.) In the function list above, A indicates a mandatory left argument, while (A) indicates an optional left argument. The right argument B of ⎕MX[X] is always mandatory.

• Determinant
• Cross Product
• Vector Angle
• Eigenvector
• Eigenvalue
• Ident
• Rotation
• Homogeneous transform
• Norm
• Randoms
• Covariance
• Histograms
• Print

      mx[1] 3 3⍴5 3 1 9 7j2 6 2 8 4j5
¯164J76

      ⊢B←3 4⍴ 1 2 3j6 4 5 6 7 8 9 10 11 12
1  2  3J6  4
5  6  7    8
9 10 11   12
      ⎕MX[2] B
0J48 0J¯72 0 0J24

      1 2 3j6 ⎕MX[2] 4 5 6
¯3J¯30 6J24 ¯3

      (⍳8) ⎕MX[3] 4j4+⍳8
0.2515670406J0.09930859532

      B
¯1  1 ¯1 1
¯8  4 ¯2 1
27  9  3 1
64 16  4 1
      ⎕PP←4
      ⎕MX[4] B
  0.09988            0.1113          ¯0.2925          ¯0.9445
  0.04308J0.009687  ¯0.07091J0.1389   0.5166J¯0.01601  0.8396J0.04139
  0.04308J¯0.009687 ¯0.07091J¯0.1389  0.5166J0.01601   0.8396J¯0.04139
 ¯0.1449             0.3566           0.9194           0.08118

      E←⎕MX[4] B
      E[⎕IO;]                   ⍝ first vector
0.09988 0.1113 ¯0.2925 ¯0.9445
      E[⎕IO+1;]                 ⍝ second vector
0.04308J0.009687 ¯0.07091J0.1389 0.5166J¯0.01601 0.8396J0.04139
      etc.

      ⊢F←⎕MX[5] B
¯6.414 5.546J3.085 5.546J¯3.085 2.323

             E[k;] ⬄ F[k]

      |F
6.41391 6.346285607 6.346285607 2.3228

      4 ⎕CR (⊂⍉(1,⍴F)⍴F),(⊂E)
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓┃
┃↓¯6.414       ┃ ↓  0.09988           0.1113         ¯0.2925          ¯0.9445          ┃┃
┃┃ 5.546J3.085 ┃ ┃  0.04308J0.00969  ¯0.07091J0.139   0.5166J¯0.01601  0.8396J0.04139  ┃┃
┃┃ 5.546J¯3.085┃ ┃  0.04308J¯0.00969 ¯0.07091J¯0.139  0.5166J0.01601   0.8396J¯0.04139 ┃┃
┃┃ 2.323       ┃ ┃ ¯0.1449            0.3566          0.9194           0.08118         ┃┃
┃┗━━━━━━━━━━━━━┛ ┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛┃
┗ϵ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
   Values            Vectors

      ⎕MX[6] 4  
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

      ⎕MX[7] ○30÷180
0.866 ¯0.5
0.5    0.866

      1 2 3 ⎕MX[8] ○(30 45 60)÷180
 0.6124 0.2803  0.7392 0
 0.3536 0.7392 ¯0.5732 0
¯0.7071 0.6124  0.3536 0
 1      2       3      1

      ⎕PP←10
      ⎕MX[9] 1 2 3 4
0.1825741858 0.3651483717 0.5477225575 0.7302967433
      (⎕MX[9] 1 2 3 4)*2
0.03333333333 0.1333333333 0.3 0.5333333333
      +/(⎕MX[9] 1 2 3 4)*2
1
     
      ⊢B←(3 3⍴⍳9)-⎕IO
0 1 2
3 4 5
6 7 8
      ⎕MX[9] B
0            0.0700140042 0.1400280084
0.2100420126 0.2800560168 0.350070021
0.4200840252 0.4900980294 0.5601120336
      +/,(⎕MX[9] B)*2
1

Next: Covariance, Previous: Norm, Up: ⎕MX - Matrix/Statistics   [Contents]

2.35.10 Randoms

The nomadic randoms operation, ⎕MX[10] or ⎕MX[10 x] where x is an integer specifying the distribution of the randoms generated. At present, the distibutions supported are:

• x=0: Normal distribution (default)
• x=1: Lognormal distribution
• x=2: Chi Squared distribution
• x=3: Student T distribution

If x is not specified then the normal distribution is used.

The right argument B of ⎕MX B is a real or complex scalar that specifies the desired standard deviation. If B is complex, the value returned is also complex with random real and imaginary components.

If the optional left argument A is specified, then A must be a scalar integer and Z←A ⎕MX B and the operation returns a random vector Z with A = ⍴Z.

If a mean value is applicable for the selected distribution, then it is set to 0.

      ⎕MX[10] 2
¯1.0504575511
      ⎕MX[10] 2
¯2.9337180373
      5 ⎕MX[10] 2
0.05154476627 3.80631021 2.455474045 3.165260688 0.6792784063
      ⎕PP←4
      3 ⎕MX[10] 1j1
¯1.134J¯1.453 1.111J¯0.1916 ¯1.106J¯0.9383

(This image is a plot of three 200-bucket histograms of 200000 samples each of a normal distribution, a chi-square distribution, and a Student T distribution. See Histograms below.)

An initial seed may be specified with

        ⎕MX[14] <seed>

where <seed> is a single integer. This allows repeatable random sequences:

      ⊣⎕MX[14] 3
      4 ⎕MX[10] 1
¯1.358348261 ¯1.75175064 ¯2.009430519 ¯0.5167968281
      ⊣⎕MX[14] 3
      4 ⎕MX[10] 1
¯1.358348261 ¯1.75175064 ¯2.009430519 ¯0.5167968281

---

      ⊢B←3 4⍴?12⍴10
5 8  1 9
7 6  2 9
6 5 10 4
      ⎕PP←10
      ⎕MX[11] B
12.91666667   9.333333333 ¯9.25
 9.333333333  8.666666667 ¯7.333333333
¯9.25        ¯7.333333333  6.916666667

      1 2j6 3 4 ⎕MX[11] 4 3 2j5 1
¯1.666666667J¯2.5

      A← 6 7 10 13 14 14 18 19 22 24
      A ⎕MX[11] A ⍝ or ⎕MX[11]⍨ A
36.67777778 

      4 ⎕MX[12] (⍳101)-⎕IO
25 25 25 26

      200 ⎕MX[12] 200000 ⎕MX[10 0] 1

      "samples.txt" ⎕MX[13] 34
      ">samples.txt" ⎕MX[13] 3 4
      ">samples.txt" ⎕MX[13] 3 4⍴⍳12
      ">samples.txt" ⎕MX[13] "example line"
      ">samples.txt" ⎕MX[13] "example\nline"

34
3 4
 1  2  3  4
 5  6  7  8 
 9 10 11 12 
example line
example
line

    ∇
[0]   distro;v
[1]   v←⍉1 200⍴200 ⎕MX[12] 200000 ⎕MX[10 0] 1
[2]   v←v,⍉1 200⍴200 ⎕MX[12] 200000 ⎕MX[10 2] 8
[3]   v←v,⍉1 200⍴200 ⎕MX[12] 200000 ⎕MX[10 3] 8
[4]   v ⎕MX[13] 'MX_distribution.data'
    ∇

   1   48    0
   0  119    0 
   0  187    0 
   1  331    0
   0  500    0
   1  728    0
   0  977    0
   2 1161    0
   1 1475    0
      .
      .
      .
  for 200 lines

      Heading←⎕CR.string_to_UTF8 "Print: 10+⍳10\n\n"   ⍝ a heading
      Value←10 + ⍳10                                   ⍝ a value
      Handle←"w" ⎕FIO.fopen "/tmp/test"                ⍝ open file for writing
      Heading ⎕FIO.fwrite Handle                       ⍝ write the heading
      Handle ⎕MX.print Value                           ⍝ write the value
      ⎕FIO.fclose Handle                               ⍝ close the file

Print: 10+⍳10

11 12 13 14 15 16 17 18 19 20 

2.36 ⎕PLOT - Plot Data

⎕PLOT is a function for visualizing numerical APL values. The values to be plotted are provided as the right argument of ⎕PLOT while the optional left argument controls details of the output, such as the plot window size, colors for plot lines, points, and grids, etc.

The general syntax of ⎕PLOT is:

   ⎕PLOT ⍬       ⍝ show a list of attributes and their default values
   H←⎕PLOT B     ⍝ plot B with all attributes set to their default values
   H←A ⎕PLOT B   ⍝ plot B with specified default attributes overridden
   ⎕PLOT H       ⍝ close the plot window with handle H
   ⎕PLOT  0      ⍝ verbosity: OFF (no debug output)
   ⎕PLOT ¯1      ⍝ verbosity: ON (print attribute values before plotting)
   ⎕PLOT ¯2      ⍝ verbosity: DEBUG (also print debug information)
   ⎕PLOT ¯3      ⍝ close all plot windows
   ⎕PLOT ¯6      ⍝ return all open plot window handles H

• The Plot Data B
• The Plot Attributes A
• Plot Window Handling
• Output File Format
• GUI driver

Next: The Plot Attributes A, Up: ⎕PLOT - Plot Data   [Contents]

2.36.1 The Plot Data B

The plot data B can be a vector (for a single plot line to be drawn) or a matrix (in that case one plot line per matrix row is drawn, by default in different colors). Each data item has to be numeric and represents one point in the plot. The points that are adjacent in a row of the matrix are connected by lines. By default plotted points are black and the lines connecting them are green (for the first plot line). However, all colors used, all diameters of points, and all thicknesses of the lines can be fine-tuned by overriding the default values (see dyadic A ⎕PLOT B below).

• Complex Plot Data B
• Real Plot Data B

---

Next: Real Plot Data B, Up: The Plot Data B   [Contents]

2.36.1.1 Complex Plot Data B

If a data item B[N] or B[row;N] is complex, say B[N] = x + iy. then it is placed (after some scaling) at position (x, y) of the plot. The X-range [Xmin ... Xmax] of the plot is then determined by the real parts of B, and the Y-range [Ymin ... Ymax] of the plot is determined by the imaginary parts of B. That is:

Xmin ← ⌊/,9○B    ⍝ smallest real part of B
Xmax ← ⌈/,9○B    ⍝ largest real part of B
Ymin ← ⌊/,11○B   ⍝ smallest imaginary part of B
Ymax ← ⌈/,11○B   ⍝ largest imaginary part of B

Example (plot a circle (actually: a regular 20-gon)):

      ⎕PLOT +⌿1 0J1×[1]1 2 ∘.○ (0,⍳2×N) × ○÷N←10

produces this plot window (only visible in the HTML version of this document; in text mode see file doc/PLOT_circle.png):

---

Previous: Complex Plot Data B, Up: The Plot Data B   [Contents]

2.36.1.2 Real Plot Data B

On the other hand, if a data item B[N] or B[row;N] is real, then it is placed at position (N, x) of the plot. The X-range is then [⎕IO ... ⎕IO + ¯1↑⍴B] and the Y-range [Y_min ... Y_max] of the plot is determined by the values of B. That is:

Xmin ← ⎕IO             ⍝ smallest real part of B
Xmax ← ¯1 + ⎕IO + ⍴B   ⍝ largest real part of B
Ymin ← ⌊/,B            ⍝ smallest value in B
Ymax ← ⌈/,B            ⍝ largest value in B

Example:

      ⎕PLOT 0 1 ¯1 2 ¯2 3 ¯3

produces this plot window (only visible in the HTML version of this document; in text mode see file doc/PLOT_zigzag.png):

NOTE: In theory one can also mix real and complex values, even though doing so makes little sense. If at least one item of the plot data B is complex, then all real items in B are taken as complex with imaginary part 0.

---

      ⎕PLOT ''

   ⎕PLOT Usage:

   ⎕PLOT B     with ⍴⍴B > 0: plot B with default attribute values
   ⎕PLOT B     with integer scalar B: special ⎕PLOT functions
   A ⎕PLOT B   plot B with attribute overrides specified by A
           ├────────  0: verbosity OFF
           ├──────── ¯1: show X events
           ├──────── ¯2: show data
           ├──────── ¯3: close all windows
           ├──────── ¯4: show rendering
           ├─────── ¯6: show open handles
           └──── N > 0: close window N

   A is a nested vector of strings.
   Each string A[i] has the form "Attribute: Value"
   Colors are specified either as #RGB or as #RRGGBB or as RR GG BB)

   The attributes understood by ⎕PLOT and their default values are:

   1. Global (plot window) Attributes:

caption:            ⎕PLOT          (plot window caption)
output_filename:                   (output file name)
gui_driver:                        (GUI driver: GTK, XCB, or ASCII)
auto_close:         0              (= do not close X window automatically)
                   (1)             (= close if file was written successfully)
                   (2)             (= always close X window automatically)
with_border:        1              (= write plot area and window borders)
                   (0)             (= write only plot area to output file)
pw_pos_X:           50 pixel       (plot window position X)
pw_pos_Y:           50 pixel       (plot window position Y)
border_width:       10 pixel       (width of the window border)
pa_width:           600 pixel      (plotarea width)
pa_height:          400 pixel      (plotarea height)
pa_border_L:        50 pixel       (pixels left of the plotarea)
pa_border_R:        20 pixel       (pixels right of the plotarea)
pa_border_T:        25 pixel       (pixels above the plotarea)
pa_border_B:        25 pixel       (pixels below the plotarea)
gridX_style:        1              (X grid style = ──────── )
                   (2)             (             = ╴╴╴╴╴╴╴╴ )
                   (3)             (             = ─╴─╴─╴─╴ )
axisX_arrow:        0              (X-axis arrow)
axisX_label:        X              (X-axis label)
axisY_arrow:        0              (Yaxis arrow)
axisY_label:        Y              (Y-axis label)
axisZ_arrow:        0              (Z-axis arrow)
axisZ_label:        Z              (Z-axis label)
gridX_pixels:       44 pixel       (pixels between X grid lines)
gridX_variable:     0              (draw X grid (only) at plot points)
gridX_line_width:   1 pixel        (thickness of the X-grid lines)
gridX_color:        #000000        (color of the X-grid lines)
gridY_style:        1              (Y grid style, see gridX_style above)
gridY_pixels:       33 pixel       (pixels between Y grid lines)
gridY_line_width:   1 pixel        (thickness of the Y-grid lines)
gridY_color:        #000000        (color of the Y-grid lines)
gridZ_style:        1              (Z grid style, see gridX_style above)
gridZ_pixels:       33 pixel       (pixels between Z grid lines)
gridZ_line_width:   1 pixel        (thickness of the Z-grid lines)
gridZ_color:        #000000        (color of the Z-grid lines)
canvas_color:       #FFFFFF        (background color of the plot window)
legend_color:       #F0F0F0        (background color of the legend)
legend_X:           50 pixel       (the X position of the legend)
legend_Y:           50 pixel       (the Y position of the legend)
legend_dY:          15 pixel       (the distance between legend lines)
legend_lX:          50 pixel       (the length of the legend lines)
rangeX_min:         0.0            (the start of the X range to be plotted)
rangeX_max:         0.0            (the end of the X range to be plotted)
rangeY_min:         0.0            (the start of the Y range to be plotted)
rangeY_max:         0.0            (the end of the Y range to be plotted)
rangeZ_min:         0.0            (the start of the Z range to be plotted)
rangeZ_max:         0.0            (the end of the Z range to be plotted)
origin_X:           100 pixel      (X position offset of the origin)
origin_Y:           100 pixel      (Y position offset of the origin)
format_X:                          (format for X-axis ticks (GTK only))
                    %sT1%T2...%Tn  (static texts for ticks)
                    %G, %g         grid line number (starting at 1 or )
                    %v             value
                    %S             seconds (SS)
                    %I             minutes (MM)
                    %H, %h         hours (HH or h/hh)
                    %D, %d         day (DD or d/dd)
                    %M, %m         month (MM or m/mm)
                    %q             quarter (0..3)
                    %Q             quarter (1..4)
                    %Y, %y         year (YYYY or yy)
format_Y:                          (format for Y-axis (dito))
format_Z:                          (format for Z-axis (dito))

color_level-P:      (none)         (color gradient at P% (surface plots only))

   2. Local (plot line N) Attributes:

line_color-N:       #00FF00        (the color of plot line N)
line_style-N:       1              (line style, see gridX_style above)
line_width-N:       3 pixel        (the thickness of plot line N)
point_color-N:      #000000        (the color of the plot points)
point_style-N:      1              (= plot_points: ● )
                   (2)             (= plot_points: ▲ )
                   (3)             (= plot_points: ▼ )
                   (4)             (= plot_points: ◆ )
                   (5)             (= plot_points: ■ )
                   (6)             (= plot_points: ✚   (GTK only)
                   (7)             (= plot_points: ✖   (GTK only)
point_size-N:       8 pixel        (the outer diameter of the plot points)
point_size2-N:      0 pixel        (the inner diameter of the plot points)
legend_name-N:                     (the name of plot line N in the legend)

      Data ← ?2 10⍴10   ⍝ two rows of random data
      Attributes  ← """
legend_name-1: Random Row 1
legend_name-2: Random Row 2
                    """
      Attributes ⎕PLOT Data

Attributes.legend_name_1 ← "Random Row 1"
Attributes.legend_name_2 ← "Random Row 2"
      Attributes ⎕PLOT Data

      DATA ← 1○.1×⍳100
      ⍝ structured variable A1
      ⍝
      A1.point_style_1 ← 2             ⍝ separator: _ (underscore)
      A1.gui_driver    ← 'ASCII'       ⍝ string value quoted
      A1 ⎕PLOT DATA

   
      ⍝ multi-line string A2
      A2←«««
      point_style-1:    2             ⍝ separator: - (minus)
      gui_driver:       ASCII         ⍝ string value not quited
         »»»
      A2 ⎕PLOT DATA

      A2 ⎕PLOT 1○.1×⍳100
A ⎕PLOT 1○.1×⍳100

      ⊣ ( ⊂ "output_filename: /tmp/bitmap.png" ) ⎕PLOT 1 3 1 4 2

Attributes.line_width   ← 2   ⍝ all plot lines:     2 pixels thick,
Attributes.line_width_2 ← 4   ⍝ except plot line 2: 4 pixels thick

Q1  ← ⎕FIO.secs_epoch 2023 2 15   ⍝ Feb. 15 (middle of Q1) 00:00:00

      )CLEAR

SPQ ← 91×24×60×60                 ⍝ seconds per quarter
Q1  ← ⎕FIO.secs_epoch 2023 2 15   ⍝ Feb. 15 (middle of Q1) 00:00:00

X ← Q1 + SPQ×0 1 2 3              ⍝ middles of Q1, Q2, Q3, and Q4
Y ← 1 3 2 4                       ⍝ values for Q1, Q2, Q3, and Q4

ATT.format_X ← "Q-%Q/%y"          ⍝ X-axis ticks: quarter and year

ATT ⎕PLOT X + 0J1×Y

2.37 ⎕PNG - Portable Network Graphics

Portable Network Graphics is a file format for images, defined in RFC 2083. To quote the RFC:

The PNG format provides a portable, legally unencumbered, well-compressed, well-specified standard for lossless bitmapped image files.

• The GNU APL Color Model
• Monadic ⎕PNG
• Dyadic ⎕PNG
• APL Examples

      ⍝ load PNG file image.png
      Image ← ⎕PNG 'image.png'

      ⍝ display the image
      Handle ← ⎕PNG Image

      ⍝ close the window that displays the image
      ⎕PNG Handle

      ⍝ display PNG file image.png without storing it in a variable
      ⎕PNG ⎕PNG 'image.png'

      ⍝ Simple color conversions...

      ⍝ convert a monochrome image into the equivalent RGB image
      RGB_Gray ← 3 ⌿ Mono

      ⍝ paint all pixels of a monochrome image red
      RGB_Red ← 1 0 0 ⍀ Mono

      ⍝ paint all pixels of a monochrome image green
      RGB_Green ← 0 1 0 ⍀ Mono

      ⍝ paint all pixels of a monochrome image blue
      RGB_Blue ← 0 0 1 ⍀ Mono

2.38 ⎕PS - Print Style

⎕PS is an integer vector that currently (read: as of SVN 982) contains two integers which control some details of how APL values are printed. The default value of ⎕PS is 0 0. ⎕PS is a session variable which survives the )LOADing of workspaces.

⎕PS[1] = 0: print rational quotients as normal floating point numbers (digits, fractional point, possibly an exponent).

⎕PS[1] = 1: print rational quotients as Numerator÷Denominator.

Note: ⎕PS[1] has no effect if rational numbers were not ./configure’d.

⎕PS[2] = 0: no "boxing" of APL values

⎕PS[2] > 0: "boxing" of APL values according to ⎕PS[2].

Setting ⎕PS[2] has the same effect as the debug command ]BOXING and uses the same values.

Note: For compatibility with older workspaces, assigning a single value to ⎕PS assigns that value to ⎕PS[2] and sets ⎕PS[1] to 0.

2.39 ⎕RE - Regular Expressions

⎕RE is a function that provides access to a subset of libpcre2, which is a powerful regular expression matching library. "pcre" is an acronym for "Perl compatible regular expressions". libpcre2 is Copyright (c) 1997-2017 University of Cambridge, England.

• Preconditions
• ⎕RE Syntax
• The Regular Expression A
• The String(s) B Being Matched
• The Flags X
• Complex Matches

Z ← A ⎕RE B      (short form, no axis)
Z ← A ⎕RE[X] B   (long form, with axis X)

    A ⎕RE B   ←→   A ⎕RE[''] B

    A ⎕RE B   ←→   A ⎕RE ¨ B

      ⍝ return (first) matched string
      4 ⎕CR 'f..' ⎕RE[''] '__foo___fun____fox'
┏→━━┓
┃foo┃
┗━━━┛

      ⍝ return (first) pair (position, length)
      4 ⎕CR 'f..' ⎕RE['↓'] '__foo___fun____fox'
┏→━━┓
┃2 3┃
┗━━━┛

      ⍝ return left argument A of A ⊂ B
      4 ⎕CR 'f..' ⎕RE['⊂'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      ⍝ return left argument A of A / B
      4 ⎕CR 'f..' ⎕RE['/'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      ⍝ return first match as string
      4 ⎕CR 'f..' ⎕RE[''] '__foo___fun____fox'
┏→━━┓
┃foo┃
┗━━━┛

      ⍝ return all matches as strings
      4 ⎕CR 'f..' ⎕RE['g'] '__foo___fun____fox'
┏→━━━━━k━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━┓ ┏→━━┓┃
┃┃foo┃ ┃fun┃ ┃fox┃┃
┃┗━━━┛ ┗━━━┛ ┗━━━┛┃
┗∊━━━━━━━━━━━━━━━━┛

      ⍝ return first match as pair (position, length)
      4 ⎕CR 'f..' ⎕RE['↓'] '__foo___fun____fox'
┏→━━┓
┃2 3┃
┗━━━┛

      ⍝ return all matches as pair (position, length)
      4 ⎕CR 'f..' ⎕RE['↓g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━┓
┃┏→━━┓ ┏→━━┓ ┏→━━━┓┃
┃┃2 3┃ ┃8 3┃ ┃15 3┃┃
┃┗━━━┛ ┗━━━┛ ┗━━━━┛┃
┗∊━━━━━━━━━━━━━━━━━┛

      ⍝ return first match as left argument of ⊂ (aka. partition)
      4 ⎕CR 'f..' ⎕RE['⊂'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      ⍝ return all matches as left argument of ⊂ (aka. partition)
      4 ⎕CR 'f..' ⎕RE['⊂g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 2 2 2 0 0 0 0 3 3 3┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      ⍝ return first match as left argument of / (aka. compress)
      4 ⎕CR 'f..' ⎕RE['/'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      ⍝ return all matches as left argument of / (aka. compress)
      4 ⎕CR 'f..' ⎕RE['/g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 1┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      A ⌷RE['/'] B   ←→   0 ≠ A  A ⌷RE['⊂'] B

       4 ⎕CR 'g..' ⎕RE[''] '__foo___fun____fox'
┏⊖┓
┃0┃
┗━┛
       'g..' ⎕RE['E'] '__foo___fun____fox'
DOMAIN ERROR+
      'g..' ⎕RE['E']'__foo___fun____fox'
      ^            ^
      )MORE
No match

      4 ⎕CR 'g..' ⎕RE['Eg'] '__foo___fun____fox'
┏⊖┓
┃0┃
┗━┛

      4 ⎕CR 'f(.)(.)' ⎕RE[''] '__foo___fun____fox'
┏→━━━━━━━━━━━━┓
┃┏→━━┓ ┏→┓ ┏→┓┃
┃┃foo┃ ┃o┃ ┃o┃┃
┃┗━━━┛ ┗━┛ ┗━┛┃
┗∊━━━━━━━━━━━━┛

      4 ⎕CR 'f(.)(.)' ⎕RE['g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━┓┃
┃┃┏→━━┓ ┏→┓ ┏→┓┃ ┃┏→━━┓ ┏→┓ ┏→┓┃ ┃┏→━━┓ ┏→┓ ┏→┓┃┃
┃┃┃foo┃ ┃o┃ ┃o┃┃ ┃┃fun┃ ┃u┃ ┃n┃┃ ┃┃fox┃ ┃o┃ ┃x┃┃┃
┃┃┗━━━┛ ┗━┛ ┗━┛┃ ┃┗━━━┛ ┗━┛ ┗━┛┃ ┃┗━━━┛ ┗━┛ ┗━┛┃┃
┃┗∊━━━━━━━━━━━━┛ ┗∊━━━━━━━━━━━━┛ ┗∊━━━━━━━━━━━━┛┃
┗∊∊━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

      4 ⎕CR 'f(.)(.)' ⎕RE['↓'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━┓
┃2 3 ┏→━━┓ ┏→━━┓┃
┃    ┃3 1┃ ┃4 1┃┃
┃    ┗━━━┛ ┗━━━┛┃
┗∊━━━━━━━━━━━━━━┛
      4 ⎕CR 'f(.)(.)' ⎕RE['↓g'] '__foo___fun____fox'
┏→━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃┏→━━━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━━━━┓ ┏→━━━━━━━━━━━━━━━━━┓┃
┃┃2 3 ┏→━━┓ ┏→━━┓┃ ┃8 3 ┏→━━┓ ┏→━━━┓┃ ┃15 3 ┏→━━━┓ ┏→━━━┓┃┃
┃┃    ┃3 1┃ ┃4 1┃┃ ┃    ┃9 1┃ ┃10 1┃┃ ┃     ┃16 1┃ ┃17 1┃┃┃
┃┃    ┗━━━┛ ┗━━━┛┃ ┃    ┗━━━┛ ┗━━━━┛┃ ┃     ┗━━━━┛ ┗━━━━┛┃┃
┃┗∊━━━━━━━━━━━━━━┛ ┗∊━━━━━━━━━━━━━━━┛ ┗∊━━━━━━━━━━━━━━━━━┛┃
┗∊∊━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛

2.40 ⎕RVAL - Random APL value

The standard way to produce random numbers is the primitive function ? aka. Roll. Roll returns a simple random array whose ravel elements are positive integers chosen in a (pseudo-) random fashion. While this is sufficient for many applications, one sometimes needs APL values whose randomness not only affects the ravel elements themselves, but also other aspects like:

• the rank of the value,
• the shape of the value,
• the data type of the ravel elements, and
• the depth (nesting) of the values.

Such values can be produced with system function ⎕RVAL. The main use case for ⎕RVAL is the production of test data for the interpreter, but ⎕RVAL might be useful for other purposes as well.

• General
• Dyadic ⎕RVAL

Rank:  0
Shape: 1 1 1 1 1 1 1 1 (initially irrelevant since Rank = 0)
Type:  0 1 0 0 0       (only integer random values)
Max. Depth: 4          (initially irrelevant since type = 0 1 0 0 0)

      STATE ← 0 ⎕RVAL ''

      0 ⎕RVAL STATE

1 ⎕RVAL 2        ⍝ produce matrices
2 ⎕RVAL 3 ¯10    ⍝ the first axis of every matrix will have length 3, and
                 ⍝ the last axis of every matrix will vary between 0 and 10

      1 ⎕RVAL 2         ⍝ produce matrices
      2 ⎕RVAL 3 3       ⍝ produce 3×3 matrices
      3 ⎕RVAL 0 50 50   ⍝ produce a mix of integer and real values
      ⎕RVAL 0
¯975954163190248487                   0.5892132425
7677327503669503253 2152001972871424768

      1 ⎕RVAL ⍬    ⍝ rank (scalar)
0

      2 ⎕RVAL ⍬    ⍝ shape (not used since rank = 0)
1 1 1 1 1 1 1 1

      3 ⎕RVAL ⍬    ⍝ types (integer)
0 1 0 0 0

      4 ⎕RVAL ⍬    ⍝ max. depth (not used since probability of NESTED = 0)
4

   Z←⎕RVAL B does essentially:

   B[1]←1 ⎕RVAL B[1]
   B[2]←2 ⎕RVAL B[2]
   B[3]←3 ⎕RVAL B[3]
   B[4]←4 ⎕RVAL B[4]

   Z←⎕RVAL 0

   B[1]←1 ⎕RVAL B[1]
   B[2]←2 ⎕RVAL B[2]
   B[3]←3 ⎕RVAL B[3]
   B[4]←4 ⎕RVAL B[4]

⎕RVAL 1 (,¯4) (0 1)    ⍝ return a random 0-4 element integer vector
⎕RVAL 2 (2 2) (1 0)    ⍝ return a random 2×2 character array

      0 ⎕RVAL B  ←→  'state' ⎕RVAL B  ←→  ⎕RVAL.state B
      1 ⎕RVAL B  ←→   'rank' ⎕RVAL B  ←→  ⎕RVAL.rank  B
      2 ⎕RVAL B  ←→  'shape' ⎕RVAL B  ←→  ⎕RVAL.shape B
      3 ⎕RVAL B  ←→   'type' ⎕RVAL B  ←→  ⎕RVAL.type  B
      4 ⎕RVAL B  ←→  'depth' ⎕RVAL B  ←→  ⎕RVAL.depth B

2.41 ⎕SI - State Indicator

⎕SI returns aspects of the current State Indicator, similar to the standard command )SI. This can be used, for example, to create debug functions similar to the assert() macro in C/C++:

∇Assert B;COND;LOC;VAR
 →(1≡B)⍴0
 ' '
 COND←7↓,¯2 ⎕SI 4
 LOC←,¯2 ⎕SI 3
 '************************************************'
 ' '
 '*** Assertion (', COND, ') failed at ',LOC
 ''

 ⍝ show stack
 ⍝
 ' '
 'Stack:'
 7 ⎕CR ⊃¯1↓⎕SI 3
 ' '
 '************************************************</pre>'
 →
∇

The right argument of ⎕SI specifies which aspect of the State Indicator shall be returned:

• ⎕SI 1: The name of the context. That name is either:
  • the name of a defined function, or
  • ◊ for an immediate execution context, or
  • ⍎ for an execute context.
• ⎕SI 2: The line number (of a defined function) or 0 for immediate execution and execute contexts.
• ⎕SI 3: The function name and line number in square brackets, for example: "FOO[3]"
• ⎕SI 4: Either the statement text of the function line or the error text of an error that has occurred on the line
• ⎕SI 5: The program counter (= token number counted from the start of the function text)
• ⎕SI 6: the parse mode of the context (immediate execution, execute, or defined function).

If no left argument is provided then the result of ⎕SI is a vector with one entry per State Indicator level (and hence ⍴⎕SI B is the depth of the SI stack).

If the optional left argument A is provided then it specifies a particular level of the SI instead of the entire SI. A should be an integer scalar. If A is positive then the level is counted from the oldest entry to the latest, while negative A counts from the latest to the oldest level.

For example, ¯1 ⎕SI refers to the currently executing context, ¯2 ⎕SI is the caller, and so on.

2.42 ⎕SQL - SQL Database Interface

⎕SQL was kindly contributed by Elias Mårtenson.

As of GNU APL 1.6, the native function SQL has been replaced by the system function ⎕SQL, described below. ⎕SQL has an axis argument that selects a subfunction of ⎕SQL.

• ⎕SQL[0] ⍬ : display subfunction numbers
• ⎕SQL[0] '' : display subfunction names
• ref ← A ⎕SQL[1] B : open database
• ⎕SQL[2] B : close database
• Z ← A ⎕SQL[3, ref] B : database query (with result)
• Z ← A ⎕SQL[4, ref] B : database query (w/o result)
• ⎕SQL[5] B : start transaction
• ⎕SQL[6] B : commit transaction
• ⎕SQL[7] B : rollback transaction
• Z←⎕SQL[8] B : table names
• Z←⎕SQL[9] B : column names
• Z ← ⎕SQL[10] B : library version number
• Z ← ⎕SQL[11] B : library version string
• SQLite Quickstart
• Deleting Rows From a Database Table

    Valid (sub-)function numbers for ⎕SQL:

    Legend: Fs - database file name (path)
            Ty - database type ('sqlite' or 'postgres')
            Db - database handle (small integer)
            Qs - SQL query string
            Pv - query parameters (APL values, to be bound to Qs)
            Ts - name of a table in the database (string)
            Vi - DB provider (library) version (integer)
            Vs - DB provider (library) version (string)

            ⎕SQL[0] ''        list the ⎕SQL (sub-)function names
            ⎕SQL[0] ⍬         list the ⎕SQL (sub-)function numbers
    Db ← Ty ⎕SQL[1] Fs        open database Fs & return a handle for it
            ⎕SQL[2] Db        close database handle Db
         Qs ⎕SQL[3, Db] Pv    perform SQL query Qs
         Qs ⎕SQL[4, Db] Pv    perform SQL update Qs
            ⎕SQL[5] Db        begin a transaction
            ⎕SQL[6] Db        commit the current transaction
            ⎕SQL[7] Db        roll the current transaction back
            ⎕SQL[8] Db        list the tables in database Db
         Db ⎕SQL[9] Ts        list the column names and types of table Tn
    Vi ←    ⎕SQL[10] Ty       the provider version number for type Ty
    Vs ←    ⎕SQL[11] Ty       the provider version string for type Ty

    With a small performance penalty, ⎕SQL also accepts the following
    subfunction names instead of subfunction numbers as axis argument:

    Legend: Fs - database file name (path)
            Ty - database type ('sqlite' or 'postgres')
            Db - database handle (small integer)
            Qs - SQL query string
            Pv - query parameters (APL values, to be bound to Qs)
            Ts - name of a table in the database (string)
            Vi - DB provider (library) version (integer)
            Vs - DB provider (library) version (string)

       ⎕SQL[5]         ←→  ⎕SQL.begin         ⍝ begin a transaction
       ⎕SQL[9]         ←→  ⎕SQL.columns       ⍝ show the columns of a table
       ⎕SQL[6]         ←→  ⎕SQL.commit        ⍝ end a transaction
       ⎕SQL[2]         ←→  ⎕SQL.close         ⍝ close a database handle
       ⎕SQL[0]         ←→  ⎕SQL.list ''       ⍝ ⎕SQL function names/numbers
       ⎕SQL[1] Fs      ←→  ⎕SQL.open          ⍝ open the database file
       ⎕SQL[3, Db] Pv  ←→  ⎕SQL.query Db Pv   ⍝ SQL database query
       ⎕SQL[7] Db      ←→  ⎕SQL.rollback      ⍝ roll a transaction back
       ⎕SQL[8] Db      ←→  ⎕SQL.tables        ⍝ show all tables
    Qs ⎕SQL[4, Db] Pv  ←→  ⎕SQL.update Db Pv  ⍝ SQL database update
       ⎕SQL[10]        ←→  ⎕SQL.version       ⍝ SQL provider version number
       ⎕SQL[11]        ←→  ⎕SQL.vstring       ⍝ SQL provider version string

      type  ⎕SQL[1] file

      ⎕SQL[2] ref

      query ⎕SQL[3, ref] params

      query ⎕SQL[4, ref] params

      ⎕SQL[5] ref

      ⎕SQL[6] ref

      ⎕SQL[7] ref

      ⎕SQL[8] ref

      ref  ⎕SQL[9] table

      )CLEAR
CLEAR WS

      ⍝ delete any existing database
      ⍝
      Filename ← "db.sqlite"                ⍝ name of the database to be deleted
      ⎕FIO.strerror ⎕FIO.unlink Filename
Success

      ⍝ verify that it was deleted
      ⍝
      ⎕FIO.strerror ⎕FIO.unlink Filename
No such file or directory

      )CLEAR
CLEAR WS

      ⍝ create a new (and empty) database
      ⍝
      Filename ← 'db.sqlite'            ⍝ name of the database to be created
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ create the database

      ⎕SQL.close DB                     ⍝ and close it

      )CLEAR
CLEAR WS

      Filename ← 'db.sqlite'            ⍝ name of the database to be used
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ open the database

      ⍝ define an SQL Query string Qs that creates a table
      ⍝
      Qs ← ,⊃«««
      CREATE TABLE EMPLOYEES(
             ID      INT  NOT NULL PRIMARY KEY,
             NAME    TEXT NOT NULL,
             AGE     INT  NOT NULL,
             ADDRESS CHAR(50),
             SALARY  REAL);
             »»»

      ⍝ perform the SQL query according to query string Qs.
      ⍝
      Qs ⎕SQL.query DB ⍬

      ⍝ show the tables in the database
      ⍝
      ⎕SQL.tables DB
 EMPLOYEES 

      DB ⎕SQL.columns 'EMPLOYEES'
 ID      INT      
 NAME    TEXT     
 AGE     INT      
 ADDRESS CHAR(50) 
 SALARY  REAL     

      ⎕SQL.close DB                     ⍝ and close DB

      ⍝ dynamic query string Qs
      ⍝
      Qs ← ,⊃«««
      INSERT INTO EMPLOYEES(ID, NAME, AGE, ADDRESS, SALARY
               VALUES(?,  ?,    ?,   ?,       ?);
           »»»

      ⍝ query string parameters
      ⍝
      PARAMS ← 1 "Paul" 32 "California" 20000

      ⍝ number of placeholders in Qs
      ⍝
      +/Qs = '?'
5

      ⍝ number of parameters (shall be equal to the number of placeholders)
      ⍝
      ⍴PARAMS
5

      ⍝ the (intermediate) result of replacing the placeholders in the dynamic
      ⍝ query string Qs would then be this static query string:
      ⍝
      RESULT ← ,⊃«««
      INSERT INTO EMPLOYEES(ID, NAME, AGE, ADDRESS, SALARY
               VALUES(1, "Paul", 32, "California", 20000);
                 »»»

      Qs ⎕SQL.query DB PARAMS       ⍝ ⎕SQL subfunction name
      Qs ⎕SQL[3, DB] PARAMS         ⍝ ⎕SQL subfunction number

      Qs ⎕SQL.update DB PARAMS      ⍝ ⎕SQL subfunction name
      Qs ⎕SQL[4, DB] PARAMS         ⍝ ⎕SQL subfunction number

      )CLEAR
CLEAR WS

      MATRIX ← 4 5⍴0
      MATRIX[1;] ← 1 'Paul'   32 'California' 20000.00
      MATRIX[2;] ← 2 'Allan'  25 'Texas'      15000.00
      MATRIX[3;] ← 3 'Teddy'  23 'Norway'     20000.00
      MATRIX[4;] ← 4 'Gunnar' 26 'Sweden'     18000.00

      ⍝ show MATRXIX
      ⍝
      8 ⎕CR MATRIX
┌→─────────────────────────────┐
↓┌→───┐   32 ┌→─────────┐ 20000│
││Paul│      │California│      │
│└────┘      └──────────┘      │
│┌→────┐  25 ┌→────┐      15000│
││Allan│     │Texas│           │
│└─────┘     └─────┘           │
│┌→────┐  23 ┌→─────┐     20000│
││Teddy│     │Norway│          │
│└─────┘     └──────┘          │
│┌→─────┐ 26 ┌→─────┐     18000│
││Gunnar│    │Sweden│          │
│└──────┘    └──────┘          │
└ϵ─────────────────────────────┘

      )CLEAR
CLEAR WS

      Filename ← 'db.sqlite'            ⍝ name of the database to be used
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ open the database

      Qs ← ,⊃«««
      INSERT INTO EMPLOYEES(ID, NAME, AGE, ADDRESS, SALARY)
                     VALUES(?,  ?,    ?,   ?,       ?);
           »»»
     Qs ⎕SQL.query DB MATRIX

      ⎕SQL.close DB                     ⍝ and close DB

      )CLEAR
CLEAR WS

      Filename ← 'db.sqlite'            ⍝ name of the database to be used
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ open the database

      Qs ← ,⊃«««
      SELECT * FROM EMPLOYEES;
             »»»

      RESULT ← Qs ⎕SQL.query DB ⍬
      RESULT
 1 Paul   32 California 20000 
 2 Allan  25 Texas      15000 
 3 Teddy  23 Norway     20000 
 4 Gunnar 26 Sweden     18000 

      ⍴RESULT
4 5

      ⎕SQL.close DB                     ⍝ and close DB

      )CLEAR
CLEAR WS

      Filename ← 'db.sqlite'            ⍝ name of the database to be used
      Type ← 'sqlite'                   ⍝ database type
      DB ← Type ⎕SQL.open Filename      ⍝ open the database

      Qs ← ,⊃«««
      UPDATE EMPLOYEES SET SALARY=? WHERE ID=?;
             »»»

      Qs ⎕SQL.query DB (22000 3)        ⍝ update column SALARY in row 3

      ⍝ check result
      Qs ← ,⊃«««
      SELECT * FROM EMPLOYEES;
             »»»

      RESULT ← Qs ⎕SQL.query DB ⍬
      RESULT
 1 Paul   32 California 20000 
 2 Allan  25 Texas      15000 
 3 Teddy  23 Norway     22000 
 4 Gunnar 26 Sweden     18000 

      ⍴RESULT
4 5

      ⎕SQL.close DB                     ⍝ and close DB

      UPDATE COMPANY SET AGE=?, SALARY=? WHERE ID IN (?, ?, ?);

      Qs←«DELETE FROM COMPANY WHERE ID=?;»                ⍝ delete one row
      Qs←«DELETE FROM COMPANY WHERE ID IN (?, ?, ?);»     ⍝ delete 3 rows
      Qs←«DELETE FROM COMPANY;»                           ⍝ delete all (!) rows

      Qs ⎕SQL.query DB B

2.43 ⎕SYL - System limits

System variable ⎕SYL shows a number of system limits when referenced:

      ⎕SYL
 SI depth limit          (0 = no limit)                    0 
 number of values limit  (0 = no limit)                    0 
 total ravel bytes limit (0 = no limit)                    0 
 current SI depth                                          1 
 current number of values                                 54 
 current total ravel bytes                              9072 
 max. rank for APL values                                  8 
 min. ⎕PW                                                 30 
 max. ⎕PW                                               1000 
 min. ⎕PP                                                  1 
 max. ⎕PP                                                 16 
 max. input line length                                 2000 
 hash table size (obsolete)                            65536 
 max. shared variable name length                         64 
 max. length of filenames (paths)                       4096 
 max. # of shared variables (obsolete)                    64 
 max. number of APs                                       16 
 max. operators per statement                             16 
 largest integer                         9000000000000000000 
 smallest integer                       ¯9000000000000000000 
 largest numeric exponent                                308 
 max. shared variable size (bytes)                     65000 
 max. cores (per ./configure)                              0 
 max. cores (per max_cores())                              1 
 current cores (per core_count())                          1 
 print length limit (0 = no limit)                         0 

Most system limits are read-only; attempting to assign ⎕SYL will result in a SYNTAX ERROR. Indexed assignment to ⎕SYL will result in an INDEX ERROR unless the limit is writable. Some read-only limits can be changed at compile-time via ./configure

• ⎕SYL[1 2 3 4 5 6;]
• ⎕SYL[7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23;]
• ⎕SYL[24 25 26;]
• ⎕SYL[27;]

      ⎕SYL[1;2]←20

      SH←⍴VALUE
      while (ravel-length(SH) ≥ ⎕SYL[27;2]) { divide longest axis in SH by 2 }

2.44 ⎕XML - XML Parsing

• Purpose
• Some XML Terminology
• How ⎕XML maps XML documents to APL Values
• Monadic ⎕XML
• Dyadic ⎕XML
• XML Queries

      XML_string ← "<A><B><C>Hello</C></B></A>"   ⍝ input data (usually from an .xml file)
      APL ← ⎕XML XML_string       ⍝ convert XML_string to associative array APL
      ABC ← APL.a.b.c             ⍝ retrieve node a.b.c in APL
      APL.a.b.c ← ABC, " World"   ⍝ modify node a.b.c in APL
      Z ← ⎕XML APL                ⍝ convert associative array APL to XML string Z

      ABC   ⍝ the original value of XML.a.b.c
Hello
      Z     ⍝ the modified XML string
<A><B><C>Hello World</C></B></A>"

      ⍙1              ⍝ the first member in every XML node. Its value is the node tag.
      ∆1declaration   ⍝ the first member in most XML documents. E.g. <?xml version= ...>
      ∆2text          ⍝ whitespace (LF) between _1∆declaration and _3∆doctype
      ∆3doctype       ⍝ DTD in XML documents that have one. E.g. <!DOCTYPE  ...
      _6Workspace     ⍝ Sub-array for <Workspace ... at position 6.
      ⍙1wsid          ⍝ first attribute wsid from e.g. <Workspace wsid=...
      ⍙2year          ⍝ second attribute year

      XML←36 ⎕CR """
<?xml version='1.0' encoding='UTF-8'?>
  <Document>
    <Tag1 name='tag1'>   <!-- first tag -->
      TEXT1
      <Subtag name='subag1.1'>   <!-- first subtag -->
        SUBTEXT1_1
      </Subtag>
    </Tag1>
  </Document>
"""

APL←⎕XML XML

2.45 Conditionals

WARNING: GNU APL conditionals are experimental. Use them with care and avoid them, wherever possible, in favour of portability. Neither the ISO APL standard, nor IBM APL2 provide conditionals.

• Design considerations.
• Syntax
• A Possible Pitfall

      ⍝ short format. Valid in immediate execution and in defined functions
      ⍝
      X←5 ◊ X≥0 →→ X 'is positive' ←→ X 'is negative' ←←
 5 is positive 


      ⍝ long format. Only possible in defined functions because immediate
      ⍝ execution works line by line (and parsing the first line would fail).

      ⍝ valid split into several lines (in a defined function)
      ⍝
      ∇FOO X
       X≥0             →→
       X 'is positive' ←→
       X 'is negative' ←←
      ∇
      FOO 5
 5 is positive 
      FOO ¯5
 ¯5 is negative 


      ⍝ invalid split into several lines: ←→ spread over 2 lines.
      ⍝ FOO can be properly parsed, but calling FOO fails.
      ⍝
      ∇FOO X
      X≥0 →→ X 'is positive' ←
           → X 'is negative' ←←
      ∇
      FOO 5
SYNTAX ERROR+
FOO[1]  X 'is positive'←
        *              *

   if (COND) { IF-statements }       // conditional C/C++ statement
   else      { ELSE-statements }

   (COND) ? IF-value : ELSE-value    // conditional C/C++ expression

      COND →→ IF-statements ←→ ELSE-statements ←←    ⍝ conditional APL statement

      IF-value ⊢[COND] ELSE-value                    ⍝ conditional APL expression

      // C/C++
      x = -5;
      Result = (x < 0) ? "negative" : "positive";                          // OK
      Result = if (x < 0) { "negative"; } else { "positive"; }         // syntax error
      if (x < 0) { Result = "negative"; } else { Result = "positive"; }   // OK

      ⍝ GNU APL conditional expression, aka. ⊢[]
      X←¯5
      Result ← "positive" ⊢[X < 0] "negative"                     ⍝ OK
      Result ← (X < 0) →→ ⊣"negative" ←→ ⊣"positive" ←←           ⍝ ???
      Result
1

      (X < 0) →→ Result ← "negative" ←→ Result ← "positive" ←←    ⍝ OK
      Result
negative

2.46 Matrix Product

In standard APL, the inner product Z of two matrices A and B as known from linear algebra, is:

      Z←A +.× B      ⍝ the most frequent special case of A f.g B

However, the inner product in APL is more general than the inner product in linear algebra. In the general case A f.g B may the functions f and g also be non-scalar APL primitives or even defined functions. This generality may occasionally be helpful, but it comes with a performance penalty since many argument checks are duplicated. At the same time A +.× B is by far the most frequent use of the inner product operator ’.’.

For this reason GNU APL also provides a slightly more efficient dyadic function ∘ (Matrix Product) for the special case +.× of the dyadic operator ’.’. This function ∘ has been optimized for numeric arguments and computes the matrix product as known from linear algebra.

For two-dimensional matrices A and B is function ∘ conceptually the same as +.×:

A∘B ←→ A +.× B

However, there are some subtle differences between the function ∘ and the operator +.×:

• A +.× B allows A and B to be of any rank, while A∘B raises a RANK ERROR if the rank of A or B is more than 2.
• The product A +.× B can be a vector (e.g. if A is a vector and B is a matrix), usually understood as row-vector. In contrast (and more similar to linear algebra) A∘B is always a matrix.
• In order to preserve the distinction between row vectors and column vectors the arguments A and B of ∘ are possibly reshaped according to the following rules:
  • if A and/or B is a scalar then the normal (element-wise) product of A and B is computed. That is, A∘B ←→ A×B.
  • if A is a APL vector of length N then it is reshaped to a row-vector before A∘B is computed. In this context, the term row-vector shall refer to a matrix of shape (1 N).
  • if B is a APL vector of length N then it is reshaped to a column-vector before A∘B is computed. In this context, the term column-vector shall refer to a matrix of shape (N 1).

  Examples:

        M←2 2⍴1 2 3 4   ⍝ a matrix
        V←10 11         ⍝ a vector
  
        M +.× V
  32 74
  
        M∘V
  32
  74
  
        V +.× M
  43 64
  
        V∘M
  43 64
  
        ⍴M +.× V   ⍝ the result is an (APL-) vector of length 2
  2
        ⍴V +.× M   ⍝ the result is an (APL-) vector of length 2
  2
        ⍴V∘M       ⍝ the result is row vector
  1 2
        ⍴M∘V       ⍝ the result is column vector
  2 1
  

• A +.× B requires that the number of columns in A is equal to the number of rows in B. In contrast, A∘B has no such requirement and will add 0-columns to A resp. 0-rows to B to match the shapes of A and B. Naturally, the multiplications of these 0-columns or 0-rows is not in fact performed, but optimized away.

2.47 Monadic ≠ (Nub Sieve)

The nub sieve Z←≠B of a value B is a Boolean vector Z with the same shape as B and:

Z[i] is 1 if B[i] is the first occurrence of B[i] in B and 0 otherwise. For vectors B this can also be expressed as:

      ≠ B ←→ (B⍳B)=(⍳⍴B)

Examples:

      ≠'Hello, World.'
1 1 1 0 1 1 1 1 0 1 0 1 1
      ≠ 1 (2 3) 4 (2 3) 5 6
1 1 1 0 1 1

3.1 System Limits

APL floating point values are 64-bit wide, thus ranging from -8.98E307 to 8.98E307.

GNU APL integers have a guaranteed range from -9200000000000000000 to 9200000000000000000, which is a slighly smaller range.

A 64-bit integer represents a value between -9223372036854775808 and 9223372036854775807.

If the result of a computation is integer by nature, for example +, -, or × with integer arguments, and falls into the guaranteed range, then the result will be an APL integer. If the result is too large for a 64-bit signed integer then the result will be automatically converted to an APL floating point value. However, if the result is outside the guaranteed range but still within the 64-bit integer range, e.g. between 9200000000000000000 and 9223372036854775807 (exclusive), then GNU APL gives no guarantees as to whether the result will be a 64-bit integer or a 64-bit floating point (with the resulting loss of precision). This decision is usually function-specific and driven by performance considerations.

APL values have a maximum rank of 8. However, the maximum range can be set by means of ./configure (see README-2-configure).

3.2 Shared Variables

The system functions and variables related to shared variables, i.e. ⎕SVO, ⎕SVR, ⎕SVC, ⎕SVS, and ⎕SVQ, are implemented.

Communication between two workspaces is supported, but with some limitations. Two workspaces can only communicate via shared variables if they (i.e. their processes) run on the same machine.

However, only two auxiliary processors, AP100 and AP210, are provided as examples of how to implement auxiliary processors in GNU APL.

Shared variables for auxiliary processors are provided for some backward compatibility. In the past - without access to the source code of the APL interpreter - shared variables were the only method available to extend the functionality of the interpreter.

With this interpreter - and access to its source code - it is often more convenient to add your own commands or your own system variables to the APL interpreter rather than adding auxiliary processors.

5.1 History

Vector Notation is as old as APL. The famous Book APL \360: An interactive approach by Leonard Gilman and Allen J. Rose introduces vector notation already on page 2, but without calling it vector notation.

Quote:

Or we can assign a string of numbers to a variable called X, and ask the
computer to execute the command shown, + / X , with the response 17.1 :
      X+-3 4 1.1 3 6
      +/X
17.1

The terms vector notation (used by IBM APL and by GNU APL) was apparently coined by Kenneth Iverson (A Programming Lagnuage, p. 163) and the equivalent term strand notation is used by other APL vendors.

5.2 Standardization

The current ISO standard ISO/IEC 13751 Programming Language APL, extended, in the following abbreviated as iso, does not formally define vector notation, but uses it all over the place in its examples.

The IBM APL2 Language Reference Manual, in the following abbreviated as lrm, was the blueprint for GNU APL. The lrm describes vector notatation briefly.

5.3 Quotes from the lrm

Different aspects of vector nortation are distributed over the lrm. To get a compressed view of the matter we quote the relevant pieces:

1. lrm page 14: The juxtaposition of two or more arrays in an expression results in a vector whose items are the arrays. Representing a vector in this manner is called vector notation.
2. lrm page 14: Characters in a vector consisting only of characters can be listed within one set of single quotation marks: ’FACE’ ←→ (’F’ ’A’ ’C’ ’E’)
3. lrm page 33: The hierarchy of binding strengths is listed below in descending order.

           Binding Strength         What Is Bound
        ──────────────────────      ────────────────────────────────────
        1. Brackets                 Brackets to what is on their left
        2. Specification left       Left arrow to what is on its left
        3. Right operand            Dyadic operator to its right operand
        4. Vector                   Array to an array
        5. Left operand             Operator to its left operand
        6. Left argument            Function to its left argument
        7. Right argument           Function to its right argument
        8. Specification right      Left arrow to what is on its right
        ════════════════════════════════════════════════════════════════
   

   The binding strength 4. above is the one for vector notation. It defines how the items of a vector notation are bound together (as opposed to being bound to other tokens). For example: 1 2 3[2] becomes 1 2 (3[2]) because [2] binds stronger to 3 left of it than 3 binds to 2 left of it (which then raises a RANK ERROR because 3 is a scalar).

5.4 Some APL2 Tests

Before explaining the details of vector notation in GNU APL we would like to share the results of some tests performed with IBM APL2 (again the demo version for PCs).

• Test 1: Binding strength of []
• Test 2: Binding strength of the left operand

      ⍝ IBM APL2: Binding strength of []...

      ⍝ Test 1a.
      ⍝
      1 2 3[2]
RANK ERROR
      1 2 3[2]
          ^^

      ⍝ Test 1b.
      ⍝
      'a' 'b' 'c'[2]
RANK ERROR
      'ab' 'c'[2]
           ^  ^

      ⍝ Test 1c.
      ⍝
      'abc'[b][2]
b

      ⍝ display the name class of B¯B
      ⍝
      ∇Z←SHOW_NC B¯B;N
       N←8↑B,':'
       →(¯1 0 1 2 3 4 = ⎕NC B¯B)/I U L V F O ◊ ++
      I: →0, Z←(8↑B), ': Invalid Name'
      U: →0, Z←(8↑B), ': Unused Name'
      L: →0, Z←(8↑B), ': Label'
      V: →0, Z←(8↑B), ': Value', ⍎B¯B
      F: →0, Z←(8↑B), ': Function'
      O: →0, Z←(8↑B), ': Operator'
      ∇

      ⍝ monadic operator
      ⍝
      ∇Z←A (LO OPER1) B
       SHOW_NC 'A'
       SHOW_NC 'LO'
       SHOW_NC 'OPER1'
       SHOW_NC 'B'
      ∇

      ⍝ dyadic operator
      ⍝
      ∇Z←A (LO OPER2 RO) B
       SHOW_NC 'A'
       SHOW_NC 'LO'
       SHOW_NC 'OPER1'
       SHOW_NC 'RO'
       SHOW_NC 'B'
      ∇

      ⍝ Test 2a: binding of the left operand (literal LO)
      ⍝
      11 22 33 OPER1 1 2 3
A       : Unused Name
LO      : Value  11 22 33
OPER1   : Operator
B       : Value  1 2 3

      ⍝ Test 2b: binding of the left operand (parser LO)
      ⍝
      (UU VV WW)←11 22 33
      UU VV WW OPER1 1 2 3
A       : Unused Name
LO      : Value  11 22 33
OPER1   : Operator
B       : Value  1 2 3

      ⍝ Test 2c. binding of the left operand (in parentheses)
      ⍝
      11 22 (33 OPER1) 1 2 3
A       : 11
LO      : Value  22 33
OPER1   : Operator
B       : Value  1 2 3

      ⍝ Test 2d. binding of the right operand
      ⍝
      11 22 33 'lop' OPER2 'rop' 1 2 3
A       : Unused Name
LO      : Value  11 22 33 lop
OPER2   : Operator
RO      : Value  rop
B       : Value  1 2 3

      ⍝ example from ISO/IEC 13751 p.126
      ⍝
      0 1 2⌽∘0 1 'ABC'
ABC
BCA
CAB

      ⍝ Test 2e. (GNU APL):
      ⍝
      0 1 2(⌽⍤0 1) 'ABC'
ABC
BCA
CAB

      ⍝ Test 2f. (GNU APL):
      ⍝ 
      0 1 2(⌽⍤0) 1 'ABC'
LENGTH ERROR
      0 1 2(⌽⍤0 0 0)1 'ABC'
      ^            ^

5.5 Definitions

Definition: A (complete) strand consists of two (!) or more adjacent values in a defined function, or in a string being ⍎’ed, or on a line entered in immediate execution mode. Such values can be literals (like in 1 2 3), but also values of variables or results of niladic functions. A strand is always a vector and we call the values in a strand the items of the strand. Strand items can be simple (i.e. numbers or single characters) or nested (strings or other strands). Strand items are always (possibly nested) APL scalars.

Definition: A literal strand is a strand whose items are only literals (including nested literals like in 1 2 (3 4) 5). Literal strands are created when an APL expression is tokenized. At that point in time the literal strand is incomplete and more strand items may be added to it when the statement is executed.

Note: Strands are, in general, subject to what is known as unexpected stranding, which occurs when an operator accept values as operands. This is the case for the power operatror ⍣ and for the rank operator ⍤; both can have values as right operands. The clean way of avoiding unexpected stranding is to put the operatorr and its operands into parentheses, but the iso does not do that and in order to make the examples in iso work, GNU APL has dedicated code for the rank and power operators that prevents unexpected stranding for their right operand and their right function value. For defined operators this is not possuble because at tokenization time a defined operator is only a symbol whose name class at execution time is unknown at tokenization time of its caller. For the same reason. the binding strengths defined above can not cure this kind of unexpected stranding.

Definition: An extended literal strand is a strand whose items are either literals (including nested literals) or particular simple primitive function is with particular literal arguments.

Definition: A mixed strand is a strand which is not a literal strand.

Definition: A strand is open if more items can be appended to it.

Definition: A strand is closed if it is not open.

Definition: A strand is nested if at lease one of its items is nested.

Examples:

      1 2 3        ⍝ open literal strand

      1 2 (3⍴4) 5     ⍝ open extended literal strand ←→ 1 2 4 4 4 5
      1 2 (3/4) 5     ⍝ open extended literal strand ←→ 1 2 4 4 4 5
     'ab' (⍳3) 'ef'   ⍝ open extended literal strand ←→ 'ab' (1 2 3) 'ef' 

      VAR←1 2 3       ⍝ mixed strand item
      ∇Z←FOO          ⍝ mixed strand item
       Z←1 2 3
      ∇

      1 VAR 3         ⍝ nested open mixed strand
      1 FOO 3         ⍝ nested open mixed strand

      (1 2 3)       ⍝ closed literal strand
      'abc'         ⍝ closed literal strand

5.6 Unexpected Stranding

Unexpected stranding (or better: false stranding) refers to a situation where the stranding of adjacent values leads to a wrong result. Unexpected stranding is not unexpected at all, as we will see in this section.

In non-nested APL interpreters, such as APL \360, the binding strength of strand items was the highest. That was a wise decision which had two consequences:

• Expressions like 1 2 3[2] were always valid in APL \360 and friends. The current APL2 and, (for the sake of compatibility) also GNU APL cases, where 1 2 3[2] raises a RANK ERROR, are merely an unfortunate consequence of binding expressions in brackets stronger than strand items.
• Similarly, the IBM APL2 design decision to bind right operand stronger than strand items, together with the decision to allow values as operands, create the possibility of false stranding.

Corollary 1: unexpected stranding was not an issue before binding strength in APL2 was introduced.

Corollary 2: The only binding strengths for which unexpected stranding can occur are those for that bind stronger than vector (i.e. strand item), i.e. 1. bracket, 2. specification left, and 3. right operand. Of these 3 binding strengths, only 3. right operand conflicts with strand notation (because neither [...] nor ← can be strand items.

To go into a little more detail we consider the following token sequences:

                                                        ⍝ at tokenizatio time...
    PRIMITIVE_FUNCTION         VALUE_A VALUE_B          ⍝  ├── expected
    PRIMITIVE_DYADIC_OPERATOR  VALUE_A VALUE_B          ⍝  ├── unexpected
    VALUE                      VALUE_A VALUE_B          ⍝  ├── expected
    SYMBOL                     VALUE_A VALUE_B          ⍝  └── unknown

                                                        ⍝ at execution time...
    VALUE                      VALUE_A VALUE_B          ⍝  ├── expected
    FUNCTION                   VALUE_A VALUE_B          ⍝  ├── expected
    DYADIC_OPERATOR            VALUE_A VALUE_B          ⍝  └── unexpected

The fact that VALUE_A and VALUE_B are adjacent constitues what is known as a shift/reduce confict in compiler terminology. Let’s add some parentheses to see how these shift/reduce conficts are, by means of the APL2 binding strength, supposed to be resolved:

                                                        ⍝ at tokenizatio time...
    PRIMITIVE_FUNCTION       ( VALUE_A   VALUE_B )      ⍝  ├── expected
  ( PRIMITIVE_DYADIC_OPERATOR  VALUE_A ) VALUE_B        ⍝  ├── unexpected
    VALUE                    ( VALUE_A   VALUE_B )      ⍝  ├── expected
  ? SYMBOL                   ? VALUE_A ? VALUE_B ?      ⍝  └── unknown

                                                        ⍝ at execution time...
    VALUE                    ( VALUE_A   VALUE_B )      ⍝  ├── expected
    FUNCTION                 ( VALUE_A   VALUE_B )      ⍝  ├── expected
  ( DYADIC_OPERATOR            VALUE_A ) VALUE_B        ⍝  └── unexpected

The first four sequences occurs at tokenization time, the others at execution time. The significant difference between primitive functions and primitive operators on the one hand and defined functions and defined operators on the other hand is that the parser "knows" in the first three cases if VALUE_A and VALUE_B shall be stranded together or not. In the fourth case the parser cannot decide whether SYMBOL refers to a defined function (including functions derived from monadic operators), to a variable, or to a dyadic defined operator. Therefore the parser can resolve the first three cases at tokenization time, i.e. bind VALUE_A to PRIMITIVE_DYADIC_OPERATOR or ito VALUE if so, and to VALUE_B otherwise. Only the binding of SYMBOL is left over and will be decided at execution time.

At execution time, the first thing that happens is that SYMBOL is resolved to its current value (so its token class changes from SYMBOL to VALUE (for variables), or to FUNCTION (for defined functions), or to OPERATOR for defined operators. At the time when the parser sees the pattern VALUE_A VALUE_B it can, with a lookahead of 1 token decide if the token VALUE_A shall be bound to it PRIMITIVE_DYADIC_OPERATOR, or stranded to its adjacxent value VALUE_B. To do this a so-called LALR-1 parser suffices. In simple terms: if the parser sees a PRIMITIVE_DYADIC_OPERATOR left of VALUE_A VALUE_B then it shifts VALUE_A (which will later on become the right operand of that operator) and otherwise it starts or continues strand building with VALUE_A. In iso terminology: the phrase evaluator for pattern VALUE_A VALUE_B stops the strand building (in compiler terminology it shifts VALUE_A) when the lookahead token is DYADIC_OPERATOR), and otherwise it continues the strand building (in compiler terminology it reduces the pattern VALUE_A VALUE_B).

As an aside: The ISO standard provides examples with strand notation that would be illegal in IBM APL2 (rank operator, page 126):

      0 1 2 ⌽⍤0 1 'ABC'
ABC
BCA
CAB

In IBM APL2, the rightmost 0 would bind to ⍤ and not to the rightmost 1, like in (parentheses added according to the APL bindig strength:

      (0 1 2) (⌽ ⍤ 0) (1 'ABC')
LENGTH ERROR
      0 1 2(⌽⍤0 0 0)1 'ABC'
      ^            ^

What saves IBM here is only the fact that the the two primitive operators that accept values as right operands (the rank operator ⍤ and the power operator ⍣) are not implemented in IBM APL2. In contrast GNU APL has implemented both operators and the iso examples work as shown in the standard.

The same iso example in Dyalog APL (who seem to be the inventor of these operators) does not work either (at least not without parentheses:

      ⍝ Dyalog APL
      ⍝
      0 1 2 ⌽⍤0 1 'ABC'
SYNTAX ERROR: Missing right argument
 0 1 2⌽⍤0 1 'ABC'
      ∧

      ⍝ Dyalog APL
      ⍝
      0 1 2 (⌽⍤0 1) 'ABC'
ABC
BCA
CAB

and 1 2 3[2] is valid in Dyalog APL, which suggests that Dyalog has not adapted the idea of binding strength but rather remains backward compatible with APL \360.

• Performance considerations

      ∇Z←FOO N
       Z←100⍴0 ◊ J←1
       LOOP: Z[J]←J+1 2 3 4 5 6 7 8 9 10 ÷ →(100 < J←J+1)/LOOP
      ∇

      ⍪3 4⍴"Hello"

5.7 GNU APL Vector Notation

With the definitions from the previous section we can explain how GNU APL has implemented vector notation:

1. GNU APL distinguishes between literal and mixed strands. literal strands are combined into a single APL value at tokenization time. This is a very effective optimization (which can be disabled at ./configure time). In contrast, mixed strands are created at phrase matching time, i.e. when an APL expression is evaluated. The tokenization of APL expressions happens only once, while phrase matching occurs every time when a statement is being executed and therefore moving major parts of the strand building from phrase matching time to tokenization time can bring substantial performance benefits.

2. The tokenization expands simple APL primitives with literal arguments and short results into their literal equivalent. Currently the APL primitives N⍴B, N/B, N⌿B, ,B, ⍪B, and ⊂B are handled this way.

3. The tokenization of literal strands has dedicated code for literal arguments of the power (⍣) and rank operators (⍤) to make make it working like described in the ISO standard. This is, in contrast to defined operators, possible because the ISO standard defines in very detail how its right operand needs to look like and how deviations shall be handled (DOMAIN ERROR, LENGTH ERROR) etc.

4. The phrase matching of mixed strands builds the strand step-by-step according to the following rules:

• two adjacent APL scalars A and B create a new open strand (with two strand items). Each of the two scalars A or B can be simple (like numbers or single characters) or nested (like strings or closed strands).
• An APL scalar A adjacent to an open strand B preprends A to B. That is, B←A, B.
• An open strand A adjacent to an an APL scalar B appends B to A. That is, A←A, B.
• the previous two rules prevent two adjacent open strands, because the strand items of a potential open left strand A are prepended item by item to an open right strand B (which prevents the creation of a new open left strand while an open right strand exists,
• Like the normal APL evaluation, this procedure is preformed from the right to the left.
• if an open strand hits a closing parenthesis then a new phrase matching context is created in which the current procedure is abandoned until the corresponding opening parentheses has been processed. The result of the expression in parentheses is then taken as the next strand item for the abandoned strand (if any).
• Any symbols encountered during this procedure are being resolved into scalars (possibly enclosing them if they are not). Niladic functions are called to obtain their result.
• Likewise, any literal strands that were created at tokenization time are considered closed (which cause them to be enclosed when preprended or appended to an open mixed strand).
• If a strand is open then any other token (functions, operators, brackets, end-of-statement, etc.) closes it. The result is a single value for the strand.

5. The operators ⍣ (power operator) and ⍤ (rank operator), whose right operand can be identified by the rules given in iso) have dedicated code that "unstrands" the right operand from a strand that may have been mistakenly created in 3. above.

• Example

      ∇FOO
       "Time is now: " ⎕TS
      ∇

6.1 The subdirectory src

The subdirectory src contains all C++ source files that are needed to build either the GNU APL interpreter itself, or libapl (a library that contains the essential part of GNU APL, but without the interactive REPL loop of the interpreter). During the build, the object files produced by the compiler are also placed in this directory.

The subdirectory src contains further subdirectories for components of GNU APL that are optional and typically require the installation of additional libraries on the build machine. The ./configure script in the top-level directory determines, which optional components shall be included in the build.

    apl-1.9/
    ├── src
        ├── APs                     APs for ⎕SVO and friends
        ├── emacs_mode              dito
        ├── Gtk                     ⎕GTK (graphical user interface)
        ├── native                  templates for native functions
        ├── sql                     ⎕SQL
        ├── testcases               dito

GNU APL comes with an automated test suite, the testcases are contained in subdirectory src/testcases

6.2 The subdirectory support-files

The subdirectory support-file contains a number of configuration files that aim at making a standard keyboard produce APL characters. Details are explained in top-level file README-3-keyboard.

6.3 The subdirectory doc

The subdirectory doc is used to create this info file in different file formats (.info, .html).

6.4 The subdirectory HOWTOs

The subdirectory HOWTOs contains some documents that try to explain how some of the non-standard features of GNU APL are intended to be used. These documents were written for two different audiences: APL programmers and GNU APL hackers. The documents for APL programmers are, by default, installed in directory /usr/local/share/doc/apl by make install while the documents for GNU APL hackers (i.e. C++ programmers or trouble-shooters) are not installed by make install, but rather remain in the source tree.

6.5 The subdirectories workspaces and wslib2 ... wslib5

These directories (and their content) are, by default, copied to /usr/local/lib/apl by make install. Some subdirectories are empty while others contain small workspaces or libraries for various purposes (mostly demonstrating the use of some non-standard APL features).

6.6 The subdirectory m4

The subdirectory m4 contains macros used by autoconf and automake. In most cases the top-level ./configure script can be used as is. If a platform on which GNU APL shall be built differs too much from the platform on which the GNU APL project tar file (e.g. apl-1.9.tar.gz) was created, then it might be necessary to re-create the ./configure script on the platform.

Re-creating the ./configure script requires:

• the installation of autoconf,
• the installation of automake,
• the installation of libtool,
• maybe running aclocal, libtoolize, and other scripts, and finally
• running autoreconf

6.7 The subdirectories debian and debian_tmp

These subdirectories are templates for building Debian packages that contain GNU APL. Subdirectory debian_tmp is initially empty and acts as a scratch directory inside which the interpreter is being compiled when a Debian package is built.

6.8 The subdirectory tools

GNU APL uses generated C++ code. The subdirectory tools contains the source code for the code generator (phrase_gen) and also some other small tools for various purposes. These tools are typically used by GNU APL hackers and are therefore not installed by make install.

7.1 The GNU APL Community Webpage

The GNU APL project maintains a web page that lists contributions made by GNU APL users:

http://www.gnu.org/software/apl/Community.html

The contributions listed there were collected from emails sent to [email protected]

Since the list was started long after [email protected] was created. It is therefore quite possible that your contribution is missing. That does not mean that your contribution was not considered worthwhile to be listed but more likely that it was overlooked. Just send an email to [email protected] to fix that.

7.2 Core Libraries

There is a small number of libraries that are considered "core" APL because, for example, the ISO standard requires their functionality, or because other APL interpreters normally support them.

These libraries are linked via the GNU APL Community web page. In addition they are stored in the GNU APL SVN repository and also included in the GNU APL packages that follows their contribution.

Currently the following core libraries exist:

GNU APL Emacs mode (for Emacs users) by Elias Mårtenson

SQL interface (SQLite and Postgres) by Elias Mårtenson

Component File System (as required by ISO 13751) by David B. Lamkins

Component File System (as required by ISO 13751) by Blake McBride

SQL

8.1 The GNU Free Documentation License

Copyright © 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
http://fsf.org/

Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.

0. PREAMBLE

   The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or non-commercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.

   This License is a kind of “copyleft”, which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.

   We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.

1. APPLICABILITY AND DEFINITIONS

   This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The “Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as “you”. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law.

   A “Modified Version” of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.

   A “Secondary Section” is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document’s overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.

   The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none.

   The “Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.

   A “Transparent” copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not “Transparent” is called “Opaque”.

   Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only.

   The “Title Page” means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, “Title Page” means the text near the most prominent appearance of the work’s title, preceding the beginning of the body of the text.

   The “publisher” means any person or entity that distributes copies of the Document to the public.

   A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as “Acknowledgements”, “Dedications”, “Endorsements”, or “History”.) To “Preserve the Title” of such a section when you modify the Document means that it remains a section “Entitled XYZ” according to this definition.

   The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License.

2. VERBATIM COPYING

   You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.

   You may also lend copies, under the same conditions stated above, and you may publicly display copies.

3. COPYING IN QUANTITY

   If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document’s license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.

   If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.

   If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.

   It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

4. MODIFICATIONS

   You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:

   1. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.
   2. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement.
   3. State on the Title page the name of the publisher of the Modified Version, as the publisher.
   4. Preserve all the copyright notices of the Document.
   5. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.
   6. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.
   7. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document’s license notice.
   8. Include an unaltered copy of this License.
   9. Preserve the section Entitled “History”, Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled “History” in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.
   10. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the “History” section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.
   11. For any section Entitled “Acknowledgements” or “Dedications”, Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.
   12. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles.
   13. Delete any section Entitled “Endorsements”. Such a section may not be included in the Modified Version.
   14. Do not retitle any existing section to be Entitled “Endorsements” or to conflict in title with any Invariant Section.
   15. Preserve any Warranty Disclaimers.

   If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version’s license notice. These titles must be distinct from any other section titles.

   You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements of your Modified Version by various parties—for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.

   You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

   The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

5. COMBINING DOCUMENTS

   You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.

   The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

   In the combination, you must combine any sections Entitled “History” in the various original documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled “Endorsements.”

6. COLLECTIONS OF DOCUMENTS

   You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

   You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

7. AGGREGATION WITH INDEPENDENT WORKS

   A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the copyright resulting from the compilation is not used to limit the legal rights of the compilation’s users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.

   If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

8. TRANSLATION

   Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.

   If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.

9. TERMINATION

   You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License.

   However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

   Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

   Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it.

10. FUTURE REVISIONS OF THIS LICENSE

    The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.

    Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License “or any later version” applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy’s public statement of acceptance of a version permanently authorizes you to choose that version for the Document.

11. RELICENSING

    “Massive Multiauthor Collaboration Site” (or “MMC Site”) means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A “Massive Multiauthor Collaboration” (or “MMC”) contained in the site means any set of copyrightable works thus published on the MMC site.

    “CC-BY-SA” means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization.

    “Incorporate” means to publish or republish a Document, in whole or in part, as part of another Document.

    An MMC is “eligible for relicensing” if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008.

    The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing.

ADDENDUM: How to use this License for your documents

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

  Copyright (C)  year  your name.
  Permission is granted to copy, distribute and/or modify this document
  under the terms of the GNU Free Documentation License, Version 1.3
  or any later version published by the Free Software Foundation;
  with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
  Texts. A copy of the license is included in the section entitled ``GNU
  Free Documentation License''.

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with…Texts.” line with this:

    with the Invariant Sections being list their titles, with
    the Front-Cover Texts being list, and with the Back-Cover Texts
    being list.

If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

8.2 The GNU GENERAL PUBLIC LICENSE

GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007

Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/> Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

Preamble

The GNU General Public License is a free, copyleft license for software and other kinds of works.

The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program–to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for them if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs, and that you know you can do these things.

To protect your rights, we need to prevent others from denying you these rights or asking you to surrender the rights. Therefore, you have certain responsibilities if you distribute copies of the software, or if you modify it: responsibilities to respect the freedom of others.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must pass on to the recipients the same freedoms that you received. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.

Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.

For the developers’ and authors’ protection, the GPL clearly explains that there is no warranty for this free software. For both users’ and authors’ sake, the GPL requires that modified versions be marked as changed, so that their problems will not be attributed erroneously to authors of previous versions.

Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users’ freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users.

Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free.

The precise terms and conditions for copying, distribution and modification follow.

TERMS AND CONDITIONS

0. Definitions.

"This License" refers to version 3 of the GNU General Public License.

"Copyright" also means copyright-like laws that apply to other kinds of works, such as semiconductor masks.

"The Program" refers to any copyrightable work licensed under this License. Each licensee is addressed as "you". "Licensees" and "recipients" may be individuals or organizations.

To "modify" a work means to copy from or adapt all or part of the work in a fashion requiring copyright permission, other than the making of an exact copy. The resulting work is called a "modified version" of the earlier work or a work "based on" the earlier work.

A "covered work" means either the unmodified Program or a work based on the Program.

To "propagate" a work means to do anything with it that, without permission, would make you directly or secondarily liable for infringement under applicable copyright law, except executing it on a computer or modifying a private copy. Propagation includes copying, distribution (with or without modification), making available to the public, and in some countries other activities as well.

To "convey" a work means any kind of propagation that enables other parties to make or receive copies. Mere interaction with a user through a computer network, with no transfer of a copy, is not conveying.

An interactive user interface displays "Appropriate Legal Notices" to the extent that it includes a convenient and prominently visible feature that (1) displays an appropriate copyright notice, and (2) tells the user that there is no warranty for the work (except to the extent that warranties are provided), that licensees may convey the work under this License, and how to view a copy of this License. If the interface presents a list of user commands or options, such as a menu, a prominent item in the list meets this criterion.

1. Source Code.

The "source code" for a work means the preferred form of the work for making modifications to it. "Object code" means any non-source form of a work.

A "Standard Interface" means an interface that either is an official standard defined by a recognized standards body, or, in the case of interfaces specified for a particular programming language, one that is widely used among developers working in that language.

The "System Libraries" of an executable work include anything, other than the work as a whole, that (a) is included in the normal form of packaging a Major Component, but which is not part of that Major Component, and (b) serves only to enable use of the work with that Major Component, or to implement a Standard Interface for which an implementation is available to the public in source code form. A "Major Component", in this context, means a major essential component (kernel, window system, and so on) of the specific operating system (if any) on which the executable work runs, or a compiler used to produce the work, or an object code interpreter used to run it.

The "Corresponding Source" for a work in object code form means all the source code needed to generate, install, and (for an executable work) run the object code and to modify the work, including scripts to control those activities. However, it does not include the work’s System Libraries, or general-purpose tools or generally available free programs which are used unmodified in performing those activities but which are not part of the work. For example, Corresponding Source includes interface definition files associated with source files for the work, and the source code for shared libraries and dynamically linked subprograms that the work is specifically designed to require, such as by intimate data communication or control flow between those subprograms and other parts of the work.

The Corresponding Source need not include anything that users can regenerate automatically from other parts of the Corresponding Source.

The Corresponding Source for a work in source code form is that same work.

2. Basic Permissions.

All rights granted under this License are granted for the term of copyright on the Program, and are irrevocable provided the stated conditions are met. This License explicitly affirms your unlimited permission to run the unmodified Program. The output from running a covered work is covered by this License only if the output, given its content, constitutes a covered work. This License acknowledges your rights of fair use or other equivalent, as provided by copyright law.

You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force. You may convey covered works to others for the sole purpose of having them make modifications exclusively for you, or provide you with facilities for running those works, provided that you comply with the terms of this License in conveying all material for which you do not control copyright. Those thus making or running the covered works for you must do so exclusively on your behalf, under your direction and control, on terms that prohibit them from making any copies of your copyrighted material outside their relationship with you.

Conveying under any other circumstances is permitted solely under the conditions stated below. Sublicensing is not allowed; section 10 makes it unnecessary.

3. Protecting Users’ Legal Rights From Anti-Circumvention Law.

No covered work shall be deemed part of an effective technological measure under any applicable law fulfilling obligations under article 11 of the WIPO copyright treaty adopted on 20 December 1996, or similar laws prohibiting or restricting circumvention of such measures.

When you convey a covered work, you waive any legal power to forbid circumvention of technological measures to the extent such circumvention is effected by exercising rights under this License with respect to the covered work, and you disclaim any intention to limit operation or modification of the work as a means of enforcing, against the work’s users, your or third parties’ legal rights to forbid circumvention of technological measures.

4. Conveying Verbatim Copies.

You may convey verbatim copies of the Program’s source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice; keep intact all notices stating that this License and any non-permissive terms added in accord with section 7 apply to the code; keep intact all notices of the absence of any warranty; and give all recipients a copy of this License along with the Program.

You may charge any price or no price for each copy that you convey, and you may offer support or warranty protection for a fee.

5. Conveying Modified Source Versions.

You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:

a) The work must carry prominent notices stating that you modified it, and giving a relevant date.

b) The work must carry prominent notices stating that it is released under this License and any conditions added under section 7. This requirement modifies the requirement in section 4 to "keep intact all notices".

c) You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.

d) If the work has interactive user interfaces, each must display Appropriate Legal Notices; however, if the Program has interactive interfaces that do not display Appropriate Legal Notices, your work need not make them do so.

A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an "aggregate" if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation’s users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.

6. Conveying Non-Source Forms.

You may convey a covered work in object code form under the terms of sections 4 and 5, provided that you also convey the machine-readable Corresponding Source under the terms of this License, in one of these ways:

a) Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by the Corresponding Source fixed on a durable physical medium customarily used for software interchange.

b) Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by a written offer, valid for at least three years and valid for as long as you offer spare parts or customer support for that product model, to give anyone who possesses the object code either (1) a copy of the Corresponding Source for all the software in the product that is covered by this License, on a durable physical medium customarily used for software interchange, for a price no more than your reasonable cost of physically performing this conveying of source, or (2) access to copy the Corresponding Source from a network server at no charge.

c) Convey individual copies of the object code with a copy of the written offer to provide the Corresponding Source. This alternative is allowed only occasionally and noncommercially, and only if you received the object code with such an offer, in accord with subsection 6b.

d) Convey the object code by offering access from a designated place (gratis or for a charge), and offer equivalent access to the Corresponding Source in the same way through the same place at no further charge. You need not require recipients to copy the Corresponding Source along with the object code. If the place to copy the object code is a network server, the Corresponding Source may be on a different server (operated by you or a third party) that supports equivalent copying facilities, provided you maintain clear directions next to the object code saying where to find the Corresponding Source. Regardless of what server hosts the Corresponding Source, you remain obligated to ensure that it is available for as long as needed to satisfy these requirements.

e) Convey the object code using peer-to-peer transmission, provided you inform other peers where the object code and Corresponding Source of the work are being offered to the general public at no charge under subsection 6d.

A separable portion of the object code, whose source code is excluded from the Corresponding Source as a System Library, need not be included in conveying the object code work.

A "User Product" is either (1) a "consumer product", which means any tangible personal property which is normally used for personal, family, or household purposes, or (2) anything designed or sold for incorporation into a dwelling. In determining whether a product is a consumer product, doubtful cases shall be resolved in favor of coverage. For a particular product received by a particular user, "normally used" refers to a typical or common use of that class of product, regardless of the status of the particular user or of the way in which the particular user actually uses, or expects or is expected to use, the product. A product is a consumer product regardless of whether the product has substantial commercial, industrial or non-consumer uses, unless such uses represent the only significant mode of use of the product.

"Installation Information" for a User Product means any methods, procedures, authorization keys, or other information required to install and execute modified versions of a covered work in that User Product from a modified version of its Corresponding Source. The information must suffice to ensure that the continued functioning of the modified object code is in no case prevented or interfered with solely because modification has been made.

If you convey an object code work under this section in, or with, or specifically for use in, a User Product, and the conveying occurs as part of a transaction in which the right of possession and use of the User Product is transferred to the recipient in perpetuity or for a fixed term (regardless of how the transaction is characterized), the Corresponding Source conveyed under this section must be accompanied by the Installation Information. But this requirement does not apply if neither you nor any third party retains the ability to install modified object code on the User Product (for example, the work has been installed in ROM).

The requirement to provide Installation Information does not include a requirement to continue to provide support service, warranty, or updates for a work that has been modified or installed by the recipient, or for the User Product in which it has been modified or installed. Access to a network may be denied when the modification itself materially and adversely affects the operation of the network or violates the rules and protocols for communication across the network.

Corresponding Source conveyed, and Installation Information provided, in accord with this section must be in a format that is publicly documented (and with an implementation available to the public in source code form), and must require no special password or key for unpacking, reading or copying.

7. Additional Terms.

"Additional permissions" are terms that supplement the terms of this License by making exceptions from one or more of its conditions. Additional permissions that are applicable to the entire Program shall be treated as though they were included in this License, to the extent that they are valid under applicable law. If additional permissions apply only to part of the Program, that part may be used separately under those permissions, but the entire Program remains governed by this License without regard to the additional permissions.

When you convey a copy of a covered work, you may at your option remove any additional permissions from that copy, or from any part of it. (Additional permissions may be written to require their own removal in certain cases when you modify the work.) You may place additional permissions on material, added by you to a covered work, for which you have or can give appropriate copyright permission.

Notwithstanding any other provision of this License, for material you add to a covered work, you may (if authorized by the copyright holders of that material) supplement the terms of this License with terms:

a) Disclaiming warranty or limiting liability differently from the terms of sections 15 and 16 of this License; or

b) Requiring preservation of specified reasonable legal notices or author attributions in that material or in the Appropriate Legal Notices displayed by works containing it; or

c) Prohibiting misrepresentation of the origin of that material, or requiring that modified versions of such material be marked in reasonable ways as different from the original version; or

d) Limiting the use for publicity purposes of names of licensors or authors of the material; or

e) Declining to grant rights under trademark law for use of some trade names, trademarks, or service marks; or

f) Requiring indemnification of licensors and authors of that material by anyone who conveys the material (or modified versions of it) with contractual assumptions of liability to the recipient, for any liability that these contractual assumptions directly impose on those licensors and authors.

All other non-permissive additional terms are considered "further restrictions" within the meaning of section 10. If the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term. If a license document contains a further restriction but permits relicensing or conveying under this License, you may add to a covered work material governed by the terms of that license document, provided that the further restriction does not survive such relicensing or conveying.

If you add terms to a covered work in accord with this section, you must place, in the relevant source files, a statement of the additional terms that apply to those files, or a notice indicating where to find the applicable terms.

Additional terms, permissive or non-permissive, may be stated in the form of a separately written license, or stated as exceptions; the above requirements apply either way.

8. Termination.

You may not propagate or modify a covered work except as expressly provided under this License. Any attempt otherwise to propagate or modify it is void, and will automatically terminate your rights under this License (including any patent licenses granted under the third paragraph of section 11).

However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, you do not qualify to receive new licenses for the same material under section 10.

9. Acceptance Not Required for Having Copies.

You are not required to accept this License in order to receive or run a copy of the Program. Ancillary propagation of a covered work occurring solely as a consequence of using peer-to-peer transmission to receive a copy likewise does not require acceptance. However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so.

10. Automatic Licensing of Downstream Recipients.

Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License.

An "entity transaction" is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party’s predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts.

You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it.

11. Patents.

A "contributor" is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor’s "contributor version".

A contributor’s "essential patent claims" are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, "control" includes the right to grant patent sublicenses in a manner consistent with the requirements of this License.

Each contributor grants you a non-exclusive, worldwide, royalty-free patent license under the contributor’s essential patent claims, to make, use, sell, offer for sale, import and otherwise run, modify and propagate the contents of its contributor version.

In the following three paragraphs, a "patent license" is any express agreement or commitment, however denominated, not to enforce a patent (such as an express permission to practice a patent or covenant not to sue for patent infringement). To "grant" such a patent license to a party means to make such an agreement or commitment not to enforce a patent against the party.

If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. "Knowingly relying" means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient’s use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid.

If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it.

A patent license is "discriminatory" if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007.

Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law.

12. No Surrender of Others’ Freedom.

If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program.

13. Use with the GNU Affero General Public License.

Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such.

14. Revised Versions of this License.

The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License "or any later version" applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation.

If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy’s public statement of acceptance of a version permanently authorizes you to choose that version for the Program.

Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version.

15. Disclaimer of Warranty.

THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

16. Limitation of Liability.

IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

17. Interpretation of Sections 15 and 16.

If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee.

END OF TERMS AND CONDITIONS

How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found.

<one line to give the program’s name and a brief idea of what it does.> Copyright (C) <year> <name of author>

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see <http://www.gnu.org/licenses/>.

Also add information on how to contact you by electronic and paper mail.

If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode:

<program> Copyright (C) <year> <name of author> This program comes with ABSOLUTELY NO WARRANTY; for details type ‘show w’. This is free software, and you are welcome to redistribute it under certain conditions; type ‘show c’ for details.

The hypothetical commands ‘show w’ and ‘show c’ should show the appropriate parts of the General Public License. Of course, your program’s commands might be different; for a GUI interface, you would use an "about box".

You should also get your employer (if you work as a programmer) or school, if any, to sign a "copyright disclaimer" for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see <http://www.gnu.org/licenses/>.

The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read <http://www.gnu.org/philosophy/why-not-lgpl.html>.