Drafts by Avraham Trakhtman
The paper deals with the hypothesis of the migration of Slavonic ancestors in 4th century to the ... more The paper deals with the hypothesis of the migration of Slavonic ancestors in 4th century to the area at Peipsi Lake in the connection with campaigns of the Goths. The study is supported by some data from historical chronicles, linguistics and archeology. Some other Slavonic migrations are also considered. The topic of the appearance of the term "Slovene" is associated with events in the Danube basin of 6th century AD.
Papers by Avraham Trakhtman
Lecture Notes in Computer Science, 2011
A word w is called synchronizing (recurrent, reset, magic, directable) word of deterministic nite... more A word w is called synchronizing (recurrent, reset, magic, directable) word of deterministic nite automaton (DFA) if w sends all states of the automaton to a unique state. In 1964 Jan Cerny found a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n 1) 2. He conjectured that it is an upper bound on the length of such words for complete DFA. Nevertheless, the best upper bound (n 3 n)=6 was found almost 30 years ago. We reduce the upper bound on the length of the minimal synchronizing word to n(7n 2 + 6n 16)=48. An implemented algorithm for nding synchronizing word with restricted upper bound is described. The work presents the distribution of all synchronizing automata of small size according to the length of an almost minimal synchronizing word.
ArXiv, 2007
A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite ... more A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Černý conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n−1). We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n − 1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Černý conjecture holds true.
A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered... more A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. The positive solution of the road coloring problem is presented. Some consequences on the length of the synchronizing word are discussed.
Linear Visualization of a Road Coloring
We provide a counterexample to a lemma used in a recent tentative improvement of the Pin-Frankl b... more We provide a counterexample to a lemma used in a recent tentative improvement of the Pin-Frankl bound for synchronizing automata. This example naturally leads us to formulate an open question, whose answer could fix the line of the proof, and improve the bound.
Computers Ieee Transactions on, 1991
We investigate the local testability problem of deterministic finite automata. A locally testable... more We investigate the local testability problem of deterministic finite automata. A locally testable language is a language with the property that, for some nonnegative integer FE, whether or not a word w is in the language depends on 1) the prefix and sufix of w of length IC, and 2) the set of substrings of w of length k + 1, without regard to the order in which these substrings occur. The local testability problem is, given a deterministic finite automaton, to decide whether or not it accepts a locally testable language. Until now, no polynomial time algorithm for this problem has appeared in the literature. We present an 0 (n2) time algorithm for the local testability problem based on two simple properties that characterize locally testable automata.
The Identities of Local Threshold Testability
Birthday, 2003
Theoretical Computer Science, 2004
A locally threshold testable language L is a language with the property that for some non-negativ... more A locally threshold testable language L is a language with the property that for some non-negative integers k and l and for some word u from L, a word v belongs to L iff: (1) the prefixes [suffixes] of length k − 1 of words u and v coincide, (2) the number of occurrences of every factor of length k in both words u and v are either the same or greater than l − 1. A deterministic finite automaton is called locally threshold testable if the automaton accepts a locally threshold testable language for some l and k. New necessary and sufficient conditions for a deterministic finite automaton to be locally threshold testable are found. On the basis of these conditions, we modify the algorithm to verify local threshold testability of the automaton, and to reduce the time complexity of the algorithm. The algorithm is implemented as a part of the C/C ++ package TESTAS. http://www.cs.biu.ac.il/ trakht/Testas.html.
Jcst, 2008
A word w is called synchronizing (recurrent, reset, directable) word of deterministic finite auto... more A word w is called synchronizing (recurrent, reset, directable) word of deterministic finite automata (DFA) if w brings all states of the automaton to a unique state. According to the famous conjecture ofČerný from 1964, every n-state synchronizing automaton possesses a synchronizing word of length at most (n − 1) 2. The problem is still open. It will be proved that theČerný conjecture holds good for synchronizing DFA with transition monoid having no involutions and for every n-state (n > 2) synchronizing DFA with transition monoid having only trivial subgroups the minimal length of synchronizing word is not greater than (n − 1) 2 /2. The last important class of DFA involved and studied by Schȗtzenberger is called aperiodic; its automata accept precisely star-free languages. Some properties of an arbitrary synchronizing DFA were established. See
Linear Visualization of a Road Coloring
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Drafts by Avraham Trakhtman
Papers by Avraham Trakhtman