Papers by Stefano Mazzanti
Fixed Point Iteration
Iteration seems to play a fundamental role in learning theory - as everywhere else. Since neurons... more Iteration seems to play a fundamental role in learning theory - as everywhere else. Since neurons need a rather long time to reach a stable state, Caianiello's Paradox suggests that special actions cause the iteration of state transitions until a stable neural state is reached. Now, a stable state is a fixed point for any transition function; so we may call the iteration above, we want to discuss, "fixed point iteration"
Electronic Notes in Theoretical Computer Science, 2016
We investigate the computing power of function algebras defined by means of unbounded recursion o... more We investigate the computing power of function algebras defined by means of unbounded recursion on notation. We introduce two function algebras which contain respectively the regressive logspace computable functions and the non-size-increasing logspace computable functions. However, such algebras are unlikely to be contained in the set of logspace computable functions because this is equivalent to L = P. Finally, we introduce a function algebra based on simultaneous recursion on notation for the non-size-increasing functions computable in polynomial time and linear space.
Abstract. We consider the function class E generated by the constant functions, the projection fu... more Abstract. We consider the function class E generated by the constant functions, the projection functions, the predecessor function, the substi-tution operator, and the recursion on notation operator. Furthermore, we introduce regressive machines, i.e. register machines which have the division by 2 and the predecessor as basic operations. We show that E is the class of functions computable by regressive machines and that the sharply bounded functions of E coincide with the sharply bounded logspace computable functions.
New substitution bases for complexity classes
Mathematical Logic Quarterly
Fundamenta Informaticae
Function complexity classes are defined as the substitution closure of finite function sets by im... more Function complexity classes are defined as the substitution closure of finite function sets by improving a method of elimination of concatenation recursion from function algebras. Consequently, the set of AC 0 functions and other canonical complexity classes are defined as the substitution closure of a finite function set.
Fundamenta Informaticae
Function classes closed with respect to substitution and concatenation recursion on notation are ... more Function classes closed with respect to substitution and concatenation recursion on notation are defined as the substitution closure of finite function sets. Consequently, the sets of T C 0 , N C 1 and L computable functions are inductively characterized as the substitution closure of a finite function set.
Regressive computations characterize logarithmic space
Information and Computation, 2016
We consider the function class E generated by the constant functions, the projection functions, t... more We consider the function class E generated by the constant functions, the projection functions, the predecessor function, the substitution operator and the recursion on notation operator. Furthermore, we introduce regressive register machines, which have division by 2 and the predecessor on natural numbers as basic operations. We show that E is the class of functions computable by regressive machines and that the sharply bounded functions (functions with logarithmic size values) of E coincide with the sharply bounded logspace computable functions.
The set of logspace computable functions is inductively characterized by means of an enhanced for... more The set of logspace computable functions is inductively characterized by means of an enhanced form of concatenation recursion on notation. The new characterization avoids the use of bounded and safe recursion schemata.
Iteration on notation and unary functions
Mathematical Logic Quarterly, 2013
In the first half of the 1990s, Clote and Takeuti characterized several function complexity class... more In the first half of the 1990s, Clote and Takeuti characterized several function complexity classes by means of the concatenation recursion on notation operators. In this paper, we borrow from computability theory well-known techniques based on pairing functions to show that , , and functions can be characterized by means of concatenation iteration on notation. Indeed, a function class satisfying simple constraints and defined by using concatenation recursion on notation is inductively characterized by means of concatenation iteration on notation. Furthermore, , , and unary functions are inductively characterized using addition, composition, and concatenation iteration on notation.
We consider the function class E generated by the constant functions, the projection functions, t... more We consider the function class E generated by the constant functions, the projection functions, the predecessor function, the substitution operator, and the recursion on notation operator. Furthermore, we introduce regressive machines, i.e. register machines which have the division by 2 and the predecessor as basic operations. We show that E is the class of functions computable by regressive machines and that the sharply bounded functions of E coincide with the sharply bounded logspace computable functions.
Acs, 2001
Traditional closure theory discusses the closure operations on orders with graph-theoretic method... more Traditional closure theory discusses the closure operations on orders with graph-theoretic methods, or the reflectors on skeletal categories with category-theoretic methods. Both approaches are confined, like most of classical mathematics, to total and deterministic operations. So traditional closure theory makes it possible to define the semantics of the while-do commands only for terminating and deterministic programming. This paper outlines a closure theory for relations which transcend totality and determinism. For the sake of conciseness, the language used is that of graph theory but the methods are category-theoretic and some hints are offered for a possible translation into the language of category theory. Our basic idea is that closure relations consist of universal arrows in the sense of category theory. The new closure theory is appropriate for defining a semantics of the while-do commands both for terminating, deterministic programming and for non-terminating, non-deterministic programming. : 06A15, 18A10, 68Q10, 68Q55.
Applied Categorical Structures, 2003
Within the framework of category theory, Cantor diagrams are introduced as the common structure o... more Within the framework of category theory, Cantor diagrams are introduced as the common structure of the self-reference constructions by Cantor, Russell, Richard, Gödel, Péter, Turing, Kleene, Tarski, according to the so-called Cantor diagonal method. Such diagrams consist not only of diagonal arrows but also of idempotent, identity and shift arrows. Cantor theorem states that no Cantor diagram is commutative. From this theorem, all the constructions above can be systematically retrieved. We do this by grouping them into two main classes: the class based on Cantor diagrams with a numerical shift function and the class based on Cantor diagrams with a Boolean shift function. : 03C35, 03D10, 03D20, 03D70, 03E17, 18D15.
Theoretical Computer Science, May 1, 1991
Germano, GM., and S. Mazzanti, Closure functions and general iterates as reflectors ( Fundamental
Loose Diagrams, Semigroupoids, Categories, Groupoids and Iteration
Csl, Oct 12, 1987
Without Abstract
An iterative characterization of computable functions on hyperwords of natural numbers
Partial closures and semantics of while: Towards an iteration-based theory of data types
The present paper proposes first a generalization of closure theory and revisits Moore's theo... more The present paper proposes first a generalization of closure theory and revisits Moore's theory in this framework. Afterwards closures of non cyclic functions are introduced and a method is given to transform cyclic into non cyclic functions. Eventually semantics of the while construct is found to be the closure of a function. Computability on inductive and non inductive data types is then studied with iterative means.
Loose Diagrams, Semigroupoids, Categories, Groupoids and Iteration
Computer Science Logic, 1987
Without Abstract
Within the framework of category theory, Cantor diagrams are introduced as the common structure o... more Within the framework of category theory, Cantor diagrams are introduced as the common structure of the self-reference constructions by Cantor, Russell, Richard, Gödel, Péter, Turing, Kleene, Tarski, according to the so-called Cantor diagonal method. Such diagrams consist not only of diagonal arrows but also of idempotent, identity and shift arrows. Cantor theorem states that no Cantor diagram is commutative. From this theorem, all the constructions above can be systematically retrieved. We do this by grouping them into two main classes: the class based on Cantor diagrams with a numerical shift function and the class based on Cantor diagrams with a Boolean shift function. : 03C35, 03D10, 03D20, 03D70, 03E17, 18D15.
Theoretical Computer Science, 1991
Germano, GM., and S. Mazzanti, Closure functions and general iterates as reflectors ( Fundamental
A setting for generalized computability
Computation theory and logic, 1987
... (... O00 am... a 1 ao .... O00 bn... b 1 bo) or as the number pair (Y'i ai . ki, Ei bi .... more ... (... O00 am... a 1 ao .... O00 bn... b 1 bo) or as the number pair (Y'i ai . ki, Ei bi . ki ), where we assume that the 'letters' ai and bi belong to the 'alphabet' { 0, 1 ..... k -I }, see I/VIM]. Set top:xl--~x ~ k . [x : k ] , pop:xl-~ Ix:k] pushf:x I--~ k.x+fx , forf:N--~ { 0, 1 ..... k-1 } , Page 11. 164 ...
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Papers by Stefano Mazzanti