Describing Periodicity in Two-Way Deterministic Finite Automata Using Transformation Semigroups
نویسندگان
چکیده
A framework for the study of periodic behaviour of twoway deterministic finite automata (2DFA) is developed. Computations of 2DFAs are represented by a two-way analogue of transformation semigroups, every element of which describes the behaviour of a 2DFA on a certain string x. A subsemigroup generated by this element represents the behaviour on strings in x. The main contribution of this paper is a description of all such monogenic subsemigroups up to isomorphism. This characterization is then used to show that transforming an n-state 2DFA over a one-letter alphabet to an equivalent sweeping 2DFA requires exactly n+1 states, and transforming it to a one-way automaton requires exactly max06`6n G(n − `) + ` + 1 states, where G(k) is the maximum order of a permutation of k elements.
منابع مشابه
On Deterministic Finite Automata and Syntactic Monoid Size, Continued
We continue our investigation on the relationship between regular languages and syntactic monoid size. In this paper we confirm the conjecture on two generator transformation semigroups. We show that for every prime n ≥ 7 there exist natural numbers k and ` with n = k + ` such that the semigroup Uk,` is maximal w.r.t. its size among all (transformation) semigroups which can be generated with tw...
متن کاملGroups and Semigroups Defined by Colorings of Synchronizing Automata
In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the corresponding DFA. Moreover, we present a decidabl...
متن کاملOn some problems in the theory of partial automata
The paper shows how some semigroup-theory methods can be used in automata problems. The theory of automata is closely connected with the theory of semigroups (or, to be more exact, with the theory of representations of semigroups by transformations). The concept of automaton without outputs is equivalent to the concept of representa tion of free semigroup by transformations. If we consider eve...
متن کاملLower Bounds on Words Separation: Are There Short Identities in Transformation Semigroups?
The words separation problem, originally formulated by Goralcik and Koubek (1986), is stated as follows. Let Sep(n) be the minimum number such that for any two words of length 6 n there is a deterministic finite automaton with Sep(n) states, accepting exactly one of them. The problem is to find the asymptotics of the function Sep. This problem is inverse to finding the asymptotics of the length...
متن کاملGAME OF COORDINATION FOR BACTERIAL PATTERN FORMATION: A FINITE AUTOMATA MODELLING
In this paper, we use game theory to describe the emergence of self-organization and consequent pattern formation through communicative cooperation in Bacillus subtilis colonies. The emergence of cooperative regime is modelled as an n-player Assurance game, with the bacterial colonies as individual players. The game is played iteratively through cooperative communication, and mediated by exchan...
متن کامل