On Varieties of Automata Enriched with an Algebraic Structure (Extended Abstract)

Eilenberg correspondence, based on the concept of syntactic monoids, relates varieties of regular languages with pseudovarieties of finite monoids. Various modifications of this correspondence related more general classes of regular languages with classes of more complex algebraic objects. Such generalized varieties also have natural counterparts formed by classes of finite automata equipped with a certain additional algebraic structure. In this survey, we overview several variants of such varieties of enriched automata.Comment: In Proceedings AFL 2014, arXiv:1405.527

Symmetric Groups and Quotient Complexity of Boolean Operations

The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L' are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively. Denote by o any binary boolean operation that is not a constant and not a function of one argument only. For m,n >= 2 with (m,n) not in {(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn if and only either (a) m is not equal to n or (b) m=n and the bases (ordered pairs of generators) of S_m and S_n are not conjugate. For (m,n)\in {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory

A novel cryptosystem based on Gluškov product of automata

The concept of Gluškov product was introduced by V. M. Gluškov in 1961. It was intensively studied by several scientists (first of all, by Ferenc Gécseg and the automata-theory school centred around him in Szeged, Hungary) since the middle of 60’s. In spite of the large number of excellent publications, no application of Gluškov-type products of automata in cryptography has arisen so far. This paper is the first attempt in this direction