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We initiate the theory and applications of biautomata. A biautomaton can read a word alternately from the left and from the right. We assign to each regular language L its canonical biautomaton. This structure plays, among all biautomata recognizing the language L, the same role as the minimal deterministic automaton has among all deterministic automata recognizing the language L. We expect that from the graph structure of this automaton one could decide the membership of a given language to certain significant classes of languages. We present the first result of this kind: a language L is piecewise testable if and only if the canonical biautomaton of L is acyclic. From this result the famous Simon's characterization of piecewise testable languages easily follows.

We study the computational complexity of checking identities in a fixed finite monoid. We find the smallest monoid for which this problem is coNPcomplete and describe a significant class of finite monoids for which the problem is tractable.

Abstract. We consider the problem of testing whether a given system of equations over a fixed finite semigroup S has a solution. For the case where S is a monoid, we prove that the problem is computable in polynomial time when S is commutative and is the union of its subgroups but is NP-complete otherwise. When S is a monoid or a regular semigroup, we obtain similar dichotomies for the restricted version of the problem where no variable occurs on the right-hand side of each equation. We stress connections between these problems and constraint satisfaction problems. In particular, for any finite domain D and any finite set of relations Γ over D, we construct a finite semigroup S Γ such that CSP (Γ ) is polynomial-time equivalent to the equation satifiability problem over S.

Abstract. It is proved that in every concatenation hierarchy of regular languages, decidability of one of its half levels, obtained by polynomial closure, implies decidability of the intersection of the following half level with its complement. In terms of the quantifier-alternation hierarchy of sentences in the first-order logic of finite words, this means that decidability of (definability in) the Σn fragment implies that of ∆n+1. In particular, the decidability of ∆5 is obtained.