An automata-theoretic approach to the word problem for ω-terms over R

The main contribution of this paper is the solution of a word problem over R, the pseudovariety of all R-trivial semigroups which corresponds, in Eilenberg’s correspondence, to disjoint unions of languages of the form A0a1A ∗ 1a2 . . . anA ∗ n, where the ai’s are letters and ai / ∈ Ai−1 for 1 6 i 6 n. They are also the divisors of transition semigroups of the so-called extensive automata, that is, automata whose state set is linearly ordered and the transition function always increases the state. Given two terms built from letters of an alphabet A using the concatenation and the ω-power, we show that deciding whether these terms coincide over all A-generated elements of R (with the usual interpretation of the ωpower in semigroups) can be solved in linear time. We also characterize the set of pseudowords—also known as implicit operations—over R which are such ω-terms. The motivation of this work is the κ-tameness property for R, where κ = { · , ω−1} is the implicit signature consisting of the semigroup multiplication and the unary (ω − 1)-power. Historically, the notion of tameness was discovered in attempting to find general decidability properties of pseudovarieties which might be preserved under taking semidirect products [6]. It remains open whether it does indeed play such a role although under certain finiteness hypotheses it has been been shown to do so [2]. Proving the κ-tameness of a pseudovariety V consists in solving two subproblems. The first one is the κ-word problem, for which this paper gives an efficient solution. Informally, the second question is whether equation systems with rational constraints having a solution in any semigroup of V also have a uniform solution in κ-terms, satisfying the same constraints. This property has proven to be robust and helpful for the solution of the membership problem (see e.g. [4], where the κ-tameness of R is used to decide joins involving R.) Moreover, if V enjoys it, then V has decidable pointlikes, an important property of pseudovarieties [6, 19].