Duality and Recognition

Abstract

Abstract. The fact that one can associate a finite monoid with universal properties to each language recognised by an automaton is central to the solution of many practical and theoretical problems in automata theory. It is particularly useful, via the advanced theory initiated by Eilenberg and Reiterman, in separating various complexity classes and, in some cases it leads to decidability of such classes. In joint work with Jean-Éric Pin and Serge Grigorieff we have shown that this theory may be seen as a special case of Stone duality for Boolean algebras extended to a duality between Boolean algebras with additional operations and Stone spaces equipped with Kripke style relations. This is a duality which also plays a fundamental role in other parts of the foundations of computer science, including in modal logic and in domain theory. In this talk I will give a general introduction to Stone duality and explain what this has to do with the connection between regular languages and monoids. Stone dualityStone type dualities is the fundamental tool for moving between linguistic specification and spatial dynamics or transitional unfolding. As such, it should come as no surprise that it is a theory of central importance in the foundations of computer science where one necessarily is dealing with syntactic specifications and their effect on physical computing systems. In 1936, M. H. Stone initiated duality theory by presenting what, in modern terms, is a dual equivalence between the category of Boolean algebras and the category of compact Hausdorff spaces having a basis of clopen sets, so-called Boolean spaces [13]. The points of the space corresponding to a given Boolean algebra are not in general elements of the algebra -just like states of a system are not in general available as entities in a specification language but are of an entirely different sort. In models of computation these two different sorts, specification expressions and states, are given a priori but in unrelated settings. Via Stone duality, the points of the space may be obtained from the algebra as homomorphisms into the two-element Boolean algebra or equivalently as ultrafilters of the algebra. In logical terms these are valuations or models of the Boolean algebra. In computational terms they are possible states of the system. Each element of the Boolean algebra corresponds to the set of all models in which it is true, or all states in which it holds, and the topology of the space is generated by these sets. A main insight of Stone is that one may recover the original algebra as the Boolean algebra of clopen subsets of the resulting space.

The dual equivalence of equations and coequations for automata

“…This is explained in some detail in Section 10. This last section moreover discusses how our work connects with the duality results appearing in Almeida [Alm89,Alm94], Pippenger [Pip97], Gehrke [Geh09,Geh11] and Gehrke, Grigorieff and Pin [GGP08]. Based on that discussion, Section 10 presents also some ideas for future research.…”

“…We begin by relating our duality result to the work of Gehrke [Geh11]). There all automata A = (Q, A, δ, I, F ) are finite.…”

“…Recall that the profinite monoid A * can be constructed both as the completion of an ultrametric defined on A * or as the projective limit of all finite monoids whose generators are in A (See [GGP08] and [Alm03], respectively). Indeed, our results in the monoid side refer to objects (A * /C, for some congruence C on A * ) and the results on [Pip97], [GGP08], [Geh09] and [Geh11] refer to limit constructions (profinite monoid). Indeed, the monoid A * cannot be written as A * /C for some congruence C on A * and, therefore, our results per se do not apply.…”

“…(Thus, aperiodic finite semigroups, for example, satisfy the identity x k = x k+1 for sufficiently large k). The modern rendering of such intuitions appear in the work done by Gehrke [Geh09,Geh11] and Pin [GGP08], who take a further step in this direction showing that lattices of languages are precisely those classes of regular languages being defined by profinite identities [GGP08]. It follows from these works the strong connection between classes of regular languages, finite monoids and sets of profinite identities.…”

Formations of Monoids, Congruences, and Formal Languages

“…One of the most interesting consequences of their work is a theorem stating that lattices of regular languages, that is sets of regular languages closed under finite intersection and finite union can be defined by sets of profinite inequations [9,Theorem 5.2]. It is intimately based on the connection between duality theory and the algebraic theory of finite state automaton presented in detail in [7,8]. This result is an instantiation of the duality between sublattices in the set Reg(A * ) of all regular languages over an alphabet A and preorders on its dual space A * , the relatively free profinite monoid.…”