A Topological Approach to Recognition

We propose a new approach to the notion of recognition, which departs from the classical definitions by three specific features. First, it does not rely on automata. Secondly, it applies to any Boolean algebra (BA) of subsets rather than to individual subsets. Thirdly, topology is the key ingredient. We prove the existence of aminimum recognizer in a very general setting which applies in particular to any BA of subsets of a discrete space. Our main results show that this minimum recognizer is a uniform space whose completion is the dual of the original BA in Stone-Priestley duality; in the case of a BA of languages closed under quotients, this completion, called the syntactic space of the BA, is a compact monoid if and only if all the languages of the BA are regular. For regular languages, one recovers the notions of a syntactic monoid and of a free profinite monoid. For nonregular languages, the syntactic space is no longer a monoid but is still a compact space. Further, we give an equational characterization of BA of languages closed under quotients, which extends the known results on regular languages to nonregular languages. Finally, we generalize all these results from BAs to lattices, in which case the appropriate structures are partially ordered. Recognizability is one of the most fruitful concepts in computer science [16, 3]. Originally introduced for finite words, it has been successfully extended to infinite words, general algebras, finite and infinite trees and to many other structures. Roughly speaking, a set is recognizable if it is saturated by a congruence of finite index. A desirable property for any notion of recognition is the existence of a minimum recognizer. Here, the word “minimum” does not refer to algorithmic properties, like the minimal number of states of an automaton, but rather to a final object in a suitable category. For instance, there is a well defined notion of morphism of deterministic automata. Final objects exist in this category and are just the usual minimal automata. But no such notion is known for automata on infinite words, even for ω-regular languages. This problem has been overcome by using another type of recognizer. For languages of finite words, automata can be replaced by monoids, for which there is a minimal recognizer, the syntactic monoid. For ω-regular languages, ω-semigroups are a category in which minimal recognizers do exist again [9, 17]. ⋆ The authors acknowledge support from the AutoMathA programme of the European Science Foundation and from the programme Research in Paris. The aim of this paper is to propose a general definition of recognition for which each object has a unique minimum recognizer. Our approach departs from the classical definitions of recognition by three specific features: (1) it does not rely at all on automata; (2) it applies to Boolean algebras or more generally to lattices of subsets rather than to individual subsets; (3) topology, and in particular Stone-Priestley duality, is the key ingredient. Our most general definition is given in the category of uniform spaces, an abstraction of the notion of metric spaces well-known to topologists. We predominantly consider discrete spaces where the topology itself carries no valuable information. However, an appropriate uniformity gives rise by completion to a compact space, a common practice in mathematics, where it is often said that “compactness is the next best thing to finiteness”. We develop a notion of recognition in this general setting and prove that any Boolean algebra of subsets of a uniform space admits a minimum recognizer, which is again a uniform space, whose completion is the dual space of the original Boolean algebra in the sense of Stone-Priestley duality. When the uniform space carries an algebraic structure for which the operations are at least separately uniformly continuous, it is natural to require that the recognizing maps be morphisms for the algebraic structure as well. In the case of a monoid, this amounts to working in the category of semiuniform monoids and imposes closure under quotients of the Boolean algebra. The minimum recognizer is then a semiuniform monoid whose completion is called the syntactic space of the Boolean algebra with quotients. We prove that this syntactic space is a compact monoid if and only if all the subsets of the Boolean algebra are recognizable. For a regular language, one recovers the classical notion of a syntactic monoid. For a variety of regular languages, one obtains the free profinite monoid of the corresponding variety of monoids. For nonregular languages, the syntactic space is no longer a monoid but is still a compact space. We also prove that any Boolean algebra of languages closed under quotient has an equational description. Finally, we generalize all these results from Boolean algebras to lattices and recover in this way the notion of a syntactic ordered monoid. 1 The topological setting In this section, we recall the notions of topology needed to read this paper. We invite the reader to look at Wikipedia for an introduction and suitable references (notably the entries uniform spaces and Stone-Čech compactification). Let X be a set. We denote by L the complement of a subset L of X . The subsets of X × X can be viewed as relations on X . Given a relation U , the transposed relation of U is the relation U = { (x, y) ∈ X ×X | (y, x) ∈ U } . We denote by UV the composition of two relations U and V on X . Thus UV = { (x, y) ∈ X ×X | there exists z ∈ X, (x, z) ∈ U and (z, y) ∈ V } . Finally, if x ∈ X and U ⊆ X×X , we write U(x) for the set {y ∈ X | (x, y) ∈ U}.