Profinite Methods in Automata Theory

This survey paper presents the success story of the topological approach to automata theory. It is based on profinite topologies, which are built from finite topogical spaces. The survey includes several concrete applications to automata theory. In mathematics, p-adic analysis is a powerful tool of number theory. The p-adic topology is the emblematic example of a profinite topology, a topology that is in a certain sense built from finite topological spaces. The aim of this survey is to convince the reader that profinite topologies also play a key role in automata theory, confirming once again the following quote of Marshall Stone [38, p.814]: ’A cardinal principle of modern mathematical research may be stated as a maxim: “One must always topologize” ’. Unfortunately, this topic is rather abstract and not really intuitive. In particular, the appropriate framework to present the whole theory, namely uniform spaces, is unlikely to be sufficiently familiar to the average participant to STACS. To thwart this “user unfriendly” aspect, I downgraded from uniform spaces to metric spaces in this survey. This is sufficient to address most of the theory and it certainly makes the presentation easier to follow. When uniform spaces are really needed, I simply include a short warning addressed to the more advanced readers, preceded by the sign . More details can be found in specialized articles [1, 2, 3, 5, 27, 30, 40]. Profinite topologies for free groups were explored by M. Hall in [13]. However, the idea of profinite topologies goes back at least to Birkhoff [8, Section 13]. In this paper, Birkhoff introduces topologies defined by congruences on abstract algebras and states that, if each congruence has finite index, then the completion of the topological algebra is compact. Further, he explicitly mentions three examples: p-adic numbers, Stone’s duality of Boolean algebras and topologization of free groups. The duality between Boolean algebras and Stone spaces also appears in [1], [2, Theorem 3.6.1] and [31]. It is also the main ingredient in [12], where the extended duality between lattices and Priestley spaces is used. This duality approach is so important that it would deserve a survey article on its own. But due to the lack of space, I forwent, with some regrets, from presenting it in the present paper. The interested reader will find duality proofs of the results of Sections 4 and 5 in [12]. The survey is organised as follows. Section 1 is a brief reminder on metric spaces. Profinite words are introduced in Section 2 and used to give equational descriptions of varieties of finite monoids in Section 3 and of lattices of regular languages in Sections 4 and 5. We discuss various extensions of the profinite metric in Section 6 and we conclude in Section 7. The author acknowledge support from the AutoMathA programme of the European Science Foundation. LIAFA, Université Paris-Diderot and CNRS, Case 7014, 75205 Paris Cedex 13, France.