and Andreas Krebs; licensed under Creative Commons License CC-BY 26th EACSL Annual Conference on Computer Science Logic

Silke Célia Borlido, Mai Czarnetzki, Gehrke
2017 Leibniz International Proceedings in Informatics Schloss Dagstuhl-Leibniz-Zentrum für Informatik   unpublished
In this paper we relate two generalisations of the finite monoid recognisers of automata theory for the study of circuit complexity classes: Boolean spaces with internal monoids and typed monoids. Using the setting of stamps, this allows us to generalise a number of results from algebraic automata theory as it relates to Büchi's logic on words. We obtain an Eilenberg theorem, a substitution principle based on Stone duality, a block product principle for typed stamps and, as our main result, a
more » ... ur main result, a topological semidirect product construction, which corresponds to the application of a general form of quantification. These results provide tools for the study of language classes given by logic fragments such as the Boolean circuit complexity classes. 1 Introduction Complexity theory and the theory of regular languages are intimately connected through logic. As with classes of regular languages, many computational complexity classes are model classes of appropriate logic fragments on finite words [12]. For example, AC 0 = FO[arb], ACC 0 = (FO + MOD)[arb], and TC 0 = MAJ[arb] where arb is the set of all predicates on the positions of a word, FO is first-order logic, and MOD and MAJ stand for the modular and majority quantifiers, respectively. On the one hand, the presence of arbitrary (numerical) predicates, and on the other hand, the presence of the majority quantifier is what brings one far beyond the scope of the profinite algebraic theory of regular languages. Most results in complexity theory are proved with combinatorial, probabilistic, and algorithmic methods [18]. However, there are a few connections with the topo-algebraic tools for regular languages. A famous result of Barrington, Compton, Straubing, and Thérien [2] states that a regular language is in AC 0 if and only if its syntactic homomorphism is quasi-aperiodic. Although this result relies on [5] and no purely algebraic proof is known, being able to characterise the class of regular languages in AC 0 gives some hope that the non-uniform classes might be amenable to treatment by the generalised topo-algebraic methods. Indeed, the hope is that one can generalise the tools of algebraic automata theory. In the paper Logic Meets Algebra: the case of regular languages [17], Thérien and Tesson lay *