Using Duality in Circuit Complexity

Abstract

We investigate in a method for proving separation results for abstract classes of languages. A well established method to characterize varieties of regular languages are identities. We use a recently established generalization of these identities to non-regular languages by Gehrke, Grigorieff, and Pin: so called equations, which are capable of describing arbitrary Boolean algebras of languages.While the main concern of their result is the existence of these equations, we investigate in a general method that could allow to find equations for language classes in an inductive manner.Thereto we extend an important tool -the block product or substitution principle -known from logic and algebra, to non-regular language classes. Furthermore, we abstract this concept by defining it directly as an operation on (non-regular) language classes. We show that this principle can be used to obtain equations for certain circuit classes, given equations for the gate types.Concretely, we demonstrate the applicability of this method by obtaining a description via equations for all languages recognized by circuit families that contain a constant number of (inner) gates, given a description of the gate types via equations.

The Schützenberger product for syntactic spaces

“…Theorem 21 on equational completeness is by no means the final word, but rather a first stepping stone in this direction. In the regular setting, as well as in the special cases treated in [9] and [4], much smaller subsets of E(B♦ +2) have been shown to provide complete axiomatisations. We expect that a notion akin to the derived categories of profinite monoid theory [23] have to be developed, and we expect the remainder of the Stone-Čech compactification to play a key rôle in this.…”

A crevice on the Crane Beach: Finite-degree predicates

“…For instance, Koucký et al [7] foray into descriptive complexity and use model-theoretic tools to obtain Parity / ∈ AC 0 , but assert that "contrary to [their] original hope, [their] Ehrenfeucht-Fraïssé game arguments are not simpler than classical lower bounds." More recent promising approaches, especially the topological ones of [8], [9], have yet to yield strong lower bounds.…”