Linear circuits, two-variable logic and weakly blocked monoids

Abstract

Following recent works that connect two-variable logic to circuits and monoids, as well as recent techniques for obtaining algebraic and logical characterizations of threshold circuits, we present an algebraic and logical characterization of the language recognition power of uniform linear threshold circuits. More specifically, for numerical predicate sets P satisfying a certain closure property, we characterize the class of languages recognized by FO[<,P]-uniform linear threshold circuits as exactly those which are recognized by a particular class of two-variable formulas that use majority quantifiers and P predicates, and are exactly those which can be recognized by a particular class of weakly blocked products of typed monoids. This correspondence will hold for any numerical predicate set which is FO[<]-closed and whose predicates do not depend on the input length.