Linear Circuits, Two-Variable Logic and Weakly Blocked Monoids

Following recent works connecting two-variable logic to circuits and monoids, we establish, for numerical predicate sets P satisfying a certain closure property, a one-to-one correspondence between FO[<, P]-uniform linear circuits, two-variable formulas with P predicates, and weak block products of monoids. In particular, we consider the case of linear TC, majority quantifiers, and finitely 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.