On Second-Order Monadic Monoidal and Groupoidal Quantifiers

We study logics defined in terms of second-order monadic monoidal and groupoidal quantifiers. These are generalized quantifiers defined by monoid and groupoid word-problems, equivalently, by regular and context-free languages. We give a computational classification of the expressive power of these logics over strings with varying built-in predicates. In particular, we show that ATIME(n) can be logically characterized in terms of second-order monadic monoidal quantifiers.

💡 Research Summary

The paper introduces a novel family of second‑order monadic quantifiers that are defined by algebraic structures—monoids and groupoids—rather than by arbitrary predicates. A “monoidal quantifier” corresponds to a regular language (the word problem of a finite monoid), while a “groupoidal quantifier” corresponds to a context‑free language (the word problem of a finite groupoid). By embedding these quantifiers into first‑order logic (FO) the authors obtain enriched logics FO + Q_Monoid and FO + Q_Groupoid, and they systematically study the expressive power of these logics over finite strings under various built‑in predicates.

The technical development proceeds in several stages. First, the authors formalize the syntax and semantics of the new quantifiers. A monoidal quantifier Q_M on a structure interprets a set of positions as a word over a fixed alphabet and accepts exactly those assignments that form a word belonging to the regular language L(M). Similarly, a groupoidal quantifier Q_G accepts assignments whose induced word lies in a context‑free language L(G). This construction yields a clean algebraic characterization: regular languages ↔ monoid word problems, context‑free languages ↔ groupoid word problems.

Next, the paper classifies the computational complexity of the resulting logics under different collections of built‑in predicates:

• Only the linear order <: FO