Characterizing Polynomial Ramsey Quantifiers

TL;DR

This paper investigates the computational complexity of Ramsey quantifiers, demonstrating the existence of intermediate cases and establishing a dichotomy that characterizes polynomial-time computable quantifiers as constant-log-bounded within a natural class.

Contribution

It introduces the concept of intermediate Ramsey quantifiers and provides a dichotomy theorem classifying polynomial-time Ramsey quantifiers as constant-log-bounded under common complexity assumptions.

Findings

Existence of intermediate Ramsey quantifiers.

Dichotomy theorem for a natural class of Ramsey quantifiers.

Polynomial-time Ramsey quantifiers are exactly the constant-log-bounded ones.

Abstract

Ramsey quantifiers are a natural object of study not only for logic and computer science, but also for the formal semantics of natural language. Restricting attention to finite models leads to the natural question whether all Ramsey quantifiers are either polynomial-time computable or NP-hard, and whether we can give a natural characterization of the polynomial-time computable quantifiers. In this paper, we first show that there exist intermediate Ramsey quantifiers and then we prove a dichotomy result for a large and natural class of Ramsey quantifiers, based on a reasonable and widely-believed complexity assumption. We show that the polynomial-time computable quantifiers in this class are exactly the constant-log-bounded Ramsey quantifiers.