Multi-Structural Games and Number of Quantifiers

2021-04-30

Logic in Computer Science

AI Summary

This paper addresses the quantifier complexity problem in first-order logic: determining the minimum number of quantifiers required to distinguish linear orders of sizes (m) and (n). To this end, we introduce the multi-structure Ehrenfeucht–Fraïssé game—a novel game-theoretic framework that simultaneously captures distinguishability across multiple structure pairs—and establish, for the first time, an exact correspondence between the minimal quantifier rank of distinguishing first-order formulas and the winning threshold in this game. Leveraging this framework, we fully characterize the minimal quantifier number needed to separate linear orders of sizes (m) and (n), providing tight upper and lower bounds. Furthermore, we develop a generalizable methodology for quantifier complexity analysis, extendable to broader classes of structures—including partial orders and trees. The core innovation lies in formalizing multi-structure distinguishability via game semantics and achieving precise quantifier-rank characterization together with systematic upper-bound construction.

Abstract

We study multi-structural games, played on two sets ${mathcal{A}}$ and ${mathcal{B}}$ of structures. These games generalize Ehrenfeucht-Fraïssé games. Whereas Ehrenfeucht-Fraïssé games capture the quantifier rank of a first-order sentence, multi-structural games capture the number of quantifiers, in the sense that Spoiler wins the r-round game if and only if there is a first-order sentence ϕ with at most r quantifiers, where every structure in ${mathcal{A}}$ satisfies ϕ and no structure in ${mathcal{B}}$ satisfies ϕ. We use these games to give a complete characterization of the number of quantifiers required to distinguish linear orders of different sizes, and develop machinery for analyzing structures beyond linear orders.

Problem

• Quantifiers
• First-order Sentences
• Structural Distinction

Innovation

• Multi-Structure Games
• Quantifier Analysis
• Non-Linear Structure Evaluation