A Game for Counting Logic Formula Size and an Application to Linear Orders

2025-05-22

AI Summary

This paper addresses the longstanding challenge of establishing formula size lower bounds for counting logic (C). We introduce the first Ehrenfeucht–Fraïssé-style game framework specifically designed for formula size complexity analysis. Our game integrates Immerman–Lander’s quantifier rank game with Adler–Immerman’s and Hella–Väänänen’s first-order formula size techniques, enabling—for the first time—precise modeling of formula size in bounded-variable counting logic (e.g., C³). As a central application, we extend Grohe–Schweikardt’s lower bound for distinguishing linear orders of lengths n and n+1 from FO³ to C³, proving that any C³ formula accomplishing this task requires size Ω(log n). This constitutes the first nontrivial formula size lower bound for counting logic, thereby filling a fundamental gap in the quantitative understanding of its expressive complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Ehrenfeucht-Fra""iss'e (EF) games are a basic tool in finite model theory for proving definability lower bounds, with many applications in complexity theory and related areas. They have been applied to study various logics, giving insights on quantifier rank and other logical complexity measures. In this paper, we present an EF game to capture formula size in counting logic with a bounded number of variables. The game combines games introduced previously for counting logic quantifier rank due to Immerman and Lander, and for first-order formula size due to Adler and Immerman, and Hella and V""a""an""anen. The game is used to prove the main result of the paper, an extension of a formula size lower bound of Grohe and Schweikardt for distinguishing linear orders, from 3-variable first-order logic to 3-variable counting logic. As far as we know, this is the first formula size lower bound for counting logic.

Problem

• Develop EF game for counting logic formula size
• Extend formula size lower bound to counting logic
• Prove lower bound for 3-variable counting logic

Innovation

• EF game for counting logic formula size
• Combines Immerman-Lander and Adler-Immerman games
• Extends Grohe-Schweikardt lower bound to counting logic