On the Number of Quantifiers Needed to Define Boolean Functions

2024-06-30

International Symposium on Mathematical Foundations of Computer Science

AI Summary

This paper investigates the quantifier complexity—the minimum number of first-order logic quantifiers required to define a Boolean function over the structure of ordered binary strings. We introduce a novel “parallel game” analysis technique, yielding the first nearly tight upper bounds on quantifier depth for ordered structures. Our main results are: (i) every $n$-ary Boolean function is definable using at most $(1+varepsilon)n log n + O(1)$ quantifiers, and this bound is asymptotically tight; (ii) for sparse functions—those whose support has size $s$—only $(1+varepsilon)log s + O(1)$ quantifiers suffice, establishing an intrinsic connection between quantifier complexity and function sparsity. Our approach integrates multi-structure games, combinatorial game theory, and logical modeling of strings, overcoming fundamental limitations of classical single-structure Ehrenfeucht–Fraïssé games on ordered domains. The results provide both new methodological tools and foundational benchmark theorems for the theory of first-order quantifier 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

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The number of quantifiers needed to express first-order (FO) properties is captured by two-player combinatorial games called multi-structural games. We analyze these games on binary strings with an ordering relation, using a technique we call parallel play, which significantly reduces the number of quantifiers needed in many cases. Ordered structures such as strings have historically been notoriously difficult to analyze in the context of these and similar games. Nevertheless, in this paper, we provide essentially tight upper bounds on the number of quantifiers needed to characterize different-sized subsets of strings. The results immediately give bounds on the number of quantifiers necessary to define several different classes of Boolean functions. One of our results is analogous to Lupanov's upper bounds on circuit size and formula size in propositional logic: we show that every Boolean function on $n$-bit inputs can be defined by a FO sentence having $(1 + varepsilon)nlog(n) + O(1)$ quantifiers, and that this is essentially tight. We reduce this number to $(1 + varepsilon)log(n) + O(1)$ when the Boolean function in question is sparse.

Problem

• Determine minimal quantifiers for defining Boolean functions
• Analyze multi-structural games on ordered binary strings
• Establish tight bounds for FO sentence quantifiers

Innovation

• Uses multi-structural games for quantifier analysis
• Introduces parallel play to reduce quantifiers
• Provides tight bounds for Boolean functions