Classification of Non-redundancy of Boolean Predicates of Arity 4

2026-03-22

AI Summary

This study addresses the classification of non-redundant decidability (NRD) for 4-ary Boolean predicates, which concerns determining the maximum instance size satisfying all but one constraint. By integrating techniques from algorithmic classification, extremal combinatorics, logic, and lattice theory, the authors achieve a complete classification for 397 out of 400 non-trivial 4-ary predicates. The remaining two cases are resolved using advanced extremal combinatorial methods, leaving only a single open problem. Notably, this work uncovers, for the first time, Boolean predicates exhibiting non-polynomial asymptotic behavior in their non-redundant instances, thereby substantially expanding the theoretical boundaries of non-redundancy within the framework of constraint satisfaction problems (CSPs).

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification

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: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Given a constraint satisfaction problem (CSP) predicate $P \subseteq D^r$, the non-redundancy (NRD) of $P$ is maximum-sized instance on $n$ variables such that for every clause of the instance, there is an assignment which satisfies all but that clause. The study of NRD for various CSPs is an active area of research which combines ideas from extremal combinatorics, logic, lattice theory, and other techniques. Complete classifications are known in the cases $r=2$ and $(|D|=2, r=3)$. In this paper, we give a near-complete classification of the case $(|D|=2, r=4)$. Of the 400 distinct non-trivial Boolean predicates of arity 4, we implement an algorithmic procedure which perfectly classifies 397 of them. Of the remaining three, we solve two by reducing to extremal combinatorics problems -- leaving the last one as an open question. Along the way, we identify the first Boolean predicate whose non-redundancy asymptotics are non-polynomial.

Problem

• non-redundancy
• Boolean predicates
• arity 4
• constraint satisfaction problem
• classification

Innovation

• non-redundancy
• Boolean predicates
• constraint satisfaction problem
• extremal combinatorics
• arity-4 classification