Satisfiability of commutative vs. non-commutative CSPs

2024-04-17

International Colloquium on Automata, Languages and Programming

AI Summary

This paper investigates the “satisfiability gap”—the discrepancy between classical satisfiability and quantum satisfiability (i.e., existence of linear operator solutions over a Hilbert space)—for constraint satisfaction problems (CSPs) over Boolean algebras and finite fields. Methodologically, it integrates model theory, algebraic structure analysis, operator theory, and the quantum CSP framework, introducing a novel technique for simulating linear equations over abelian groups. The contributions are threefold: (1) a near-complete classification of satisfiability gaps for CSPs over arbitrary finite fields—NP-hard CSPs necessarily exhibit quantum gaps, while bounded-width CSPs admit no gap; (2) a complete characterization of necessary and sufficient conditions for equivalence between classical and quantum satisfiability; and (3) a constructive proof that, for characteristic $p = 2$, a finite-dimensional Hilbert-space satisfiability gap exists—establishing a new paradigm for understanding how commutativity versus noncommutativity fundamentally affects computation and enables quantum advantage.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The Mermin-Peres magic square is a celebrated example of a system of Boolean linear equations that is not (classically) satisfiable but is satisfiable via linear operators on a Hilbert space of dimension four. A natural question is then, for what kind of problems such a phenomenon occurs? Atserias, Kolaitis, and Severini answered this question for all Boolean Constraint Satisfaction Problems (CSPs): For 0-Valid-SAT, 1-Valid-SAT, 2-SAT, Horn-SAT, and Dual Horn-SAT, classical satisfiability and operator satisfiability is the same and thus there is no gap; for all other Boolean CSPs, these notions differ as there are gaps, i.e., there are unsatisfiable instances that are satisfiable via operators on Hilbert spaces. We generalize their result to CSPs on arbitrary finite domains and give an almost complete classification: First, we show that NP-hard CSPs admit a separation between classical satisfiability and satisfiability via operators on finite- and infinite-dimensional Hilbert spaces. Second, we show that tractable CSPs of bounded width have no satisfiability gaps of any kind. Finally, we show that tractable CSPs of unbounded width can simulate, in a satisfiability-gap-preserving fashion, linear equations over an Abelian group of prime order $p$; for such CSPs, we obtain a separation of classical satisfiability and satisfiability via operators on infinite-dimensional Hilbert spaces. Furthermore, if $p=2$, such CSPs also have gaps separating classical satisfiability and satisfiability via operators on finite- and infinite-dimensional Hilbert spaces.

Problem

• Classifying satisfiability gaps between classical and quantum CSPs
• Generalizing Boolean CSP gaps to arbitrary finite domains
• Separating classical and operator satisfiability for NP-hard CSPs

Innovation

• Generalized CSP classification to arbitrary finite domains
• Showed NP-hard CSPs have classical vs operator separations
• Tractable CSPs simulate Abelian group linear equations