Generalized Satisfiability Problems via Operator Assignments

TL;DR

This paper extends Schaefer's framework for Boolean satisfiability by incorporating operator assignments and Fourier transforms, providing a complete characterization of when quantum-inspired relaxations differ from classical solutions.

Contribution

It introduces an algebraic approach using Fourier transforms and operator assignments to analyze generalized satisfiability problems, extending classical CSP theory to quantum-inspired relaxations.

Findings

Characterizes Boolean relations with satisfiability gaps between classical and operator-based solutions.

Shows pp-definability with operator variables does not increase expressive power.

Provides gadget reductions that preserve satisfiability gaps.

Abstract

Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer's framework in which the Fourier transform is used to represent constraints by multilinear polynomials in a unique way. The polynomial representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For the case of constraints given by a system of linear equations over the two-element field, this relaxation has received considerable attention in the foundations of quantum mechanics, where such constructions as the Mermin-Peres magic square show that there are systems that have no solutions in the Boolean domain, but have solutions via operator assignments on some finite-dimensional Hilbert space. We obtain a complete characterization of the classes of Boolean relations for which there is a gap between satisfiability in the Boolean domain and the relaxation of satisfiability via operator assignments. To establish our main result, we adapt the notion of primitive-positive definability (pp-definability) to our setting, a notion that has been used extensively in the study of constraint satisfaction problems. Here, we show that pp-definability gives rise to gadget reductions that preserve satisfiability gaps. We also present several additional applications of this method. In particular and perhaps surprisingly, we show that the relaxed notion of pp-definability in which the quantified variables are allowed to range over operator assignments gives no additional expressive power in defining Boolean relations.