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Satisfiability problems and algebras of boolean constraint system games

Connor Paddock, William Slofstra

TL;DR

The paper develops BC S algebras $\mathscr{A}(B)$ for boolean constraint systems and links them to contextuality and nonlocal games, introducing a projection-based presentation equivalent to the standard one. It proves a suite of equivalences between perfect strategies and $*$-homomorphisms into appropriate operator algebras, and constructs a $C^*$-satisfiable but not tracially satisfiable example to separate these notions. By establishing functorial reductions that induce $*$-homomorphisms, it strengthens Schaefer-type dichotomies and connects satisfiability questions to hyperlinear group questions via $\mathcal{R}^{\mathcal{U}}$-satisfiability. The work clarifies when tracial, $C^*$, and algebraic satisfiability coincide (notably for linear systems) and shows how definability of constraint languages transfers separations across languages, illuminating the computational and foundational landscape of quantum satisfiability and MIP$^*$-related questions.

Abstract

Mermin and Peres showed that there are boolean constraint systems (BCSs) which are not satisfiable, but which are satisfiable with quantum observables. This has led to a burgeoning theory of quantum satisfiability for constraint systems, connected to nonlocal games and quantum contextuality. In this theory, different types of quantum satisfying assignments can be understood as representations of the BCS algebra of the system. This theory is closely related to the theory of synchronous games and algebras, and every synchronous algebra is a BCS algebra and vice-versa. The purpose of this paper is to further develop the role of BCS algebras in this theory, and tie up some loose ends: We give a new presentation of BCS algebras in terms of joint spectral projections, and show that it is equivalent to the standard definition. We construct a constraint system which is $C^*$-satisfiable but not tracially satisfiable. We show that certain reductions between constraint systems lead to $*$-homomorphisms between the BCS algebras of the systems, and use this to streamline and strengthen several results of Atserias, Kolaitis, and Severini on analogues of Schaefer's dichotomy theorem. In particular, we show that the question of whether or not there is a non-hyperlinear group is linked to dichotomy theorems for $\mathcal{R}^{\mathcal{U}}$-satisfiability.

Satisfiability problems and algebras of boolean constraint system games

TL;DR

The paper develops BC S algebras for boolean constraint systems and links them to contextuality and nonlocal games, introducing a projection-based presentation equivalent to the standard one. It proves a suite of equivalences between perfect strategies and -homomorphisms into appropriate operator algebras, and constructs a -satisfiable but not tracially satisfiable example to separate these notions. By establishing functorial reductions that induce -homomorphisms, it strengthens Schaefer-type dichotomies and connects satisfiability questions to hyperlinear group questions via -satisfiability. The work clarifies when tracial, , and algebraic satisfiability coincide (notably for linear systems) and shows how definability of constraint languages transfers separations across languages, illuminating the computational and foundational landscape of quantum satisfiability and MIP-related questions.

Abstract

Mermin and Peres showed that there are boolean constraint systems (BCSs) which are not satisfiable, but which are satisfiable with quantum observables. This has led to a burgeoning theory of quantum satisfiability for constraint systems, connected to nonlocal games and quantum contextuality. In this theory, different types of quantum satisfying assignments can be understood as representations of the BCS algebra of the system. This theory is closely related to the theory of synchronous games and algebras, and every synchronous algebra is a BCS algebra and vice-versa. The purpose of this paper is to further develop the role of BCS algebras in this theory, and tie up some loose ends: We give a new presentation of BCS algebras in terms of joint spectral projections, and show that it is equivalent to the standard definition. We construct a constraint system which is -satisfiable but not tracially satisfiable. We show that certain reductions between constraint systems lead to -homomorphisms between the BCS algebras of the systems, and use this to streamline and strengthen several results of Atserias, Kolaitis, and Severini on analogues of Schaefer's dichotomy theorem. In particular, we show that the question of whether or not there is a non-hyperlinear group is linked to dichotomy theorems for -satisfiability.
Paper Structure (10 sections, 17 theorems, 32 equations)

This paper contains 10 sections, 17 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Finitely-presented $*$-algebras
  4. The joint spectrum and representations of $\mathbb{C}\mathbb{Z}_2^k$
  5. Boolean constraint systems
  6. Boolean constraint system algebras and games
  7. Contextuality scenarios and nonlocal games
  8. Synchronous non-local games and algebras
  9. Separations between types of satisfiability
  10. Constraint languages and definability

Key Result

Lemma 3.5

Suppose $B = (X, \{(U_i,V_i)\}_{i=1}^{\ell})$ is a BCS with contexts. Given a constraint $C = (S,R) \in V_i$, $1 \leq i \leq \ell$ and an assignment $\phi$ to $X \cap S$, let Then $\mathscr{A}(B)$ is the quotient of $\mathscr{A}_{con}(B)$ by the relations $\Pi_{C,\phi} = 0$ for all $C = (S,R) \in V_i$, $1 \leq i \leq \ell$, and $\phi$ an assignment to $X \cap S$ such that $\phi(S) \not\in R$.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Example 3.4
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • proof
  • Example 3.7
  • ...and 36 more