Complexity of Satisfiability in Kochen-Specker Partial Boolean Algebras

2026-02-27

AI Summary

This study investigates the computational complexity of the propositional satisfiability problem in Kochen–Specker-type partial Boolean algebras, which arise from the structure of projection operators in quantum mechanics. By introducing Kochen–Specker sets as a novel reduction tool and integrating real algebraic geometry (via the existential theory of the reals), semantic properties of partial Boolean algebras, and complexity-theoretic analysis, the work systematically characterizes the problem’s hardness across Hilbert space dimensions: it is NP-complete in nontrivial cases; complete for the existential theory of the reals when the real dimension exceeds 2 or the complex dimension exceeds 3; and undecidable over the class of all Hilbert spaces. These results establish a profound connection between quantum logical satisfiability and classical complexity classes.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The Kochen-Specker no-go theorem established that hidden-variable theories in quantum mechanics necessarily admit contextuality. This theorem is formally stated in terms of the partial Boolean algebra structure of projectors on a Hilbert space. Each partial Boolean algebra provides a semantics for interpreting propositional logic. In this paper, we examine the complexity of propositional satisfiablity for various classes of partial Boolean algebras. We first show that the satisfiability problem for the class of non-trivial partial Boolean algebras is NP-complete. Next, we consider the satisfiability problem for the class of partial Boolean algebras arising from projectors on finite dimensional Hilbert spaces. For real Hilbert spaces of dimension greater 2 and any complex Hilbert spaces of dimension greater than 3, we demonstrate that the satisfiablity problem is complete for the existential theory of the reals. Interestingly, the proofs of these results make use of Kochen-Specker sets as gadgets. As a corollary, we conclude that deciding quantum homomorphism in these fixed dimensions are also complete for the existential theory of the reals. Finally, we show that the satisfiability problems for the class of all Hilbert spaces and all finite-dimensional Hilbert spaces is undecidable.

Problem

• Satisfiability
• Partial Boolean Algebras
• Kochen-Specker theorem
• Computational Complexity
• Hilbert Spaces

Innovation

• Kochen-Specker theorem
• partial Boolean algebras
• satisfiability complexity
• existential theory of the reals
• quantum contextuality