SAT + NAUTY: Orderly Generation of Small Kochen-Specker Sets Containing the Smallest State-independent Contextuality Set

2026-04-21

AI Summary

This work addresses the problem of identifying the smallest three-dimensional Kochen-Specker (KS) set that contains the complete 25-ray state-independent contextuality (SI-C) set. To this end, the authors propose a novel framework integrating SAT solving with NAUTY-based graph isomorphism detection, introducing recursive canonical labeling (RCL) for the first time to enable efficient and ordered generation of candidate sets. This approach substantially mitigates the exponential performance degradation inherent in traditional lexicographic ordering for large instances, achieving millisecond-scale isomorph rejection while incorporating DRAT proof certificates to ensure verifiability. Using 1,641 CPU hours, the study exhaustively searches all KS sets up to 33 rays, formally proving that Schütte’s 33-ray set is indeed the minimal three-dimensional KS set containing the 25-ray SI-C set and providing a formal non-existence proof for any smaller such configuration.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present a search for small Kochen-Specker (KS) sets in dimension 3, specifically targeting extensions of the 13-ray Yu-Oh set, which has been proven to be the minimal witness to state-independent contextuality. To enable this search, we introduce a novel SAT-based orderly generation framework integrating recursive canonical labeling (RCL) with the graph isomorphism tool NAUTY. We demonstrate that previous SAT approaches relying on lexicographical canonicity suffer from exponential scaling on canonical graphs. This limitation renders them intractable on the large instances (25 to 33 vertices) encountered in our search, whereas our RCL check maintains consistent millisecond-level performance, effectively eliminating the bottleneck. Overcoming this bottleneck allows us to perform the first exhaustive enumeration of all KS sets with up to 33 rays containing the complete 25-ray state-independent contextuality (SI-C) set obtained by rigid extensions of the Yu-Oh set in 1,641 CPU hours. We found and verified that the 33-ray set discovered by Schütte is the smallest three-dimensional KS set containing the complete 25-ray SI-C set. All non-existence results are backed by independently verifiable proof certificates via an extension of the DRAT proof format.

Problem

• Kochen-Specker sets
• state-independent contextuality
• Yu-Oh set
• orderly generation
• graph isomorphism

Innovation

• SAT-based orderly generation
• Recursive Canonical Labeling
• NAUTY
• Kochen-Specker sets
• state-independent contextuality