Satisfiability of Commutative vs. Non-Commutative CSPs
Bibliographic record
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.018 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".