Modular Counting CSP: Reductions and Algorithms
Bibliographic record
Abstract
The Constraint Satisfaction Problem (CSP) is ubiquitous in various areas of mathematics and computer science. Many of its variations have been studied including the Counting CSP, where the goal is to find the number of solutions to a CSP instance. The complexity of finding the exact number of solutions of a CSP is well understood (Bulatov, 2013, and Dyer and Richerby, 2013) and the focus has shifted to other variations of the Counting CSP such as counting the number of solutions modulo an integer. This problem has attracted considerable attention recently. In the case of CSPs based on undirected graphs Bulatov and Kazeminia (STOC 2022) obtained a complexity classification for the problem of counting solutions modulo p for arbitrary prime p. In this paper we report on the progress made towards a similar classification for the general CSP, not necessarily based on graphs. We identify several features that make the general case very different from the graph case such as a stronger form of rigidity and the structure of automorphisms of powers of relational structures. We provide a solution algorithm in the case p = 2 that works under some additional conditions and prove the hardness of the problem under some assumptions about automorphisms of the powers of the relational structure. We also reduce the general CSP to the case that only uses binary relations satisfying strong additional conditions.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".