Orthogonal Latin squares of order 10 with two relations: A SAT investigation
Bibliographic record
Abstract
A [Formula: see text]-net([Formula: see text]) is a combinatorial design equivalent to [Formula: see text] mutually orthogonal Latin squares of order [Formula: see text]. A relation in a net is a linear dependency over [Formula: see text] in the incidence matrix of the net. A computational enumeration of all orthogonal pairs of Latin squares of order 10 whose corresponding nets have at least two nontrivial relations was achieved by Delisle in 2010 and verified by an independent search of Myrvold. In this paper, we confirm the correctness of their exhaustive enumerations with a satisfiability (SAT) solver approach instead of using custom-written backtracking code. Performing the enumeration using a SAT solver has at least three advantages. First, it reduces the amount of trust necessary, as SAT solvers produce independently-verifiable certificates that their enumerations are complete. These certificates can be checked by formal proof verifiers that are relatively simple pieces of software, and therefore easier to trust. Second, it is typically more straightforward and less error-prone to use a SAT solver over writing search code. Third, it can be more efficient to use a SAT-based approach, as SAT solvers are highly optimized pieces of software incorporating backtracking-with-learning for improving the efficiency of the backtracking search. For example, the SAT solver completely enumerates all orthogonal pairs of Latin squares of order 10 with two nontrivial relations in under 2[Formula: see text]h on a desktop machine, while Delisle’s 2010 search used 11,700 CPU hours. Although computer hardware was slower in 2010, this alone cannot explain the improvement in the efficiency of our SAT-based search.
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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 | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.016 | 0.002 |
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".