Strategy synthesis for linear arithmetic games
Why this work is in the frame
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Bibliographic record
Abstract
Many problems in formal methods can be formalized as two-player games. For several applications—program synthesis, for example—in addition to determining which player wins the game, we are interested in computing a winning strategy for that player. This paper studies the strategy synthesis problem for games defined within the theory of linear rational arithmetic. Two types of games are considered. A satisfiability game , described by a quantified formula, is played by two players that take turns instantiating quantifiers. The objective of each player is to prove (or disprove) satisfiability of the formula. A reachability game , described by a pair of formulas defining the legal moves of each player, is played by two players that take turns choosing positions—rational vectors of some fixed dimension. The objective of each player is to reach a position where the opposing player has no legal moves (or to play the game forever). We give a complete algorithm for synthesizing winning strategies for satisfiability games and a sound (but necessarily incomplete) algorithm for synthesizing winning strategies for reachability games.
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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 | 0.001 | 0.011 |
| 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 | 0.000 | 0.000 |
| Open science | 0.006 | 0.001 |
| 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 it