On emulational equivalence of impartial games and the game Hackenforb
Bibliographic record
Abstract
We introduce a variant of the game Hackenbush, called Hackenforb. It is a class of games, each of which is determined by two parameters: a given graph, and a given set of connected graphs (called forbidden graphs). The significance of this game within the realm of impartial combinatorial games is reflected in the fact that, as we show in this article, various known combinatorial games, such as Nim, Subtraction game, Notakto, Treblecross, Chomp, are emulationally equivalent to an instance of Hackenforb (an emulational equivalence of two games is a concept stronger than Grundy-equivalence, but weaker than the isomorphism between games' structures; our belief is that this version of equivalence is what really captures the core of the intuitive perception of what it means for two games to be ``basically the same game"). At the end of our article, we show that Hackenforb is, unfortunately, not ``almighty," that is, we describe a game that is not emulationally equivalent to an instance of Hackenforb.
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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 | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 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".