<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -sets and finite automata
Bibliographic record
Abstract
It is observed that Derksen’s Skolem–Mahler–Lech theorem is a special case of the isotrivial positive characteristic Mordell-Lang theorem due to the second author and Scanlon. This motivates an extension of the classical notion of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -automatic subset of the natural numbers to that of an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -automatic subset of a finitely generated abelian group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> equipped with an endomorphism <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> . Applied to the Mordell–Lang context, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> is the Frobenius action on a commutative algebraic group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> over a finite field, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> is a finitely generated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -invariant subgroup of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , it is shown that the “ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -subsets” of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> introduced by the second author and Scanlon are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -automatic. It follows that when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> is semiabelian and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> is a closed subvariety then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -automatic. Derksen’s notion of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -normal subset of the natural numbers is also here extended to the above abstract setting, and it is shown that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -subsets are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -normal. In particular, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> appearing in the Mordell-Lang problem are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -normal. This generalises Derksen’s Skolem–Mahler–Lech theorem to the Mordell–Lang context.
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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 | 0.001 | 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 | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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".