Affine extensions of integer vector addition systems with states
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Bibliographic record
Abstract
We study the reachability problem for affine ℤ-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in general, and we therefore restrict ourselves to affine ℤ-VASS with the finite-monoid property (afmp-ℤ-VASS). The latter have the property that the monoid generated by the matrices appearing in their affine transformations is finite. The class of afmp-ℤ-VASS encompasses classical operations of counter machines such as resets, permutations, transfers and copies. We show that reachability in an afmp-ℤ-VASS reduces to reachability in a ℤ-VASS whose control-states grow linearly in the size of the matrix monoid. Our construction shows that reachability relations of afmp-ℤ-VASS are semilinear, and in particular enables us to show that reachability in ℤ-VASS with transfers and ℤ-VASS with copies is PSPACE-complete. We then focus on the reachability problem for affine ℤ-VASS with monogenic monoids: (possibly infinite) matrix monoids generated by a single matrix. We show that, in a particular case, the reachability problem is decidable for this class, disproving a conjecture about affine ℤ-VASS with infinite matrix monoids we raised in a preliminary version of this paper. We complement this result by presenting an affine ℤ-VASS with monogenic matrix monoid and undecidable reachability relation.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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