Bibliographic record
Abstract
Abstract Leroux has proved that unreachability in Petri nets can be witnessed by a Presburger separator, i.e. if a marking $$\boldsymbol{m}_\text {src}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mi>src</mml:mi> </mml:msub> </mml:math> cannot reach a marking $$\boldsymbol{m}_\text {tgt}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mi>tgt</mml:mi> </mml:msub> </mml:math> , then there is a formula $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> of Presburger arithmetic such that: $$\varphi (\boldsymbol{m}_\text {src})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mi>src</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> holds; $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> is forward invariant, i.e., $$\varphi (\boldsymbol{m})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$\boldsymbol{m} \rightarrow \boldsymbol{m}'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> imply $$\varphi (\boldsymbol{m}'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> ); and $$\lnot \varphi (\boldsymbol{m}_\text {tgt})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>¬</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mi>tgt</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> holds. While these separators could be used as explanations and as formal certificates of unreachability, this has not yet been the case due to their (super-)Ackermannian worst-case size and the (super-)exponential complexity of checking that a formula is a separator. We show that, in continuous Petri nets, these two problems can be overcome. We introduce locally closed separators, and prove that: (a) unreachability can be witnessed by a locally closed separator computable in polynomial time; (b) checking whether a formula is a locally closed separator is in NC (so, simpler than unreachablity, which is P-complete).
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.007 | 0.004 |
| Research integrity | 0.000 | 0.002 |
| 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".