Towards Completeness via Proof Search in the Linear Time mu-Calculus: The case of Büchi inclusions
Bibliographic record
Abstract
Modal µ-calculus is one of the central languages of logic and verification , whose study involves notoriously complex objects: au-tomata over infinite structures on the model-theoretical side; infinite proofs and proofs by (co)induction on the proof-theoretical side. Nevertheless, axiomatizations have been given for both linear and branching time µ-calculi, with quite involved completeness arguments. We come back to this central problem, considering it from a proof search viewpoint, and provide some new completeness arguments in the linear time µ-calculus. Our results only deal with restricted classes of formulas that closely correspond to (non-alternating) ω-automata but, compared to earlier proofs, our completeness arguments are direct and constructive. We first consider a natural circular proof system based on sequent calculus, and show that it is complete for inclusions of parity automata directly expressed as formulas, making use of Safra's construction directly in proof search. We then consider the corresponding finitary proof system, featuring (co)induction rules, and provide a partial translation result from circular to finitary proofs. This yields completeness of the finitary proof system for inclusions of sufficiently determin-istic parity automata, and finally for arbitrary Büchi automata.
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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 | 0.013 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.005 | 0.004 |
| 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 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".