Translating Multi-Agent Modal Logics of Knowledge and Belief into Decidable First-Order Fragments
Bibliographic record
Abstract
Translation-based modal theorem proving has been studied for decades. By reducing modal formulae to fragments of first-order logic, methods developed for first-order reasoning can be applied to modal inference problems. However, the existing translation approaches are insufficient for modal systems with specific frame properties, such as transitivity or Euclideanity, since they result in formulae not in a decidable first-order fragment. With a revisit of the set-based possible-worlds semantics, we propose a new translation for multi-agent modal systems of knowledge and belief, such as K(D)45_n and S5_n. We prove that the resulting formulae of the translation are in the two-variable guarded fragment. Therefore the decidability of the general satisfiability problem is preserved and it paves the way for translation-based reasoning in these modal systems. We also extend our approach to first-order modal logic and consider a decidable fragment.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| 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".