Basics of Modal Semirings and of Kleene / Omega Algebras
Bibliographic record
Abstract
Algebraic structures, such as modal idempotent semirings or Kleene algebras, offer a large variety of applications, while requiring only a small set of operators and axioms.Such algebras abstractly capture so-called Kripke structures, i.e., access relations over a set of worlds or states.In addition they provide the associated multi-modal operators box and diamond that allow reasoning, e.g., about possible actions of agents in a system or about state transitions in general.Particular instances of modal semirings are provided by the algebra of homogeneous binary relations and by abstract relation algebras.This setting allows many general considerations and results, ranging from epistemic logics with knowledge and belief [55] to propositional dynamic Hoare logic and resource-based settings such as separation logic [15].Moreover, many further applications are covered, like abstract reasoning about bisimulations for model refinement [29], formal concept analysis, simple and concise correctness proofs for the optimisation of database preference queries [56], Petri nets [16] or generally applicable models of module hierarchies in a feature oriented software development process [6].A large collection of such examples is treated in the forthcoming book Modal semirings and applications by the two authors, of which this report presents the first three chapters with the basic algebraic definitions and essential theorems about them.It serves as a reference for the current state of the theory.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.005 | 0.010 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".