Many-valued coalgebraic dynamic logics: Safety and strong completeness via reducibility

2025-12-28

AI Summary

This paper addresses the generalization of multi-valued dynamic logics—such as Propositional Dynamic Logic (PDL) and game logic—to semantics over FLew-algebras, a class of residuated lattices modeling fuzzy and many-valued reasoning. Method: We introduce a coalgebraic framework based on *A-valued predicate liftings*, modeling programs and games as *F-coalgebras*, where the functor *F* encodes *A*-weighted transition systems. Crucially, we define the novel notion of *reducible coalgebraic operations*, ensuring that modalities for composite actions decompose safely into algebraic combinations of sub-action modalities. Contribution/Results: We establish a uniform strong completeness theorem, yielding strong completeness of two-valued iteration-free PDL over finite chains and novel strong completeness results for Łukasiewicz multi-valued game logic. We prove that reducible operations preserve behavioral equivalence and bisimilarity. This framework provides the first generic, semantically sound, and compositionally guaranteed coalgebraic foundation for multi-valued dynamic logics.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a coalgebraic framework for studying generalisations of dynamic modal logics such as PDL and game logic in which both the propositions and the semantic structures can take values in an algebra $mathbf{A}$ of truth-degrees. More precisely, we work with coalgebraic modal logic via $mathbf{A}$-valued predicate liftings where $mathbf{A}$ is a $mathsf{FL}_{mathrm{ew}}$-algebra, and interpret actions (abstracting programs and games) as $mathsf{F}$-coalgebras where the functor $mathsf{F}$ represents some type of $mathbf{A}$-weighted system. We also allow combinations of crisp propositions with $mathbf{A}$-weighted systems and vice versa. We introduce coalgebra operations and tests, with a focus on operations that are reducible in the sense that modalities for composed actions can be reduced to compositions of modalities for the constituent actions. We prove that reducible operations are safe for bisimulation and behavioural equivalence, and prove a general strong completeness result, from which we obtain new strong completeness results for $2$-valued iteration-free PDL with $mathbf{A}$-valued accessibility relations when $mathbf{A}$ is a finite chain, and for many-valued iteration-free game logic with many-valued strategies based on finite Lukasiewicz logic.

Problem

• Develops a coalgebraic framework for many-valued dynamic modal logics.
• Studies reducible operations ensuring safety for bisimulation and behavioral equivalence.
• Proves strong completeness for many-valued PDL and game logic variants.

Innovation

• Coalgebraic framework for many-valued dynamic logics
• Reducible operations ensuring bisimulation safety
• Strong completeness proofs for many-valued modal logics