Non-Expansive Fuzzy Coalgebraic Logic

2025-10-13

AI Summary

Fuzzy modal logic faces a fundamental trade-off between expressivity and decidability: the Zadeh basis is decidable but weakly expressive, whereas the Łukasiewicz basis is highly expressive yet often undecidable. This paper introduces a non-expansive fuzzy coalgebraic logic framework, using non-expansiveness of propositional bases as a decidability criterion, and uniformly models behavioral distance logics for quantitative systems—including probabilistic and metric transition systems—at the coalgebraic level. Methodologically, it integrates fuzzy logic, coalgebraic modal logic, and behavioral distance theory, employing non-expansive conjunctions and fixed-point semantics to characterize computational complexity. The key contributions are: (i) the first PSpace-decidability result for fuzzy modal logic within this framework, overcoming undecidability in the Łukasiewicz setting; and (ii) PSpace upper bounds for non-expansive fuzzy ALC and several probabilistic and metric modal logics. This yields a unified logical foundation for quantitative systems that balances expressivity with computational feasibility.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Fuzzy logic extends the classical truth values "true" and "false" with additional truth degrees in between, typically real numbers in the unit interval. More specifically, fuzzy modal logics in this sense are given by a choice of fuzzy modalities and a fuzzy propositional base. It has been noted that fuzzy modal logics over the Zadeh base, which interprets disjunction as maximum, are often computationally tractable but on the other hand add little in the way of expressivity to their classical counterparts. Contrastingly, fuzzy modal logics over the more expressive Lukasiewicz base have attractive logical properties but are often computationally less tractable or even undecidable. In the basic case of the modal logic of fuzzy relations, sometimes termed fuzzy ALC, it has recently been shown that an intermediate non-expansive propositional base, known from characteristic logics for behavioural distances of quantitative systems, strikes a balance between these poles: It provides increased expressiveness over the Zadeh base while avoiding the computational problems of the Lukasiewicz base, in fact allowing for reasoning in PSpace. Modal logics, in particular fuzzy modal logics, generally vary widely in terms of syntax and semantics, involving, for instance, probabilistic, preferential, or weighted structures. Coalgebraic modal logic provides a unifying framework for wide ranges of semantically different modal logics, both two-valued and fuzzy. In the present work, we focus on non-expansive coalgebraic fuzzy modal logics, providing a criterion for decidability in PSpace. Using this criterion, we recover the mentioned complexity result for non-expansive fuzzy ALC and moreover obtain new PSpace upper bounds for various quantitative modal logics for probabilistic and metric transition systems.

Problem

• Develops non-expansive coalgebraic fuzzy modal logics for quantitative systems
• Balances expressiveness and computational tractability in fuzzy modal logics
• Establishes PSpace decidability for probabilistic and metric transition logics

Innovation

• Non-expansive coalgebraic framework for fuzzy modal logics
• Decidability criterion ensuring PSpace complexity bounds
• Balanced expressiveness between Zadeh and Lukasiewicz bases