Non-Expansive Fuzzy Coalgebraic Logic
Stefan Gebhart, Lutz Schröder, Paul Wild
TL;DR
The paper develops a unifying, non-expansive fuzzy coalgebraic framework that encompasses probabilistic, metric, and relational modal logics with real-valued truth degrees in $[0,1]$. By introducing a robust one-step logic and a rectangularity (independence) condition, it derives a PSPACE decidability criterion that applies across multiple instantiated logics, including non-expansive fuzzy $ALC$, the logic of generally, and fuzzy metric modal logics. The key technical contribution is a vector-based representation of one-step models (via Carathéodory-style decompositions) that reduces full satisfiability to a finite, space-bounded check, together with a tableau calculus and a propositional reduction that keeps space complexity in check. The results yield new PSPACE upper bounds for several quantitative modal logics and provide a coherent framework for analyzing tractability in a broad spectrum of coalgebraic, fuzzy, and probabilistic systems, with open questions remaining for the logic of probably and global assumptions.
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.