Fuzzy Description Logics with General Concept Inclusions

Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived

Approximate State Reduction of Fuzzy Finite Automata

In this paper we introduce a new type of approximate state reductions where the behaviors of the reduced and the original automaton do not have to be identical, but they must match on all words of length less than or equal to some given natural number. We provide four methods for performing such reductions.Comment: In Proceedings AFL 2023, arXiv:2309.0112

Robust Linear Temporal Logic

Although it is widely accepted that every system should be robust, in the sense that "small" violations of environment assumptions should lead to "small" violations of system guarantees, it is less clear how to make this intuitive notion of robustness mathematically precise. In this paper, we address this problem by developing a robust version of Linear Temporal Logic (LTL), which we call robust LTL and denote by rLTL. Formulas in rLTL are syntactically identical to LTL formulas but are endowed with a many-valued semantics that encodes robustness. In particular, the semantics of the rLTL formula $\varphi \Rightarrow \psi$ is such that a "small" violation of the environment assumption $\varphi$ is guaranteed to only produce a "small" violation of the system guarantee $\psi$. In addition to introducing rLTL, we study the verification and synthesis problems for this logic: similarly to LTL, we show that both problems are decidable, that the verification problem can be solved in time exponential in the number of subformulas of the rLTL formula at hand, and that the synthesis problem can be solved in doubly exponential time