Undecidability of Fuzzy Description Logics

Fuzzy description logics (DLs) have been investigated for over two decades, due to their capacity to formalize and reason with imprecise concepts. Very recently, it has been shown that for several fuzzy DLs, reasoning becomes undecidable. Although the proofs of these results differ in the details of each specific logic considered, they are all based on the same basic idea. In this report, we formalize this idea and provide sufficient conditions for proving undecidability of a fuzzy DL. We demonstrate the effectiveness of our approach by strengthening all previously-known undecidability results and providing new ones. In particular, we show that undecidability may arise even if only crisp axioms are considered

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

Towards a Tableau Algorithm for Fuzzy ALC with Product T-norm

Very recently, the tableau-based algorithm for deciding consistency of general fuzzy DL ontologies over the product t-norm was shown to be incorrect, due to a very weak blocking condition. In this report we take the first steps towards a correct algorithm by modifying the blocking condition, such that the (finite) structure obtained through the algorithm uniquely describes an infinite system of quadratic constraints. We show that this procedure terminates, and is sound and complete in the sense that the input is consistent iff the corresponding infinite system of constraints is satisfiable

Consistency in Fuzzy Description Logics over Residuated De Morgan Lattices

Fuzzy description logics can be used to model vague knowledge in application domains. This paper analyses the consistency and satisfiability problems in the description logic SHI with semantics based on a complete residuated De Morgan lattice. The problems are undecidable in the general case, but can be decided by a tableau algorithm when restricted to finite lattices. For some sublogics of SHI, we provide upper complexity bounds that match the complexity of crisp reasoning

From fuzzy to annotated semantic web languages

The aim of this chapter is to present a detailed, selfcontained and comprehensive account of the state of the art in representing and reasoning with fuzzy knowledge in Semantic Web Languages such as triple languages RDF/RDFS, conceptual languages of the OWL 2 family and rule languages. We further show how one may generalise them to so-called annotation domains, that cover also e.g. temporal and provenance extensions

Gödel Description Logics

In the last few years there has been a large effort for analysing the computational properties of reasoning in fuzzy Description Logics. This has led to a number of papers studying the complexity of these logics, depending on their chosen semantics. Surprisingly, despite being arguably the simplest form of fuzzy semantics, not much is known about the complexity of reasoning in fuzzy DLs w.r.t. witnessed models over the Gödel t-norm. We show that in the logic G-IALC, reasoning cannot be restricted to finitely valued models in general. Despite this negative result, we also show that all the standard reasoning problems can be solved in this logic in exponential time, matching the complexity of reasoning in classical ALC

I can get some satisfaction: Fuzzy ontologies for partial agreements in blockchain smart contracts

This paper proposes a novel extension of blockchain systems with fuzzy ontologies. The main advantage is to let the users have flexible restrictions, represented using fuzzy sets, and to develop smart contracts where there is a partial agreement among the involved parts. We propose a general architecture based on four fuzzy ontologies and a process to develop and run the smart contracts, based on a reduction to a well-known fuzzy ontology reasoning task (Best Satisfiability Degree). We also investigate different operators to compute Pareto-optimal solutions and implement our approach in the Ethereum blockchain

The Complexity of Fuzzy Description Logics over Finite Lattices with Nominals

The complexity of reasoning in fuzzy description logics (DLs) over finite lattices usually does not exceed that of the underlying classical DLs. This has recently been shown for the logics between L-IALC and L-ISCHI using a combination of automata- and tableau-based techniques. In this report, this approach is modified to deal with nominals and constants in L-ISCHOI. Reasoning w.r.t. general TBoxes is ExpTime-complete, and PSpace-completeness is shown under the restriction to acyclic terminologies in two sublogics. The latter implies two previously unknown complexity results for the classical DLs ALCHO and SO