Reasoning in Fuzzy Description Logics using Automata

Abstract

Automata-based methods have been successfully employed to prove tight complexity bounds for reasoning in many classical logics, and in particular in Description Logics (DLs). Very recently, the ideas behind these automata-based approaches were adapted for reasoning also in fuzzy extensions of DLs, with semantics based either on finitely many truth degrees or the Gödel t-norm over the interval [0, 1]. Clearly, due to the different semantics in these logics, the construction of the automata for fuzzy DLs is more involved than for the classical case. In this paper we provide an overview of the existing automata-based methods for reasoning in fuzzy DLs, with a special emphasis on explaining the ideas and the requirements behind them. The methods vary from deciding emptiness of automata on infinite trees to inclusions between automata on finite words. Overall, we provide a comprehensive perspective on the automata-based methods currently in use, and the many complexity results obtained through them.

Algorithms for reasoning in very expressive description logics under infinitely valued Gödel semantics

“…A G-SROIQ ontology is consistent if it has a witnessed G-model. Other common reasoning problems for FDLs, such as concept satisfiability and subsumption can be reduced to consistency in linear time [14]. For instance, the subsumption between C and D to degree p w.r.t.…”

“…The study of reasoning algorithms for expressive Gödel FDLs that can handle infinitely many truth degrees started in [10,14]. The main contribution of that work was the development of automata-based methods for testing the existence of (potentially infinite) models of a fuzzy ontology.…”

Fuzzy Description Logics – A Survey

“…Automata-based algorithms generalize similar techniques from classical DLs. The basic idea behind these algorithms is to use tree automata to decide the existence of a forest-shaped model of the ontology [36,41,45]. In some cases, e.g.…”

“…For example, more complex assertions like Tall(a) > Tall(b) allow to compare fuzzy degrees between different individuals a and b. Usually, such extensions do not affect the complexity of consistency [31,[44][45][46]. Going one step further, one can also allow comparisons inside concepts like Tall > ∀friend.Tall, representing the set of all people that are taller than all their friends.…”

Similarity-based relaxed instance queries

“…However, on the one hand reasoning in fuzzy DLs easily becomes undecidable [6][7][8] and on the other hand depending on the user and on the request, different ways of relaxing the query concept are needed. For instance, for a request to a car rental company to rent a particular car model in Beijing, it might be acceptable to get an offer for a similar car model to be rented in Beijing, instead of getting the offer to rent the requested car model in London.…”