A Fuzzy Description Logic

From: AAAI-98 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved. A Fuzzy Description Logic Umberto Straccia I.E.I - C.N.R. Via S. Maria, 46 I-56126 Pisa (PI) ITALY [email protected] Abstract is such a concept: we may say that an individual tom is an instance of the concept Tall only to a certain degree Description Logics (DLs, for short) allow reasoning n ∈ [0, 1] depending on tom’s height. about individuals and concepts, i.e. set of individuals with common properties. Typically, DLs are limited to Fuzzy logic directly deals with the notion of vague- dealing with crisp, well dened concepts. That is, con- ness and imprecision using fuzzy predicates. Therefore, cepts for which the problem whether an individual is it oers an appealing foundation for a generalisation of an instance of it is a yes/no question. More often than DLs in order to dealing with such vague concepts. not, the concepts encountered in the real world do not The aim of this work is to present a general fuzzy DL, have a precisely dened criteria of membership: we which combines fuzzy logic with DLs. In particular we may say that an individual is an instance of a concept will extend DLs by allowing expressions of the form only to a certain degree, depending on the individual’s hC(a) ni (n ∈ [0, 1]), e.g. hTall(tom) .7i, with intended properties. Concepts of this kind are rather vague than precise. As fuzzy logic directly deals with the notion of meaning “the membership degree of individual a being vagueness and imprecision, it oers an appealing foun- an instance of concept C is at least n”. dation for a generalisation of DLs to vague concepts. Extending DLs with fuzzy features has already be done in the past. For instance, in (Yen 1991) the In this paper we present a general fuzzy DL, which combines fuzzy logic with DLs. We dene its syntax, very limited DL FL (Brachman & Levesque 1984) semantics and present constraint propagation calculi has been extended with some fuzzy features. In par- for reasoning in it. ticular, it allows the denition of fuzzy concepts and the only supported reasoning mechanism is determin- Introduction ing subsumption2. Unfortunately, it does not allow reasoning in presence of assertions. Recently, (Meghini, Description Logics (DLs, for short) provide a logical Sebastiani, & Straccia 1997) proposed a fuzzy DL as reconstruction of the so-called frame-based knowledge a tool for modelling multimedia document retrieval3. representation languages1. Concepts, roles and individ- But this work was rather at a preliminary stage and no uals are the basic building blocks of these logics. Con- reasoning algorithm was given. cepts are expressions which collect the properties, de- We present a more general framework in the sense scribed by means of roles, of a set of individuals. From a ALC rst order point of view, concepts can be seen as unary that it is based both on the DL , a signicant and predicates, whereas roles are interpreted as binary pred- expressive representative of the various DLs, and on icates. A knowledge base (KB) typically contains a set sound and complete constraint propagation calculi for of assertions. An assertion states either that an indi- reasoning in it. This allows us to adapt it easily to vidual a is an instance of a concept C (written C(a)), the dierent DLs presented in the literature. Moreover, or that two individuals a and b are related by means of we will show that the additional expressive power has a role R (written R(a, b)). A basic inference task with no impact from a computational complexity point of knowledge bases is entailment and amounts to verify view. This is important as the nice trade-o between whether the individual a is an instance of the concept computational complexity and expressive power of DLs C w.r.t. the KB (written |= C(a)). contributes to their popularity. Typically, DLs are limited to dealing with crisp con- Finally, note that most existing work in extend- cepts. However, many useful concepts that are needed ing DLs for uncertainty management lie in the cate- by an intelligent system do not have well dened bound- gory of probabilistic extension like e.g. (Heinsohn 1994; aries. That is, often it happens that the concepts en- Jager 1994; Koller, Levy, & Pfeer 1997) with some countered in the real world do not have a precisely de- exceptions like (Hollunder 1994). Even though these ned criteria of membership, i.e. they are vague concepts rather than precise concepts. For instance, Tall 2Roughly, a concept D subsumes a concept C i from a rst order point of view, ∀x.C(x) → D(x) is logically valid. Copyright c 1998, American Association for Articial 3The idea to use DLs in the context of multimedia docu- Intelligence (www.aaai.org). All rights reserved. ment retrieval has been proposed in (Gobel, Haul, & Bech- 1See the DL Web Home Page: http://dl.kr.org/dl. hofer 1996) too. probabilistic extensions enlarge the applicability of DLs |=(∃Friend.Tall)(tim), i.e. tim has a tall friend. they do not directly address the issue of reasoning about Notice that |= R(a, b)iR(a, b) ∈ . individuals and vague concepts. Moreover, reasoning in a probabilistic framework is generally a harder task, Fuzzy ALC from a computational point of view, than the rela- ALC tive non probabilistic case (see e.g. (Roth 1996) for an From a syntax point of view, in fuzzy we are overview) and thus, the computational problems have dealing with fuzzy assertions (denoted with ), i.e. expressions of type hni, where is an ALC assertion to be addressed carefully like in (Koller, Levy, & Pfeer ∈ 1997). and n [0, 1]. In the following sections we rst introduce crisp ALC, From a semantics point of view, we will follow then we extend it to the fuzzy case. Thereafter, we will Zadeh’s semantics. According Zadeh’s work about present constraint propagation calculi for reasoning in fuzzy sets (Zadeh 1965), a fuzzy set X with respect it. to a set S is characterized by a membership function X : S → [0, 1], assigning a X-membership degree, A quick look to ALC X (s), to each element s in S. This membership degree gives us an estimation of the belonging of s to The specic DL we will extend with fuzzy capabilities X. Typically, if (s) = 1 then s denitely belongs ALC X is , a signicant representative of the best-known to X, while (x)=.7 means that s is “likely” to AL X and most important family of DLs, the family. be an element of X. Moreover, according to Zadeh, the We assume three alphabets of symbols, called prim- membership function has to satisfy three well-known re- itive concepts (denoted by A), primitive roles (denoted strictions. For all s ∈ S and for all fuzzy sets X, Y with by R) and individuals (denoted by a and b). The con- respect to S: X∩Y (s) = min{X (s),Y (s)}, X∪Y (s) cepts (denoted by C and D) of the language ALC are { } = max X (s),Y (s) , and X (s)=1 X (s), where formed out of primitive concepts according to the fol- \ 5 lowing syntax rules4: X is the complement of X in S, i.e. S X . In fuzzy ALC, a concept is interpreted as a fuzzy C, D → > (top concept) set. Therefore, concepts and roles become imprecise (or ⊥ (bottom concept) vague). According to this view, the intended meaning A| (primitive concept) of e.g. hC(a) ni we will adopt is: “the membership de- C u D| (concept conjunction) gree of individual a being an instance of concept C is at t | C D (concept disjunction) least n”. Similarly for roles. Hence, e.g. hTall(tom) .7i ¬C| (concept negation) ∀ | means that the degree of tom being Tall is at least .7, R.C (universal quantication) i.e. tom is likely tall; hTall(tom)1i means that tom is ∃R.C (existential quantication) tall, whereas h¬Tall(tom)1i means that tom is not tall. I I An interpretation I is a pair I =(I , I ) consisting A fuzzy interpretation is now a pair I =(, ), I where I is, as for the crisp ALC case, the domain, of a non empty set (called the domain) and of an I I whereas is an interpretation function mapping (i) interpretation function mapping dierent individuals ALC into dierent elements of I , primitive concepts into individuals as for the crisp case; (ii) concepts I → subsets of I and primitive roles into subsets of I into a membership degree function [0, 1], and I (iii) ALC roles into a membership degree function . The interpretation of complex concepts is dened I I >I I ⊥I ∅ u I I ∩ → [0, 1]. Therefore, if C is a concept then in the usual way: = , = ,(C D) = C I DI ,(C t D)I = CI ∪ DI ,(¬C)I =I \ CI ,(∀R.C)I C will naturally be interpreted as the membership de- I 0 0 I 0 I gree function of the fuzzy concept (set) C w.r.t. I, i.e. if = {d ∈ : ∀d . (d, d ) ∈ R implies d ∈ C }, and I I I d ∈ is an object of the domain then C (d) gives (∃R.C)I = {d ∈ I : ∃d0. (d, d0) ∈ RI and d0 ∈ CI }. u us the degree of being the object d an element of the For instance, the concept Tall Student denotes the fuzzy concept C under the interpretation I.

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