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 de ned concepts. That is, con- ness and imprecision using fuzzy predicates. Therefore, cepts for which the problem whether an individual is it o ers 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 de ned 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 o ers 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 de ne 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 de nition of fuzzy concepts and the only supported reasoning mechanism is determin- Introduction ing subsumption2. Unfortunately, it does not allow rea- soning 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 signi cant 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 di erent 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 de ned bound- gory of probabilistic extension like e.g. (Heinsohn 1994; aries. That is, often it happens that the concepts en- J ager 1994; Koller, Levy, & Pfe er 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 con- cepts 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 Arti cial 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)i R(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. ex- pressions of type h ni, where is an ALC assertion to be addressed carefully like in (Koller, Levy, & Pfe er ∈ 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 de- gree gives us an estimation of the belonging of s to The speci c DL we will extend with fuzzy capabilities X. Typically, if (s) = 1 then s de nitely belongs ALC X is , a signi cant 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 quanti cation) i.e. tom is likely tall; hTall(tom)1i means that tom is ∃R.C (existential quanti cation) 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 di erent individuals ALC into di erent 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 de ned 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. Similarly set of tall students. I for roles. Additionally, has to satisfy the following An assertion (denoted by ) is an expression of type ∈ I C(a)(a is an instance of C), or an expression of type equations: for all d R(a, b)(a is related to b by means of R). For instance, I > (d)=1 (Tall u Student)(tom) asserts that tom is a tall stu- I ⊥ (d)=0 dent, whereas Friend(tim,tom) asserts that tom is a I (C u D) (d) = min{CI (d),DI (d)} friend of tim. I (C t D) (d) = max{CI (d),DI (d)} The semantics of assertions is speci ed by saying that I I the assertion C(a) (resp. R(a, b)) is satis ed by I i (¬C) (d)=1 C (d) I I I I I ∀ I { { I 0 I 0 }} a ∈ C (resp. (a ,b ) ∈ R ). A set of assertions ( R.C) (d) = mind0∈ I max 1 R (d, d ),C (d ) ∃ I { { I 0 I 0 }} will be called a knowledge base (KB). An interpretation ( R.C) (d) = maxd0∈ I min R (d, d ),C (d ) . I I satis es (is a model of)aKB i satis es each ele- I ment in . A KB entails an assertion (written |= Just note that w.r.t. the ∀ connective, (∀R.C) (d)is ) i every model of also satis es . For instance, if the result of viewing ∀R.C as the rst order formula is{(Tall u Student)(tom), Friend(tim,tom)} then 5Other membership functions have been proposed in the 4Through this work we assume that every metavariable literature. The interested reader can consult e.g. (Dubois & has an optional subscript. Prade 1980; Kundu & Chen 1994).
∀y.R(x, y) → C(y), where F → G is ¬F ∨ G6 and (crisp) KB ={ : h ni∈ }. Since every “crisp” the universal quanti er ∀ is viewed as a conjunction interpretation is a fuzzy interpretation, the following over the elements of the domain. Similarly, for the ∃ proposition is easily veri ed. I connective (∃R.C) (d) is the result of viewing ∃R.C Proposition 1 Let be a fuzzy KB and let be an ∃ ∧ as y.R(x, y) C(y), where the existential quanti er assertion. For all n>0,if | h ni then |= . a ∃ is considered a disjunction over the elements of the domain (see e.g. (Lee 1972)). Proposition 1 states that there cannot be fuzzy en- It is easily veri ed that for all interpretations I and tailment without entailment. For instance, w.r.t. Ex- I I | h i | individuals d ∈ I ,(¬(C u D)) (d)=(¬C t¬D) (d) ample 1 we have 1 C(i1) .8 and 1 = C(i1). Unfortunately, the converse of Proposition 1 is ¬ ∀ I ∃ ¬ I and ( ( R.C)) (d)=( R. C) (d). not true in the general case. For instance, An interpretation I satis es a fuzzy assertion {h i h ¬ t i} 6| h i I I C(a) .3 , ( C D)(a) .6 D(a) n for all n>0, hC(a) ni (resp. hR(a, b) ni)i C (a ) n (resp. whereas {C(a), (¬C t D)(a)}|= D(a). RI (aI ,bI ) n). An interpretation I satis es (is a A simple result concerning the “converse” relation model of) a set of fuzzy assertions , i.e. a fuzzy KB,i between | and |= is the following. Let be a crisp I satis es each element of . A fuzzy KB fuzzy en- KB: we de ne =˜ {h 1i : ∈ }. tails a fuzzy assertion (written | ) i every model of also satis es . Given a fuzzy KB and an Proposition 2 If |= then ˜| h 1i. a assertion , we de ne the maximal degree of truth of A closer relationship holds whenever we consider nor- with respect to (written Maxdeg( , )) to be malized fuzzy KBs. We will say that a fuzzy assertion max{n>0: | h ni} (max ∅ = 0). Notice that h ni is normalized if n>.5. A fuzzy KB is normalized | h ni i Maxdeg( , ) n. if every fuzzy assertion in it is. If we consider normal- Example 1 Suppose we have two images i1 and ized fuzzy KBs only, then from (Lee 1972) it follows i2 regarding tim, tom and joe. i1 and i2 have that been indexed as follows: i1 = {hAbout(i1, tim) .9i, Proposition 3 If is normalized then there is n .5 h i h i h i} Tall(tim) .8 , About(i1, tom) .6 , Tall(tom) .7 , such that | h ni i |= . a i2 = {hAbout(i1, joe) .6i, hTall(joe) .9i}. More- {h i h i For instance, {hC(a) .6i, h(¬C t D)(a) .7i}| hD(a) .7i over, let B = Student(tim)1 , Student(tom)1 , { ¬ t }| hStudent(joe)1i, hImage(i1)1i, hImage(i2)1i}.We and and C(a), ( C D)(a) = D(a) hold. The rea- ∪ ∪ son relies on the fact that for m,n>.5, the condition de ne 1 = i1 B and 2 = i2 B. Our inten- tion is to retrieve all images in which there is a tall m>1 n in (2) – (4) is always true. student. This can be formalised by means of the query concept C = Image u∃About. (Student u Tall). It can Deciding fuzzy entailment easily veri ed that Maxdeg( 1,C(i1)) = .8, whereas Deciding whether | h ni requires a calculus. We will Maxdeg( 1,C(i2)) = .6. Therefore, we will retrieve develop a calculus in the style of the constraint propaga- both images and rank i1 before i2. tion method, as this method is usually proposed in the context of DLs (see, e.g. (Buchheit, Donini, & Schaerf Some properties 1993)). The calculus extends the propositional frame- The following properties are easily veri ed: for work described in (Chen & Kundu 1996) to the DL all concepts C, D and for all n, m ∈ [0, 1], case. {hC(a) ni, h¬C(a) mi} is satis able i n 1 m and Consider a new alphabet of variables. An Inter- pretation is extended to variables by mapping these into elements of the interpretation domain. An ob- Maxdeg(∅, (¬C t C)(a)) = .5 (1) ject (written w) is either an individual or a variable. {hC(a) mi, h(¬C t D)(a) ni}| hD(a) ni, if m>1 n (2) A constraint (written ) is an expression of the form w:C or (w1,w2):R, where w, w1,w2 are objects, C is an ALC concept and R is a role. A fuzzy constraint Relation (2) is a sort of modus ponens over concepts. (written ) is an expression having one of the follow- Similarly for ∀, the semantics of the ∀ connective gives ing forms: h ni, h >ni, h ni, h
(∃ ) hw:∃R.C ni→h(w, x):R ni, hx:C ni if x new variable ∀ h ∀ i→h i h i | h C(a) ni i S ∪{ha:C
(7b) hx:C<.6i (u<) : (6) where and are sequences of fuzzy constraints and (8b) clash (5), (7b). is a condition. Both rules re only if the condition holds and if the current set S of fuzzy constraints con- tains fuzzy constraints matching . After execution, the rst deletes the fuzzy constraints matching from Soundness, completeness and complexity S, while the second keeps them. Both forms add the It is easily veri ed that the above rules are sound, i.e. if constraints from to S after ring. In order to pre- S1 is satis able then there is a satis able completion vent in nite application of the second type of rules, we S2 of S1 and, thus, S2 contains no clash. Vice-versa, assume that each instantiation of the rules is applied completeness, i.e. if there is a completion S2 of S1 con- only once. The rules are the following: taining no clash then S1 is satis able, can be shown by building an interpretation I from S2 satisfying S1. ¬ h ¬ i→h i ( ) w: C n w:C 1 n Roughly, given a clash-free completion S2 of S1 we con- ¬ h ¬ i→h i ( ) w: C n w:C 1 n sider N1[ ] = max{n : h ni∈S2}, and N2[ ]= max{n : h >ni∈S2}. Since S2 is clash-free, it fol- (u ) hw:C u D ni→hw:C ni, hw:D ni t h t i→h i h i lows that there is >0 such that the interpretation ( ) w:C D n w:C n , w:D n I,(i) with domain I being the set of objects ap- I ∈ I (t ) hw:C t D ni→hw:C ni|hw:D ni pearing in S2,(ii) w = w for all w and (iii) u h u i→h i|h i I I I I I I ( ) w:C D n w:C n w:D n > (w )=1,⊥ (w )=0,A (w ) = max{N1[w:A], } I I I { c N2[w:A]+ , R (w1 ,w2 ) = max N1[(w1,w2):R], (∀ ) hw1:∀R.C ni, ⇒hw2:C ni N2[(w1,w2):R]+ }, satis es both S2 and S1. It can if is h(w1,w2):R 1 ni c be shown that (∃ ) hw1:∃R.C ni, ⇒hw2:C ni if is h(w1,w2):R ni Proposition 4 A set of fuzzy constraints S is satis - able i there exists a clash free completion of S. a
From a computational complexity point of view, it is is a sport car which is owned by a car fanatic. When we easily veri ed that termination of the above algorithm switch to the fuzzy case, the simple form of fuzzy termi- is guaranteed. Moreover, from Proposition 2 and from nological axiom we allow is hC ⇒ Dni, where n ∈ [0,n]. PSPACE-completeness of the entailment problem in The semantics is given coherently to the above rst crisp ALC (Schmidt-Schau & Smolka 1991), PSPACE- order view of C ⇒ D: an interpretation I satis es I hardness of the fuzzy entailment problem follows. It h ⇒ i I { ¬ t } C Dn i mind∈ ( C D) (d) n. As for can be veri ed that trace rules as in (Schmidt-Schau the ∀ connective, F → G is viewed as ¬F ∨ G.Itis & Smolka 1991) can be de ned. Therefore, easily veri ed that {hC(a) mi, hC ⇒ Dni} | hD(a) ni Proposition 5 Let be a fuzzy KB and and let if m>1 n, which is similar to (2). be a fuzzy assertion. Determining whether | is a We will say that D subsumes C with degree n w.r.t. PSPACE-complete problem. a (written | h C ⇒ Dni) i all models of are mod- els of hC ⇒ Dni. Maxdeg( ,C ⇒ D) is the maxi- Computing the maximal degree of truth mal degree n such that | h C ⇒ Dni. For instance, if is{hA ⇒ C.6i, hB ⇒ D.7i}, then it can be veri ed The problem of determining Maxdeg( , ) is impor- that Maxdeg( ,Au B ⇒ C u D)=.6. Notice that tant, as computing Maxdeg( , ) is in fact the way Maxdeg(∅,C ⇒ C)=.5, according to (1). to answer a query of type “to which degree is (at least) true, given the (vague) facts in ?”. An easy algorithm can be given in terms of a sequence of fuzzy Example 3 Consider Example 1. Suppose entailment tests. It is based on the observation that we add {hStudent u (Male t Tall) ⇒ TallStudent .7i, h i h i h i} Maxdeg( , ) ∈{0,.5, 1}∪N , where N = {n : Male(tim)1 , Male(tom)1 , Male(joe)1 to the hAni∈ }. The algorithm is described below. background KB B. Suppose the query concept C is Image u∃About. TallStudent. It can be Algorithm Max( , ) veri ed that Maxdeg( ,C(i1)) = .7, whereas Let be a set of ALC fuzzy assertions, let be an assertion. 1 Set Min := 0 and Max := 2. Maxdeg( 1,C(i2)) = .6. ∈ ∪{ } 1. Pick n N .5, 1 such that Min
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Another interest- concept descriptions with complements. Arti cial Intelli- ing point is to understand the impact of fuzziness on gence 48:1–26. the computational complexity: is it always true that Straccia, U. 1997. A four-valued fuzzy propositional logic. the upper bound of the complexity is the same as in In Proc. of the 15th Int. Joint Conf. on Arti cial Intelli- the crisp case? gence (IJCAI-97), 128–133. Yen, J. 1991. Generalizing term subsumption languages Acknowledgements to fuzzy logic. In Proc. of the 12th Int. Joint Conf. on This work is funded by the European Community ES- Arti cial Intelligence (IJCAI-91), 472–477. PRIT project FERMI 8134. Thanks go to C. Meghini, Zadeh, L. A. 1965. Fuzzy sets. Information and Control F. Sebastiani and the reviewers for their suggestions 8(3):338–353. and comments.