A Crisp Representation for Fuzzy SHOIN with Fuzzy Nominals and General Concept Inclusions

A Crisp Representation for Fuzzy SHOIN with Fuzzy Nominals and General Concept Inclusions Fernando Bobillo Miguel Delgado Juan Gómez-Romero Department of Computer Science and Artificial Intelligence University of Granada, Spain 2th Workshop on Uncertainty Reasoning for the Semantic Web Athens, GE, USA, November 2006 F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 1 / 45

Agenda 1 Introduction 2 A Quick Survey on SHOIN 3 Fuzzy SHOIN 4 A Crisp Representation for Fuzzy SHOIN 5 Conclusions and Future Work F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 2 / 45

Preliminaries Ontologies are a core element in the layered architecture of the Semantic Web Description Logics (DLs) are a family of logics for representing structured knowledge DLs have been proved to be very useful as ontology languages Standard for ontology representation: OWL Web Ontology Language: OWL Lite OWL DL, the most used, nearly equivalent to SHOIN (D) DL OWL Full, undecidable F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 3 / 45

Motivation Classical ontologies/dls are not appropriate for uncertain or imprecise knowledge Examples: German is generally spoken in Germany, Austria and Switzerland An inn is a cheap and small hotel Vagueness is inherent to a lot of real-world application domains The Semantic Web will not be fully operative as long as it does not provide means to manage it One solution: Fuzzy DLs (DLs extended with fuzzy sets theory) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 4 / 45

Motivation (2) OWL may be extended to a fuzzy DL-based language e.g. FuzzyOWL The large number of resources available should be adapted In particular, we need reasoners Reasoning within expressive DLs has a very high worst-case complexity Significant gap between the design of a decision procedure and the achievement of a practical implementation The experience with crisp DLs induces us to think that developing highly optimized implementations will be a hard task where ad-hoc mechanisms should be used for every particular fuzzy DL In fact, there is no implemented reasoner for f SHOIN Nothing is known about the efficiency of existing reasoners: fuzzydl (f KD 1 SHIF(D), f LSHIF(D)) Fire (f KD SHIN ) 1 subscript stands for the implication used, which specifies the other operators F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 5 / 45

Motivation (3) Alternative way to obtain fuzzy ontologies facing these problems: To represent fuzzy DLs using crisp DLs To reduce reasoning within fuzzy DLs to reasoning within crisp DLs This way it would be possible: To translate them automatically into a crisp ontology language To use currently available and highly optimized reasoners Not a lot of work following this line U. Straccia showed a reasoning preserving procedure for f ALCH F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 6 / 45

Motivation (4) On the other hand, current fuzzy DLs still present some limitations which we think that should be overcome: Some works on fuzzy DLs deal with nominals (named individuals) They choose not to fuzzify the nominal construct arguing that a fuzzy singleton set does not represent any real concept world. Hence, only crisp concepts can be defined extensively, as nominals either have to fully belong to them or not There have been proposed fuzzy general concept inclusions which allow to constrain the truth value of a general concept inclusion Current reasoning algorithms do not allow them F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 7 / 45

Contributions 1 We propose a different definition of f SHOIN including: A fuzzy nominal construct Reasoning with fuzzy GCIs 2 We reduce reasoning in f KD SHOIN to reasoning in SHOIN, extending the existing work on f KD ALCH F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 8 / 45

Syntax: Knowledge base ABox: finite set of assertions about individuals: Concept assertions a : C (a is an instance of C) Role assertions (a, b) : R ((a, b) is an instance of R) Individual assertions: a b (a and b are different individuals) a = b (a and b refer to the same individual) TBox: a finite set of concept axioms: General concept inclusions (GCI) C D (C is more specific than D) Concept definitions C D (C D and D C) RBox: a finite set of role axioms: Role inclusions R R (R is more specific than R ) Role definitions R R (R R and R R) Transitive role axioms trans(r) (R is transitive) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 10 / 45

Syntax: Complex concepts and roles The concepts of the language can be built inductively: C, D A (atomic concept) (top concept) (bottom concept) C D (concept conjunction) C D (concept disjunction) C (concept negation) R.C (universal quantification) R.C (full existential quantification) {o 1,..., o m } (nominals) ( n S) (at-least unqualified number restriction) ( n S) (at-most unqualified number restriction) The roles of the language can be built using this syntax rule: R R A (atomic role) R (inverse role) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 11 / 45

Semantics (1) An interpretation I is a pair ( I, I) consisting of: A non empty set I (the interpretation domain) An interpretation function I mapping: Every individual onto an element of I Every atomic concept A onto a set A I I Every atomic role R onto a binary relation R I I I. We do not impose unique name assumption (UNA), i.e. two nominals might refer to the same individual F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 12 / 45

Semantics (2) The interpretation is extended to complex concepts and roles by the following inductive definitions: I = I I = (C D) I = C I D I (C D) I = C I D I ( C) I = I \ C I ( R.C) I = {a : b, (a, b) / R I or b C I } ( R.C) I = {a : b, (a, b) R I and b C I } {o 1,..., o m } I = {o1 I,..., oi m} ( 0 S.C) I = I = I ( m 2 S.C) I = {a : {b : (a, b) S I and b C I } m} ( n S.C) I = {a : {b : (a, b) S I and b C I } n} (R ) I = {(b, a) I I (a, b) R I } 2 m > 0 F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 13 / 45

Semantics (3) An interpretation I satisfies (is a model of): a : C iff a I C I (a, b) : R iff (a, b) I R I a b iff a I b I a = b iff a I = b I C D iff C I D I C D iff C I = D I R R iff R I R I R R iff R I = R I trans(r) iff (R) I is transitive ABox K A iff I satisfies each element in K A TBox K T iff I satisfies each element in K T RBox K R iff I satisfies each element in K R KB K = K A, K T, K R iff I satisfies all K A, K T and K R F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 14 / 45

Reasoning A DL not only stores axioms and assertions, but also offers some reasoning services, such as: KB satisfiability Concept satisfiability Subsumption between concepts Instance checking If a DL is closed under negation then all the basic reasoning services are reducible to KB satisfiability F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 15 / 45

Syntax f SHOIN extends SHOIN to the fuzzy case by letting: Concepts denote fuzzy sets of individuals Roles denote fuzzy binary relations between individuals Our logic is similar to other approaches adding: Fuzzy nominals Constraints on fuzzy GCIs A fuzzy Knowledge Base (fkb) contains: A fuzzy ABox fk A A fuzzy TBox fk T A fuzzy RBox fk R F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 17 / 45

Syntax: fuzzy ABox Novelty: the truth value of constrained concept and role assertions Concepts and role assertions: Ψ α Ψ > β Φ β Φ < α where: Ψ is an assertion of the form a : C or (a, b) : R Φ is an assertion of the form a : C α (0, 1] (excludes 0) β [0, 1) (excludes 1) Some assertions are not allowed: (a, b) : R β, (a, b) : R < α : relate to negated roles ( SHOIN ) a : C > 1, a : C < 0, (a, b) : R > 1 : trivially unsatisfiable a : C 0, a : C 1, (a, b) : R 0 : trivially satisfiable Individual assertions: a b a = b F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 18 / 45

Syntax: fuzzy TBox and RBox Novelty: the truth value of GCIs may be constrained Fuzzy GCIs: C D α C D > β C D β C D < α C D is an abbreviation of C D 1 and D C 1 Fuzzy role inclusions R R Fuzzy role definitions R R (R R and R R) Transitive role axiom trans(r) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 19 / 45

Syntax: complex concepts and roles Concepts can be built inductively: C, D A (atomic concept) (top concept) (bottom concept) C D (concept conjunction) C D (concept disjunction) C (concept negation) R.C (universal quantification) R.C (full existential quantification) {(o 1, α 1 ),..., (o m, α m )} (nominals) ( n S) (at-least number restriction) ( n S) (at-most number restriction) Complex roles can be built using this syntax rule: R R A R F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 20 / 45

Semantics (1) A fuzzy interpretation I is a pair ( I, I) consisting of: A non empty set I (the interpretation domain) A fuzzy interpretation function I mapping: Every individual onto an element of I Every concept C onto a function C I : I [0, 1] Every role R onto a function R I : I I [0, 1] C I : membership degree function of the fuzzy concept C w.r.t. I R I : membership degree function of the fuzzy role R w.r.t. I In fuzzy DLs most reasoning services are reducible to fkb satisfiability, so here in after we will only consider this task We do not impose unique name assumption (UNA) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 21 / 45

Semantics (2) The fuzzy interpretation function is extended: I (a) = 1 I (a) = 0 (C D) I (a) = C I (a) D I (a) (C D) I (a) = C I (a) D I (a) ( C) I (a) = C I (a) ( R.C) I (a) = inf b I{R I (a, b) C I (b)} ( R.C) I (a) = sup b I{R I (a, b) C I (b)} {(o i, α i )} I (a) = sup i a {o I i } α i ( 0) I (a) = I (a) = 1 ( m) I (a) = sup b1,...,b m I[ m i=1 S I (a, b i ) i<j {b i b j }] ( n S) I (a) = ( n+1 S) I (a) (R ) I (a, b) = R I (b, a) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 22 / 45

Fuzzy nominals Example: Country where German is a widely spoken language: C {germany, austria, switzerland} The classical semantics forces switzerland to fully belong to the concept or not: {o i } I (a) = 1 if a {o I i } or 0 otherwise With fuzzy nominals: {(germany, 1), (austria, 1), (switzerland, 0.67)} It does represent a real-life concept: a fuzzy set defined extensively Recall that the semantics is sup i a {o I i } α i a : C 0.8 prevents a of being germany or austria Different from a fuzzy disjunction of nominals. We consider equality between individuals (a = o i ) to be crisp The definition generalizes the previous definition for nominals F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 23 / 45

Semantics (3) A fuzzy interpretation I satisfies (is a model of): a : C α 3 iff C I (a I ) α (a, b) : R α 3 iff R I (a I, b I ) α a b iff a I b I a = b iff a I = b I C D α 3 iff inf a I {C I (a) D I (a)} α C D iff C I = D I R R iff R I R I R R iff R I = R I trans(r) iff a, b I, R I (a, b) sup c I R I (a, c) R I (c, b) ABox K A iff I satisfies each element in K A TBox K T iff I satisfies each element in K T RBox K R iff I satisfies each element in K R fkb K A, K T, K R iff I satisfies all K A, K T and K R 3 Definitions are similar for > β, β and < α F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 24 / 45

Fuzzy GCIs The definition of fuzzy GCIs allows concept subsumption to hold to a certain degree in [0, 1] Example: Inn Hotel 0.5 Translating universal quantification and GCIs to First Order Logic leads to implication functions It seems natural to let both (or neither) of them be fuzzy This does not hold for role inclusion axioms Asymmetry in the expressivity The implication function would require the subjacent DL to have negated roles and role disjunction We have preferred to consider SHOIN, underlying OWL DL F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 25 / 45

Some properties (1) Lemma Fuzzy interpretations coincide with crisp interpretations if we restrict to the membership degrees of 0 and 1 Here in after we concentrate on f KD SHOIN : Gödel t-norm (minimum) α β = min{α, β} Gödel t-conorm (maximum) α β = max{α, β} Lukasiewicz negation α = 1 α Kleene-Dienes implication α β = max{1 α, β} This choice of the t-norm and the t-conorm eases the translation F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 26 / 45

Some properties (2) Lemma f KD SHOIN allows some sort of modus ponens over concepts and roles, even with the new semantics of fuzzy GCIs: For α, β, γ [0, 1], = {, >} and α 1 β ( = <, > = ), the following properties are verified: (i) a : C α and C D β imply a : D β (ii) (a, b) : R γ and R R imply (a, b) : R γ (iii) (a, b) : R α and a : R.C β imply b : C β F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 27 / 45

Some properties (3) The use of Kleene-Dienes implication in the semantics of fuzzy GCIs brings about two counter-intuitive effects: 1 A concept does not fully subsume itself: C C inf a I max{1 C I (a), C I (a)} = 0.5 2 Crisp concept subsumption forces fuzzy concepts to be crisp: C D 1 inf a I max{1 C I (a), D I (a)} 1 which is true iff for each element of the domain D I (a) = 1 or C I (a) = 0 Need of further investigation involving alternative fuzzy operators! A residuum based implication would fix 1: a b = 1 if a b Lukasiewicz implication would fix 2: a b = min{1, 1 a + b} F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 28 / 45

Intuitive idea U. Straccia presents a reasoning preserving transformation for f KD ALCH into crisp ALCH We extend this work to f KD SHOIN Example: transform a : A 0.8 into a : A 0.8 (0.8-cut of A) Procedure: 1 Define some new atomic concepts and roles 2 Add some new axioms to preserve the semantics of the fkb 3 Map separately ABox, TBox and RBox F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 30 / 45

New elements (1) It has been shown that in f KD ALC, the set of the degrees which must be considered for any reasoning task can be computed as: N fk = X fk {1 α α X fk } where X fk is the set of degrees appearing in the fkb: X fk = {0, 0.5, 1} {α Ψ α fk A } {β Ψ > β fk A } {1 β Φ β fk A } {1 α Φ < α fk A } {α Ω α fk T } {β Ω > β fk T } {1 β Ω β fk T } {1 α Ω < α fk T } This also holds in f KD SHOIN When other fuzzy operators are considered this is no longer true We may calculate all possible degrees in [0, 1] with a given precision, but further investigation is required F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 31 / 45

New elements (2) For every atomic concept and role in the fkb and for each degree N fk we create: Four new atomic concepts A α, A >β, A β, A <α Two new atomic roles R α, R >β Informally, A α represents the crisp set of individuals which are instance of A with degree higher or equal than α (α-cut of A) A <0, A >1, R >1 are not considered (they are always empty sets) A 1, A 0, R 0 are not considered (they are equivalent to ) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 32 / 45

New elements (3) The semantics of these newly introduced atomic concepts and roles is preserved by some terminological and role axioms New concept axioms: A γi+1 A >γi A >γi A γi A <γj A γj A γi A <γi+1 A γj A <γj A >γi A γi A γj A <γj A >γi A γi New role axioms: R γi+1 R >γi R >γi R γi (a, b) : R β, (a, b) : R < α would need additional role constructs: R, R, R, R F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 33 / 45

Mapping the ABox A fuzzy ABox is mapped into using a mapping σ: σ( a : C γ ) = a : ρ(c, γ) σ( (a, b) : R γ ) = (a, b) : ρ(r, γ) σ( a b ) = a b σ( a = b ) = a = b Fuzzy assertions are mapped into crisp assertions ρ is inductively defined on the structure of concepts and roles F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 34 / 45

Mapping concepts and roles (1) x y ρ(x, y) A γ A γ if γ 0, otherwise A > γ A >γ, if γ 1, otherwise A γ A γ if γ 0, otherwise A < γ A <γ, if γ 1, otherwise R γ R γ if γ 0, otherwise R > γ R >γ, if γ 1, otherwise γ > γ if γ 1, otherwise γ if γ = 1, otherwise < γ γ if γ = 0, otherwise > γ γ < γ if γ 0, otherwise F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 35 / 45

Mapping concepts and roles (2) x y ρ(x, y) C D {, >} γ ρ(c, {, >} γ) ρ(d, {, >} γ) C D {, <} γ ρ(c, {, <} γ) ρ(d, {, <} γ) C D {, >} γ ρ(c, {, >} γ) ρ(d, {, >} γ) C D {, <} γ ρ(c, {, <} γ ρ(d, {, <} γ) C {, >} γ ρ(c, {, <} 1 γ) C {, <} γ ρ(c, {, >} 1 γ) R.C {, >} γ ρ(r, {, >} γ).ρ(c, {, >} γ) R.C {, <} γ ρ(r, {>, } γ).ρ(c, {, <} γ) R.C {, >} γ ρ(r, {>, } 1 γ).ρ(c, {, >} γ) R.C {, <} γ ρ(r, {, >} 1 γ).ρ(c, {, <} γ) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 36 / 45

Mapping concepts and roles (3) x y ρ(x, y) {(o 1, α 1 ),..., (o m, α m )} γ {o i α i γ, 1 i n} γ 0 S γ ρ(, γ) m S {, >} γ m ρ(s, {, >} γ) m S {, <} γ m 1 ρ(s, {>, } γ) n S {, >} γ n ρ(s, {>, } 1 γ) n S {, <} γ n+1 ρ(s, {, >} 1 γ) R γ ρ(r, γ) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 37 / 45

Mapping the TBox A positive GCI (, >) is reduced into a GCI: k( C D γ ) = ρ(c, > 1 γ) ρ(d, γ) k( C D > γ ) = ρ(c, 1 γ) ρ(d, > γ) A negative GCI (, <) is reduced into an assertion about a new individual x: A( C D γ ) = x : ρ(c, 1 γ) ρ(d, γ) A( C D < γ ) = x : ρ(c, > 1 γ) ρ(d, < γ) The natural reduction would be to a negated GCI, but it is not part of crisp SHOIN How to deal with alternative implication functions? F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 38 / 45

Mapping the RBox A fuzzy RBox is reduced using a function k(fk, fk R ) = Ω fk R k(ω) Role axioms are reduced using a function k(ω): k(r R ) = γ N fk, {,>} ρ(r, γ) ρ(r, γ) k(trans(r)) = γ N fk, {,>} trans(ρ(r, γ)) F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 39 / 45

Complexity A fkb fk is reduced into a KB K(fK): The complexity is quadratic: ABox is linear TBox and RBox are quadratic fkb K(fK) fk A σ(fk A ) A(fK T ) fk T T (N fk ) k(fk, fk T ) fk R R(N fk ) k(fk, fk R ) In the previous work, a fuzzy GCI is reduced into a set of crisp GCIs Our semantics for fuzzy GCIs allows to reduce each axiom into either an axiom or an assertion This reduction in the size of the TBox is very interesting since reasoning with GCIs is a source of computational complexity F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 40 / 45

Reasoning Theorem A f KD SHOIN fkb fk is satisfiable iff K(fK) is satisfiable Firstly, it has to be proved that the translation preserves the satisfiability of every single statement of the fkb If there exists a fuzzy interpretation satisfying a fuzzy statement, then a crisp interpretation satisfying the result of its translation can be built Secondly, it has to be proved that the translation preserves the satisfiability of the whole fkb Clashes produced by pairs of conjugated axioms are preserved by the new concept axioms: A γj A <γj, A >γi A γi F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 41 / 45

Conclusions A sound fuzzy extension of SHOIN including: Fuzzy nominals, enabling to define fuzzy sets extensively Reasoning with fuzzy GCIs, allowing to constrain the truth value of a GCI Reasoning preserving procedure into a crisp KB. Alternative approach to achieve fuzzy ontologies, reusing currently existing crisp ontology languages and reasoners The semantics of fuzzy GCIs: Allows fuzzy GCIs holding to some degree Reduces the size of the resulting TBox w.r.t. Zadeh implication However, it imposes some counter-intuitive effects F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 43 / 45

Future work Empirical evaluation to test the translation Consider different fuzzy operators In particular, avoid the counter-intuitive effects of the Kleene-Dienes implication Include a crisp representation for fuzzy datatypes Since OWL does not currently allow to define customised datatypes, consider OWL Eu Consider more expressive DLs and, in particular, SROIQ: Subjacent DL of OWL 1.1 The additional expressivity in roles may help to overcome the asymmetry in fuzzy concept and role inclusion axioms F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 44 / 45

Questions? Thank you very much for your attention F. Bobillo et al. (University of Granada) Fuzzy SHOIN + fuzzy nominals + GCIs URSW 2006 45 / 45