on fuzzy description logics · from description logics to fuzzy description logics dls (alc) fol...

On Fuzzy Description Logics

Francesc Esteva

Institut d’Investigació en Intel·ligència Artificial-CSIC

DIPLEAP workshop, Wien November 2010

Francesc Esteva DIPLEAP workshop, Wien, November 2010

Network-based structures plus logical systems

The antecedents of DL languages are Network-based structures carrying the intuitionthat, by exploiting the notion of hierarchical structure, one could gain in terms of both

ease of representation

the efficiency of reasoning


Network-based structures


Network-based structures plus logical systems

The basic elements of the representation are characterized as

unary predicates, denoting sets of individuals, and

binary predicates, denoting relationships between individuals.

Such a characterization does not capture the constraints of network-based structureswith respect to logic.

They did not require all the machinery of FOL,

but could be regarded as fragments of it [Brachman and Levesque, 1985].

In addition, different features of the representation language would lead todifferent fragments of FOL.

Important consequence:

recognition that the typical forms of reasoning used in structure-basedrepresentations could be accomplished by specialized reasoning techniques,

without necessarily requiring FOL theorem provers.

Moreover, reasoning in different fragments of FOL leads to computationalproblems of differing complexity.


Terminological systems / Concept languages

First name: Terminological Systems

To emphasize that the representation language was used to establish the basicterminology adopted in the modeled domain.

Second name: Concept Languages

The emphasis was on the set of concept-forming constructs admitted in the language.

Third name: Description Logics

In more recent years, after attention was further moved towards the properties of theunderlying logical systems, the term Description Logics became popular.


Description Logics: The language

Terminology

Denote a hierarchical structure built to provide an intensional representation of thedomain of interest.

DL language

Concepts are interpreted semantically as sets and in FOL as unary predicates

Roles are interpreted as binary relations and in FOL as binary predicates

the Description language is built from concepts and roles by means ofconstructors

Different DLs depend on constructors (more or less expressive)


Example: the language AL

AL (= attributive language) introduced by [Schmidt-Schauß and Smolka, 1991] as aminimal language that is of practical interest.

attributive language: a language to express, concisely, certain natural constructionsthat are commonly used to build other concepts from simpler ones.

Given A ∈ NA, and R ∈ NR , an AL-description formula is defined in accordance withthe following syntactic rules (C,D are metavariables for descriptions of concepts):


Description Logics: concept constructors


Description Logics: a basic hierarchy of AL-languages

6

6

6

66

��

��

��

��

��XX

XXXXXy

XXXXX

XXy

XXXXX

XXy

XXXXX

XXy

XXXXy

AL

ALU

ALE

ALUE = ALC

ALN

ALUN

ALEN

ALUEN = ALCN

r

rr

r

r

rr

r

+N

+E

+U


Description formulas as FOL formulas

Description signature: D = 〈NI ,NC ,NR〉 First order signature: ΣD = 〈CD,PD〉

CD = NI , PD = NC ∪ NR , where we read

each c ∈ NI as an object constant,

each A ∈ NC as a unary predicate symbol,

each R ∈ NR as a binary predicate symbol.

Instance of an ALC-description

⊥ := 0̄, > := 1̄

A(t)

(¬C)(t) :=∼ C(t)

(C1 t C2)(t) := C1(t) Y C2(t)

(C1 u C2)(t) := C1(t)&C2(t)

(C1 = C2)(t) := C1(t)→ C2(t)

(∀R.C)(t) := (∀y)(R(t, y)→ C(y))

(∃R.C)(t) := (∃y)(R(t, y)&C(y))

R(t1, t2)

An interpretation for D:

I = 〈M, (cI)c∈NI , (AI)A∈NC, (RI)R∈NR 〉

is also an interpretation for ΣD.

The following are equivalent:

a ∈ CI

CI(a) = 1

‖C(x)‖I,v [x→a] = 1


Description Logics: knowledge base (KB)

Terminology

Denote a hierarchical structure built to provide an intensional representation of thedomain of interest.

DL knowledge base (KB)

TBox: definitions and hierarchies of the relevant domain concepts

ABox: specifications of properties of the domain instances

Issues

The statements in the KB can be identified with formulas in FOL

Tools from FOL can be used to obtain implicit knowledge from the explicitknowledge in the KB by means of deductive reasoning


The Robots dataset

r1 r5r4r3r2

. . . . . . . . . .

r6

. .

r9

. .. .

r7

. .

r8


Classical TBox

Friendly ≡ Robot & (∃hasObject.FriendlyObject)& (Happy Y Homogeneous)Unfriendly ≡ Robot & ∼ Friendly

FriendlyObject v ObjectUnfriendlyObject ≡ Object & ∼ FriendlyObject

Robot & Object v 0̄1̄ v Robot Y Object

Homogeneous v RobotHappy v Robot

WearsTie v Robot

Flower v FriendlyObjectBalloon v FriendlyObject

Flag v FriendlyObjectSword v UnfriendlyObject

Ax v UnfriendlyObject


Classical ABox

For each i, 1 ≤ i ≤ 9, Robot(ri ), hasObject(ri , oi )

Homogeneous(r1),Balloon(o1),Happy(r1),WearsTie(r1)

Homogeneous(r2),Flag(o2),Happy(r2),WearsTie(r2)

∼ Homogeneous(r3),Sword(o3),Happy(r3),WearsTie(r3)

∼ Homogeneous(r4),Flower(o4),∼ Happy(r4),∼ WearsTie(r4)

∼ Homogeneous(r5),Sword(o5),∼ Happy(r5),∼ WearsTie(r5)

∼ Homogeneous(r6),Flag(o6),∼ Happy(r6),∼ WearsTie(r6)

Homogeneous(r7),Ax(o7),∼ Happy(r7),WearsTie(r7)

∼ Homogeneous(r8),Ax(o8),∼ Happy(r8),WearsTie(r8)

Homogeneous(r9),Balloon(o9),∼ Happy(r9),∼ WearsTie(r9)


Knowledge bases: TBox and ABox

Knowledge base = Terminological Box + Assertional Box

TBox (T ): definitions and hierarchies of the relevant domain concepts

ABox (A): specifications of properties of the domain instances

K = 〈T ,A〉

TBox: finite set of (general) concept inclusion axioms

C v D : correspond to sentences of the form (∀x)(C(x)→ D(x))

Given I = 〈M, (.)I〉, I |= C v D iff CI ⊆ DI

C ≡ D: abbreviation for the two axioms C v D and D v C

ABox: finite set of assertion axioms

a : C or (a, b) : R : correspond to sentences of the form C(a) or R(a, b)

Given I = 〈M, (.)I〉,

I |= a : C iff aI ∈ CI ; I |= (a, b) : R iff 〈aI , bI〉 ∈ RI


Reasoning tasks for concepts

Reasoning algorithms: Structural subsumption algorithms

In order to obtain decision procedures for any of the four inferences:it is sufficient to develop algorithms that decide the satisfiability of concepts,

provided the language supports conjunction as well as negation of arbitrary concepts.

Reasoning algorithms: specialized TABLEAU CALCULI


Description Logics

Classical Description Logics

From logical point of view:They are decidable fragments of FOLRelation with modal logic

Development of DLs

More complex languages (related to constructors and expresivity) OWLEfficient algorithms for satisfiability (validity and subsumption)A good balance between expressivity and efficiency.Application to ontologies and semantic web


From Description Logics to Fuzzy Description Logics

Fuzzy Description Logic

Gradual concepts and roles

Take the same language and constructors

Interpret concepts by Fuzzy Sets and roles as fuzzy relations

Patient with high fever Person living near Paris

First papers on Fuzzy Description Logic

Generalizing term subsumption languages to fuzzy logic, John Yen.Proceedings of the IJCAI’91.

A Description Logic for Vague Knowledge, Christopher B. Tresp and RalfMolitor. Techreport of the Aachen University of Technology (1998). Proceedingsof the ECAI’98.

Reasoning within Fuzzy Description Logics, Umberto Straccia. Journal ofArtificial Intelligence Research (2001).


Fuzzy Description Logics: General approach

Mathematical Fuzzy Logic

Developed in the last decade (propositional and first order logics).Fuzzy logic as residuated multiple-valued logic with semantic based on t-norms andtheir residua.Propositional and first order logic of a t-norm and its residuum.

FDLs in the setting of Mathematical Fuzzy logic

FDLs as fragments of first order Fuzzy logics with [0,1]-semanticsDifferences with the classical DLs in the setting of classical logic



DLs(ALC)

FOL Interpretation asfuzzy sets

Fuzzy DLs(ALC)

Fuzzy FOL

P. Hájek, Making fuzzy logic description more general, Fuzzy Sets and Systems, 2005

Dealing with FDLs taking as basis t-norm based fuzzy logics is proposed

Description logic can profit from advanced fuzzy logic getting richer expressivepossibilities (but still decidable)

P. Hájek, Computational complexity of t-norm based propositional fuzzy logics withrational truth constants, Fuzzy Sets and Systems, 2006



À. García-Cerdaña, E. Armengol, F. Esteva; Fuzzy Description Logics and t-normbased fuzzy logics, I.J. of Approximate Reasoning, 2010

Analysis of the proof-theoretical counterpart for FDLs and their KBs

Introduction of an involutive negation and truth constants in the languages

Graduated notions for reasoning tasks

Development of an illustrative example

M. Cerami, À. García-Cerdaña, F. Esteva; From Classical Description Logic ton-graded Fuzzy Description Logic, Proceedings FUZZ-IEEE’10

Discussion on the basic constructors

Proposal for a hierarchy of basic AL-languages (adequate to interpretations intofuzzy FOL)

First step in a technical report on n-graded FDLs

U. Straccia F. Bobillo (also Stoilos et al.) have many papers on satisfiability algorithmsfor different languages and t-norm semantics.


Constructors for AL-like DLs

C,C1,C2 A | (atomic concept)R | (atomic role)⊥ | (empty concept)> | (universal concept)¬A | (restricted compl. concept) (atomic negation)

C1 u C2 | (concept intersection)∀R.C | (universal quantification) (value restriction)∃R.> | (limited existential quantif.)

C1 t C2 | (concept union) (U )∃R.C | (existential quantification) (E)¬C | (complementary concept) (C)

Having (U ) and (C) we can define the implication of concepts: C1 = C2 := ¬C1 t C2

In ALC the implication can be also also defined as: C1 = C2 := ¬(C1 u ¬C2)


FDLs: a basic hierarchy of AL-languages

6

6

6

6

6

6

6

6

6

6

��

��

��

��

��

��

��XX

XXXXXy

XXXXX

XXy

XXXXX

XXy

XXXXXX

Xy

XXXXXX

Xy

XXXXXX

Xy

XXXXy AL

ALU

ALUC

ALE

ALUE

ALUEC

IAL

IALU

IALUC

IALE

IALUE

IALUEC

r

r

r

r

r

r

r

r

r

r

r

r

+I

+E

+U

+C


Interpretation for complex ALC-descriptions

1 Fix a continuous t-norm ∗2 Let C̃∗ be the standard algebra [0, 1]∗ extended with Łukasiewicz negation

3 Let S be the universe of a countable subalgebra of C̃∗

Using fuzzy sets, continuous t-norms, and their residua CIM :→ [0, 1]

⊥I(a) = 0>I(a) = 1r̄I(a) = r r ∈ S

(∼C)I(a) = N(CI(a)) N(x) = 1− x(C � D)I(a) = CI(a) ∗ DI(a) continuous t-norm ∗(C = D)I(a) = CI(a)→∗ DI(a) residuum of ∗

(∀R.C)I(a) = inf{RI(a, b)→∗ CI(b) : b ∈ M}(∃R.C)I(a) = sup{RI(a, b) ∗ CI(b) : b ∈ M}

C u D := C � (C = D), C t D := (C = (C = D) u (D = (D = C), ¬C := C = ⊥C � D := ∼((∼C) � (∼D))

(C � D)I(a) = 1− [(1− CI(a)) ∗ (1− DI(a)] dual t-conorm of ∗


Knowledge bases in classical DLs

Knowledge base = Terminological Box + Assertional Box

TBox (T ): definitions and hierarchies of the relevant domain concepts

ABox (A): specifications of properties of the domain instances

K = 〈T ,A〉

TBox: finite set of (general) concept inclusion axioms

C v D : correspond to sentences of the form (∀x)(C(x)→ D(x))

Given I = 〈M, (.)I〉, I |= C v D iff CI ⊆ DI

C ≡ D: abbreviation for the two axioms C v D and D v C

ABox: finite set of assertion axioms

a : C or (a, b) : R : correspond to sentences of the form C(a) or R(a, b)

Given I = 〈M, (.)I〉,

I |= a : C iff aI ∈ CI ; I |= (a, b) : R iff 〈aI , bI〉 ∈ RI


Knowledge bases in n-graded and fuzzy DLs

fuzzy FOL sentence fuzzy axiomr̄ → (∀x)(C(x)→ D(x)) 〈C v D,< r̄〉(∀x)(C(x)→ D(x))→ r̄ 〈C v D,4 r̄〉r̄ ↔ (∀x)(C(x)→ D(x)) 〈C v D,≈ r̄〉

r̄ → C(a) 〈a : C,< r̄〉C(a)→ r̄ 〈a : C,4 r̄〉r̄ ↔ C(a) 〈a : C,≈ r̄〉

r̄ → R(a, b) 〈(a, b) : R,< r̄〉

} TBox: fuzzy concept inclusion axioms

} ABox: fuzzy assertion axioms

Reasoning with fuzzy KBs in our languages involves evaluated formulas

Evaluated formulas

r̄ → ϕ ϕ→ r̄ ϕ↔ r̄

ϕ is a formula with no occurrences of truth constants

x →∗ y = 1 iff x ≤ y


The Robots dataset

r1 r5r4r3r2

. . . . . . . . . .

r6

. .

r9

. .. .

r7

. .

r8


Fuzzy TBox

Friendly ≡ Robot & (∃hasObject.FriendlyObject)& (Happy Y Homogeneous)

〈Robot & Object v 0̄,≈ 1̄〉〈1̄ v Robot Y Object,≈ 1̄〉

〈Homogeneous v Robot,≈ 1̄〉〈Happy v Robot,≈ 1̄〉〈WearsTie v Robot,≈ 1̄〉

〈FriendlyObject v Object,≈ 1̄〉〈Flower v FriendlyObject,≈ 1̄〉〈Balloon v FriendlyObject,≈ 0.75〉〈Flag v FriendlyObject,≈ 0.50〉〈Sword v FriendlyObject,≈ 0.25〉〈Ax v FriendlyObject,≈ 0̄〉


Fuzzy ABox

For each i, 1 ≤ i ≤ 9, 〈ri : Robot,≈ 1̄〉, 〈(ri , oi ) : hasObject,≈ 1̄〉

〈r1 : Homogeneous,≈ 1̄〉, 〈o1 : Balloon,≈ 1̄〉, 〈r1 : Happy,≈ 1̄〉, 〈r1 : WearsTie,≈ 1̄〉

〈r2 : Homogeneous,≈ 1̄〉, 〈o2 : Flag,≈ 1̄〉, 〈r2 : Happy,≈ 1̄〉, 〈r2 : WearsTie,≈ 1̄〉

〈r3 : Homogeneous,≈ 0.75〉, 〈o3 : Sword,≈ 1̄〉, 〈r3 : Happy,≈ 1̄〉, 〈r3 : WearsTie,≈ 1̄〉

〈r4 : Homogeneous,≈ 0.50〉, 〈o4 : Flower,≈ 1̄〉, 〈r4 : Happy,≈ 0̄〉, 〈r4 : WearsTie,≈ 0̄〉

〈r5 : Homogeneous,≈ 0.50〉, 〈o5 : Sword,≈ 1̄〉, 〈r5 : Happy,≈ 0̄〉, 〈r5 : WearsTie,≈ 0̄〉

〈r6 : Homogeneous,≈ 0.75〉, 〈o6 : Flag,≈ 1̄〉, 〈r6 : Happy,≈ 0.50〉, 〈r6 : WearsTie,≈ 0̄〉

〈r7 : Homogeneous,≈ 1̄〉, 〈o7 : Ax,≈ 1̄〉, 〈r7 : Happy,≈ 0.50〉, 〈r7 : WearsTie,≈ 1̄〉

〈r8 : Homogeneous,≈ 0.75〉, 〈o8 : Ax,≈ 1̄〉, 〈r8 : Happy,≈ 0.50〉, 〈r8 : WearsTie,≈ 1̄〉

〈r9 : Homogeneous,≈ 1̄〉, 〈o9 : Balloon,≈ 1̄〉, 〈r9 : Happy,≈ 0.50〉, 〈r9 : WearsTie,≈ 1̄〉


Reasoning in FDLs

Reasoning notions in Fuzzy DL:

C is ∗-satisfiable iff exists an interpretation I∗ such that I∗ |= C(a)

C is ∗-valid iff I∗ |= (∀x)C(x), for every interpretation I∗C is ∗-subsumed by D iff I∗ |= (∀x)(C(x)→ D(x)), for every interpretation I∗ (i.e.,C → D is ∗-valid)

Reasoning notions with truth constants:

C is ∗-satisfiable in a degree greater or equal than r iff the concept r̄ → C is ∗-satisfiable

C is ∗-valid in a degree greater or equal than r iff r̄ → C is ∗-valid.

C is ∗-subsumed by D in a degree greater or equal than r iff r̄ → (C → D) is ∗-valid.


∗-subsumption with respect to the KB

Homogeneous v Happy

Interpretation: infx{Homogeneous(x)→∗ Happy(x)} = r?

Robot Homogeneous Happy →∗r1 1 1 1r2 1 1 1r3 0.75 1 1r4 0.50 0 0.50→∗ 0r5 0.50 0.50 0.50→∗ 0r6 0.75 0.50 0.75→∗ 0.50r7 1 0.50 1→∗ 0.50r8 0.75 0.50 0.75→∗ 0.50r9 1 0.50 1→∗ 0.50

• 1→∗ 0.50 = 0.50

• 0.50→∗ 0Min, Prod: 0Ł: min(1, 1-0.50+0)= 0.50

• 0.75→∗ 0.50Min: 0.50Prod: 0.50

0.75 = 2/3Ł: min(1, 1-0.75+0.50) = 0.75

Min : infx{1, 0, 0.50} = 0Prod: infx{1, 0, 2/3} = 0Ł: infx{1, 0.50, 0.75} = 0.50

It is well known that:∗min ≥ ∗prod ≥ ∗Ł

→∗min≤→∗prod≤→∗Ł


∗-entailment

A fuzzy assertion α is ∗-entailed by K if every interpretation I∗ satisfying K also satisfies α

K ` 〈Friendly(r6),< 0.50〉

ABox ` α = 〈ϕ,< r̄〉

ABox + {β = 〈ϕ,< s̄1〉} ` α = 〈ϕ,< s̄2〉


Conclusions

FDL is a natural extension of DL for dealing with graded (vague) concepts,commonly present in real applications.

As Hájek proposed, we take t-norm based fuzzy logics as basis to deal with FDL.In that way we can take profit of recent developments in mathematical fuzzylogics.

We propose the logics L∗∼(S)∀ as a framework for FDL (many-valued or graded).The languages in the family ALC(S∗) are fragments of such logics.

Having truth constants in the language allow us to have a syntactical counterpartin FDLs for the KB axioms.

Truth constants also allow to define graded notions of validity, satisfiability andsubsumption from a syntactic perspective.


Conclusions

The most natural FDLs for applications are n-graded FDLs

We know (Hájek) that the satisfiability problem for ABox in an ALC-like languageover Lukasiewicz logic (and in any n-graded FDL) is decidable (via witnessedmodels)

Cerami et alt. extended the last result to Product logic (via quasi-witnessedmodels)

We know that satisfiability in Modal logic over the fuzzy logic corresponding ton-valued residuated chain is in PSPACE (like in the classical case)

The complexity issues for deduction from a KB is unknown as far as I know.


A research plan in FDLs

Define complex more expressive languages with the goal to define a standard Fuzzy OWL

Study the fragment of first order logic associated to these languages (from logic andcomputational point of view) depending on the t-norm used and on the constructors.

Study possible axiomatizations of these fragments (Modal over multiple-valued)

build efficient satisfiability algorithms for each of these FDLs.

Application to fuzzy ontologies (special attention given to the meaning of the values inrelation with graded concepts and notions).


on fuzzy description logics · from description logics to fuzzy description logics dls (alc) fol...

Documents