Web ontology representation and reasoningvia fragments of set theory

Domenico Cantone Cristiano LongoMarianna Nicolosi-Asmundo Daniele Francesco Santamaria

Dipartimento di Matematica ed InformaticaUniversita` di Catania - ITALY

{cantone, longo, nicolosi}@dmi.unict.it, daniele.f.santamaria@gmail.com

CILC 2015 - Genova, 3 July 2015

This paper will appear in:Web Reasoning and Rule Systems, 978-3-319-22001-7, LNCS Volume 9209, Chapter

6, pp. 1-16, Springer International Publishing, 2015.

1

Overview of the talk

Motivation DL4D: a description logic representable in 4LQSR

Expressiveness of DL4D Translating SWRL rules in 4LQSR

Conclusions Hints of future work

2

Motivation

Computable set theory studies the decision problem for fragmentsof set theory (Cantone, Omodeo, Policriti 2001):

Integration in tnaNova/Referee; theoretic results.

Most of the decidability results and applications concernone-sorted multi-level fragments of set theory.

One-sorted multi-level syllogistics knowledge representation: Cantone, Longo, Pisasale, 2010; Cantone, Longo, Nicolosi-Asmundo, 2011; Cantone, Longo, 2014; ... .

3

Multi-sorted stratified syllogistics

Multi-sorted stratified syllogistics are set-theoretic languagesadmitting variables of different sorts (sort 0, sort 1,.. and so on)

Assignments for such variables are based on collections of objects(natural numbers, real numbers, possible worlds...)

Variables of sort 0 objects of the considered domain Variables of sort 1 sets of such objects Variables of sort 2 collections of sets ...

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Multi-sorted stratified syllogistics

The study of set-theoretic languages admitting variables ofdifferent sorts is motivated by the fact that most of the formulaein the statements and proofs of theorems in many fields ofmathematics and computer science involve variables of differentsorts

Example:In description logics there are entities of different types: individualelements, concepts, roles

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Multi-sorted stratified syllogistics

Less investigated than one-sorted multi-level syllogistics

Some results: Two-level syllogistics, 2LS

(A. Ferro and E. Omodeo, 1978)

Extension of 2LS with singleton and cartesian product(D. Cantone, V. Cutello, 1990)

Three-level syllogistics, extended with powerset, general union, singleton(D. Cantone, V. Cutello 1993)

Recently, decidability of the satisfiability problem for a four-level stratified

syllogistic, 4LQSR, admitting a restricted form of quantification over variables

of three types was proved in (D. Cantone, M. Nicolosi-Asmundo 2011)

6

This work

The description logic DL4D supporting: datatypes,

concept constructs (i.e., concept domain and range),

role constructs (i.e., union, complement, role chains).

Decidability of the consistency problem for DL4D-knowledgebases via a reduction to the satisfiability problem for4LQSR-formulae.

NP-completeness of the consistency problem for h-restrictedDL4D-knowledge bases. Expressivity of DL4D. Representation of SWRL rules in 4LQSR.

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Syntax of 4LQS

Pairing operator , Predicate symbols = and

(i) variables of sort 0 : x, y, z, . . .

(ii) variables of sort 1 : X1, Y 1, Z1, . . .

(iii) variables of sort 2 : X2, Y 2, Z2, . . .

(iv) variables of sort 3 : X3, Y 3, Z3, . . .

4LQS quantifier-free atomic formulae

level 0: x = y, x X1level 1:X1 = Y 1, X1 X2level 2:X2 = Y 2, x, y = X2, x, y X3, X2 X3

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Syntax of 4LQS

4LQS quantified atomic formulae

level 1: (z1) . . . (zn)0, 0 propositional combination ofquantifier-free atomic formulae

level 2: (Z11) . . . (Z1m)1, 1 propositional combination ofquantifier-free atomic formulae and of quantified atomicformulae of level 1

level 3: (Z21) . . . (Z2p)2, 2 propositional combination ofquantifier-free atomic formulae and of quantified atomicformulae of levels 1 and 2

4LQS-FormulaePropositional combinations of quantifier-free atomic formulae oflevels 0, 1, 2, and of quantified atomic formulae of levels 1, 2, 3

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Semantics of 4LQS

A 4LQS-interpretation is a pair M = (D,M), where

Mx DMX1 pow(D)MX2 pow(pow(D))MX3 pow(pow(pow(D)))

We put Mx, y = {{Mx}, {Mx,My}}

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Semantics of 4LQS

Formulae are interpreted as usual. In particular

M |= (z1) . . . (zn)0 iff M[z1/u1, . . . , zn/un] |= 0, forall u1, . . . , un D

M |= (Z11) . . . (Z1m)1 iffM[Z11/U11 , . . . , Z1m/U1m] |= 1,for all U11 , . . . , U

1m pow(D)

M |= (Z21) . . . (Z2p)2 iffM[Z21/U21 , . . . , Z2p/U2p ] |= 2,for all U21 , . . . , U

2p pow(pow(D))

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Characterizing 4LQSR

4LQSR is the subcollection of the formulae of 4LQS such that

Restriction INestings of quantifiers over variables of sort 0 into quantifiers overvariables of sort 1 are allowed if the former are linked to thecorresponding variables of sort 1[0 ni=1 mj=1 zi Z1j ]

Example

(Z1)(Z1 X2 (z) (z Z1 z X1) )

IfM |= (z Z1 z X1) thenM |= z Z112

Characterizing 4LQSR

Restriction IIEvery quantified atomic formula of level 3 is

either of type (Z21), . . . , (Z2p)2, where 2 is a propositionalcombination of quantifier-free atomic formulae

or of type (Z2)(Z2 X3 (z1)(z2)(z1, z2 = Z2)

Examples

(Z2)(Z2 X3 (Z2 X31 Z2 X32)) (intersection)

(Z2)(Z2 X3 (Z2 X31 (Z2 X32))) (set difference)13

The description logic DL4D

D = (ND, NC, NF , D), datatype map,RA, RD, C, I, sets of abstract role names, concrete role

names, concept names, individual names.

(a) t1, t2 dr | t1 | t1 u t2 | t1 unionsq t2 | {ed} ,(b)C1, C2 A | > | | C1 | C1 unionsq C2 | C1 u

C2 | {a} | R.Self |R.{a}|P.{ed} ,(c)R1, R2 S | U | R1 | R1 | R1 unionsqR2 | R1 u

R2 | RC1| | R|C1 | RC1 | C2 | id(C) ,(d) P T | P | PC1| | P|t1 | PC1|t1 ,

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The description logic DL4DA DL4D-TBox is a set of statements of the types:

-C1 C2, C1 v C2, C1 v R1.C2, R1.C1 v C2,nR1.C1 v C2, C1 v nR1.C2,

- t1 t2, t1 v t2, C1 v P1.t1, P1.t1 v C1, nP1.t1 v C1,C1 v nP1.t1,

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The description logic DL4D

A DL4D-RBox is a collection of statements of type:R1 R2, R1 v R2, R1 . . . Rn v Rn+1, Sym(R1), Asym(R1),Ref(R1), Irref(R1), Dis(R1, R2), Tra(R1), Fun(R1), P1 P2,P1 v P2, Fun(P1).

A DL4D-ABox is a set of assertions of the forms:a : C1, (a, b) : R1, (a, b) : R1, a = b, a 6= b, ed : t1,(a, ed) : P1, (a, ed) : P1.

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Semantics of terms and axioms specific to DL4D

Name Syntax Semanticsdata range dr dr drD D

negative datatype term t1 (t1)D = D \ tD1datatype terms

intersectiont1 u t2 (t1 u t2)D = tD1 tD2

datatype terms union t1 unionsq t2 (t1 unionsq t2)D = tD1 tD2constant in NC(d) ed e

Dd dD

valued exist.quantification

R.a (R.a)I = {x I : x, aI RI}datatyped exist.

quantif.T.ed (T.ed)I = {x I : x, eDd T I}

abstract rolecomplement

R (R)I = (I I) \RIabstract role union R1 unionsqR2 (R1 unionsqR2)I = RI1 RI2

abstract roleintersection

R1 uR2 (R1 uR2)I = RI1 RI2abstract role domain

restr.RC| (RC|)I = {x, y RI : x CI}

concrete role domainrestr.

TC| (TC|)I = {x, y T I : x CI}concrete role range

restr.T|t (T|t)I = {x, y T I : y tD}

concrete rolerestriction

TC1|t (TC1|t)I = {x, y T I : x CI1 y tD}

datatype termsequivalence

t1 t2 I |=D t1 t2 tD1 = tD2datatype terms diseq. t1 6 t2 I |=D t1 6 t2 tD1 6= tD2

datatype termssubsum.

t1 v t2 I |=D (t1 v t2) tD1 tD2

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Expressivity of DL4D

Existential quantification and at-least number restriction (resp.,universal quantification and at-most number restriction) only onthe left- (resp., right-) hand side of inclusion axioms.

More liberal than SROIQ(D) in: construction of role inclusion axioms (roles involved not

subject to any ordering relationship),

simple roles are not needed to define role inclusion axiomsand axioms involving number restrictions,

Boolean operators on roles are admitted.

Derived datatypes (inside inclusion axioms involving concreteroles).

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Decidability of the consistency problem for DL4D-knowledge bases

Theorem 1. Let K be a DL4D-knowledge base. Then, one canconstruct a 4LQSR-formula K s.t. K is satisfiable if and onlyif K is consistent.

Idea of the proof:

function from DL4D-statements into 4LQSR-formulaeExample: (C1 {a}) =Def (z)(z X1C1 z = xa), 4LQSR-formula K expressing the consistency of K, equivalence between the consistency problem for K and the

satisfiability problem for K.

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NP-completeness of the consistency problem forh-restricted DL4D-knowledge bases

DL4D-knowledge base K is h-restricted if axioms of types:R1 . . . Rn1 v R, n2R.C1 v C2, n3P.t1 v t2, C1 v n4R.C2,t1 v n5P.t2 occur in K only if n1, n2, n3, n4, n5 h.

Reasoning as in Theorem 1, we construct K such that

(i) K belongs to (4LQSR)h, whose satisfiability problem is

NP-complete (Cantone, Nicolosi-Asmundo 2011), and

(ii) the size of K is polynomially related to that of K.

Quite expressive: Ontoceramic representable in (4LQSR)3.

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Translating SWRL-rules into 4LQSR-formulae

An SWRL-rule r has the form (x1, . . . , xn)(B H), where:

- B and H are conjunctions of atoms of the following types:x C, y t, x, y R, x, y P, x = y, x 6= y, and

- Var (H) Var (B) = {x1, . . . , xn}.

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Translating SWRL-rules into 4LQSR-formulae

For space reasons we do not provide here a formal translationfunction. However, it is not hard to see that it could beconstructed by modifying the map introduced in the proof of theprevious theorem.

Type of Rule Rule

SWRL-rulehasParent(X, Y ), hasBrother(Y, Z) :

hasUncle(X,Z).4LQSR-rule

(x)(y)(z)(x, y X3hasParenty, z X3hasBrother x, z X3hasUncle)

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Conclusions

DL4D: a description logic expressible in 4LQSR

Decidability of the consistency problem for DL4D-knowledgebases

NP-Completeness.

Expressiveness of SROIQ(D)

Expressing SWRL rules in 4LQSR

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Hints of future work

DL4D: a description logic expressible in 4LQSR

Concrete roles union and intersection.

Datatype groups

Metaroles and Metaconcepts

Implementing reasoner, available on Protege.

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