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					Homogeneous or Hybrid?
Jing Mei
Information Science Department Peking University

2006-06-09

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

Contents

1

Homogeneous DLP & SWRL & DL2DD ASP & CLP & DR

2

Hybrid AL-log & CARIN DL+log & HEX

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

Contents

1

Homogeneous DLP & SWRL & DL2DD ASP & CLP & DR

2

Hybrid AL-log & CARIN DL+log & HEX

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

Overview

Monotonic Logics
DLP: Description Logic Programs SWRL: Semantic Web Rule Language Reducing DL to Disjunctive Datalog Extending DL Tableaux with DL-safe Rules

Non Monotonic Logics
ASP: Answer Set Programing CLP: Conceptual Logic Programming DR: Defeasible Reasoning

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

DL ⊕ LP

The Intersection: DLP
Decidable Less Expressive Full Supported: Datalog Rules Published in Proc. of WWW 2003
By: Benjamin N. Grosof, Ian Horrocks, Raphael Volz, and Stefan Decker

The Union: SWRL
Undecidable More Expressive Partial Supported: First Order Formulae Published in Proc. of WWW 2004
By: Ian Horrocks and Peter F. Patel-Schneider

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

DL2DD: Reducing DL to Disjunctive Datalog
Translating a DL KB K to an equisatisfiable set of clauses Ξ(K ) Deciding satisfiability of Ξ(K ) by an ordered resolution calculus R: Saturation of Ξ(K ) Function-free version FF(K ) of K : Deleting the clauses containing function symbols A disjunctive Datalog program DD(K ): Moving positive literals into the rule head, and negative into the body
Referring to:
1

Reasoning for Description Logics around SHIQ in a Resolution Framework By: Ullrich Hustadt and Boris Motik and Ulrike Sattler FZI Technical Report 3-8-04, 2004 http://kaon2.semanticweb.org/ Query Answering for OWL-DL with Rules By: Boris Motik and Ulrike Sattler and Rudi Studer Journal of Web Semantics, 3(1):41-60, 2005
Jing Mei Homogeneous or Hybrid?

2

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

DL2DD: Taking An Example (P. 52)
A DL KB K = {C ∃R.D}

The translation: R(x, f (x)) ← C (x) and D(f (x)) ← C (x) An equisatisfiable set Ξ(K ) = {¬C (x) ∨ R(x, f (x)), ¬C (x) ∨ D(f (x))} The deletion: Remove all, leaving function-free A DD KB DD(K ) = ∅ Fail to checking the satisfiability of D or R, when C (a) holds. Unfair, but acceptable for query answering – unnamed individuals cannot be used in queries or answers. Claim: The models of K and DD(K ) coincide only on positive ground facts, whereas, for unnamed individuals, the models are completely unrelated.
Jing Mei Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

DL2DD: Taking Another Example (P. 53)

A DL KB K = {A ∃R.B, B A DD KB DD(K ) consists of
C (x) ← B(x) D(x) ← R(x, y ), C (y ) D(x) ← A(x)

C , ∃R.C

D}

Claim: DD(K1 ) ∪ DD(K2 ) is not necessarily equal to DD(K1 ∪ K2 ). Such concept as ∃R.∃R.A, containing a non-atomic subconcept ∃R.A, introduces a fresh atomic concept Q standing for ∃R.A, later applying rule unfolding to resolve Q (See Page 54)

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

DL2DD: Correspondence
Theorem Let K be an ALCHIQ− KB and DD(K ) be its reduction to disjunctive datalog a K is unsatisfiable iff DD(K ) is unsatisfiable K |= α iff DD(K ) |=C α where α = A(a), R(a, b) and A is an atomic concept K |= C (a) iff DD(K ∪ {C non-atomic concept
a

Q}) |=C Q(a) where C is a

Cautious entailment P |=C α: α is entailed in every minimal model of P

Notes: Eliminating transitivity axioms happens to very simple DL properties, resulting in ALCHIQ− rather than ALC R + HIQ− i.e. SHIQ
Jing Mei Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

Extending DL Tableaux with DL-safe Rules
First Order Semantics for DL-safe KBs (K, P)
Mapping the DL KB K to first order logic π(K) Treating rules in P as logical implications
Rule: H ← B1 , · · · , Bn Implication: H ∨ ¬B1 ∨ · · · ∨ ¬Bn

Tableaux Algorithm extended with R-rule
Binding: Only named individuals not introduced by the tableaux expansion rules are considered New Expansion R-Rule
For each ground literal C (a) in H or ¬Bi if C ∈ L(a) then L(a) = L(a) ∪ {C } For each ground literal P(a, b) in H or ¬Bi if P ∈ L(a, b) then L(a, b) = L(a, b) ∪ {P} For each ground literal ¬P(a, b) in H or ¬Bi create a new branch where L(a) = L(a) ∪ {∀P.¬{b}} Referring to: 1 Extending the SHOIQ(D) Tableaux with DL-safe Rules: First Results By: Vladimir Kolovski and Bijan Parsia and Evren Sirin In Proc. of DL 2006
Jing Mei Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

Deduction in Ontologies via ASP

Defining First Order Semantics for DL class expressions { without consideration of TBox } Grounding for the FO counterpart of DL expressions { via coding in XSB } Reducing FO formulae to Answer Set Programs { ¬ becomes not fed into Smodels or DLV }
Referring to: 2004 Deduction in Ontologies via Answer Set Programming By: Terrance Swift In Proc. of LPNMR (Logic Programming and Nonmonotonic Reasoning)

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

Deduction in Ontologies via ASP: An Example
DL Class: ∃R0 .A1 ∃R0 .(∃R0 .¬A1 ) ∀R0 .A0

FO Sentence: rel(d0 , R0 , d1 ) ∧ elt(d1 , A1 ) ∧rel(d0 , R0 , d2 ) ∧ rel(d2 , R0 , d3 ) ∧ ¬elt(d3 , A1 ) ∧(¬rel(d0 , R0 , d1 ) ∨ (rel(d0 , R0 , d1 ) ∧ elt(d1 , A0 ))) ∧(¬rel(d0 , R0 , d2 ) ∨ (rel(d0 , R0 , d2 ) ∧ elt(d2 , A0 ))) ASP Program
sat0:- rel(d0 , R0 , d1 ), elt(d1 , A1 ), rel(d0 , R0 , d2 ), rel(d2 , R0 , d3 ), not elt(d3 , A1 ), sat1, sat2. sat1:- not rel(d0 , R0 , d1 ). sat1:- rel(d0 , R0 , d1 ), elt(d1 , A0 ). sat2:- not rel(d0 , R0 , d2 ). sat2:- rel(d0 , R0 , d2 ), elt(d2 , A0 ).

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

Guarded Open ASP with Generalized Literals
CLP: Conceptual Logic Programs Extended Logic Programs having infinite answer sets Guardedness for regaining decidability { A positive atom contains every variable in the rule } Support for DLs with number restrictions: CLP { An extension of ASP (i.e. Datalog¬,∨ ) towards infinite domains }
Referring to: 1 Guarded Open Answer Set Programming with Generalized Literals By: Stijn Heymans, Davy Van Nieuwenborgh, Dirk Vermeir In Proc. of FoIKS 2006 (Foundations of Information and Knowledge) G-hybrid Knowledge Bases By: Stijn Heymans, Livia Predoiu, Cristina Feier, Jos de Bruijn and Davy Van Nieuwenborgh. In Proc. of ALPSWS 2006 (Applications of Logic Programming in the Semantic Web and Semantic Web Services) Integrating Semantic Web Reasoning and Answer Set Programming By: Stijn Heymans and Dirk Vermeir In Proc. of RuleML 2004

2

3

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

DLP & SWRL & DL2DD ASP & CLP & DR

Defeasible Reasoning
Rule-based reasoning with incomplete and inconsistent information (Facts, Rules, Priorities among rules) Defeasible Theory: (R, >) where R is a finite set of rules and < is a superiority relation on R
Strict Rules: A → p Defeasible Rules: A ⇒ p

Translation: p is defeasible provable in a defeasible theory D iff p is included in the well-founded model of Tr (D)
Referring to:
1

A System for Nonmonotonic Rules on the Web By: Grigoris Antoniou and Antonis Bikakis and Gerd Wagner In Proc. of RuleML 2004 DR-Prolog: A System for Reasoning with Rules and Ontologies on the Semantic Web By: Antonis Bikakis and Grigoris Antoniou. In Proc. of AAAI 2005
Jing Mei Homogeneous or Hybrid?

2

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

Contents

1

Homogeneous DLP & SWRL & DL2DD ASP & CLP & DR

2

Hybrid AL-log & CARIN DL+log & HEX

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

Overview

Monotonic Logics
AL-log: Backward CARIN: Forward OWL-log: From AL-log/CARIN to SWRL

Non Monotonic Logics
HEX: Module DL+log: Partition

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

DL + Datalog
p0 (X0 ) ← p1 (X1 ), · · · , pm (Xm ), q1 (Y1 ), · · · , qn (Yn ) where each pi (Xi ) is a Datalog atom and each qj (Yj ) is a DL atom First Datalog Last DL: AL-log
SLD-resolution for Datalog rules DL satisfiability checking for concept constraints Published in Journal of Intelligent Information Systems [1998]
By: Francesco M. Donini and Maurizio Lenzerini and Daniele Nardi and Andrea Schaerf

First DL Last Datalog: CARIN
Tableaux existential entailment for DL concepts and roles Bottom-up evaluation for Datalog rules Published in Proc. of ECAI [1996]
By: Alon Y. Levy and Marie-Christine Rousset

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

OWL-log
From AL-log/CARIN to SWRL
Restricting Datalog predicates to OWL DL atomic classes and properties Extending DL constraints to OWL DL Class and Datatype predicates

Decision Procedure: A combination of DL and Datalog reasoners, adopting DL-safeness condition
Dynamic: Like AL-log, using constrained SLD resolution Precompilation: Like CARIN, using entailment of DL atoms
Referring to:
1

Integrating Datalog with OWL: Exploring the AL-log Approach By: Edna Ruckhaus, Vladimir Kolovski, Bijan Parsia and Bernardo Cuenca In Proc. of ICLP 2006 (International Conference on Logic Programming)

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

HEX Programs
α1 ∨ · · · ∨ αk ← β1 , · · · , βm , notβn+1 , · · · , βm (Higher-Order) Atom αi : A tuple (Y0 , Y1 , · · · , Yn ) where each Yj is a term, specially Y0 is the predicate name (External) Atom βi : An external atom is &g [Y1 , · · · , Yn ](X1 , · · · , Xm ) where Y1 , · · · , Yn and X1 , · · · , Xm are two lists of terms (called input list and output list resp.) and &g is an external predicate name dlvhex for evaluating HEX programs: external atoms are embedded in plug-ins
Referring to: 1 Effective Integration of Declarative Rules with External Evaluations for Semantic-Web Reasoning By: Thomas Eiter, Giovambattista Ianni, Roman Schindlauer, Hans Tompits In Proc. of ESWC 2006 2 Combining Answer Set Programming with Description Logics for the SW By: Thomas Eiter, Thomas Lukasiewicz, Roman Schindlauer, Hans Tompits In Proc. of KR 2004 (A previous work known as dl-programs)
Jing Mei Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

DL + log
Three mutually disjoint predicate alphabets: ΣC DL concept names ΣR DL property names ΣD Datalog predicates Definition A DL+log KB B is a pair (K, P) where K is a DL KB, i.e., a pair (T , A) P is a set of Datalog¬,∨ where each rule R has the form p1 (X1 ) ∨ · · · ∨ pn (Xn ) ← r1 (Y1 ), · · · , rm (Ym ), s1 (Z1 ), · · · , sk (Zk ), not u1 (W1 ), · · · , not uh (Wh )
pi ∈ ΣC ∪ ΣR ∪ ΣD ri ∈ ΣD and ui ∈ ΣD s i ∈ ΣC ∪ ΣR
Jing Mei Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

DL + log: Safeness
Datalog Safeness
{X1 , · · · , Xn } ∪ {Y1 , · · · , Ym } ∪ {Z1 , · · · , Zk } ∪ {Wh , · · · , Wh } ⊆ {Y1 , · · · , Ym } ∪ {Z1 , · · · , Zk }

DL Safeness
{X1 , · · · , Xn } ∪ {Y1 , · · · , Ym } ∪ {Z1 , · · · , Zk } ∪ {Wh , · · · , Wh } ⊆ {Y1 , · · · , Ym }

Weak Safeness
{X1 , · · · , Xn } ⊆ {Y1 , · · · , Ym }

Datalog Safeness

DL Safeness

Weak Safeness

For example, Person ∃hasParent.Person Datalog Predicates: hasUncle, hasBrother DL Predicate: hasParent (1) hasUncle(x, z) ← hasParent(x, y ), hasBrother(y , z) √ Datalog Safeness ( ); DL Safeness (x); Weak Safeness (x) (2) happy(x) ← hasParent(x, y ) √ √ Datalog Safeness ( ); DL Safeness (y ); Weak Safeness( )
Jing Mei Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

DL + log: Reasoning Services
Query Answering: q(x) ← ∃y .conj(x, y ) UCQ: ∃z.conj1 (z) ∨ · · · ∨ conjn (z) where each conji (z) is a set of atoms like p1 (z) ∧ · · · pm (z) Boolean CQ/UCQ containment problem: The problem of deciding T |= Q1 ⊆ Q2 Existential entailment problem: The problem of deciding Q1 ∪ T |= Q2 Boolean unions of conjunctive queries (UCQ) Boolean conjunctive queries (CQ) when n = 1 Above, Q1 a Boolean CQ while Q2 a Boolean CQ. It means, for every model I of T , if Q1 is satisfied in I then Q2 is satisfied in I
Jing Mei Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

DL + log: First-Order Semantics

Regarding A ← B, not C and A ∨ C ← B the same Definition A FOL-model of a DL+log KB B = (K, P) is an interpretation I of ΣC ∪ ΣR ∪ ΣD such that I satisfies FO(K) and FO(P) FOL-satisfiability can always be reduced (in linear time) to NM-satisfiability, by moving every negated atom in the body to the rule head

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

DL + log: Non-monotonic Semantics
Definition A NM-model of a DL+log KB B = (K, P) is an interpretation I of ΣC ∪ ΣR ∪ ΣD such that
1 2

IΣC ∪ΣR satisfies K IΣD is a stable model for Π(gr (P, C), IΣC ∪ΣR )
IΣ is the projection of I to Σ, interpreting only predicates in Σ gr (P, C) is the ground instantiation of P w.r.t. C Π(Pg , I) is the projection of (a ground program) Pg to I, deleting those DL-related issues

According to the NM semantics, DL-predicates are still interpreted under the classical open-world assumption (OWA), while Datalog predicates are interpreted under a closed-world assumption (CWA)

Jing Mei

Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

DL + log: Algorithm
Table of the Deletion IΣC ∪ΣR Head Body Partition Satisfied The Rule DL atom itself GP Unsatisfied DL atom itself The Rule GN Algorithm NMSAT-DL + log(B) Input: DL + log-KB B = (K, P) with K = (T , A) Output: true if B is NM-satisfiable, false otherwise begin if there exists partition (GP , GN ) of grp (P) such that P(GP , GN ) has a stable model T CQ(A ∪ GP ) ⊆ UCQ(GN ) then return true else return faslse end
Referring to:
1

DL+log: Tight Integration of Description Logics and Disjunctive Datalog By: Riccardo Rosati In Proc. of KR 2006 http://www.dis.uniroma1.it/˜osati/publications/Rosa06b.htm r
Jing Mei Homogeneous or Hybrid?

Homogeneous Hybrid

AL-log & CARIN DL+log & HEX

MKNF-DL Rules
Definition H1 ∨ · · · ∨ Hn ← B1 , · · · , Bm Hi Atoms of the form A or KA Bj Atoms of the form A, KA, or notA where A denotes arbitrary first-order function-free atoms, concerning DL concepts, properties and rule predicates Referring to:
1

Closing Semantic Web Ontologies By: Boris Motik and Riccardo Rosati Technical Report 2006-05-22 ˜ http://www.cs.man.ac.uk/bmotik/publications/paper.pdf (A MKNF extension to DL+log in KAON2)

Jing Mei

Homogeneous or Hybrid?


				
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