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A Principled Framework for Modular Web Rule Bases and its Semantics

A Principled Framework for Modular Web Rule Bases and its Semantics. Anastasia Analyti Institute of Computer Science, FORTH-ICS, Greece Grigoris Antoniou Dept. of Computer Science, Univ. of Crete, Greece Carlos Viegas Damásio CENTRIA, Depart. De Informatica, Univ. Nova de Lisboa, Portugal.

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A Principled Framework for Modular Web Rule Bases and its Semantics

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  1. A Principled Framework for Modular Web Rule Bases and its Semantics Anastasia Analyti Institute of Computer Science, FORTH-ICS, Greece Grigoris Antoniou Dept. of Computer Science, Univ. of Crete, Greece Carlos Viegas Damásio CENTRIA, Depart. De Informatica, Univ. Nova de Lisboa, Portugal KR-2008

  2. Presentation Overview • Motivation & Requirements • Open-World and Closed-World Reasoning • MWeb Modular Rule Bases • syntax • validity • example • MWebAS & MWebWFSmodel-theoretic semantics • properties • Conclusions - Future work KR-2008

  3. Motivation & Requirements • Merging knowledge over rule bases on the Web is an important problem. • Modularity, encapsulation, information hiding, and access control mechanisms are required. • Closed-world and open-world reasoning should be supported. • Proposed mechanisms should guarantee monotonicity of reasoning on all predicates,in the case that new rule bases are added to the modular base. • The proposed modularity framework should have clear model-theoretic semantics. KR-2008

  4. Requirements (Cont.) • Rule bases defining a predicate p need mechanisms to express: • if reasoning on p is open, closed, or normal • that p is hidden, can only be imported, or redefined by other rule bases • that p is exported to all requesting rule bases or only to particular ones • Rule bases importing knowledge about a predicate p need mechanisms to express: • the reasoning mode that p is requested • if knowledge on p is imported from all providing rule bases or from particular ones KR-2008

  5. Open-World & Closed-World Reasoning • Open-World Reasoning • Founded on First Order Logic • Adopted in Description Logics, OWL, and SWRL • Appropriate for predicates “fully-shared” in the Semantic Web • Sometimes too conservative • Closed-World Reasoning • Founded on Nonmonotonic Logics • Adopted in Logic Programming and WRL • Appropriate for Deductive Databases • Sometimes too brave KR-2008

  6. Our Approach for OWAs & CWAs • Open-world and closed-world assumptions can be already combined in Extended Logic Programming! • Two forms of negation are required: • strong or explicit • weak, default, or negation as failure • The two forms of negation are available in: • Well-founded semantics with explicit negation (WFSX) • Answer Set Semantics (AS) • The proposed solution is the same for both semantics KR-2008

  7. Example • Consider the following list of facts: % All current EU countries CountryEU(Austria) … CountryEU(UK) % Some non EU countries (not all…)  CountryEU(China)  CountryEU(Djibuti) KR-2008

  8. OWAs & CWAs with the help of Extended LP • Closed world reasoning:  CountryEU(?c) ← ~ CountryEU(?c) • Open world reasoning:  CountryEU(?c) ← ~ CountryEU(?c) CountryEU(?c) ← ~ CountryEU(?c) KR-2008

  9. A Little Geography …. • Is Austria an EU country ? • YES, due to the fact CountryEU(Austria) • Is China an EU country ? • NO, due to the fact  CountryEU(China) • Is Montenegro an EU country ? • NO, because it is not listed and the list is complete (CLOSED WORLD REASONING) • DON’T KNOW, because it is not listed and the list is not complete (OPEN WORLD REASONING) KR-2008

  10. MWeb Modular Rule Bases • An (MWeb) rule bases is a triple<Names, Ps, Ints> where: • Names is the name of s (an IRI reference) • Psis the logic program of s (an MWeb logic program) • Ints is the interface of s • DEFINES declarations • USES decarations • An (MWeb) modular rule baseS is a set of rule bases KR-2008

  11. The Logic Program of a Rule Base • A term is an IRI reference, an RDF literal, or a variable • An atom is: • a simple atom p(t1, …, tk) or • a qualified atom p@Namet(t1, …, tk) • An (MWeb) ruler is a formula of the form L ← L1 , …, Lm, ~Lm+1 , …,~ Ln where L is a simple atomA or the strong negation of a simple atom  A Li is an atomA or the strong negation of an atom  A If m=n, rule r is called objective • An (MWeb) logic programP is a set of rules KR-2008

  12. Predicate Reasoning Modes • According to their reasoning modes, predicates are declared as: • Definite predicates • Open predicates • Closed predicates • positively closed • negatively closed • Normal predicates • Definite, open, and closed predicates are limited to be defined by rules without weak negation • Normal predicates can use the full language KR-2008

  13. Example: <http://geography.int> defines … definite geo:Country Rule Bases geo:Country(Egypt). geo:Country(Canada). Definite Predicates • If a predicate is declared definite in a rule base s then p is defined by objective rules, only. • Similar to Definite Logic Programming, but allowing for explicit negation in the head and body of the rules • Open and closed predicates appearing in the defining rules of p are interpreted, as if they had been declared definite. KR-2008

  14. Open Predicates • If a predicate is declared open in a rule base s w.r.t. a context cxtthen p is defined • not only by a set of objective rules, • but also through the following rules, called, contextual OWA rules of p p(?x1,…,?xn) ← cxt(?x1,…,?xn), ~p(?x1,…,?xn) p(?x1,…,?xn) ← cxt(?x1,…,?xn), ~p(?x1,…,?xn) • Context cxt is optional, and if defined, it should be a definite predicate • Closed predicates appearing in the defining rules of p are interpreted, as if they had been declared open KR-2008

  15. Rule Bases <http://europa.eu> defines … open eu:CountryEU wrt context geo:Country. defines … definite geo:Country. eu:CountryEU(Austria). eu:CountryEU(Greece). … geo:Country(Egypt). geo:Country(Canada). … Example of Open Predicates OWA rules:  eu:CountryEU(?c) ← geo:Country(?c), ~ eu:CountryEU(?c) eu:CountryEU(?c) ← geo:Country(?c), ~eu:CountryEU(?c) KR-2008

  16. Positively/Negatively Closed Predicates • If a predicate is declared positively or negativelyclosed in a rule base s w.r.t. a context cxtthen p is defined • not only by a set of objective rules, • but also through a positive contextual CWA rule p(?x1,…,?xn) ← cxt(?x1,…,?xn), ~p(?x1,…,?xn) or a negative contextual CWA rule p(?x1,…,?xn) ← cxt(?x1,…,?xn), ~p(?x1,…,?xn) • Context cxt is optional, and if defined, it should be a definite predicate KR-2008

  17. Rule Bases <http://europa.eu> defines … posClosed eu:CountryEU wrt context geo:Country. defines … definite geo:Country. eu:CountryEU(Austria). euCountryEU(Greece). … geo:Country(Egypt). geo:Country(Canada). … Example of Positively Closed Predicates CWA rule:  eu:CountryEU(?c) ← geo:Country(?c), ~ eu:CountryEU(?c) KR-2008

  18. Predicate Scope • Let S be a modular rule base. • The scope of a predicate p defined in a rule base s S can take the following values: • Global:p is visible outside s and can be defined by any other rule base s’ Sin global or internal scope only. The reasoning mode of global predicates must be always definite or open. • Local:p is visible outside s and can be defined by any other rule base s’ Sin internal scope only. • Internal:p is visible inside s , only. • No constraint is imposed on the reasoning mode of local and internal predicates KR-2008

  19. Interface of a Rule Base s • DEFINES declarations: • Determine which predicates p are defined in s, their scope, and defining reasoning mode • Provide the rule bases to which s is willing to export p • USES declarations: • Determine which predicates p are requested by s and their requesting reasoning mode in s i.e., the reasoning mode in which s requests p • definite, open, closed, normal • Provide the rule bases from which s requests p KR-2008

  20. The syntax of Defines and Uses Declarations • DEFINES declarations: defines global | local |internal definite | open | posClosed | negClosed |normal PredicateInd [ wrt contextPredicateInd] [visible toRuleBaseList] • USES declarations: uses definite | open |closed | normal PredicateInd [ fromRuleBaseList] KR-2008

  21. Validity of a Modular Rule Base • A modular rule base is valid if it satisfies a number of validity constraints: • most of these constraints have already been presented • Additionally: • the complete list can be found in the paper • In our work, we consider valid modular rule bases, only KR-2008

  22. Example of a Modular Rule Base <http://security.int> defineslocal open sec:citizenOf.definesglobal open sec:Suspect. sec:citizenOf(Anne,Austria). sec:citizenOf(Boris,Croatia). sec:Suspect(Peter). Rule base s2 <http://gov.countryY> defineslocal normal gov:Enter visible to <http://security.int>.defineslocal negClosed gov:RequiresVisa.definesinternal open sec:citizenOf.usesopen eu:CountryEU from <http://europa.eu>.usesdefinite sec:citizenOf from <http://security.int>. usesdefinite sec:Suspect from <http://security.int>. gov:Enter(?p)  eu:Country(?c), sec:citizenOf(?p,?c), ~sec:Suspect@<http://security.int>(?p). gov:Enter(?p)  eu:Country(?c), sec:citizenOf(?p,?c),gov:RequiresVisa(?c),~sec:Suspect@<http://security>(?p). gov:RequiresVisa(Croatia). sec:citizenOf(Peter,Greece). Rule base s3 <http://europa.eu> defineslocal posClosed eu:CountryEU. eu:CountryEU(Austria). eu:CountryEU(Greece). Rule base s1 KR-2008

  23. Model-theoretic Semantics • Let S be a modular rule base • We define the MWebAS & MWebWFS of S • model-theoretically • through a uniform definition • extending the AS and WFSX semantics on ELPs • keeping all of their computational characteristics • equivalent transformational semantics can be provided • through generated ELPs that can be evaluated through AS and WFSX semantics, respectively KR-2008

  24. Herbrand Base of a Rule Base • Let S be a modular rule base • The Herbrand Base of rule base s S wrt S consists of: • p(c1, …., ck), where • p is defined in s • ci is a constant appearing in S • p@rbase(c1, …., ck), where • p is requested by s from rbase • ci is a constant appearing in S KR-2008

  25. Normal & Extended Interpretation of a Modular Rule Base • Let S be a modular rule base • A simple normal (resp. extended) interpretation of a rule base s wrt S is a 2-valued (resp. 3-valued) interpretation of the Herbrand Base of s wrt S • A normal (resp. extended) interpretation of S is a setI = {Is | s  S}, where: • Is = < Isd, Iso, Isc, Isn> • Isi is a simple (resp.extended) interpretation of s wrt S • Isd, Iso, Isc, Isn correspond to the 4 reasoning modes of s KR-2008

  26. MWebAS & MWebWFS entailment • Let S be a modular rule base • A normal (resp. extended) answer set of S is a normal (resp. extended) interpretation of S that satisfies certain constraints provided in the paper • extending the definition of answer set (resp. extended stable model) of AS and WFSX semantics on ELPs • Let p be a predicate defined in a rule base s  S at reasoning modem then: KR-2008

  27. Properties of MWeb Semantics • Let p be a predicate defined in a rule base s currentreasoning mode of s defining reasoning mode of p in s KR-2008

  28. Properties of MWeb Semantics (cont.) • Let p be a predicate • defined in a rule base s and • imported by a rule base s’ requesting reasoning mode of p in s’ (importer) defining reasoning mode of p in s (exporter) KR-2008

  29. Example <http://security.int> defineslocal open sec:citizenOf.definesglobal open sec:Suspect. sec:citizenOf(Anne,Austria). sec:citizenOf(Boris,Croatia). sec:Suspect(Peter). Rule base s2 <http://gov.countryY> defineslocal normal gov:Enter visible to <http://security.int>.defineslocal negClosed gov:RequiresVisa.definesinternal open sec:citizenOf.uses open eu:CountryEU from <http://europa.eu>.usesdefinite sec:citizenOf from <http://security.int>. usesdefinite sec:Suspect from <http://security.int>. gov:Enter(?p)  eu:Country(?c), sec:citizenOf(?p,?c), ~sec:Suspect@<http://security.int>(?p). gov:Enter(?p)  eu:Country(?c), sec:citizenOf(?p,?c),gov:RequiresVisa(?c),~sec:Suspect@<http://security>(?p). gov:RequiresVisa(Croatia). sec:citizenOf(Peter,Greece). Rule base s3 S={s1, s2, s3} <http://europa.eu> defines local posClosed eu:CountryEU. eu:CountryEU(Austria). eu:CountryEU(Greece). Rule base s1 KR-2008

  30. Example (cont.) <http://security.int> defineslocal open sec:citizenOf.definesglobal open sec:Suspect. sec:citizenOf(Anne,Austria).sec:citizenOf(Boris,Croatia). sec:Suspect(Peter). Rule base s2 <http://gov.countryY> defineslocal normal gov:Enter visible to <http://security.int>.defineslocal negClosed gov:RequiresVisa.definesinternal open sec:citizenOf.usesopen eu:CountryEU from <http://europa.eu>.usesdefinite sec:citizenOf from <http://security.int>. usesdefinite sec:Suspect from <http://security.int>. gov:Enter(?p)  eu:Country(?c), sec:citizenOf(?p,?c), ~sec:Suspect@<http://security.int>(?p). gov:Enter(?p)  eu:Country(?c), sec:sitizenOf(?p,?c),gov:RequiresVisa(?c),~sec:Suspect@<http://security>(?p). gov:RequiresVisa(Croatia). sec:citizenOf(Peter,Greece). Rule base s3 S={s1, s2, s3} <http://europa.eu> defineslocal posClosed eu:CountryEU. eu:CountryEU(Austria). eu:CountryEU(Greece). Rule base s1 KR-2008

  31. <http://security.int> defineslocal open sec:sitizenOf.definesglobal open sec:Suspect. sec:citizenOf(Anne,Austria). sec:citizenOf(Boris,Croatia). sec:Suspect(Peter). Rule base s2 <http://gov.countryY> defineslocal normal gov:Enter visible to <http://security.int>.defineslocal negClosed gov:RequiresVisa.definesinternal open sec:citizenOf.usesopen eu:CountryEU from <http://europa.eu>.usesdefinite sec:citizenOf from <http://security.int>. usesdefinite sec:Suspect from <http://security.int>. gov:Enter(?p)  eu:Country(?c), sec:sitizenOf(?p,?c), ~sec:Suspect@<http://security.int>(?p). gov:Enter(?p)  eu:Country(?c), sec:sitizenOf(?p,?c),gov:RequiresVisa(?c),~sec:Suspect@<http://security>(?p). gov:RequiresVisa(Croatia). sec:citizenOf(Peter,Greece). Rule base s3 Case-based Analysis of MWebAS S={s1, s2, s3} <http://europa.eu> defineslocal posClosed eu:CountryEU. eu:CountryEU(Austria). eu:CountryEU(Greece). Rule base s1 KR-2008

  32. <http://security.int> defineslocal open sec:sitizenOf.definesglobal open sec:Suspect. sec:citizenOf(Anne,Austria). sec:citizenOf(Boris,Croatia). sec:Suspect(Peter). eu:CountryEU(Austria). Rule base s2 <http://gov.countryY> defineslocal normal gov:Enter visible to <http://security.int>.defineslocal negClosed gov:RequiresVisa.definesinternal open sec:citizenOf.usesopen eu:CountryEU from <http://europa.eu>.usesdefinite sec:citizenOf from <http://security.int>. usesdefinite sec:Suspect from <http://security.int>. • s1 is inconsistent according to all simple and extended answer sets of S • s3 is affected from theinconsistency • s2 remains consistent gov:Enter(?p)  eu:Country(?c), sec:sitizenOf(?p,?c), ~sec:Suspect@<http://security.int>(?p). gov:Enter(?p)  eu:Country(?c), sec:sitizenOf(?p,?c),gov:RequiresVisa(?c),~sec:Suspect@<http://security>(?p). gov:RequiresVisa(Croatia). sec:citizenOf(Peter,Greece). Rule base s3 Restricted Propagation of Local Inconsistencies S={s1, s2, s3} <http://europa.eu> defineslocal posClosed eu:CountryEU. eu:CountryEU(Austria).eu:CountryEU(Greece). Rule base s1 KR-2008

  33. Properties of MWeb Semantics (cont.) • Similarly to AS and WFSX on ELPs, MWebAS is more informative than MWebWFS • Entailment under MWebAS is data-complete for co-NP • Entailment under MWebWFS has polynomial data complexity • Both MWeb semantics guarantee monotonicity of reasoning on all predicates, in the case that: • new rule bases are added to the modular rule base S, and • the importing rule base list of the predicates of the old rule bases stays the same. KR-2008

  34. Conclusions – Future Work • Our MWeb framework for modular rule bases supports: • Reasoning in four modes: • Definite:weak negation is not accepted at all • Open:only OWAs are accepted • Closed:both CWAs and OWAs are accepted • Normal:weak negation is fully accepted • Local closed-world and open-world assumptions • Scoped negation-as-failure and scoped literal evaluation • Restricted propagation of local inconsistencies • Future work concerns: • the definition of the transformational semantics • the definition of modular Extended RDF ontologies KR-2008

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