a principled framework for modular web rule bases and its semantics

KR-2008 1

A Principled Framework for Modular Web Rule Bases and its Semantics

Anastasia AnalytiInstitute of Computer Science, FORTH-ICS, Greece

Grigoris Antoniou Dept. of Computer Science, Univ. of Crete, Greece

Carlos Viegas DamásioCENTRIA, Depart. De Informatica, Univ. Nova de Lisboa,

Portugal

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• Motivation & Requirements

• Open-World and Closed-World Reasoning

• MWeb Modular Rule Bases

• syntax• validity• example

• MWebAS & MWebWFS model-theoretic semantics

• properties

• Conclusions - Future work

Presentation Overview

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• 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.

Motivation & Requirements

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• 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

Requirements (Cont.)

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• 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

Open-World & Closed-World Reasoning

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• 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

Our Approach for OWAs & CWAs

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• Consider the following list of facts:

% All current EU countriesCountryEU(Austria) … CountryEU(UK)

% Some non EU countries (not all…) CountryEU(China) CountryEU(Djibuti)

Example

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• Closed world reasoning:

CountryEU(?c) ← ~ CountryEU(?c)

• Open world reasoning:

CountryEU(?c) ← ~ CountryEU(?c) CountryEU(?c) ← ~ CountryEU(?c)

OWAs & CWAs with the help of Extended LP

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• 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)

A Little Geography ….

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• An (MWeb) rule base s is a triple <Names, Ps, Ints>

where:

• Names is the name of s (an IRI reference)

• Ps is the logic program of s (an MWeb logic program)

• Ints is the interface of s

DEFINES declarations USES decarations

• An (MWeb) modular rule base S is a set of rule bases

MWeb Modular Rule Bases

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• 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) rule r is a formula of the form

L ← L1 , …, Lm , ~ Lm+1 , …,~ Ln

where L is a simple atom A or the strong negation of a

simple atom A

Li is an atom A or the strong negation of an atom A

If m=n, rule r is called objective

• An (MWeb) logic program P is a set of rules

The Logic Program of a Rule Base

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• 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

Predicate Reasoning Modes

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• 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.

Definite Predicates

<http://geography.int>

defines … definite geo:Country

geo:Country(Egypt). geo:Country(Canada).Rule Base s

Example:

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• If a predicate is declared open in a rule base s w.r.t. a context cxt then 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

Open Predicates

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Example of Open Predicates

<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). …

Rule Base s

OWA rules:

eu:CountryEU(?c) ← geo:Country(?c), ~ eu:CountryEU(?c)


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• If a predicate is declared positively or negatively closed in a rule base s w.r.t. a context cxt then 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

Positively/Negatively Closed Predicates

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Example of Positively Closed Predicates

<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). …

Rule Base s

CWA rule:


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• 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’ S in 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’ S in internal scope only.

• Internal: p is visible inside s , only.

• No constraint is imposed on the reasoning mode of local and internal predicates

Predicate Scope

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• 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

Interface of a Rule Base s

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• 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

Validity of a Modular Rule Base

normal allowed allowed allowed allowed

closed allowed allowed allowed error

open allowed allowed allowed error uses

p

(im

port

er)

objective allowed allowed allowed error objective open closed normal

defines p (exporter)

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Example of a Modular Rule Base

<http://europa.eu>

defines local posClosed eu:CountryEU.

eu:CountryEU(Austria).

eu:CountryEU(Greece). Rule base s1

<http://security.int>

defines local open sec:citizenOf.defines global open sec:Suspect.

sec:citizenOf(Anne,Austria). sec:citizenOf(Boris,Croatia). sec:Suspect(Peter). Rule base s2

<http://gov.countryY>

defines local normal gov:Enter visible to <http://security.int>. defines local negClosed gov:RequiresVisa.defines internal open sec:citizenOf.

uses open eu:CountryEU from <http://europa.eu>.uses definite sec:citizenOf from <http://security.int>.uses definite 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

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• 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

Model-theoretic Semantics

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• 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

Herbrand Base of a Rule Base

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• 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 set

I = {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

Normal & Extended Interpretation of a Modular Rule Base

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• 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 mode m then:

MWebAS & MWebWFS entailment

1))(( , of set answer extendedeach for if )(|

1))(( , of set answer normaleach for if )(|

M

M

cpms

MScpmWFSS

s

cpms

MScpmASS

s

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normal definite open closed normal

closed definite open closed

open definite open open

definite definite definite definite definite open closed normal

currentreasoning mode of s

• Let p be a predicate defined in a rule base s

Properties of MWeb Semantics

defining reasoning mode of p in s

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normal definite open closed normal

closed definite open closed

open definite open open

definite definite definite definite definite open closed normal

requesting reasoning

mode of p in s’ (importer)

• Let p be a predicate

• defined in a rule base s and

• imported by a rule base s’

Properties of MWeb Semantics (cont.)

defining reasoning mode of p in s (exporter)

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Example

<http://europa.eu>











gov:Enter(?p) eu:Country(?c), sec:citizenOf(?p,?c), gov:RequiresVisa(?c), ~sec:Suspect@<http://security>(?p).


S={s1, s2, s3}

},{

for

)(:|

while

, )(:|

3

1

mWFSmASSEM

CroatiaCountryEUeuSEMS

s

CroatiaCountryEUeuSEMS

s

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<http://europa.eu>











gov:Enter(?p) eu:Country(?c), sec:sitizenOf(?p,?c), gov:RequiresVisa(?c), ~sec:Suspect@<http://security>(?p).


S={s1, s2, s3}

Example (cont.)

},{

for

)(:~|

and

)(:|

3

3

mWFSmASSEM

PeterEntergovSEMS

s

AnneEntergovSEMS

s

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<http://europa.eu>





defines local open sec:sitizenOf.defines global open sec:Suspect.





gov:Enter(?p) eu:Country(?c), sec:sitizenOf(?p,?c), ~sec:Suspect@<http://security.int>(?p).



S={s1, s2, s3}

Case-based Analysis of MWebAS

)(:|

while

, )(:|

3

3

BorisEntergovmWFSS

s

BorisEntergovmASS

s

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<http://europa.eu>





defines local open sec:sitizenOf.defines global open sec:Suspect.





gov:Enter(?p) eu:Country(?c), sec:sitizenOf(?p,?c), ~sec:Suspect@<http://security.int>(?p).



S={s1, s2, s3}

Restricted Propagation of Local Inconsistencies

• s1 is inconsistent according to all simple and extended answer sets of S

• s3 is affected from the inconsistency

• s2 remains consistent


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• 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.

Properties of MWeb Semantics (cont.)

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• 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

Conclusions – Future Work

a principled framework for modular web rule bases and its semantics

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