logics and reasoning - sesa tutorial · reasoning is an act of deriving a conclusion from certain...

Logics and Reasoning - SESA Tutorial

Darko [email protected]

Digital Enterprise Research Institute (DERI)Leopold-Franzens Universität

Innsbruck, Austria

Seoul – September 10-13, 2007

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IntroductionOntologies and Reasoning

Ontology Languages & WSMLIRIS Reasoner

Hands-on Session

Agenda

Logics and Reasoning

Agenda for day 2nd: Logics and Reasoning

1 Ontologies and Reasoning2 The WSML Language3 IRIS Reasoner4 Hands-on Session

Darko Anicic SESA Tutorial - Logics and Reasoning

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Introduction

Ontology Definition

Figure: Ontology Definition


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Introduction

Ontology Example


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Introduction

Reasoning

Reasoning is an act of deriving a conclusion from certainpremises.The process of inferring implicit knowledge about somedomain of discourse from a given (formal) knowledge ofthat domain.Reasoning is a critical success factor for Semantic Webapplications.Reasoning problems: satisfiability, model finding,deduction...


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Introduction

Practical ReasonsGiven key role of ontologies in e-Science and Semantic Web, it is essential toprovide tools and services to help users:

Design and maintain high quality ontologies, e.g.:

Meaningful - all named classes can have instancesCorrect - captured intuitions of domain expertsMinimally redundant - no unintended synonymsRichly axiomatised - (sufficiently) detailed descriptions

Store (large numbers) of instances of ontology classes, e.g.:

Annotations from web pages (or gene product data)Answer queries over ontology classes and instances, e.g.:

Find more general/specific classesRetrieve annotations/pages matching a given description

Integrate and align multiple ontologies


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Ontology LanguagesRDF (S)OWLFirst Order LogicWSML

Ontology Languages

Requirements:- Expressivity

knowledge representation and ontology theory supportDimensions to Consider: Open vs. Closed WorldSemantics, Classical vs. Default neg, performance etc.

- Reasoning support

sound (unambiguous, decidable)support reasoners / inference engines (query answering,classification, logical ent., consistency check...)

Semantic Web languages:- Web compatibility- Existing W3C Recommendations: XML, RDF, OWL


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Layer Cake 2005

Figure: Tim Berners Lee, International Semantic Web Conference,2005


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Simplified Stack (from a logicians point of view)


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Open-World Assumption vs. Closed-WorldAssumption

Logic Programming

hasFather(peter,john)hasFather(john,michael)grandchild(x)←hasFather(x,y), hasFather(y,z),Person(x).

Every person is a grandchild is not deducible due to closeddomain of interpretation!Domain bounded to known objects: peter is a grandchild.No tree model property required


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Open-World Assumption vs. Closed-WorldAssumption

Description LogicPerson(peter ) Peter is a person.Person v ∃ father.Person Each person has a father

who is a person.∃ father.(∃ father.Person) v Grandchild Things having a father of

a father who is a person aregrandchildren.

Every person is a grandchild is deducible due to opendomain of interpretation!hasAunt(x,y) ← hasParent(x,z),hasSibling(z,y), Female(y)Arbitrary rule is not possible to express due to the need fortree model property


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Classical vs. Default Negation

Default Negation(i.e., negation as failure)

Rules:male(x) :- father(x).female(x) :- person(x), ¬male(x).grandchild(x)←hasFather(x,y), hasFather(y,z),Person(x).

Facts:person(joe), person(bill), father(joe)

Value Computation:male(joe)female(bill)


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Classical vs. Default Negation

Classical Negation

Rules:male(x) :- father(x).female(x) :- person(x), ¬male(x).grandchild(x)←hasFather(x,y), hasFather(y,z),Person(x).

Facts:person(joe), person(bill), father(joe)

Value Computation:male(joe)


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Basic Model of RDF (S)

RDF triples:

A model is a set of statementsStatement := (predicate,subject,object)Predicate is a resourceSubject is a resourceObject is either a resource or a literal


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RDF(S) - Core Classes and Properties

RDFS CLASS represents the generic concept of a type or category andcan be defined to represent almost everything, e.g. Web pages, people,document types...

RDFS PROPERTY represents the subset of RDFS resources that areproperties

RDFS SubClassOf defines a subset or superset relation betweenclasses. This property is transitive!

RDFS SubPropertyOf is used to indicate that one property is aspecialization of another property

RDFS RANGE is used to define that the values of a property areinstances of one or more stated classes

RDFS DOMAIN is used to state that any resource that has a givenproperty is an instance of one or more classes


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RDF(S) - Schema Example


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RDF(S) - Summary

Simple Triple Data ModelBasic Ontology Modeling PrimitivesURIs for IdentificationRDFS too weak to describe resources in sufficient detail

No localised range and domain constraintsNo existence/cardinality constraintsNo transitive, inverse or symmetrical properties

Difficult to provide reasoning support (No "native"reasoners for non-standard semantics)


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OWL - DL based ontology language

DLs are a family of logic based KR formalismsDLs - languages mainly characterised by:

Set of constructors for building complex concepts and rolesfrom simpler onesSet of axioms for asserting facts about concepts, roles andindividuals

DL is a monotonic LogicThis is due to the open world assumption.


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OWL - Language

Three species of OWL[3]OWL full is union of OWL syntax and RDFOWL DL restricted to DL fragment (1/4 DAML+OIL)OWL Lite is "simpler" subset of OWL DL

Semantic layeringOWL DL based on OWL full within DL fragment

OWL DL based on SHOIN (D) Description LogicOWL DL benefits from many years of DL research

Well defined semanticsFormal properties well understood (complexity, decidability)Known reasoning algorithmsImplemented systems (highly optimized)


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DL Concept and Role Constructors

Range of other constructors found in DLs, including:

Number restrictions (cardinality constraints) on roles, e.g.,>3 hasChildQualified number restrictions, e.g., >2 hasChild.FemaleNominals (singleton concepts), e.g., KoreaConcrete domains (datatypes), e.g., hasAge.(21)Inverse roles, e.g., hasChildp (hasParent)Transitive roles, e.g., locatedInRole composition, e.g., hasParent o hasBrother (uncle)


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OWL Class Constructors


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OWL as DL: Axioms

Axioms (mostly) reducible to inclusion (v):C ≡ D iff both C v D and D v C


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OWL - Inference Tasks

Knowledge is correct (captures intuitions)Does C subsume D w.r.t. ontology O? (in every model Iof O, CI ⊆ DI)

Knowledge is minimally redundant (no unintendedsynonyms)

Is C equivalent to D w.r.t. O? (in every model I of O, CI= DI)

Knowledge is meaningful (classes can have instances)Is C satisfiable w.r.t. O? (there exists some model I of O,s.t. CI 6= ∅)


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OWL - Inference Tasks (2)

Querying knowledgeIs x an instance of C w.r.t. O? (in every model I of O, xI∈ CI)Is an instance of R w.r.t. O? (in every model I of O,(xI , yI) ∈ RI)

Knowledge base consistencyA KB K is consistent iff there exists some model I of K

All inference tasks reducible to KB satisfiability or conceptsatisfiability w.r.t. a KB


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OWL DL Architecture


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First Order Logic - Basic Facts

In opposition to DLs and HLs it provides significant moreflexibility in writing down required axioms.This language is not easy to support in terms of scalablereasoning serviceMany interesting theorems can be proven and the theoremprover community has made significant progress over thelast years.First order language could define the common umbrella,where DLs and HLs are sublanguages and unified.


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Web Service Modeling Language (WSML)

1 Aims to provide a language (or a set of interoperablelanguages) for representing the elements of WSMO:Ontologies, Web services, Goals, Mediators

2 WSML provides a formal language for the conceptualelements of WSMO, based on:

Description LogicsLogic ProgrammingFirst-Order LogicF-Logic


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Rationale of WSML

1 Provide a Web Service Modeling Language based on theWSMO conceptual model

Concrete syntaxSemantics

2 Provide a Rule Language for the Semantic Web3 Many current Semantic Web languages have

undesirable computational propertiesunintuitive conceptual modeling featuresinappropriate language layering


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Variants of WSML

Figure: Variants of WSML


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Syntaxes for WSML

Human-readable syntax:- Modular syntax

WSMO-syntax functions as "umbrella"Modules for different WSML variants with clear layering

- Syntax:Inspired by OIL/OWL and F-LogicConceptual syntaxLogical Expression Syntax

- Semantics is fixed in WSML variantsXML syntax- provides syntax transport layerRDF syntax- provides basic relational language and simple ontologicalprimitives


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Syntaxes for WSML


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WSML- Example


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WSML Logical Expressions

Frame and first order based concrete syntaxElements:

Function symbols (e.g. f())Molecules (e.g. Human subClassOf Animal, JohnmemberOf Human, John[name hasValue "John Smith"]).Predicates (e.g. distance(?x,?y,?z))Logical connectives (or, and, not, implies, equivalent,impliedBy, forall, exists)

Example: axiom WomanDefinitiondefinedBy ?x memberOf Woman equivalent?x memberOf Female and ?x memberOf Human.


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WSML-Core

Intersection of Description Logics and Horn Logic:description logic programsFeatures:

classesattributesbinary relationsinstancesclasses and relation hierarchiesdatatype and datatype predicates

Compatible with OWLSome limitations in conceptual modeling of Ontologies:

no cardinality constraintsonly "inferring" range of attributesno meta-modeling


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Example WSML-Core


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Example WSML-Core (2)


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WSML-DL

Extension of WSML-Core capturing the Description Logic SHIQ(D)entailment is decidableclose to DL species of Web Ontology Language OWLmany efficient subsumption reasoners (FaCT++, Pellet,Racer)

Some limitations in conceptual modeling of Ontologies:no cardinality constraintsonly "inferring" range of attributesno meta-modeling

Limitations in logical expressions: -From Logic Programmingpoint-of-view, there is a lack of:

n-ary predicateschaining variables over predicates(default) negation


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WSML-Flight

Extension of WSML-Core, equivalent to Datalog withinequality and (stratified) default negationFeatures:

powerful rule-based language (n-ary predicates allowed)meta-modelingconstraintsnonmonotonic negation (conclusions can be retracted atlater points of time)

Query answering supported by WSML2Reasoner andunderlying Datalog reasoner


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Example WSML-Flight


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WSML-Rule

Extension of WSML-Flight in the direction of LogicProgrammingFeatures:

function symbolsunsafe rules

Query answering is undecidable


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WSML-Full

WSML-Full unifies WSML-DL and WSML-Rule under aFirst-Order umbrellaVery expressiveQuery answering is undecidableSpecification of WSML-Full is now in progress


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IntroductionEvaluation TechniquesThe Magic-SetsThe Bottom-Up Evaluation in IRIS

Mission

WSML GoalEfficient and extensible reasoning engine for expressiverule-based languages: WSML Core/Flight/Rule.

Research GoalFramework consisting of a collection of components whichcover various aspects of reasoning with formally representedknowledge. Development of effective optimization algorithms aswell as memory and storage management strategies forreasoning with large data sets.


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Research Methodology

WSML Core/Flight Reasoner

Datalog extended with locally stratified negation.

1 Full Datalog support2 Support for (stratifed) default negation3 Built-in predicates4 Integrity constraints (for checking datatypes)


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Architecture

Figure: Evaluation strategies


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Integration of IRIS with WSML


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Query-Sub-Query Recursive Framework

1 Top-down, direct evaluation [1].2 Avoid the calculation of tuples that are not used for deriving

answer.3 Begin with constants in a query "pushing" them from goals

to subgoals.4 Use "sideways information passing" to pass constant

binding information from one atom to the next in subgoals.


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Adornments

Consider the rule:rsg(X,Y) :- up(X,X1), rsg(Y1,X1), down(Y1,Y).

Suppose that a subquery involving rsgbf is invoked (e.g.rsg(’a’,Y)):

rsgbf (X,Y) :- up(X,X1), rsgfb(Y1,X1), down(Y1,Y).rsgfb(X,Y) :- up(X,X1), rsgfb(Y1,X1), down(Y1,Y).


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Supplementary Relations

Goal: to store information during intermediate stages ofevaluation.

Consider the rule:rsgbf (X,Y) :- up(X,X1), rsgfb(Y1,X1), down(Y1,Y).

Supplementary relations, for head adorned by rsgbf , are:sup0[X]sup1[X,X1]sup2[X,Y1]sup3[X,Y]


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QSQR Evaluation - Examplersg(X,Y) :- flat(X,Y).rsg(X,Y) :- up(X,X1), rsg(Y1,X1), down(Y1,Y).?- rsg(’a’,Y).

Figure: ProgramDarko Anicic SESA Tutorial - Logics and Reasoning

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QSQR Evaluation - Example

Figure: QSQR evaluationDarko Anicic SESA Tutorial - Logics and Reasoning

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Magic Sets

Bottom-up technique whose efficiency rivals with top-downapproaches [1, 2].Given a query, the magic set rewriting method generates amodified program in order to take advantage of boundvariables from the query.A bottom-up evaluation procedure is used to evaluate thenew program (e.g., seminaive evaluation).The bottom-up evaluation produces only the set of factsproduced by the top-down approaches such as QSQ.


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Relational Algebra and Logical Rules

Relational algebra

Compute relations for logical rules and evaluate them usingrelational algebra operations [4].

Relational algebra expressions:

1 Take given relations as arguments and produce relationsas results;

2 Can be combined to form complex expressions;3 Can be efficiently evaluated using RDBMS.


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Computing the Relations

Input: Datalog rulesq(X,Y) :- p(X,b) & X=Y.q(X,Y) :- p(X,Z) & s(Z,Y).

Output: Relational algebra expressionsπX (σZ=b(P(X,Z)))σX=Y (πX (σZ=b(P(X,Z))) × πY (P(Y,W)))

Q(X,Y)=σX=Y (πX (σZ=b(P(X,Z))) × πY (P(Y,W)))∪ πX ,Y (P(X,Z) ./ S(Z,Y))


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Rectification

Non-rectified rules:q(’a’,X,Y) :- r(X,Y).q(X,Y,X) :- r(Y,X).

Making heads equal:q(U,V,W) :- r(X,Y) & U=a & V=X & W=Y.q(U,V,W) :- r(Y,X) & U=X & V=Y & W=X.

Rectified rules:q(U,V,W) :- r(V,W) & U=a.q(U,V,W) :- r(V,U) & W=U.


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Semi-Naive Algorithm

Computes the least fixed point of the equations to which it isapplied, with respect to the given EDB relations [5]:

for i := 1 to m doPi := ∅;

repeatfor i := 1 to m doQi := Pi ; //save old values

for i := 1 to m doPi := EVAL(pi , R1,...,Rk ,Q1,...,Qm);

until Pi = Qi for all i, 1≤i≤m;output Pi ’s

Figure: Naive evaluationDarko Anicic SESA Tutorial - Logics and Reasoning

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Algorithm

Computes the least fixed point based on incremental relationsfor the IDB predicates [5]:for i := 1 to m do

∆Pi := EVAL(pi , R1,...,∅,...,∅);Pi := ∆Pi ;

end;repeat

for i := 1 to m do∆Qi := ∆Pi ; //save old ∆P’s

for i := 1 to m do begin∆Pi := EVAL-INCR(pi ,R1,...,Rk ,P1,...,Pm,∆Q1,∆Qm);∆Pi := ∆Pi - Pi // Removes "new" tuples that actually appeared before

endfor i := 1 to m do

Pi := Pi ∪ ∆Pi ;until ∆Pi = ∅ for all i; output Pi ’s

Figure: Semi-Naive evaluationDarko Anicic SESA Tutorial - Logics and Reasoning

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Negation

Consider the rules:p(X) :- r(X) & ¬ q(X).q(X) :- r(X) & ¬ p(X).

For R={1} the least fixed point is not unique:S1: P=1, Q=∅S2: Q=1, P=∅

IRIS supports safe, stratified rules! Algorithms for checkingsafeness and stratification are implemented.


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Handling Negation

Consider the rule:trueCousin(X,Y) :- cousin(X,Y) & ¬ sibling(X,Y).

The rule may be evaluated as:C(X,Y) ./ S (X,Y)

DOM - union of the symbols appearing in the EDB relationsand in the rules themselves.

DOM × DOM - S(X,Y)

In general for ¬ s(X1,...,Xn):DOM × ... × DOM (n times) - S


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The IRIS Roadmap

1 Query optimization2 Performance and scalability3 Storage management4 Support for function symbols5 Wellfounded semantics implementation


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Combining Reasoning with Search

1 Goal: Provide an infrastructure that scales for realisticsemantic computing applications

2 What: A plugable & distributed infrastructure forWeb-reasoning and search

3 Project: LarCK4 Like: Search for Extraterrestrial Intelligence (SETI),

Folding@Home and Google techniques for large scaleparallelized computing


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IRIS Code Access

IRIS is an open source project developed under LGPL andavailable at:

http://sourceforge.net/projects/iris-reasoner


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Serge Abiteboul, Richard Hull, and Victor Vianu.Foundations of Databases.Addison-Wesley, 1995.

Y. Chen.Magic sets and stratified databases.Int. Journal of Intelligent Systems, 12(3):203–231, 1997.

Ian Horrocks and Alan Rector.The semantic web: Ontologies and owl.

Jeffrey D. Ullman.Principles of Database and Knowledge-Base Systems,Volume I.Computer Science Press, 1988.

Jeffrey D. Ullman.Darko Anicic SESA Tutorial - Logics and Reasoning

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Principles of Database and Knowledge-Base Systems,Volume II.Computer Science Press, 1989.


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Thank you!Questions, comments, suggestions please...


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Examples Using WSMT

1 Set up WSMT;2 Browsing ontologies using WSMT GUI3 Editing ontologies using WSML Text Editor4 Execute a query;


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Examples Using Java API

1 Set up required WSMO4J factories;2 Parse and load an ontology (written in WSML) in the

reasoner;3 Create a query;4 Execute a query;5 Print the loaded ontology and results after the query

execution.


IntroductionAgenda

Ontologies and ReasoningIntroduction

Ontology Languages & WSMLOntology LanguagesRDF (S)OWLLogic Programming / F-Logic (Horn)First Order LogicWSML

IRIS ReasonerIntroductionEvaluation TechniquesThe Magic-SetsThe Bottom-Up Evaluation in IRIS

Hands-on Session

logics and reasoning - sesa tutorial · reasoning is an act of deriving a conclusion from certain...

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