DEDUCTION PRINCIPLES DEDUCTION PRINCIPLES AND STRATEGIES AND STRATEGIES

FOR SEMANTIC WEBFOR SEMANTIC WEB

Chain resolution and its fuzzyfication

Dr. Hashim Habiballa Dr. Hashim Habiballa University of OstravaUniversity of Ostrava

IntroductionIntroduction

Semantic web - logical foundations Description logic First-order logic (FOL) – undecidability,

effective Automated Theorem Proving (ATP)

Fuzzyfication in the frame of Fuzzy FOL vs. Fuzzy DL

Logical foundations Logical foundations for Semantic Webfor Semantic Web

Specialized knowledge base of DL (FOL) Specialized inference rules and strategies for DL (FOL)

Is native logical framework of SW the only way? (syntactic methods are also effective w.r.t. searching)

Logic Language transformation Requirement of a good inference engine

remains

Description LogicDescription Logic

Proved methods and properties of FOL– resolution, tableaux– decidable classes

Relatively narrowed quantifier usage (consider FOL vs. PROLOG)

Fuzzyfication in the frame of Fuzzy FOL Known resolution strategies for FOL may

be used in DL Furthermore exist high-speed techniques

for ontologies (e.g. chain resolution)

First-order logicFirst-order logic

Automated Theorem proving – well studied branch.

http://www.cs.miami.edu/~tptp/ (theorem proving web site)– high-speed theorem provers based on

various techniques– Thousands of Problems for Theorem

Provers– CADE ATP System Competition.

First-order logicFirst-order logicresolution principleresolution principle

http://rpc25.cs.man.ac.uk/manchester/handbook-ar/ (Handbook of Automated Reasoning)

http://www.mpi-sb.mpg.de/~hg/

(Resolution Theorem Proving) Resolution strategies– SOS (set of support)– Filtration s.– Orderings

FuzzyficationFuzzyfication

http://ac030.osu.cz/irafm/ps/rep47.ps

(Fuzzy general resolution) Fuzzy Description Logic Special strategies for Fuzzy FOL (Fuzzy

DL)

Research framework for IRAFM– Fuzzy Logics for SW– Resolution principles and strategies– Implementation

Chain resolution Chain resolution motivationmotivation

Tammet, T.: Extending Classical Theorem Proving for the Semantic Web

http://km.aifb.uni-karlsruhe.de/ws/psss03/proceedings/tammet.pdf

Chain resolution – encapsulation of simple implications (chain clauses - CC)

A B, B C, … Key problem of ATP = combinatorial

explosion (CE) during inference process Chain clauses (even simple) cause CE Ontology is full of chain clauses

e.g. person(X) mammal(X), mammal(X) animal(X), …

Chain resolution Chain resolution motivationmotivation

Chain clauses produce potentially enormous number of propositional variationse.g. person(X) animal(X), animal(X) person(X), …

Solution lies in encapsulation of variations into boolean matrix

variations are forbidden in a set of resolvents

inference algoritmhs modifications

Significant restriction of CE is obtained

Chain resolution Chain resolution background, explanationbackground, explanation

Chain clause: A(X1, …, Xn) B(X1, …, Xn),

A, B - signed predicate symbols (reduced to unary predicate symbols in this presentation)

Xi - variables

Propositional variation C’ of C:C’ is derivable by binary res. from C and set of chain clauses

Chain clauses are excluded from set of resolvents and are stored in Chain Box

Chain Box: Data structure containing for every key (signed pred. symbol) its chain of pred. symbols derivable by chain clauses from key

Chain resolution Chain resolution exampleexample

Assume following knowledge and chain box rows:

person(X)mammal(X),mammal(X)animal(X),horse(X) mammal(X)

Key: person chain: {person, mammal, animal}Key: mammal chain:{mammal, animal}Key: mammal chain:{mammal,person, horse}

The chain box could be implemented as a bit matrix of the size 4*number_of_predicates2

Chain resolution Chain resolution background, explanationbackground, explanation

Chain box stores information for A B: Both of the type A B and B A

(A B A B B A B A)

The chain box could be implemented as a bit matrix of the size 4*(number_of_predicates)2

Of course ontology may contain also complex formulas (not only chain clauses)

Chain resolution Chain resolution exampleexample

Assume same knowledge as previous:person(X)mammal(X),mammal(X)animal(X),horse(X) mammal(X)(where person(X) = P, mammal(X) = M, horse(X) = H, animal(X) = A)

MatrixP P P P M M M M H H H H A A A A

P 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0

M 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0

H 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0

A 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1

Chain resolution Chain resolution motivation and algorithmmotivation and algorithm

• During proof search it is obvious: New chain clauses are produced Some clauses are typically present Chain clauses produce high amount propositional

variations

• The algorithm of chain resolution consists of: Moving chain clauses into chain box Ordinary resolution, factorisation, subsumption using

chain box

Chain resolution Chain resolution building the chain boxbuilding the chain box

• Moving chain clauses: Initialization - key P and P contain itself Removing CC from search space and adding to chain

box (recursive function); if unit clause produced, then added into search space

Every time the chain clauses produced, it is added by the same rule as above

Unit clause p(x) is produced if key(p) consists of r, r(pr and pr means p is derivable)

Chain resolution Chain resolution using the chain boxusing the chain box

• Resolution with chain box: A(t1, …, tn), B(u1, …, un) are resolvable literals,

iff A(t1, …, tn) and A(u1, …, un) are unifiable using standard unification and B = A or B chain(A) (note that chain box is constructed as follows - AB BA)

Chain resolution Chain resolution using the chain boxusing the chain box

• Factorisation with chain box: A(t1, …, tn), B(u1, …, un) are literals in two clauses,

A(t1, …, tn) and A(u1, …, un) are unifiable using standard unification then the resulting literal should be:

1. A(t1, …, tn) if A = B2. A(t1, …, tn) if A chain(B) 3. B(t1, …, tn) if B chain(A)(note that if cond. 2. and 3. hold simultaneously then

resulting literal should be like 2. or 3. without any preference)

Chain resolution Chain resolution using the chain boxusing the chain box

• Subsumption (of literals!) with chain box: A(t1, …, tn) subsumes B(u1, …, un),

iff A(t1, …, tn) subsumes B(u1, …, un) using standard subsumption and A = B or B chain(A)

Chain resolution procedures significantly reduce proof search for FOL

Using it for DL, where ontologies contain typically large amount of simple implications (CC), it brings high-efficient technique for SW

• Chain resolution is sound and complete

Chain resolution Chain resolution strategiesstrategies

• Set of Support (SOS): Sets R (knowledge base), Q (query), Q’(new

clauses) Allows resolution only when at least one premise is

from Q or Q’ (derivations from R alone are prohibited)

In standard resolution it is complete strategy• Naive combination with chain resolution: Resolution is restricted by SOS, chain clauses are

moved from R,Q,Q’, it is allowed to use any clause from chain box

Naive combination is not complete

Chain resolution Chain resolution strategiesstrategies

• Weak combination with SOS: Resolution is restricted by SOS Chain clauses are moved to chain box only from R It is always allowed to use clause from chain box R is not allowed chain subsumption with clause

from Q or Q’ Weak combination is complete

• Ordering strategies: Orderings form modern approach in ATP Term based orderings preserve completeness in

combination with chain resolution

Chain resolution Chain resolution implementationimplementation

• Chain resolution is implemented for FOL – Gandalf TP: http://deepthought.ttu.ee/it/gandalf/

• Scheme of ATP: Compilation(analysis – serching for suitable strategy, terminating strategy, first filtering, chain box, final filtering, query – in case of repeated queries)

Perspectives for Perspectives for IRAFMIRAFM

• Fuzzyfication of DL – implementation• Research on inference strategies (theory,

implementation and testing) Chain resolution for Fuzzy FOL Other strategies for Fuzzy FOL and DL Effective inference – fuzzy selection of premises,

evolutionary search for optimal selection of premises

Syntactical means and combination with SW (formal languages, linguistic expressions of fuzzy logic)?