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Logical foundations for 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
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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 0M 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0H 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0A 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)?
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