DEDUCTION PRINCIPLES AND STRATEGIES FOR SEMANTIC WEB Chain resolution and its fuzzyfication Dr. Hashim Habiballa University of Ostrava

Introduction 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 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

Description 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 logic Automated Theorem proving – well studied branch. (theorem proving web site) –high-speed theorem provers based on various techniques –Thousands of Problems for Theorem Provers –CADE ATP System Competition.

First-order logic resolution principle (Handbook of Automated Reasoning) (Resolution Theorem Proving) Resolution strategies –SOS (set of support) –Filtration s. –Orderings

Fuzzyfication (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 motivation Tammet, T.: Extending Classical Theorem Proving for the Semantic Web 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 motivation Chain clauses produce potentially enormous number of propositional variations e.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 background, explanation Chain clause: A(X 1, …, X n )  B(X 1, …, X n ), A, B - signed predicate symbols (reduced to unary predicate symbols in this presentation) X i - 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 example 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_predicates 2

Chain resolution background, 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 example 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) Matrix PP PP PP MM MM MM HH HH HH AA AA AA  P M H A

Chain resolution motivation 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 building 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 (p  r and p  r means p is derivable)

Chain resolution using the chain box Resolution with chain box: ­A(t 1, …, t n ), B(u 1, …, u n ) are resolvable literals, iff A(t 1, …, t n ) and A(u 1, …, u n ) are unifiable using standard unification and B =  A or B  chain(  A) (note that chain box is constructed as follows -  A  B   B  A)

Chain resolution using the chain box Factorisation with chain box: ­A(t 1, …, t n ), B(u 1, …, u n ) are literals in two clauses, A(t 1, …, t n ) and A(u 1, …, u n ) are unifiable using standard unification then the resulting literal should be: 1.A(t 1, …, t n ) if A = B 2.A(t 1, …, t n ) if A  chain(B) 3.B(t 1, …, t n ) 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 using the chain box Subsumption (of literals!) with chain box: ­A(t 1, …, t n ) subsumes B(u 1, …, u n ), iff A(t 1, …, t n ) subsumes B(u 1, …, u n ) 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 strategies 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 strategies 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 implementation Chain resolution is implemented for FOL – Gandalf TP: 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 IRAFM 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)?