1. DEDUCTION PRINCIPLES AND STRATEGIES FOR SEMANTIC WEB Resolution principle, Description Logic and Fuzzy Description Logic Hashim Habiballa University of Ostrava, 30. dubna 22, Ostrava, Czech Republic Institute for Research and Applications of Fuzzy Modeling

2. 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

3. 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)

4. First-order logic Automated Theorem proving – well studied branch. http://www.cs.miami.edu/~tptp/ (theorem proving web site) http://www.cs.miami.edu/~tptp/ –high-speed theorem provers based on various techniques –Thousands of Problems for Theorem Provers –CADE ATP System Competition.

5. First-order logic resolution principle http://rpc25.cs.man.ac.uk/manchester/handbook-ar/ (Handbook of Automated Reasoning) http://rpc25.cs.man.ac.uk/manchester/handbook-ar/ http://www.mpi-sb.mpg.de/~hg/ (Resolution Theorem Proving) Resolution strategies –SOS (set of support) –Filtration s. –Orderings

6. Fuzzyfication 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

7. General resolution Simple resolution principle for general formulas of FOL (non-clausal resolution) F[G] F’[G] ____________________ F[G/false]  F’[G/true] F,F’,G formulas of FOL Selection of positive and negative premise -> more complex algorithm (vs. CNF) Polarity criteria (example from article…)

8. Non-propositional general resolution Requires good unification algorithm Notion of globally universal (existential) variable Simulate skolemization - Var. Unification Restriction (no transformations!) MGU for General Resolution When existentiality occurs equivalence is forbidden - variable can’t be both globally universal and existential Rewrite equivalence to two implications

9. Resolution strategies Proof sequence of resolution theorem prover (CNF, Non-clausal) quickly flood by resolvents -> combinatorial explosion Solution -> various techniques and strategies (orderings, filtration, SOS, chain res., etc.) Detection of consequent formulas (DCF) - diploma thesis, empirically proved DCF utilizes general self-resolution -> very simple (almost trivial) implementation

10. Implementation GERDS (GEneralized Resolution Deductive System) - non-clausal resolution, strategies High-level data structures - trees,simultaneous creation of polarities, quantifiers, linkage of variable occurrences to one quantifier all within parsing -> good time complexity of compilation Inference - Both resolvents and DCF performed by one procedure, no-need to create new tree - virtual tree -> space complexity Open to extensions…

11. Prospective applications Fuzzy FOL -> Fuzzy General Resolution (research report) Fuzzy propositional calculus (graded formal proof) - example from article… Problems -> Refutation, Non-propositional case (Fuzzy MGU?) Integration into GERDS -> seems to be simple, graded proof in [ 0, 1] (generalized truth value)

12. Prospective applications Description Logic = modified narrowed notation based on FOL Concepts (C) and their hierarchy (male->person, parent->person, male  parent -> father Roles (R) - relation between concepts (is_child_of(X,Y)) Quantifiers significantly restricted -  R.C,  R.C means  Y(R(X,Y)  C(Y)),  Y(R(X,Y)  C(Y)) (example in article…) Connectives depend on type of DL (ALC…)

13. Prospective applications Deductive systems for DL - tableaux, DL is decidable Non-clausal resolution could be used for DL -> advantages: no destruction of contextual information (vs. CNF), resolution strategies and its combination See example from article... GERDS may be extended (conversion to FOL or new system for DL based on non- clausal resolution)

14. Prospective applications Fuzzy Description Logic Prof. Hajek’s research report-”Making fuzzy description logic more general” extension based on t-norms (old notion based on poor apparatus - min,max, KD- implication) Algorithm for “witnessing” the model (translation to propositional logic) Results - decidability,validity, satisfiability, time and space complexity(bin. pred.)

15. Prospective applications Fuzzy Description Logic -> implementation within the frame of resolution theory (full inference engine based on GERDS) Integration of Fuzzy General Resolution and Description Logic resolution principle together New resolution strategies for Fuzzy DL (maybe evolutionary search? Resolution = Evolution?