Modelling And Reasoning With Fuzzy Logic Redundant Knowledge Bases

Abstract

Fuzzy Predicate Logic with evaluated syntax together with resolution principle is presented. The paper focuses mainly on modelling and treating redudancy in knowledge bases. It presents original resolution rule together with algorithm DCF Detection of Consequent Formulas developed especially for fuzzy logic with evaluated syntax. REASONING IN FUZZY PREDICATE LOGIC WITH EVALUATED SYNTAX The problem of effective modelling and reasoning on knowledge bases rises especially when dealing with many-valued logics. We would like to recall previously devised notions and methods of fuzzy resolution principle and then show original efficient methods making such a reasoning tractable. Fuzzy Predicate Logic with Evaluated Syntax (FPL) [Novak, V., Perfilieva, I., Mockoř, J., 1999] is a well-studied and wide-used logic capable of expressing vagueness. It has a lot of applications based on robust theoretical background. It also requires an efficient formal proof theory. However the most widely applied resolution principle [Dukic, N., Avdagic, Z., 2005] brings syntactically several obstacles mainly arising from normal form transformation. There are also recent attempts of similarity based resolution [Mondal, B., Raha, S., 2012], but our approach is based on classical proof theory of FPL. FPL is associating with even harder problems when trying to use the resolution principle. Solutions to these obstacles based on the nonclausal resolution [Bachmair, L., Ganzinger, H., 1997] were already proposed in [Habiballa, H., 2006]. In this article it would be presented a natural integration of these two formal logical systems into fully functioning inference system with effective proof search strategies. It leads to the refutational resolution theorem prover for FPL (RRTPFPL). Another issue addressed in the paper concerns to the efficiency of presented inference strategies developed originally for the proving system. It is showed their perspectives in combination with standard proof-search strategies. The main problem for the fuzzy logic theorem proving lies in the large amount of possible proofs with different degrees and there is presented an algorithm (Detection of Consequent Formulas DCF) solving this problem. The algorithm is based on detection of such redundant formulas (proofs) with different degrees. The article presents the method which is the main point of the work on any automated prover. There is a lot of strategies which make proofs more efficient when we use refutational proving. We consider wellknown strategies orderings, filtration strategy, set of support etc. One of the most effective strategies is the elimination of consequent formulas. It means the check if a resolvent is not a logical consequence of a formula in set of axioms or a previous resolvent. If such a condition holds it is reasonable to not include the resolvent into the set of resolvents, because if the refutation can be deduced from it, then so it can be deduced from the original resolvent, which it implies of. Resolution and Fuzzy Predicate Logic The fuzzy predicate logic with evaluated syntax is a flexible and fully complete formalism, which will be used for the below presented extension [Novak, V., Perfilieva, I., Mockoř, J., 1999]. In order to use an efficient form of the resolution principle we have to extend the standard notion of a proof (provability value and degree) with the notion of refutational proof (refutation degree). Propositonal version of the fuzzy resolution principle has been already presented in [Habiballa, H., 2002]. We suppose that set of truth values is Lukasiewicz algebra. Therefore we assume standard notions of conjunction, disjunction etc. to be bound with Lukasiewicz operators. We will assume Lukasewicz algebra to be L L = 〈[0, 1],∧,∨,⊗,→, 0, 1〉 Proceedings 27th European Conference on Modelling and Simulation ©ECMS Webjorn Rekdalsbakken, Robin T. Bye, Houxiang Zhang (Editors) ISBN: 978-0-9564944-6-7 / ISBN: 978-0-9564944-7-4 (CD) where [0, 1] is the interval of reals between 0 and 1, which are the smallest and greatest elements respectively. Basic and additional operations are defined as