A Complete Calcultis for Possibilistic Logic Programming with Fuzzy Propositional Variables

In this paper we present a propositional logic programming language for reasoning under possibilistic uncertainty and represent­ ing vague knowledge. Formulas are repre­ sented by pairs (ip, a), where ip is a many­ valued proposition and a E [0, 1] is a lower bound on the belief on ip in terms of necessity measures. Belief states are modeled by pos­ sibility distributions on the set of all many­ valued interpretations. In this framework, (i) we define a syntax and a semantics of the gen­ eral underlying uncertainty logic; (ii) we pro­ vide a modus ponens-style calculus for a sub­ language of Horn-rules and we prove that it is complete for determining the maximum de­ gree of possibilistic belief with which a fuzzy propositional variable can be entailed from a set of formulas; and finally, (iii) we show how the computation of a partial matching be­ tween fuzzy propositional variables, in terms of necessity measures for fuzzy sets, can be included in our logic programming system.