Abduction in Łukasiewicz Logic With Rational Interval Terms

Abstract

We explore abductive reasoning in contexts involving fuzzy statements (statements with truth degrees) such as ‘the lift is heavily loaded’ (meaning, e.g., that the lift is carrying more than 70% of its maximal load), ‘the symptoms are severe’, ‘it is cold outside’, etc. Such statements are both used to express observations and to explain observed events. That is, both the observed event and our explanation of it may have truth degrees.

To formalise these contexts, we use infinitely-valued Łukasiewicz fuzzy logic Ł with semantics defined over the real-valued interval [0,1]. Here, 0 and 1 are interpreted as ‘absolutely false’ and ‘absolutely true’, respectively, and the remaining values as degrees of truth. We consider two standard entailment relations for Ł: the ‘truth-preserving’ one (if the premise is absolutely true, then the conclusion is absolutely true) and the ‘degree-preserving’ (the truth degree of the premise should be at most as high as the truth degree of the conclusion).

For each of these entailment relations, we define and motivate the notions of abduction problems and explanations in the language of Ł expanded with ‘interval literals’ of the form p≥c, p≤c, and their negations that express the set of values a~variable can have. We analyse the complexity of standard abductive reasoning tasks (solution recognition, solution existence, and relevance / necessity of hypotheses) in Ł for the case of the full language and for the case of theories containing only disjunctive clauses and show that, in contrast to classical propositional logic, the abduction in the clausal fragment has lower complexity than in the general case.