Complexity of Abduction in Łukasiewicz Logic
Katsumi Inoue, Daniil Kozhemiachenko
TL;DR
This work studies abductive reasoning in infinitely-valued Łukasiewicz logic, introducing interval literals $p\ge\mathbf{c}$ and $p\le\mathbf{c}$ as tools to express ranges of truth for propositions. It develops interval terms as the solution formalism and provides a near-complete complexity classification for core tasks—solution recognition, solution existence, and relevance/necessity of hypotheses—in both the general language and clausal fragments. A key finding is that abduction in the clausal fragment can be computationally easier than in the full language, contrasting with classical propositional logic. The results advance the understanding of fuzzy abduction, enabling finite, interval-based explanations and informing algorithmic approaches for reasoning with graded truths in knowledge bases and diagnostics.
Abstract
We explore the problem of explaining observations in contexts involving statements with truth degrees such as `the lift is loaded', `the symptoms are severe', etc. To formalise these contexts, we consider infinitely-valued Łukasiewicz fuzzy logic. We define and motivate the notions of abduction problems and explanations in the language of Łukasiewicz logic expanded with `interval literals' of the form $p\geq\mathbf{c}$, $p\leq\mathbf{c}$, 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 Łukasiewicz logic 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.