Complexity of Abduction in \L{}ukasiewicz Logic

TL;DR

This paper investigates the computational complexity of abduction in ukiewicz fuzzy logic, revealing that certain restricted fragments are computationally easier than the full language, unlike classical propositional logic.

Contribution

It formalizes abduction in ukiewicz logic with interval literals and analyzes the complexity of reasoning tasks, highlighting differences from classical logic.

Findings

Abduction complexity varies between full language and disjunctive fragments.

In ukiewicz logic, the clausal fragment has lower complexity than the full language.

Complexity results differ from classical propositional logic.

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 {\L}ukasiewicz fuzzy logic. We define and motivate the notions of abduction problems and explanations in the language of {\L}ukasiewicz 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 {\L}ukasiewicz 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.