Explainability via Short Formulas: the Case of Propositional Logic with Implementation

TL;DR

This paper explores explainability in propositional logic through minimal formulas that clarify a formula's truth value across models, demonstrating computational complexity and implementing solutions via answer set programming.

Contribution

It formalizes explainability using logic and formula size, analyzes the complexity of the special explanation problem, and provides an ASP-based implementation for practical problems.

Findings

The special explanation problem is complete for the second level of the polynomial hierarchy.

An implementation in answer set programming effectively explains solutions to combinatorial problems.

The approach links explainability with computational complexity and practical reasoning tasks.

Abstract

We conceptualize explainability in terms of logic and formula size, giving a number of related definitions of explainability in a very general setting. Our main interest is the so-called special explanation problem which aims to explain the truth value of an input formula in an input model. The explanation is a formula of minimal size that (1) agrees with the input formula on the input model and (2) transmits the involved truth value to the input formula globally, i.e., on every model. As an important example case, we study propositional logic in this setting and show that the special explainability problem is complete for the second level of the polynomial hierarchy. We also provide an implementation of this problem in answer set programming and investigate its capacity in relation to explaining answers to the n-queens and dominating set problems.