Explainability via Short Formulas: the Case of Propositional Logic with Implementation
This addresses the need for formal, logic-based explanations in AI and computer science, though it is incremental as it builds on existing logical frameworks.
The paper tackles the problem of explainability in logic by defining it via minimal-size formulas that agree with an input formula on a given model and transmit truth values globally, showing that for propositional logic, this special explainability problem is complete for the second level of the polynomial hierarchy. It includes an implementation in answer set programming and tests it on n-queens and dominating set problems.
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.