Exploiting Game Theory for Analysing Justifications

Abstract

Justification theory is a unifying semantic framework. While it has its roots in non-monotonic logics, it can be applied to various areas in computer science, especially in explainable reasoning; its most central concept is a justification: an explanation why a property holds (or does not hold) in a model.In this paper, we continue the study of justification theory by means of three major contributions. The first is studying the relation between justification theory and game theory. We show that justification frameworks can be seen as a special type of games. The established connection provides the theoretical foundations for our next two contributions. The second contribution is studying under which condition two different dialects of justification theory (graphs as explanations vs trees as explanations) coincide. The third contribution is establishing a precise criterion of when a semantics induced by justification theory yields consistent results. In the past proving that such semantics were consistent took cumbersome and elaborate proofs.We show that these criteria are indeed satisfied for all common semantics of logic programming.

On Nested Justification Systems

“…In some works, justifications are formalized as graphs and the justifications as defined here are then called tree-like justifications (Marynissen et al 2020). Since we restrict attention to the latter, we shall just use the term justification.…”

“…Another question that shows up in several papers on justification theory (Denecker 1993;Marynissen et al 2018;Marynissen et al 2020) is the consistency question: is it always so that SV(∼x, I) = ∼ SV(x, I)? The same question is relevant for nested systems.…”

“…We use the formalization of justification theory of Marynissen et al . (2020). We first give and explain all necessary definitions, and afterwards illustrate them with an example.…”

“…2018; Marynissen et al . 2020) is the consistency question: is it always so that ? The same question is relevant for nested systems.…”

Tree-Like Justification Systems are Consistent

“…A notable difference between the original work of Denecker [2] and the theory of Denecker et al [3] is that in the former work, a justification is a tree where the nodes are labeled with literals, while in the latter a justification is a (directed) graph. The relationship between these two formalisms was studied among others by Marynissen et al [11]. For clarity, we will refer to tree-like and graph-like justifications when the distinction is important.…”