Causal Graph Justifications of Logic Programs

TL;DR

This paper introduces a semantic framework for logic programs that associates each true atom with causal justifications using graphs, enabling a deeper understanding of derivations and negations.

Contribution

It presents a novel multi-valued semantics for logic programs that captures causal justifications algebraically, extending to programs with negation.

Findings

Causal justifications correspond to syntactic proofs in positive programs.

Causal information is derived through algebraic operations on causal values.

The framework extends to programs with negation, defining causal stable models.

Abstract

In this work we propose a multi-valued extension of logic programs under the stable models semantics where each true atom in a model is associated with a set of justifications. These justifications are expressed in terms of causal graphs formed by rule labels and edges that represent their application ordering. For positive programs, we show that the causal justifications obtained for a given atom have a direct correspon- dence to (relevant) syntactic proofs of that atom using the program rules involved in the graphs. The most interesting contribution is that this causal information is obtained in a purely semantic way, by algebraic op- erations (product, sum and application) on a lattice of causal values whose ordering relation expresses when a justification is stronger than another. Finally, for programs with negation, we define the concept of causal stable model by introducing an analogous transformation to Gelfond and Lifschitz's program reduct. As a result, default negation behaves as "absence of proof" and no justification is derived from negative liter