Causal Graph Justifications of Logic Programs

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 correspondence 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 operations (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 literals, something that turns out convenient for elaboration tolerance, as we explain with a running example.

Enablers and inhibitors in causal justifications of logic programs

“…Definition 10 (CG Values in Cabalar et al 2014a) Given a set of labels Lb, a CG causal value is any ideal (or lower-set) for the poset G Lb , ≤ . By I CG Lb , we denote the set of CG causal values.…”

“…Theorem 11 (Theorem 2 from Cabalar et al 2014a) Let P be a (possibly infinite) positive logic program with n causal rules. Then, (i) lfp(T P ) is the least model of P, and (ii) lfp(T P ) =T P ↑ ω (0) =T P ↑ n (0).…”

“…Rather than manipulating justifications as mere syntactic objects, two recent approaches have considered extended multi-valued semantics for LP where justifications are treated as algebraic constructions: Why-not Provenance (WnP) (Damásio et al 2013) and Causal Graphs (CG) (Cabalar et al 2014a). Although these two approaches present formal similarities, they start from different understandings of the idea of justification.…”

Causal Logic Programming

“…In fact, is just a preorder, i.e., it satisfies reflexivity and transitivity, but not antisymmetry. As a counterexample with proof trees: t = a · {b, b · {c}} and t = a · {b}, where both t t and t t but t = t. The reader is referred to [18] for a corrected and substantially improved approach. The correction implies representing both proof trees and causes as a graphs of labels, and defining C 1 C 2 as C 1 ⊆ C * 2 where C * 2 is the reflexive and transitive closure of C 2 and ⊆ is the standard subgraph relation.…”

“…The correction implies representing both proof trees and causes as a graphs of labels, and defining C 1 C 2 as C 1 ⊆ C * 2 where C * 2 is the reflexive and transitive closure of C 2 and ⊆ is the standard subgraph relation. The rest of results in the current paper are subsumed by the new version [18]. Sets of causes S ∈ 2 C Lb will be represented with capital boldface letters.…”