Causal Graph Justifications of Logic Programs

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

💡 Research Summary

The paper introduces a novel multi‑valued extension of logic programs under the stable‑model semantics, where every true atom in a model is equipped with a set of justifications represented as causal graphs. A causal graph (c‑graph) is a reflexive and transitive directed graph whose vertices are rule labels; edges encode a partial ordering of rule applications in a derivation. The subgraph relation is inverted to define a “sufficient” ordering (G ≤ G′ iff G ⊇ G′), meaning that a larger graph contains at least as much causal information as a smaller one.

Two basic algebraic operations are defined on graphs. The product ‘∗’ corresponds to the union of two graphs followed by transitive closure, modelling joint causation. The concatenation ‘·’ connects every vertex of the first graph to every vertex of the second, then closes transitively, modelling sequential rule application. Concatenation is non‑commutative, while product is commutative; moreover, G·G′ ≤ G∗G′ (concatenation is sufficient for the product).

To capture alternative independent causes, the authors lift individual graphs to ideals of the partially ordered set (C_Lb, ≤). An ideal is a downward‑closed set; it can be compactly represented by its maximal elements S, written ↓S. A causal value is any such ideal, and the collection of all causal values V_Lb forms a free, completely distributive lattice. The lattice operations are: meet (∗) as set intersection, join (+) as set union, and concatenation (·) as the ideal generated by all pairwise concatenations of elements from the two operands. The lattice has bottom 0 (the empty ideal, representing falsity) and top 1 (the ideal of the empty graph, representing absolute truth).

Causal terms provide a syntactic shorthand for causal values: a term is built from labels using ∗, +, and ·, with ∗ binding tighter than +. The semantics of a term is obtained by recursively applying the algebraic operations down to labels, each label denoting its principal ideal. The paper lists key algebraic properties of these operators: associativity, absorption, identity, annihilator, idempotence, and distributivity, both for ∗ and ·, and shows how they interact with +.

For positive (negation‑free) programs, the authors prove that the causal value assigned to an atom coincides with the set of all syntactic proofs (derivations) of that atom using the program’s rules. Thus the semantics does not merely indicate existence of a proof but actually records its structure as a causal graph.

When default negation is present, the authors adapt the Gelfond‑Lifschitz reduct to the causal setting, defining a causal stable model. The reduct removes rules whose bodies contain a negative literal that is true in the candidate model, and strips away all negative literals from the remaining rules. Consequently, a default literal “not p” is interpreted as “there is no proof of p”; no causal graph is generated from a negative literal, which aligns with the intuition that falsity corresponds to absence of justification. This yields a semantics where negative information contributes no causal structure, enhancing elaboration tolerance.

The paper illustrates the approach with a running example: a program encoding two laws (driving drunk and resisting arrest) that both lead to punishment and ultimately imprisonment. Two distinct causal graphs justify the atom prison, reflecting the two alternative legal routes. The example demonstrates how product and concatenation combine rule labels to form complex justifications, and how the lattice operations automatically discard redundant causes (e.g., a graph containing an unnecessary intermediate rule).

In the discussion, the authors compare their framework with earlier causal graph work, emphasizing that they embed causality directly into the logical semantics rather than treating it as an external annotation. They argue that the algebraic treatment facilitates extensions to richer causal notions such as sufficient or necessary causes, and that the approach is well‑suited for applications in diagnosis, legal reasoning, and any domain where tracing the provenance of conclusions is essential.

Overall, the contribution is a mathematically rigorous, algebraically grounded semantics that augments answer‑set programming with explicit, manipulable causal justifications, bridging the gap between truth‑valuation and proof‑traceability in logic programming.