An Algebra of Causal Chains
This work addresses a theoretical problem in logic programming for researchers, but it appears incremental as it builds on existing stable models semantics.
The paper tackles the problem of representing justifications for true atoms in logic programs under stable models semantics by proposing a multi-valued extension with an algebra of truth values, resulting in a one-to-one correspondence between syntactic proof trees and interpretations, and enabling detection of semantic properties like redundancy and relevance.
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, in a similar spirit than a set of proof trees. The main contribution of this paper is that we capture justifications into an algebra of truth values with three internal operations: an addition '+' representing alternative justifications for a formula, a commutative product '*' representing joint interaction of causes and a non-commutative product '.' acting as a concatenation or proof constructor. Using this multi-valued semantics, we obtain a one-to-one correspondence between the syntactic proof tree of a standard (non-causal) logic program and the interpretation of each true atom in a model. Furthermore, thanks to this algebraic characterization we can detect semantic properties like redundancy and relevance of the obtained justifications. We also identify a lattice-based characterization of this algebra, defining a direct consequences operator, proving its continuity and that its least fix point can be computed after a finite number of iterations. Finally, we define the concept of causal stable model by introducing an analogous transformation to Gelfond and Lifschitz's program reduct.