An Algebraic Notion of Conditional Independence, and Its Application to Knowledge Representation (full version)

TL;DR

This paper introduces an algebraic framework for understanding conditional independence across various logics, enabling more efficient reasoning in knowledge representation and logic programming.

Contribution

It develops a language-independent algebraic notion of conditional independence within approximation fixpoint theory, applicable to any logic with fixpoint semantics.

Findings

Reduces global reasoning to local reasoning instances

Achieves fixed-parameter tractability in reasoning tasks

Applies framework to normal logic programming

Abstract

Conditional independence is a crucial concept supporting adequate modelling and efficient reasoning in probabilistics. In knowledge representation, the idea of conditional independence has also been introduced for specific formalisms, such as propositional logic and belief revision. In this paper, the notion of conditional independence is studied in the algebraic framework of approximation fixpoint theory. This gives a language-independent account of conditional independence that can be straightforwardly applied to any logic with fixpoint semantics. It is shown how this notion allows to reduce global reasoning to parallel instances of local reasoning, leading to fixed-parameter tractability results. Furthermore, relations to existing notions of conditional independence are discussed and the framework is applied to normal logic programming.