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

Jesse Heyninck

TL;DR

This work presents an algebraic, operator-based notion of conditional independence for systems with fixpoint semantics, grounded in Approximation Fixpoint Theory (AFT). By treating KR formalisms as operators over product lattices, the authors show that independence between sublattices enables decomposing global fixpoint computations into parallel local problems, yielding fixed-parameter tractability results parameterized by a CIT-size. The framework is instantiated for normal logic programs, where marginalisation yields decomposable IC-operators and WF fixpoints can be computed efficiently under the CIT paradigm. The approach unifies and extends existing notions of independence in KR and connects to stratification and treewidth results, offering a language-agnostic pathway to scalable reasoning across diverse formalisms.

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.

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

TL;DR

This work presents an algebraic, operator-based notion of conditional independence for systems with fixpoint semantics, grounded in Approximation Fixpoint Theory (AFT). By treating KR formalisms as operators over product lattices, the authors show that independence between sublattices enables decomposing global fixpoint computations into parallel local problems, yielding fixed-parameter tractability results parameterized by a CIT-size. The framework is instantiated for normal logic programs, where marginalisation yields decomposable IC-operators and WF fixpoints can be computed efficiently under the CIT paradigm. The approach unifies and extends existing notions of independence in KR and connects to stratification and treewidth results, offering a language-agnostic pathway to scalable reasoning across diverse formalisms.

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.

Paper Structure

This paper contains 13 sections, 2 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Logic Programming
  4. Lattices and sub-lattices
  5. Approximation Fixpoint Theory
  6. Conditional Independence
  7. Conditional Independence and AFT
  8. Parametrized Complexity Results Based on Conditional Independence
  9. Preliminaries on Parametrized Complexity
  10. Conditional Independence Trees
  11. Application to Logic Programs
  12. Related Work
  13. Conclusion

Key Result

Corollary 1

Let an nlp ${\cal P}$ be given for which ${\cal A}_{\cal P}$ is partitioned into ${\cal A}_1\cup {\cal A}_2\cup {\cal A}_3$ s.t. ${{\cal A}_1}\perp \!\!\! \perp_{\cal P}{\cal A}_2\mid {\cal A}_3$. $x_1\cup x_2\cup x_3$ is a supported (respectively three-valued stable) model of ${\cal P}$ iff $x_i\cu

Figures (1)

  • Figure 1: A dependency graph for the program ${\cal P}_1$ (Example 1), and its extension ${\cal P}_2$ (Example \ref{['example:cit:tree']}, atoms only occuring in ${\cal P}_2$ have gray outlines.).

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 1
  • Remark 1
  • Example 2
  • Definition 5
  • Example 3
  • Definition 6
  • ...and 14 more