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Probabilistic Linear Logic Programming with an application to Bayesian Networks computations

Matteo Acclavio, Roberto Maieli

TL;DR

The paper addresses the challenge of integrating Bayesian networks with logic programming by introducing probLO, a probabilistic extension of Linear Objects (LO) built on resource-sensitive linear logic. It uses multi-head probabilistic methods to encode conditional distributions and internal BN computations directly within LO’s proof-search framework, avoiding external semantic interpretations. The authors show how to encode Bayesian networks as probLO programs, prove boolean-consistency properties that guarantee coherent boolean assignments, and demonstrate that Derivations compute both joint and marginal probabilities via BN factorization. This operational, modular approach enables BN reasoning within a logic-programming setting and points to future work on integrating classical BN inference techniques like clique trees and belief propagation directly into probLO's proof-theoretic core.

Abstract

Bayesian networks are a canonical formalism for representing probabilistic dependencies, yet their integration within logic programming frameworks remains a nontrivial challenge, mainly due to the complex structure of these networks. In this paper, we propose probLO (probabilistic Linear Objects) an extension of Andreoli and Pareschi's LO language which embeds Bayesian network representation and computation within the framework of multiplicative-additive linear logic programming. The key novelty is the use of multi-head Prolog-like methods to reconstruct network structures, which are not necessarily trees, and the operation of slicing, standard in the literature of linear logic, enabling internal numerical probability computations without relying on external semantic interpretation.

Probabilistic Linear Logic Programming with an application to Bayesian Networks computations

TL;DR

The paper addresses the challenge of integrating Bayesian networks with logic programming by introducing probLO, a probabilistic extension of Linear Objects (LO) built on resource-sensitive linear logic. It uses multi-head probabilistic methods to encode conditional distributions and internal BN computations directly within LO’s proof-search framework, avoiding external semantic interpretations. The authors show how to encode Bayesian networks as probLO programs, prove boolean-consistency properties that guarantee coherent boolean assignments, and demonstrate that Derivations compute both joint and marginal probabilities via BN factorization. This operational, modular approach enables BN reasoning within a logic-programming setting and points to future work on integrating classical BN inference techniques like clique trees and belief propagation directly into probLO's proof-theoretic core.

Abstract

Bayesian networks are a canonical formalism for representing probabilistic dependencies, yet their integration within logic programming frameworks remains a nontrivial challenge, mainly due to the complex structure of these networks. In this paper, we propose probLO (probabilistic Linear Objects) an extension of Andreoli and Pareschi's LO language which embeds Bayesian network representation and computation within the framework of multiplicative-additive linear logic programming. The key novelty is the use of multi-head Prolog-like methods to reconstruct network structures, which are not necessarily trees, and the operation of slicing, standard in the literature of linear logic, enabling internal numerical probability computations without relying on external semantic interpretation.
Paper Structure (11 sections, 6 theorems, 21 equations, 9 figures)

This paper contains 11 sections, 6 theorems, 21 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Contributions of the paper.
  3. Paper Structure.
  4. Background
  5. Multiplicative Additive Linear Logic
  6. Probability and Bayesian Networks
  7. From Logic Programming to Linear Objects
  8. From Exploring Trees to Exploring DAGs using LO
  9. From LO to probLO
  10. Encoding Bayesian Networks in probLO
  11. Conclusion, related and future works

Key Result

proposition 1

The synthetic inference rule(s) associated to a bipole $\mathcal{F}_{}$ are derivable in $\mathsf{MLL}^{\mathsf{mix}}$.

Figures (9)

  • Figure 1: A (Boolean) Bayesian Network with five variables: Cloudy (${\mathsf{C}}$), Sprinklers (${\mathsf{S}}$), Rain (${\mathsf{R}}$), Wet Grass (${\mathsf{W}}$), and Traffic Jam (${\mathsf{T}}$).
  • Figure 2: Computation of the marginal probability of Cloudy (${\mathsf{C}}$), and Rain (${\mathsf{R}}$) being $\textsf{True}$ while Wet Grass (${\mathsf{W}}$), and Traffic Jam (${\mathsf{T}}$) being $\textsf{False}$ with respect to Sprinklers (${\mathsf{S}}$) in the Bayesian Network from \ref{['fig:BN1']}: where ${{\tt t\!t_{{\mathsf{X}}}}}$ and ${{\tt f\!f_{{\mathsf{X}}}}}$ denote, resp., the value True and the value False for a Boolean variable $X$.
  • Figure 3: Sequent calculus rules for $\mathsf{MALL}^{\mathsf{mix}}$.
  • Figure 4: A Prolog program made of three methods ${\tt M}_1$, ${\tt M}_2$ and ${\tt M}_3$, and the representation of the computation of the two queries ${\tt :- a.}$ and ${\tt :- a, c.}$.
  • Figure 5: Rules of the operational semantics of the multiplicative fragment of LO, and the corresponding synthetic inference rules.
  • ...and 4 more figures

Theorems & Definitions (20)

  • definition 1
  • proposition 1
  • proof
  • proposition 2
  • proof
  • definition 2
  • definition 3
  • theorem 1
  • definition 4
  • theorem 2
  • ...and 10 more