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Semirings for Probabilistic and Neuro-Symbolic Logic Programming

Vincent Derkinderen, Robin Manhaeve, Pedro Zuidberg Dos Martires, Luc De Raedt

TL;DR

This work presents a unified semiring-based algebraic framework for probabilistic and neural-symbolic logic programming, showing that many PLP extensions can be cast as algebraic logic programs with semiring-labeled facts and algebraic model counting (AMC). It integrates probabilistic facts, neural facts, distributional/indicator facts, and algebraic facts within a common semantics, enabling inference via logical reasoning, translation to weighted/algebraic model counting, and efficient solving using sd-DNNF-derived arithmetic circuits. A gradient semiring is introduced to enable gradient-based parameter learning across probabilistic and neural components, allowing end-to-end optimization within the algebraic framework. The approach provides a flexible, extensible foundation for coupling neural networks with logic programs and has broad applicability in domains like robotics and SR AI, where discrete and continuous uncertainties must be reasoned about jointly.

Abstract

The field of probabilistic logic programming (PLP) focuses on integrating probabilistic models into programming languages based on logic. Over the past 30 years, numerous languages and frameworks have been developed for modeling, inference and learning in probabilistic logic programs. While originally PLP focused on discrete probability, more recent approaches have incorporated continuous distributions as well as neural networks, effectively yielding neural-symbolic methods. We provide a unified algebraic perspective on PLP, showing that many if not most of the extensions of PLP can be cast within a common algebraic logic programming framework, in which facts are labeled with elements of a semiring and disjunction and conjunction are replaced by addition and multiplication. This does not only hold for the PLP variations itself but also for the underlying execution mechanism that is based on (algebraic) model counting.

Semirings for Probabilistic and Neuro-Symbolic Logic Programming

TL;DR

This work presents a unified semiring-based algebraic framework for probabilistic and neural-symbolic logic programming, showing that many PLP extensions can be cast as algebraic logic programs with semiring-labeled facts and algebraic model counting (AMC). It integrates probabilistic facts, neural facts, distributional/indicator facts, and algebraic facts within a common semantics, enabling inference via logical reasoning, translation to weighted/algebraic model counting, and efficient solving using sd-DNNF-derived arithmetic circuits. A gradient semiring is introduced to enable gradient-based parameter learning across probabilistic and neural components, allowing end-to-end optimization within the algebraic framework. The approach provides a flexible, extensible foundation for coupling neural networks with logic programs and has broad applicability in domains like robotics and SR AI, where discrete and continuous uncertainties must be reasoned about jointly.

Abstract

The field of probabilistic logic programming (PLP) focuses on integrating probabilistic models into programming languages based on logic. Over the past 30 years, numerous languages and frameworks have been developed for modeling, inference and learning in probabilistic logic programs. While originally PLP focused on discrete probability, more recent approaches have incorporated continuous distributions as well as neural networks, effectively yielding neural-symbolic methods. We provide a unified algebraic perspective on PLP, showing that many if not most of the extensions of PLP can be cast within a common algebraic logic programming framework, in which facts are labeled with elements of a semiring and disjunction and conjunction are replaced by addition and multiplication. This does not only hold for the PLP variations itself but also for the underlying execution mechanism that is based on (algebraic) model counting.
Paper Structure (15 sections, 24 equations, 4 figures)

This paper contains 15 sections, 24 equations, 4 figures.

Table of Contents

  1. Introduction
  2. From Logic Programs to Algebraic Logic Programs
  3. Logic programming
  4. Probabilistic Facts
  5. Neural Facts
  6. Distributional Facts and Indicator Facts
  7. Algebraic Facts
  8. Inference
  9. Logical inference
  10. Translation to algebraic model counting
  11. Solving model counting
  12. Learning
  13. Gradient semiring
  14. Related Work and Applications
  15. Conclusion

Figures (4)

  • Figure 1: The SLD tree for Example \ref{['ex:inference']}. The two branches represent the two separate proofs for the query problog.py:ProbLogLexer -xwet(sunday).
  • Figure 2: An sd-DNNF corresponding to the ProbLog program in Example \ref{['ex:bayesian_network']}.
  • Figure 3: A $WMC(\mathcal{T})$ representation of Figure \ref{['fig:sd_ddnnf']}.
  • Figure 4: The arithmetic circuit for Example \ref{['ex:learning']}. Each node is annotated with an element of the gradient semiring, where the two elements of the gradient represent the partial derivative of the probability with respect to the neural network, and the probabilistic parameter respectively.

Theorems & Definitions (28)

  • Definition 1: Probabilistic fact
  • Definition 2: ProbLog program
  • Example 1: Bayesian network
  • Example 2: Possible worlds
  • Definition 3: Success probability
  • Example 3
  • Definition 4: Neural fact
  • Example 4: Neural fact
  • Definition 5: Distributional Fact
  • Definition 6: Indicator Fact
  • ...and 18 more