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Declarative Probabilistic Logic Programming in Discrete-Continuous Domains

Pedro Zuidberg Dos Martires, Luc De Raedt, Angelika Kimmig

TL;DR

This work extends probabilistic logic programming to hybrid discrete-continuous domains by introducing a measure-based semantics (the measure semantics) and the DC-ProbLog language, which unifies discrete PLP, distributional clauses, and continuous distributions. A central contribution is the Infinitesimal Algebraic Likelihood Weighting (IALW) inference method, which uses knowledge compilation to handle conditioning, including zero-probability events, by operating on algebraic circuits (sd-DNNF) and infinitesimal semirings. The paper shows the semantic and algorithmic integration of random-variable dependencies with logical reasoning, enabling exact or quasi-exact inference in hybrid settings and positioning DC-ProbLog as a versatile generalization of ProbLog, Extended PRISM, PCLP, and BLOG. Empirical results demonstrate robustness of SIALW against rare events and demonstrate the practicality of the approach for complex hybrid models. The framework advances declarative hybrid probabilistic programming with a principled semantics, scalable inference, and broad compatibility with existing PLP paradigms.

Abstract

Over the past three decades, the logic programming paradigm has been successfully expanded to support probabilistic modeling, inference and learning. The resulting paradigm of probabilistic logic programming (PLP) and its programming languages owes much of its success to a declarative semantics, the so-called distribution semantics. However, the distribution semantics is limited to discrete random variables only. While PLP has been extended in various ways for supporting hybrid, that is, mixed discrete and continuous random variables, we are still lacking a declarative semantics for hybrid PLP that not only generalizes the distribution semantics and the modeling language but also the standard inference algorithm that is based on knowledge compilation. We contribute the measure semantics together with the hybrid PLP language DC-ProbLog (where DC stands for distributional clauses) and its inference engine infinitesimal algebraic likelihood weighting (IALW). These have the original distribution semantics, standard PLP languages such as ProbLog, and standard inference engines for PLP based on knowledge compilation as special cases. Thus, we generalize the state of the art of PLP towards hybrid PLP in three different aspects: semantics, language and inference. Furthermore, IALW is the first inference algorithm for hybrid probabilistic programming based on knowledge compilation

Declarative Probabilistic Logic Programming in Discrete-Continuous Domains

TL;DR

This work extends probabilistic logic programming to hybrid discrete-continuous domains by introducing a measure-based semantics (the measure semantics) and the DC-ProbLog language, which unifies discrete PLP, distributional clauses, and continuous distributions. A central contribution is the Infinitesimal Algebraic Likelihood Weighting (IALW) inference method, which uses knowledge compilation to handle conditioning, including zero-probability events, by operating on algebraic circuits (sd-DNNF) and infinitesimal semirings. The paper shows the semantic and algorithmic integration of random-variable dependencies with logical reasoning, enabling exact or quasi-exact inference in hybrid settings and positioning DC-ProbLog as a versatile generalization of ProbLog, Extended PRISM, PCLP, and BLOG. Empirical results demonstrate robustness of SIALW against rare events and demonstrate the practicality of the approach for complex hybrid models. The framework advances declarative hybrid probabilistic programming with a principled semantics, scalable inference, and broad compatibility with existing PLP paradigms.

Abstract

Over the past three decades, the logic programming paradigm has been successfully expanded to support probabilistic modeling, inference and learning. The resulting paradigm of probabilistic logic programming (PLP) and its programming languages owes much of its success to a declarative semantics, the so-called distribution semantics. However, the distribution semantics is limited to discrete random variables only. While PLP has been extended in various ways for supporting hybrid, that is, mixed discrete and continuous random variables, we are still lacking a declarative semantics for hybrid PLP that not only generalizes the distribution semantics and the modeling language but also the standard inference algorithm that is based on knowledge compilation. We contribute the measure semantics together with the hybrid PLP language DC-ProbLog (where DC stands for distributional clauses) and its inference engine infinitesimal algebraic likelihood weighting (IALW). These have the original distribution semantics, standard PLP languages such as ProbLog, and standard inference engines for PLP based on knowledge compilation as special cases. Thus, we generalize the state of the art of PLP towards hybrid PLP in three different aspects: semantics, language and inference. Furthermore, IALW is the first inference algorithm for hybrid probabilistic programming based on knowledge compilation
Paper Structure (68 sections, 28 theorems, 84 equations, 6 figures, 3 algorithms)

This paper contains 68 sections, 28 theorems, 84 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. A Panoramic Overview
  3. Panorama of the Syntax and Semantics
  4. Panorama of the Inference
  5. DF-PLP
  6. Distributional Facts and Random Variables
  7. Boolean Comparison Atoms over Random Variables
  8. Logical Consequences of Boolean Comparisons
  9. Constructing Distributional Databases
  10. DC-ProbLog
  11. Boolean Random Variables
  12. Probabilistically Selected Logical Consequences
  13. Context-Dependent Distributions
  14. Probabilistic Inference Tasks
  15. Conditioning on Zero-Probability Events
  16. ...and 53 more sections

Key Result

Proposition 12

The Boolean comparison atoms $\mathcal{F}\xspace$ induce a product sample space $\Omega\xspace_\mathcal{F}\xspace$.

Figures (6)

  • Figure 2.1: Graphical representation of the computation graph (i.e. algebraic circuit) used to compute the probability $(\text{\mintinline{problog.py:ProbLogLexer -x}{favorite}}=\top, \text{\mintinline{problog.py:ProbLogLexer -x}{large}}=\bot)$ using the IALW algorithm introduced in Section \ref{['sec:alw']}.
  • Figure 3.1: Directed acyclic graph representing the ancestor relationship between the random variables in Example \ref{['ex:dist_db']}. The ancestor set of problog.py:ProbLogLexer -xx is the empty set, the one of problog.py:ProbLogLexer -xy is $\{\text{\mintinline{problog.py:ProbLogLexer -x}{x}} \}$ and the one of problog.py:ProbLogLexer -xz is $\{\text{\mintinline{problog.py:ProbLogLexer -x}{x}}, \text{\mintinline{problog.py:ProbLogLexer -x}{y}} \}$.
  • Figure 7.1: At the bottom of the circuit we see the distributions feeding in. The problog.py:ProbLogLexer -xflip distribution feeds into its two possible (non-zero probability) outcomes. The two problog.py:ProbLogLexer -xbeta distributions feed into an observation statement each. We use the '$\doteq$' symbol to denote such an observation. Note how we identify each of the two random variables for the size by a unique identifier in their respective subscripts. The circled numbers next to the internal nodes, i.e. the sum and product nodes, will allow us to reference the nodes later on and do not form a part of the algebraic circuit.
  • Figure 7.2: Circuit representation of the SIALW algorithm for the probability $P(\text{\mintinline{problog.py:ProbLogLexer -x}{size}}\doteq 4/10 )$.
  • Figure 7.3: We queried SIALW and (non-symbolci) likelihood weighting each $100$ times for the probability $p( \text{\mintinline{problog.py:ProbLogLexer -x}{faulty}}{=}\top {\mid} \text{\mintinline{problog.py:ProbLogLexer -x}{temperature}}{\doteq} 2.0 )$. On the $y$-axis we give the ratio $\text{\#correct runs}/\text{\#runs}$. Die to the use of knowledge compilation, SIALW is insensitive to the probability of fault (on the x-axis). This is in contrast to the log-likelihood weighting (LLW) algorithm presented by nitti2016probabilistic, which necessitates a considerable number of samples to compute the queried probability reliably. The different dotted lines indicate settings with varying sample sizes ($\{10^1, 10^2, 10^3, 10^4, 10^5 \}$).
  • ...and 1 more figures

Theorems & Definitions (140)

  • Example 1: Probabilistic Logic Program
  • Example 2
  • Example 3
  • Definition 4: Reserved Vocabulary
  • Definition 5: Regular Vocabulary
  • Definition 6: Distributional Fact
  • Definition 7: Sample Space
  • Definition 8: Distributional Database
  • Example 9
  • Definition 10: Well-Defined Distributional Database
  • ...and 130 more