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Probabilistic Answer Set Programming with Discrete and Continuous Random Variables

Damiano Azzolini, Fabrizio Riguzzi

TL;DR

This work extends probabilistic answer set programming to support both discrete and continuous random variables by introducing Hybrid Probabilistic Answer Set Programming (HPASP). HPASP is solved by discretizing the hybrid program into a discrete PASP, enabling reuse of existing inference tools, and by employing four algorithms: two exact (PASTA-based projection and 2AMC-based knowledge compilation) and two approximate (sampling on discretized and on the hybrid programs). Empirical results show that exact inference via projection is feasible only for small instances, while knowledge compilation yields substantial scalability gains; approximate sampling can handle larger problems but may incur memory costs due to discretization. The framework advances relational, abductive-style reasoning under credal semantics for hybrid uncertainty and points to future work on multi-variable comparisons and lifted inference techniques to further enhance scalability and expressivity.

Abstract

Probabilistic Answer Set Programming under the credal semantics (PASP) extends Answer Set Programming with probabilistic facts that represent uncertain information. The probabilistic facts are discrete with Bernoulli distributions. However, several real-world scenarios require a combination of both discrete and continuous random variables. In this paper, we extend the PASP framework to support continuous random variables and propose Hybrid Probabilistic Answer Set Programming (HPASP). Moreover, we discuss, implement, and assess the performance of two exact algorithms based on projected answer set enumeration and knowledge compilation and two approximate algorithms based on sampling. Empirical results, also in line with known theoretical results, show that exact inference is feasible only for small instances, but knowledge compilation has a huge positive impact on the performance. Sampling allows handling larger instances, but sometimes requires an increasing amount of memory. Under consideration in Theory and Practice of Logic Programming (TPLP).

Probabilistic Answer Set Programming with Discrete and Continuous Random Variables

TL;DR

This work extends probabilistic answer set programming to support both discrete and continuous random variables by introducing Hybrid Probabilistic Answer Set Programming (HPASP). HPASP is solved by discretizing the hybrid program into a discrete PASP, enabling reuse of existing inference tools, and by employing four algorithms: two exact (PASTA-based projection and 2AMC-based knowledge compilation) and two approximate (sampling on discretized and on the hybrid programs). Empirical results show that exact inference via projection is feasible only for small instances, while knowledge compilation yields substantial scalability gains; approximate sampling can handle larger problems but may incur memory costs due to discretization. The framework advances relational, abductive-style reasoning under credal semantics for hybrid uncertainty and points to future work on multi-variable comparisons and lifted inference techniques to further enhance scalability and expressivity.

Abstract

Probabilistic Answer Set Programming under the credal semantics (PASP) extends Answer Set Programming with probabilistic facts that represent uncertain information. The probabilistic facts are discrete with Bernoulli distributions. However, several real-world scenarios require a combination of both discrete and continuous random variables. In this paper, we extend the PASP framework to support continuous random variables and propose Hybrid Probabilistic Answer Set Programming (HPASP). Moreover, we discuss, implement, and assess the performance of two exact algorithms based on projected answer set enumeration and knowledge compilation and two approximate algorithms based on sampling. Empirical results, also in line with known theoretical results, show that exact inference is feasible only for small instances, but knowledge compilation has a huge positive impact on the performance. Sampling allows handling larger instances, but sometimes requires an increasing amount of memory. Under consideration in Theory and Practice of Logic Programming (TPLP).
Paper Structure (24 sections, 4 theorems, 32 equations, 3 figures, 6 tables, 2 algorithms)

This paper contains 24 sections, 4 theorems, 32 equations, 3 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Answer Set Programming
  4. Probabilistic Answer Set Programming
  5. ProbLog and Hybrid ProbLog
  6. Hybrid Probabilistic Answer Set Programming
  7. Algorithms
  8. Exact Inference
  9. Modifying the PASTA solver.
  10. Modifying the aspcs solver.
  11. Approximate Inference
  12. Sampling the discretized program.
  13. Sampling the hybrid program.
  14. Approximate algorithm description.
  15. Experiments
  16. ...and 9 more sections

Key Result

Theorem 1

Consider an hybird probabilistic answer set program $P$. Let $n_r$ be the number of rules, $n_c$ be the number of continuous facts, $n_k$ be the number of comparison atoms in the program, $n_i$ be the maximum number of intervals for a continuous fact, $o$ be the maximum number of $outside/3$ atoms i

Figures (3)

  • Figure 1: Results for $t1$, $t4$, and $t5$ (left) and results for aspcs applied to $t3$ (right) with 2 and 5 continuous facts.
  • Figure 2: Results for the experiment $t2$ with a fixed number of discrete facts and an increasing number of continuous variables.
  • Figure 3: Results for the experiment $t3$ with a fixed number of continuous random variables and an increasing number of discrete facts. The x axis of the left plot is cut at size 20, to keep the results readable.

Theorems & Definitions (16)

  • Example 1
  • Example 2
  • Definition 1
  • Example 3
  • Example 4
  • Theorem 1
  • proof
  • Example 5
  • Theorem 2
  • proof
  • ...and 6 more