[Paper Review] Probabilistic Disjunctive Logic Programming

Liem Viet Ngo|arXiv (Cornell University)|Feb 13, 2013

Logic, Reasoning, and Knowledge10 references7 citations

TL;DR

This paper introduces a framework for probabilistic disjunctive logic programming by integrating Poole's probabilistic Horn abduction with disjunctive logic programs. It uses minimal models and the principle of indifference to derive default probability distributions, enabling the calculation of default probabilities for goals through explanation and partial explanation mechanisms, with a formal algorithm provided for computation.

ABSTRACT

In this paper we propose a framework for combining Disjunctive Logic Programming and Poole's Probabilistic Horn Abduction. We use the concept of hypothesis to specify the probability structure. We consider the case in which probabilistic information is not available. Instead of using probability intervals, we allow for the specification of the probabilities of disjunctions. Because minimal models are used as characteristic models in disjunctive logic programming, we apply the principle of indifference on the set of minimal models to derive default probability values. We define the concepts of explanation and partial explanation of a formula, and use them to determine the default probability distribution(s) induced by a program. An algorithm for calculating the default probability of a goal is presented.

Motivation & Objective

• To integrate probabilistic reasoning with disjunctive logic programming by extending Poole's probabilistic Horn abduction to handle disjunctions.
• To define a principled method for assigning default probabilities when no explicit probabilistic information is available.
• To use the set of minimal models as the basis for applying the principle of indifference to derive default probability distributions.
• To formalize the concepts of explanation and partial explanation for determining default probabilities of logical formulas.
• To present an algorithm for computing the default probability of a given goal in a disjunctive logic program.

Proposed method

• Uses hypotheses to structure the probability space in disjunctive logic programs.
• Applies the principle of indifference over the set of minimal models to assign default probabilities when no prior probabilities are given.
• Defines explanation and partial explanation of a formula as mechanisms to determine which minimal models support the formula.
• Derives default probability distributions based on the proportion of minimal models that satisfy a formula or its subformulas.
• Employs a recursive algorithm to compute the default probability of a goal by analyzing its explanations and partial explanations.
• Uses disjunctive logic programs as the knowledge representation formalism, with probabilistic semantics grounded in minimal model theory.

Experimental results

Research questions

• RQ1 How can probabilistic reasoning be systematically extended to disjunctive logic programs without prior probability distributions?
• RQ2 What is an appropriate default probability assignment mechanism when no probabilistic data is available?
• RQ3 How can the principle of indifference be applied meaningfully to minimal models in disjunctive logic programs?
• RQ4 What defines a valid explanation and partial explanation in the context of probabilistic disjunctive logic programs?
• RQ5 How can the default probability of a goal be computed algorithmically from the program's structure and minimal models?

Key findings

• The framework provides a coherent method for assigning default probabilities to disjunctive formulas using minimal models and the principle of indifference.
• The probability of a formula is determined by the fraction of minimal models that satisfy it, ensuring consistency with default reasoning principles.
• Explanation and partial explanation mechanisms allow for the decomposition of complex formulas into manageable components for probability computation.
• An algorithm is formally defined and proven to compute the default probability of a goal based on its explanatory structure.
• The approach successfully extends Poole's probabilistic Horn abduction to disjunctive logic programs, enabling richer probabilistic reasoning.
• The method is applicable even in the absence of explicit probability intervals, making it suitable for default reasoning under uncertainty.