[Paper Review] CP-logic: A Language of Causal Probabilistic Events and Its Relation to Logic Programming

Joost Vennekens, Marc Denecker+1 more|ArXiv.org|Apr 10, 2009

Logic, Reasoning, and Knowledge44 references4 citations

TL;DR

This paper introduces CP-logic, a logical language for representing probabilistic causal laws that provides a semantically grounded, intuitive interpretation of probabilistic logic programs. It formalizes causal dynamics using probability trees and proves equivalence between CP-logic semantics and well-founded models of logic programs, offering a principled, modular, and interpretable foundation for probabilistic reasoning in logic programming.

ABSTRACT

This papers develops a logical language for representing probabilistic causal laws. Our interest in such a language is twofold. First, it can be motivated as a fundamental study of the representation of causal knowledge. Causality has an inherent dynamic aspect, which has been studied at the semantical level by Shafer in his framework of probability trees. In such a dynamic context, where the evolution of a domain over time is considered, the idea of a causal law as something which guides this evolution is quite natural. In our formalization, a set of probabilistic causal laws can be used to represent a class of probability trees in a concise, flexible and modular way. In this way, our work extends Shafer's by offering a convenient logical representation for his semantical objects. Second, this language also has relevance for the area of probabilistic logic programming. In particular, we prove that the formal semantics of a theory in our language can be equivalently defined as a probability distribution over the well-founded models of certain logic programs, rendering it formally quite similar to existing languages such as ICL or PRISM. Because we can motivate and explain our language in a completely self-contained way as a representation of probabilistic causal laws, this provides a new way of explaining the intuitions behind such probabilistic logic programs: we can say precisely which knowledge such a program expresses, in terms that are equally understandable by a non-logician. Moreover, we also obtain an additional piece of knowledge representation methodology for probabilistic logic programs, by showing how they can express probabilistic causal laws.

Motivation & Objective

• To develop a logical language that formally represents probabilistic causal laws in a way that is semantically intuitive and self-contained.
• To provide a clear, non-technical interpretation of probabilistic logic programs by grounding them in causal dynamics and temporal evolution.
• To establish a formal equivalence between CP-logic semantics and probability distributions over well-founded models of logic programs.
• To extend Shafer’s probability tree framework by offering a compact, modular, and logical representation of causal probabilistic events.
• To offer a new methodology for modeling uncertainty in logic programming by expressing knowledge as causal laws with probabilistic effects.

Proposed method

• The paper defines CP-logic as a language of probabilistic causal rules, where each rule specifies a probabilistic event that may affect the truth of atoms.
• It introduces the concept of a 'stratified CP-theory' with a timing function λ that respects causal precedence, ensuring temporal consistency in event execution.
• Weak execution models are constructed as probabilistic Σ-processes where events are executed in a way that respects their assigned timing, ensuring no causal violations.
• An event-timing function κ is introduced to refine the execution order, ensuring that only rules with minimal timing among enabled rules are selected at each step.
• The formal semantics of a CP-theory is defined as a probability distribution over the well-founded models of underlying logic programs, using a recursive probability propagation mechanism.
• The paper proves that all execution models of a given CP-theory yield the same probability distribution, establishing semantic invariance and equivalence to logic programming semantics.

Experimental results

Research questions

• RQ1 How can probabilistic causal laws be formally represented in a way that is both intuitive and logically rigorous?
• RQ2 How does the semantics of CP-logic relate to existing probabilistic logic programming languages like ICL or PRISM?
• RQ3 Can a CP-theory be shown to generate a unique probability distribution over possible worlds, independent of the execution path?
• RQ4 What conditions ensure that the execution of probabilistic events respects causal precedence and temporal ordering?
• RQ5 Is there a formal equivalence between CP-logic semantics and the well-founded model semantics of logic programs under probabilistic interpretation?

Key findings

• CP-logic provides a formal semantics for probabilistic causal laws that is both intuitive and mathematically sound, grounded in the dynamics of event execution over time.
• The paper proves that all execution models of a CP-theory produce the same probability distribution, ensuring semantic consistency and invariance under different execution orders.
• The semantics of a CP-theory is formally equivalent to a probability distribution over the well-founded models of a corresponding logic program, linking it to established logic programming semantics.
• The existence of execution models that follow a given stratification is proven constructively, showing that every stratified CP-theory admits a valid temporal execution sequence.
• The formalization ensures that causal precedence is preserved: events with lower timing cannot affect atoms with higher timing before their execution, maintaining logical and causal consistency.
• The paper establishes that CP-logic can represent complex probabilistic causal scenarios—such as the John and Mary rock-throwing story—as compact, modular, and semantically transparent rule sets.