π€ AI Summary
Traditional probabilistic logic programming is constrained by discrete propositionalization, hindering precise reasoning over continuous domains. This work proposes the MT-PDCL framework, which introduces measure theory into probabilistic logic programming for the first time. By leveraging standard Borel Ο-algebras and Lebesgue integration, MT-PDCL defines logical variables directly over continuous measurable spaces and establishes a semantics based on continuous probability distributions along with a continuous immediate consequence operator. The approach preserves a purely declarative syntax while enabling exact, algebraic, and structurally differentiable probabilistic inference over continuous domains. Centered on continuous integration rather than discrete combinatorial enumeration, the framework effectively circumvents the limitations of finite domains and supports efficient, exact inference with continuous priors and observations.
π Abstract
Standard probabilistic logic programming frameworks typically rely on grounding logic programs into discrete propositional representations. This operational requirement restricts exact inference to finite domains and discrete probability distributions. In this paper, we introduce Measure-Theoretic Probabilistic Definite Clause Logic (MT-PDCL), a generalized foundational framework that eliminates this finite-domain restriction. By explicitly defining stochastic variables over bounded index domains and equipping the interpretation space with standard Borel $Ο$-algebras, MT-PDCL allows logical variables to operate natively over continuous measurable spaces. Building on Continuous Distribution Semantics, MT-PDCL models probabilistic rules as mutually independent causal events. However, rather than aggregating these derivations via finite boolean circuits, declarative entailment is formally defined through exact Lebesgue integration over the continuous measure space. We introduce a continuous immediate consequence operator that unifies the integration of continuous prior distributions with the evaluation of exact continuous observations. We demonstrate that this approach replaces the combinatorial bottleneck of discrete grounding with exact, algebraic, and structurally differentiable inference. While this transition trades discrete combinatorics for the geometric curse of dimensionality, it achieves the expressive power of continuous probabilistic models while preserving the pure declarative syntax of definite clause logic.