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This paper introduces Measure-Theoretic Probabilistic Definite Clause Logic (MT-PDCL), a novel framework that overcomes the limitations of traditional probabilistic logic programming by allowing logical variables to function over continuous measurable spaces. By utilizing standard Borel $\sigma$-algebras and defining probabilistic rules as independent causal events, MT-PDCL enables exact inference through Lebesgue integration rather than finite boolean circuits. The key result is that this approach not only eliminates the combinatorial bottleneck of discrete grounding but also retains the declarative syntax of definite clause logic, thus enhancing the expressiveness of probabilistic models in continuous domains.
MT-PDCL transforms probabilistic logic programming by enabling exact inference in continuous spaces, sidestepping the limitations of discrete representations.
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 $\sigma$-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.
