MT-PDCL Unifies Continuous Probabilistic Models with Logic Programming

Traditional probabilistic logic programming frameworks are often limited by their reliance on grounding logic programs into discrete propositional representations. This operational requirement restricts exact inference to finite domains and discrete probability distributions, hindering their application in scenarios involving continuous variables. Researchers have introduced Measure-Theoretic Probabilistic Definite Clause Logic (MT-PDCL) as a generalized foundational framework to overcome this limitation. MT-PDCL explicitly defines stochastic variables over bounded index domains and equips the interpretation space with standard Borel sigma-algebras, allowing logical variables to operate natively over continuous measurable spaces. Building on Continuous Distribution Semantics, MT-PDCL models probabilistic rules as independent causal events. Crucially, declarative entailment is defined through exact Lebesgue integration over the continuous measure space, rather than finite boolean circuits. This approach replaces the combinatorial bottleneck of discrete grounding with exact, algebraic, and structurally differentiable inference, offering the expressive power of continuous probabilistic models while retaining the declarative syntax of definite clause logic.