MT-PDCL的基础：测度论概率确定性子句逻辑

中文摘要 AI 辅助

标准概率逻辑编程框架通常依赖于将逻辑程序归约为离散命题表示，这一操作要求将精确推理限制在有限域和离散概率分布中。本文中，我们引入测度论概率确定性子句逻辑（MT-PDCL），这是一个消除了有限域限制的通用基础框架。通过在有界索引域上显式定义随机变量，并为解释空间配备标准Borel σ代数，MT-PDCL允许逻辑变量原生地在连续可测空间上操作。基于连续分布语义，MT-PDCL将概率规则建模为相互独立的因果事件。然而，与通过有限布尔电路聚合这些推导不同，声明式蕴涵是通过对连续测度空间的精确Lebesgue积分形式化定义的。我们引入了一个连续 immediate consequence 算子，该算子将连续先验分布的积分与精确连续观测的评估统一起来。我们证明，该方法用精确的、代数的且结构可微的推理替代了离散归约的组合瓶颈。尽管这种转换用几何维度灾难替代了离散组合性，但它在保留确定性子句逻辑纯声明式语法的同时，实现了连续概率模型的表达能力。

英文摘要

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 $σ$-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.

发表机构

• University of Craiova（克拉约瓦大学）

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