State Algebra for Probabilistic Logic

2026-03-13

AI Summary

This work addresses the challenge of integrating logical interpretability with probabilistic reasoning in high-stakes decision-making. It proposes a novel probabilistic state algebra that uniquely embeds logical reduction directly within purely linear algebraic operations, eliminating the need for graph traversal or circuit compilation. By mapping logical states to energy potentials and employing the Hadamard product to construct the Gibbs distribution of a Markov random field, the framework unifies symbolic rules with statistical inference. The approach supports modular rule representation using t-objects and wildcards, yielding a mathematically rigorous, auditable, and maintainable probabilistic logic system well-suited for high-risk human-AI collaboration domains such as healthcare and finance.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Logic Programming

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

This paper presents a Probabilistic State Algebra as an extension of deterministic propositional logic, providing a computational framework for constructing Markov Random Fields (MRFs) through pure linear algebra. By mapping logical states to real-valued coordinates interpreted as energy potentials, we define an energy-based model where global probability distributions emerge from coordinate-wise Hadamard products. This approach bypasses the traditional reliance on graph-traversal algorithms and compiled circuits, utilising $t$-objects and wildcards to embed logical reduction natively within matrix operations. We demonstrate that this algebra constructs formal Gibbs distributions, offering a rigorous mathematical link between symbolic constraints and statistical inference. A central application of this framework is the development of Probabilistic Rule Models (PRMs), which are uniquely capable of incorporating both probabilistic associations and deterministic logical constraints simultaneously. These models are designed to be inherently interpretable, supporting a human-in-the-loop approach to decisioning in high-stakes environments such as healthcare and finance. By representing decision logic as a modular summation of rules within a vector space, the framework ensures that complex probabilistic systems remain auditable and maintainable without compromising the rigour of the underlying configuration space.

Problem

• Probabilistic Logic
• Markov Random Fields
• Logical Constraints
• Interpretable Models
• Statistical Inference

Innovation

• Probabilistic State Algebra
• Markov Random Fields
• Energy-based Models
• Probabilistic Rule Models
• Interpretable AI