State Algebra for Probabilistic Logic

Computer Science > Artificial Intelligence [Submitted on 13 Mar 2026] State Algebra for Probabilistic Logic Dmitry Lesnik, Tobias Schäfer 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. Comments: 31 pages Subjects: Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Logic in Computer Science (cs.LO) MSC classes: 60K35 (Primary) 03B48, 68T27 (Secondary) ACM classes: I.2.3; I.2.4; G.3 Cite as: arXiv:2603.13574 [cs.AI]   (or arXiv:2603.13574v1 [cs.AI] for this version)   https://doi.org/10.48550/arXiv.2603.13574 Focus to learn more Submission history From: Dmitry Lesnik [view email] [v1] Fri, 13 Mar 2026 20:28:45 UTC (28 KB) Access Paper: HTML (experimental) view license Current browse context: cs.AI < prev   |   next > new | recent | 2026-03 Change to browse by: cs cs.LG cs.LO References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)