State Algebra for Propositional Logic

2025-09-12

AI Summary

Efficient algebraic modeling and computation for propositional logic remain challenging due to the tension between expressive flexibility and representational compactness. Method: This paper introduces State Algebra—a unified algebraic framework built upon a three-layer representation system: sets, coordinates, and row decomposition. It integrates set-theoretic and linear-structural principles, enabling algebraic engine–driven hierarchical representation and variable-ordered reduction. A novel non-normalized state vector reduction mechanism is proposed, restoring canonicity under a fixed variable order while preserving both flexibility and conciseness. Contribution/Results: Experimental evaluation demonstrates State Algebra’s natural expressiveness and scalability across propositional reasoning, knowledge compilation, and weighted model counting. By unifying logical state manipulation, search algorithms, and knowledge compilation within a single algebraic foundation, the framework provides a principled basis for integrating symbolic logic and probabilistic inference.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper presents State Algebra, a novel framework designed to represent and manipulate propositional logic using algebraic methods. The framework is structured as a hierarchy of three representations: Set, Coordinate, and Row Decomposition. These representations anchor the system in well-known semantics while facilitating the computation using a powerful algebraic engine. A key aspect of State Algebra is its flexibility in representation. We show that although the default reduction of a state vector is not canonical, a unique canonical form can be obtained by applying a fixed variable order during the reduction process. This highlights a trade-off: by foregoing guaranteed canonicity, the framework gains increased flexibility, potentially leading to more compact representations of certain classes of problems. We explore how this framework provides tools to articulate both search-based and knowledge compilation algorithms and discuss its natural extension to probabilistic logic and Weighted Model Counting.

Problem

• Representing propositional logic algebraically through hierarchical structures
• Achieving canonical form via fixed variable ordering in reduction
• Extending framework to probabilistic logic and weighted model counting

Innovation

• State Algebra framework for propositional logic
• Hierarchical representations: Set, Coordinate, Row
• Flexible non-canonical forms with variable ordering