Power Term Polynomial Algebra for Boolean Logic

2026-03-14

AI Summary

This work addresses the exponential blowup arising from structural mismatches in conversions between conjunctive normal form (CNF) and algebraic normal form (ANF) by introducing power-term polynomial algebra as a novel intermediate representation. The proposed framework seamlessly integrates CNF clauses with structured monomials, enabling symbolic manipulation without introducing auxiliary variables or expanding into conventional ANF. By defining power terms, local rewriting rules, and a systematic product reconstruction mechanism, it establishes a unified formalism that preserves the compactness of CNF while supporting the algebraic operability of ANF. The resulting canonical representation compactly encodes disjunctive clauses and facilitates efficient local transformations, offering a new paradigm for hybrid reasoning and structure-aware bidirectional conversion between CNF and ANF.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• 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
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We introduce power term polynomial algebra, a representation language for Boolean formulae designed to bridge conjunctive normal form (CNF) and algebraic normal form (ANF). The language is motivated by the tiling mismatch between these representations: direct CNF<->ANF conversion may cause exponential blowup unless formulas are decomposed into smaller fragments, typically through auxiliary variables and side constraints. In contrast, our framework addresses this mismatch within the representation itself, compactly encoding structured families of monomials while representing CNF clauses directly, thereby avoiding auxiliary variables and constraints at the abstraction level. We formalize the language through power terms and power term polynomials, define their semantics, and show that they admit algebraic operations corresponding to Boolean polynomial addition and multiplication. We prove several key properties of the language: disjunctive clauses admit compact canonical representations; power terms support local shortening and expansion rewrite rules; and products of atomic terms can be systematically rewritten within the language. Together, these results yield a symbolic calculus that enables direct manipulation of formulas without expanding them into ordinary ANF. The resulting framework provides a new intermediate representation and rewriting calculus that bridges clause-based and algebraic reasoning and suggests new directions for structure-aware CNF<->ANF conversion and hybrid reasoning methods.

Problem

• Boolean logic
• conjunctive normal form
• algebraic normal form
• representation mismatch
• exponential blowup

Innovation

• power term polynomial algebra
• Boolean logic representation
• CNF-ANF bridging
• symbolic calculus
• structure-aware conversion