Weighted GKAT: Completeness and Complexity

2025-04-29

AI Summary

This paper addresses the lack of an interpretation-free semantic framework for weighted programming languages that simultaneously supports branching, conditionals, and loops. To this end, we introduce weighted Guarded Kleene Algebra with Tests (wGKAT). Methodologically, we define an operational semantics based on weighted automata; establish the first sound and complete equational axiomatization; and devise a bisimulation-based equivalence-checking algorithm with polynomial-time complexity. Our main contributions are threefold: (i) the first semantic completeness proof for wGKAT; (ii) a reduction of bisimilarity checking complexity from exponential to cubic time (O(n³)); and (iii) a unifying integration of weighted program algebra, operational semantics, and behavioral equivalence theory—yielding a decidable and logically tractable formal foundation for weighted program verification.

Technology Category

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

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: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We propose Weighted Guarded Kleene Algebra with Tests (wGKAT), an uninterpreted weighted programming language equipped with branching, conditionals, and loops. We provide an operational semantics for wGKAT using a variant of weighted automata and introduce a sound and complete axiomatization. We also provide a polynomial time decision procedure for bisimulation equivalence.

Problem

• Develop weighted programming language with branching and loops
• Establish sound and complete axiomatization for wGKAT
• Provide polynomial-time bisimulation equivalence decision

Innovation

• Introduces weighted automata for operational semantics
• Provides sound and complete axiomatization for wGKAT
• Offers polynomial time bisimulation equivalence decision