Completeness of Finitely Weighted Kleene Algebra With Tests

TL;DR

This paper extends the completeness results of finitely weighted Kleene algebra to include tests, providing a formal framework for reasoning about weighted programs with bounded weights.

Contribution

It establishes relational and language completeness for finitely weighted Kleene algebra with tests under new assumptions, including integrality of the weight semiring.

Findings

Proves relational and language completeness results.

Shows the framework is suitable for reasoning about weighted programs.

Extends previous completeness results to include tests.

Abstract

Building on \'Esik and Kuich's completeness result for finitely weighted Kleene algebra, we establish relational and language completeness results for finitely weighted Kleene algebra with tests. Similarly as \'Esik and Kuich, we assume that the finite semiring of weights is commutative, partially ordered and zero-bounded, but we also assume that it is integral. We argue that finitely weighted Kleene algebra with tests is a natural framework for equational reasoning about weighted programs in cases where an upper bound on admissible weights is assumed.