The paper revisits, combines and extends existing results on the completeness of Kleene algebra with hypotheses. It develops a toolbox of techniques to construct reductions from one set of hypotheses to another, which can then be used to prove completeness in a modular way.
The key contributions are:
A general theory of reductions between Kleene algebra with hypotheses, building on the notion of closure under hypotheses. Reductions allow reducing completeness of one set of hypotheses to completeness of another.
A collection of primitive reductions for common sets of hypotheses, and lemmas to compose these reductions in a modular fashion.
New and modular proofs of completeness for several variants of Kleene algebra, including Kleene algebra with tests (KAT), Kleene algebra with observations (KAO), and NetKAT.
Proofs of completeness for new variants of Kleene algebra, such as KAT extended with a full relation constant or a converse operation, and a version of KAT where tests form only a distributive lattice.
The paper first introduces the general framework of Kleene algebra with hypotheses and the notion of reductions (Sections 2-3). It then showcases the tools by proving completeness for KAT (Section 4), before developing more advanced composition techniques (Section 5) and applying them to various examples (Sections 6-10).
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arxiv.org
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