On Tools for Completeness of Kleene Algebra with Hypotheses
In the literature on Kleene algebra, a number of variants have been proposed which impose additional structure specified by a theory, such as Kleene algebra with tests (KAT) and the recent Kleene algebra with observations (KAO), or make specific assumptions about certain constants, as for instance i...
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| Published in: | Logical methods in computer science Vol. 20, Issue 2 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science Association
16.05.2024
Logical Methods in Computer Science e.V |
| Subjects: | |
| ISSN: | 1860-5974, 1860-5974 |
| Online Access: | Get full text |
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| Summary: | In the literature on Kleene algebra, a number of variants have been proposed
which impose additional structure specified by a theory, such as Kleene algebra
with tests (KAT) and the recent Kleene algebra with observations (KAO), or make
specific assumptions about certain constants, as for instance in NetKAT. Many
of these variants fit within the unifying perspective offered by Kleene algebra
with hypotheses, which comes with a canonical language model constructed from a
given set of hypotheses. For the case of KAT, this model corresponds to the
familiar interpretation of expressions as languages of guarded strings. A
relevant question therefore is whether Kleene algebra together with a given set
of hypotheses is complete with respect to its canonical language model. In this
paper, we revisit, combine and extend existing results on this question to
obtain tools for proving completeness in a modular way. We showcase these tools
by giving new and modular proofs of completeness for KAT, KAO and NetKAT, and
we prove completeness for new variants of KAT: KAT extended with a constant for
the full relation, KAT extended with a converse operation, and a version of KAT
where the collection of tests only forms a distributive lattice. |
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| ISSN: | 1860-5974 1860-5974 |
| DOI: | 10.46298/lmcs-20(2:8)2024 |