The algebra of non-deterministic programs: demonic operations, orders and axioms
Hirsch, R. and Mikulas, Szabolcs and Stokes, T. (2022) The algebra of non-deterministic programs: demonic operations, orders and axioms. Logic Journal of the IGPL 30 (5), pp. 886-906. ISSN 1367-0751.
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Abstract
Demonic composition, demonic refinement and demonic union are alternatives to the usual “angelic” composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and demonic via an analysis of the beha- viour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non- deterministic programs incorporating both aspects. Zarecki ̆ı has shown that the isomorphism class of algebras of binary relations under angelic composition and inclusion is finitely axio- matised as the class of ordered semigroups. The proof can be used to establish that the same axiomatisation applies to binary relations under demonic composition and refinement, and a further modification of the proof can be used to incorporate a zero element representing the empty relation in the angelic case and the full relation in the demonic case. For the signature of angelic composition and union, it is known that no finite axiomatisation exists, and we show the analogous result for demonic composition and demonic union by showing that the same axiomatisation holds for both. We show that the isomorphism class of algebras of binary relations with the “mixed” signature of demonic composition and angelic inclusion has no finite axiomatisation. As a contrast, we show that the isomorphism class of partial algebras of binary relations with the partial operation of constellation product and inclusion (also a “mixed” signature) is finitely axiomatisable.
Metadata
| Item Type: | Article |
|---|---|
| Additional Information: | This is a pre-copyedited, author-produced PDF of an article accepted for publication following peer review. The version of record is available online at the link above. |
| School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |
| Depositing User: | Szabolcs Mikulas |
| Date Deposited: | 08 Nov 2021 13:54 |
| Last Modified: | 09 Aug 2023 12:51 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/46100 |
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