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Dynamic algebras: Examples, constructions, applications

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Abstract

Dynamic algebras combine the classes of Boolean (B ∨ ′ 0) and regular (R ∪; *) algebras into a single finitely axiomatized variety (B R ◊) resembling an R-module with “scalar” multiplication ◊. The basic result is that * is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of prepositional dynamic logic, and yet another notion of regular algebra.

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Authors and Affiliations

  1. Department of Computer Science, Stanford University, 94305, Stanford, CA, USA

    Vaughan Pratt

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  1. Vaughan Pratt
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Additional information

Dept. of Computer Science, Stanford, CA 94305 This research was supported by the National Science Foundation under NSF grant no. MCS78-04338. Preparation of the present version was supported by the National Science Foundation under NSF grant number CCR-8814921.

This paper originally appeared as Technical Memo #138, Laboratory for Computer Science, MIT, July 1979, and was subsequently revised, shortened, retitled, and published as [Pra80b]. After more than a decade of armtwisting I. Németi and H. Andréka finally persuaded the author to publish TM#138 itself on the ground that it contained a substantial body of interesting material that for the sake of brevity had been deleted from the STOC-80 version. The principal changes here to TM#138 are the addition of footnotes and the sections at the end on retrospective citations and reflections.

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Pratt, V. Dynamic algebras: Examples, constructions, applications. Stud Logica 50, 571–605 (1991). https://doi.org/10.1007/BF00370685

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