article

Abstract computability and algebraic specification

Authors:

J. I. ZuckerAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 3, Issue 2

Pages 279 - 333

https://doi.org/10.1145/505372.505375

Published: 01 April 2002 Publication History

Abstract

Abstract computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable functions on any many-sorted algebra; (ii) all functions effectively approximable by abstract computable functions on any metric algebra. We show that there exist universal algebraic specifications for all the classically computable functions on the set ℝ of real numbers. The algebraic specifications used are mainly bounded universal equations and conditional equations. We investigate the initial algebra semantics of these specifications, and derive situations where algebraic specifications precisely define the computable functions.

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Kasymov NKarimova NKhoussainov BSobocinski PLago UEsparza J(2024)Defining algorithmically presented structures in first order logicProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662120(1-13)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662120
Tucker JZucker J(2017)Infinitary initial algebra specifications for stream algebrasReflections on the Foundations of Mathematics10.1017/9781316755983.012(228-248)Online publication date: 31-Mar-2017
https://doi.org/10.1017/9781316755983.012
Thompson BTucker JZucker J(2009)Unifying computers and dynamical systems using the theory of synchronous concurrent algorithmsApplied Mathematics and Computation10.1016/j.amc.2009.04.058215:4(1386-1403)Online publication date: 1-Oct-2009
https://dl.acm.org/doi/10.1016/j.amc.2009.04.058
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Index Terms

Abstract computability and algebraic specification
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Theorem proving algorithms
2. Theory of computation
  1. Logic
    1. Proof theory
  2. Models of computation
    1. Computability

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cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 3, Issue 2

April 2002

157 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/505372

Issue’s Table of Contents

Copyright © 2002 ACM.

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Publication History

Published: 01 April 2002

Published in TOCL Volume 3, Issue 2

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Cited By

Kasymov NKarimova NKhoussainov BSobocinski PLago UEsparza J(2024)Defining algorithmically presented structures in first order logicProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662120(1-13)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662120
Tucker JZucker J(2017)Infinitary initial algebra specifications for stream algebrasReflections on the Foundations of Mathematics10.1017/9781316755983.012(228-248)Online publication date: 31-Mar-2017
https://doi.org/10.1017/9781316755983.012
Thompson BTucker JZucker J(2009)Unifying computers and dynamical systems using the theory of synchronous concurrent algorithmsApplied Mathematics and Computation10.1016/j.amc.2009.04.058215:4(1386-1403)Online publication date: 1-Oct-2009
https://dl.acm.org/doi/10.1016/j.amc.2009.04.058
Strahm TZucker J(2008)Primitive recursive selection functions for existential assertions over abstract algebrasThe Journal of Logic and Algebraic Programming10.1016/j.jlap.2008.02.00276:2(175-197)Online publication date: Jul-2008
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Yap C(2008)Theory of Real Computation According to EGCReliable Implementation of Real Number Algorithms: Theory and Practice10.1007/978-3-540-85521-7_12(193-237)Online publication date: 1-Mar-2008
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Zucker J(2006)Primitive recursive selection functions over abstract algebrasProceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers10.1007/11780342_61(595-606)Online publication date: 30-Jun-2006
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Chen HDong Y(2006)Towards practical computable functions on context-free languagesProceedings of the Third international conference on Theory and Applications of Models of Computation10.1007/11750321_53(555-565)Online publication date: 15-May-2006
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Xu JZucker J(2005)First and Second Order Recursion on Abstract Data TypesFundamenta Informaticae10.5555/2370476.237048067:4(377-419)Online publication date: 1-Dec-2005
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