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Recursive definitions of partial functions and their computations

Authors:

J. M. Cadiou,

Zohar MannaAuthors Info & Claims

ACM SIGPLAN Notices, Volume 7, Issue 1

Pages 58 - 65

https://doi.org/10.1145/942578.807072

Published: 01 January 1972 Publication History

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Abstract

The object of this paper is to present a syntactic and semantic model for recursive definitions, and to study the relation between their computed functions and their fixpoints. The recursive definitions that we consider are syntactic generalizations of those introduced in [2] by Kleene and in [5] by McCarthy.

Each recursive definition yields two classes of fixpoint partial functions (“fixpoints over D υ {ω}” and “fixpoints over D”), and a class of computed partial functions obtained by applying different computation rules to the recursive definition. In this work we first describe the relations between the computed functions and the fixpoints over D υ {ω} (Theorem 1), and between the computed functions and the fixpoints over D (Theorem 2). Our main interest is in the class of fixpoints (over D or D υ {ω}) which are also computed functions of the recursive definition.

References

[1]

Cadiou, J.M. "Recursive definitions of partial functions and their computations". Ph.D. Thesis, Computer Science Department, Stanford University, Stanford, California (to appear).

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[2]

Kleene, S.C. "Introduction to Metamathematics", Van Nostrand, New York (1952).

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[3]

Manna, Z. and McCarthy, J. "Properties of Programs and Partial Function Logic", in Machine Intelligence 5, Edinburgh University Press (1970).

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[4]

Manna, Z. and Pnueli, A. "Formalization of Properties of Functional Programs", JACM, Vol. 17, No. 3, July 1970.

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[5]

McCarthy, J. "A basis for a mathematical theory of computation", in Computer Programming and Formal Systems, (P. Braffort and D. Hirschberg, Eds.), North Holland, Amsterdam (1963).

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[6]

Morris, J. H. Jr. "Lambda Calculus Models of Programming Languages", MAC-TR-57, MIT Project MAC, Cambridge, Mass. (1968).

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Rosen, B.K. "Tree manipulating systems and Church-Rosser Theorems", Internal Report, Engineering Sciences Laboratory, Harvard University, Cambridge, Mass. (1971). {A preliminary version of this paper appeared in the Proceedings of the Second Annual ACM Symposium on Theory of Computing, Northampton, Mass., May 1970.}

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[8]

Scott, D. "A type-theoretical alternative to CUCH, ISWIM, OWHY". Unpublished notes, Oxford University (1969).

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Cited By

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Plotkin G(2004)The origins of structural operational semanticsThe Journal of Logic and Algebraic Programming10.1016/j.jlap.2004.03.00960-61(3-15)Online publication date: Jul-2004
https://doi.org/10.1016/j.jlap.2004.03.009
Padawitz P(1982)Graph grammars and operational semanticsTheoretical Computer Science10.1016/0304-3975(82)90056-119:2(117-141)Online publication date: Aug-1982
https://doi.org/10.1016/0304-3975(82)90056-1
Padawitz P(1979)Graph grammars and operational semanticsGraph-Grammars and Their Application to Computer Science and Biology10.1007/BFb0025732(350-366)Online publication date: 1979
https://doi.org/10.1007/BFb0025732
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Index Terms

Recursive definitions of partial functions and their computations
1. Computing methodologies
  1. Artificial intelligence
    1. Knowledge representation and reasoning
      1. Logic programming and answer set programming
2. Theory of computation
  1. Models of computation
    1. Computability
      1. Recursive functions

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Published In

ACM SIGPLAN Notices Volume 7, Issue 1

Proceedings of ACM conference on Proving assertions about programs

January 1972

211 pages

ISSN:0362-1340

EISSN:1558-1160

DOI:10.1145/942578

Issue’s Table of Contents

Proceedings of ACM conference on Proving assertions about programs
January 1972
215 pages
ISBN:9781450378918
DOI:10.1145/800235

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 1972

Published in SIGPLAN Volume 7, Issue 1

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Plotkin G(2004)The origins of structural operational semanticsThe Journal of Logic and Algebraic Programming10.1016/j.jlap.2004.03.00960-61(3-15)Online publication date: Jul-2004
https://doi.org/10.1016/j.jlap.2004.03.009
Padawitz P(1982)Graph grammars and operational semanticsTheoretical Computer Science10.1016/0304-3975(82)90056-119:2(117-141)Online publication date: Aug-1982
https://doi.org/10.1016/0304-3975(82)90056-1
Padawitz P(1979)Graph grammars and operational semanticsGraph-Grammars and Their Application to Computer Science and Biology10.1007/BFb0025732(350-366)Online publication date: 1979
https://doi.org/10.1007/BFb0025732
MacLennan B(1973)Fen—an axiomatic basis for program semanticsCommunications of the ACM10.1145/355609.36231616:8(468-474)Online publication date: 1-Aug-1973
https://dl.acm.org/doi/10.1145/355609.362316
Arnold A(2011)Sémantique algébrique de l'appel par valeurRAIRO. Informatique théorique10.1051/ita/197812020069112:2(69-82)Online publication date: 8-Jan-2011
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Raymond F(2011)Note sur l'algèbre des fonctionsRevue française d'automatique informatique recherche opérationnelle. Informatique théorique10.1051/ita/197509R3002519:R3(25-49)Online publication date: 8-Jan-2011
https://doi.org/10.1051/ita/197509R300251
Sangiorgi D(2011)Origins of bisimulation and coinductionAdvanced Topics in Bisimulation and Coinduction10.1017/CBO9780511792588.002(1-37)Online publication date: 5-Nov-2011
https://doi.org/10.1017/CBO9780511792588.002
Sangiorgi D(2009)On the origins of bisimulation and coinductionACM Transactions on Programming Languages and Systems10.1145/1516507.151651031:4(1-41)Online publication date: 26-May-2009
https://dl.acm.org/doi/10.1145/1516507.1516510
Lisper B(2008)Total unfolding: theory and applicationsJournal of Functional Programming10.1017/S09567968000011554:04(479-498)Online publication date: 7-Nov-2008
https://doi.org/10.1017/S0956796800001155
Indermark KNoll T(2006)Algebraic Correctness Proofs for Compiling Recursive Function Definitions with Strictness InformationActa Informatica10.1007/s00236-006-0013-043:1(1-43)Online publication date: 15-Jun-2006
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