Skip to main content
SpringerLink
Log in

Partial closures and semantics of while: Towards an iteration-based theory of data types

  • Conference paper
  • First Online:
Computation and Proof Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1104))

Abstract

The present paper proposes first a generalization of closure theory and revisits Moore's theory in this framework. Afterwards closures of non cyclic functions are introduced and a method is given to transform cyclic into non cyclic functions. Eventually semantics of the while construct is found to be the closure of a function. Computability on inductive and non inductive data types is then studied with iterative means.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Birkhoff, Lattice theory, AMS Colloq. Publ. 25, Providence 1967.

    Google Scholar 

  2. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, New York 1981.

    Google Scholar 

  3. P. M. Cohn, Universal Algebra, New York 1965.

    Google Scholar 

  4. J. de Bakker, Mathematical Theory of Program Correctness, Englewood Cliffs 1980.

    Google Scholar 

  5. R. Dedekind, Was sind und was sollen die Zahlen, Braunschweig 1888.

    Google Scholar 

  6. G. Germano and A. Maggiolo-Schettini, Equivalence of partial recursivity and computability by algorithms without concluding formulas, Calcolo 8 (1971), 273–292.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Germano and A. Maggiolo-Schettini, Sequence-to-sequence recursiveness, Inform. Processing Lett. 4 (1975), 1–6.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Germano and A. Maggiolo-Schettini, Computable stack functions for semantics of stack programs, J. Comput. System Sci. 19 (1979), 133–144.

    Article  MathSciNet  MATH  Google Scholar 

  9. G. Germano and A. Maggiolo-Schettini, Sequence recursiveness without cylindrification and limited register machines, Theor. Comput. Sci. 15 (1981), 213–221.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. D. Gladstone, Simplifications of the recursion scheme, J. Symbolic Logic 36 (1971), 653–665.

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatsh. Math. Phys. 38 (1931), 173–198.

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Gödel, On undecidable propositions of formal mathematical systems, Mimeography, Princeton 1934.

    Google Scholar 

  13. G. Grätzer, Universal Algebra, New York 1968.

    Google Scholar 

  14. H. Hermes, Aufzählbarkeit, Entscheidbarkeit, Berechenbarkeit, Berlin 1961.

    Book  MATH  Google Scholar 

  15. S. C. Kleene, General recursive functions of natural numbers, Math. Ann. 112 (1936), 727–742.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. C. Kleene, λ-definability and recursiveness, Duke Math. J. 2 (1936), 340–353.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. C. Kleene, A note on recursive functions, Bull. Amer. Math. Soc. 42 (1936), 544–546.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. C. Kleene, On notation for ordinal numbers, J. Simbolic logic 3 (1938), 150–155.

    Article  MATH  Google Scholar 

  19. S. C. Kleene, Introduction to Metamathematics, Amsterdam 1952.

    Google Scholar 

  20. S. C. Kleene, The theory of recursive functions approaching its centennial, Bull. Amer. Math. Soc. 5 (1981), 43–60.

    Article  MathSciNet  MATH  Google Scholar 

  21. Z. Manna, Mathematical Theory of Computation, New York 1974.

    Google Scholar 

  22. J. C. C. Mc Kinsey and A. Tarski, The algebra of topology, Ann. of Math. 45 (1944), 141–191.

    Article  MathSciNet  Google Scholar 

  23. J. C. C. Mc Kinsey and A. Tarski, On closed elements in closure algebras, Ann. of Math. 47 (1946), 122–162.

    Article  MathSciNet  Google Scholar 

  24. E. H. Moore, Introduction to a form of a general analysis, AMS Colloq. Publ. 2, New Haven 1910.

    Google Scholar 

  25. R. M. Robinson, Primitive recursive functions, Bull. Amer. Math. Soc. 53 (1947), 925–942.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Robinson, General recursive functions, Bull. Amer. Math. Soc. 56 (1950), 703–717.

    MathSciNet  MATH  Google Scholar 

  27. D. S. Scott. The Lattice of Flow Diagrams, Technical Monograph PRG 3, Oxford Univ. Computing Laboratory, Oxford 1970.

    Google Scholar 

  28. D. S. Scott, Lectures on a Mathematical Theory of Computation, Oxford Univ. PRG Tech. Monograph 1981.

    Google Scholar 

  29. J. E. Stoy, Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory, Cambridge, MA 1977.

    Google Scholar 

  30. A. Tarski, Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften I, Monatsh. Math. Phys. 37 (1930), 360–404.

    MathSciNet  Google Scholar 

  31. A. Tarski, A. Mostowski and R. M. Robinson, Undecidable Theories, Amsterdam 1953.

    Google Scholar 

  32. P. C. Treleaven, D. R. Brownbridge and R. P. Hopkins, Data-driven and demand-driven computer architecture, ACM Computing Surveys 14 (1982), 93–143.

    Article  Google Scholar 

  33. T. R. Walsh, Iteration strikes back-at the cyclic towers of Hanoi, Inform. Processing Lett. 16 (1983), 91–93.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipart. Informatica Università di Pisa, Corso Italia, 40, 56100, Pisa, Italia

    G. Germano & S. Mazzanti

Authors
  1. G. Germano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Mazzanti
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Egon Börger Walter Oberschelp Michael M. Richter Brigitta Schinzel Wolfgang Thomas

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Germano, G., Mazzanti, S. (1984). Partial closures and semantics of while: Towards an iteration-based theory of data types. In: Börger, E., Oberschelp, W., Richter, M.M., Schinzel, B., Thomas, W. (eds) Computation and Proof Theory. Lecture Notes in Mathematics, vol 1104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099485

Download citation

  • DOI: https://doi.org/10.1007/BFb0099485

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13901-0

  • Online ISBN: 978-3-540-39119-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation