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Sound and complete Hoare-like calculi based on copy rules

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Summary

This paper presents a uniform approach to known and new results on relative completeness of Hoare-like calculi for languages of ALGOL-like programs with procedures as procedure parameters. First the notion of a copy rule is introduced. It provides a uniform framework for dealing with different variants of semantics reaching from dynamic to static scope. Then for each copy rule ℒ a Hoare-like calculus ℋ(ℒ) is presented, the soundness of which is shown by using an approximating semantics. The key to the completeness results lies in a general completeness theorem on the calculi ℋ(ℒ) which has these results as corollaries. Finally, a new type of theorem on Hoare-like calculi is proved by which the notion of formal provability in ℋ(ℒ) is completely characterized. This characterization theorem is the main result of the paper. It covers both soundness and completeness of the calculi ℋ(ℒ) and additionally gives an idea of what the limits of presently known Hoare-like proof techniques for programming languages with procedures are.

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References

  1. Apt, K.R.: A sound and complete Hoare-like system for a fragment of PASCAL. Report IW 96/78. Mathematisch Centrum, 1978

  2. Apt, K.R.: Ten years of Hoare's logic, a survey, part I. Tech. Report, Faculty of Economics, Erasmus Univ., Rotterdam, April 1979 ACM TOPLAS (in press, 1981)

    Google Scholar 

  3. Apt, K.R., de Bakker, J.W.: Semantics and proof theory of PASCAL procedures. In: Proc. 4th Coll. Automata, Languages and Programming (A. Salomaa, M. Steinby, eds.). Lecture Notes in Computer Science Vol. 52, pp. 30–44. Berlin-Heidelberg-New York: Springer 1977

    Google Scholar 

  4. Backus, J.W., Bauer, F.L., Green, J., Katz, C., McCarthy, J., Naur, P., Perlis, A.J., Rutishauser, H., Samelson, K., Vauquois, B., Wegstein, J.H., van Wijngaarden, A., Woodger, M.: Revised report on the algorithmic language ALGOL 60 (P. Naur, ed.) Numer. Math. 4, 420–453 (1963)

  5. de Bakker, J.W.: Semantics and the foundation of program proving. In: Proc. IFIP Congress 1977, Toronto, pp. 279–284. Amsterdam: North Holland 1977

    Google Scholar 

  6. de Bakker, J.W.: Mathematical theory of programm correctness. London: Prentice Hall International 1980

    Google Scholar 

  7. Bergstra, J.A., Tucker, J.V.: Some natural structures which fail to possess a sound and decidable Hoare-like logic for their while-programs. Theor. Comput. Sci. (in press, 1981)

  8. Cartwright, R., Oppen, D.: Unrestricted procedure calls in Hoare's logic. In: Proc. of the Fith Annual ACM Symposium on Principles of Programming Languages, Tucson, Arizona, pp. 131–140. New York: ACM 1978

    Google Scholar 

  9. Clarke, E.M.: Programming language constructs for which it is impossible to obtain good Hoarelike axioms. J. Assoc. Comput. Mach. 26, 129–147 (1979)

    Google Scholar 

  10. Cook, S.A.: Soundness and completeness of an axiom system for program verification. SIAM J. Comput. 7, 70–90 (1978)

    Google Scholar 

  11. Dijkstra, E.W.: Recursive programming. Numer. Math. 2, 312–318 (1960)

    Google Scholar 

  12. Donahue, J.E.: Complementary definitions of programming language semantics. Lecture Notes in Computer Science Vol. 42. Berlin-Heidelberg-New York: Springer 1976

    Google Scholar 

  13. Gorelick, G.A.: A complete axiomatic system for proving assertions about recursive and nonrecursive programs. Tech. Rep. 75, Dept. of Computer Sci., Univ. of Toronto (1975)

  14. Grau, A.A., Hill, U., Langmaack, H.: Translation of ALGOL 60. Handbook for Automatic Computation, Vol. I, Part b. (Grundlehren der mathematischen Wissenschaften, Band 137) (K. Samelson, ed.) Berlin-Heidelberg-New York: Springer 1967

    Google Scholar 

  15. Gries, D., Levin, G.: Assignment and procedure call proof rules. ACM TOPLAS 2, 564–579 (1980)

    Google Scholar 

  16. Guttag, J.V., Horning, J.J., London, R.L.: A proof rule for EUCLID procedures. In: Formal description of Programming Concepts (E.J. Neuhold, ed.), pp. 211–220. Amsterdam: North Holland 1978

    Google Scholar 

  17. Harel, D., Pnueli, A., Stavi, J.: A complete axiomatic system for proving deductions about recursive programs. In: Proc. of the 9th ACM Symposium on Theory of Computing, pp. 249–260. New York: ACM 1977

    Google Scholar 

  18. Hoare, C.A.R.: An axiomatic basis for computer programming. Comm. ACM 12, 576–580, 583 (1969)

    Google Scholar 

  19. Hoare, C.A.R.: Procedures and parameters: An axiomatic approach. In: Symposium on semantics of algorithmic languages (E. Engeler, ed.). Lecture Notes in Mathematics 188, pp. 102–116. Berlin-Heidelberg-New York: Springer 1971

    Google Scholar 

  20. Igarashi, S., London, R.L., Luckham, D.C.: Automatic program verfication I: A logical basis and its implementation. Acta Informat. 4, 145–182 (1975)

    Google Scholar 

  21. Jensen, K., Wirth, N: PASCAL user manual and report. Berlin-Heidelberg-New York: Springer 1975

    Google Scholar 

  22. Kandzia, P.: On the “most recent” property of ALGOL-like programs. In: Proc. 2nd Coll. Automata, Languages and Programming (J. Loeckx, ed.). Lecture Notes in Computer Science Vol. 14, pp. 97–111. Berlin-Heidelberg-New York: Springer 1974

    Google Scholar 

  23. Kandzia, P., Langmaack, H.: On a theorem of McGowan concerning the “most recent” property of programs. Bericht Nr. A 74/07 des Fachbereichs Angew. Math. und Inform., Univ. d. Saarlandes (1974)

  24. Klein, H.-J.: Untersuchungen im Zusammenhang mit der “most recent” Eigenschaft von Pro-Programmen. Bericht Nr. 8/77 des Inst. f. Inform. u. Prakt. Math., Christian-Albrechts-Univ., Kiel 1977

    Google Scholar 

  25. Klein, H.-J.: Zur Charakterisierung von Programmen mit endlichen Arten. Dissertation, Christian-Albrechts-Univ., Kiel 1980

    Google Scholar 

  26. Langmaack, H.: On correct procedure parameter transmission in higher programming languages. Acta Informat. 2, 110–142 (1973)

    Google Scholar 

  27. Langmaack, H.: On procedures as open subroutines I, II. Acta Informat. 2, 311–333 (1973) and Acta Informat. 3, 227–241 (1974)

    Google Scholar 

  28. Langmaack, H.: On a theory of decision problems in programming languages. In: Proc. of the Int. Conf. on Math. Studies of Inform. Processing, Kyoto (E.K. Blum, S. Takasu, eds.). Lecture Notes in Computer Science Vol. 75, pp. 538–558. Berlin-Heidelberg-New York: Springer 1969

    Google Scholar 

  29. Langmaack, H.: On termination problems for finitely interpreted ALGOL-like programs. Bericht Nr. 7904 des Inst. f. Inform, u. Prakt. Math., Christian-Albrechts-Univ., Kiel, 1979

    Google Scholar 

  30. Langmaack, H.: A proof of a theorem of Lipton on Hoare logic and applications. Bericht Nr. 8003 des Inst. f. Inform, u. Prakt. Math., Christian-Albrechts-Univ., Kiel, 1980

    Google Scholar 

  31. Langmaack, H., Olderog, E.-R.: Present-day Hoare-like systems for programming languages with procedures: power, limits and most likely extensions. In: Proc. 7th Coll. Automata, Languages and Programming (J.W. de Bakker, J. van Leeuwen, eds.). Lecture Notes in Computer Science Vol. 85, pp. 363–373. Berlin-Heidelberg-New York: Springer 1980

    Google Scholar 

  32. Lauer, P.E.: Consistent formal theories of the semantics of programming languages. Tech. Rep. TR 25.121, IBM Laboratory, Vienna 1971

    Google Scholar 

  33. Lipton, R.J.: A necessary and sufficient condition for the existence of Hoare logics. In: 18th IEEE Symposium on Foundations of Computer Science, Providence, Rhode Island, pp. 1–6. New York: IEEE 1977

    Google Scholar 

  34. London, R.L., Guttag, J.V., Horning, J.J., Lampson, B.W., Mitchell, J.G., Popek, G.J.: Proof rules for the programming language EUCLID. Acta Informat. 10, 1–26 (1978)

    Google Scholar 

  35. McGowan, C.L.: The “most recent” error: its causes and correction. SIGPLAN Notices 7, 191–202 (1972)

    Google Scholar 

  36. O'Donnell, M.J.: A critique of the foundations of Hoare-style programming logic. Tech. Report, Computer Sci. Dept., Purdue Univ., West-Lafayette, 1980

    Google Scholar 

  37. Olderog, E.-R.: Korrektheits- und Vollständigkeitsaussagen über Hoaresche Ableitungskalküle. Diplomarbeit, Christian-Albrechts-Univ., Kiel 1979

    Google Scholar 

  38. Prawitz, D.: Natural deduction — a proof-theoretic study. Stockholm: Almqvist and Wiksell 1965

    Google Scholar 

  39. Randell, B., Russell, L.J.: ALGOL 60 implementation. London: Academic Press, 1964

    Google Scholar 

  40. Schwartz, R.L.: An axiomatic definition of ALGOL 68. Comp. Sci. Dept., UCLA-34P214-75, Univ. of California, Los Angeles, 1978

    Google Scholar 

  41. Schwartz, R.L.: On axiomatizability as a language design tool. Dept. of Applied Maths., Weizmann Inst. of Sci., Rehovot, January 1979

    Google Scholar 

  42. Schwarz, J.: Generic commands — a tool for partial correctness formalism. Comput. J. 20, 151–155 (1977)

    CAS  PubMed  Google Scholar 

  43. Sieber, K.: A new Hoare-calculus for programs with recursive parameterless procedures. Bericht A 81/02 des Fachbereichs Angew. Math. u. Inform., Univ. d. Saarlandes, Saarbrücken 1981

    Google Scholar 

  44. Wand, M.: A new incompleteness result for Hoare's system. J. Assoc. Comput. Mach. 25, 168–175 (1978)

    Google Scholar 

  45. van Wijngaarden, A., Mailloux, B.J., Peck, J.E.L., Koster, C.H.A., Sintzoff, M., Lindsey, C.H., Meertens, L.G.L.T., Fisker, R.G. (eds.): Revised report on the algorithmic language ALGOL 68. Acta Informat. 5, 1–236 (1975)

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  1. Institut für Informatik und Praktische Mathematik, Christian-Albrechts-Universität Kiel, Olshausenstr. 40-60, D-2300, Kiel 1, Germany

    Ernst-Rüdiger Olderog

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Olderog, ER. Sound and complete Hoare-like calculi based on copy rules. Acta Informatica 16, 161–197 (1981). https://doi.org/10.1007/BF00261258

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