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On Fixpoint/Iteration/Variant Induction Principles for Proving Total Correctness of Programs with Denotational Semantics

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Book cover Logic-Based Program Synthesis and Transformation (LOPSTR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12042))

Abstract

We study partial and total correctness proof methods based on generalized fixpoint/iteration/variant induction principles applied to the denotational semantics of first-order functional and iterative programs.

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Notes

  1. 1.

    Notice that with our choice of \(\mathcal {Q}\), this is not necessarily true for chains that are not \(F_{\texttt {W}}\)-iterations.

References

  1. Apt, K.R., Plotkin, G.D.: Countable nondeterminism and random assignment. J. ACM 33(4), 724–767 (1986)

    Article  MathSciNet  Google Scholar 

  2. de Bakker, J.W., Scott, D.S.: A theory of programs. IBM Seminar Vienna, Austria, August 1969. Unpublished notes

    Google Scholar 

  3. Burstall, R.M.: Program proving as hand simulation with a little induction. In: IFIP Congress, pp. 308–312. North-Holland (1974)

    Google Scholar 

  4. Clarke Jr., E.M.: Programming language constructs for which it is impossible to obtain good Hoare axiom systems. J. ACM 26(1), 129–147 (1979)

    Article  MathSciNet  Google Scholar 

  5. Cook, S.A.: Soundness and completeness of an axiom system for program verification. SIAM J. Comput. 7(1), 70–90 (1978)

    Article  MathSciNet  Google Scholar 

  6. Cook, S.A.: Corrigendum: soundness and completeness of an axiom system for program verification. SIAM J. Comput. 10(3), 612 (1981)

    Article  MathSciNet  Google Scholar 

  7. Cousot, P.: Méthodes itératives de construction et d’approximation de points fixes d’opérateurs monotones sur un treillis, analyse sémantique de programmes. Thèse d’État ès sciences mathématiques, Université de Grenoble Alpes, Grenoble, France, 21 March 1978. (in French)

    Google Scholar 

  8. Cousot, P., Cousot, R.: Constructive versions of Tarski’s fixed point theorems. Pac. J. Math. 82(1), 43–57 (1979)

    Article  MathSciNet  Google Scholar 

  9. Cousot, P., Cousot, R.: Induction principles for proving invariance properties of programs. In: Néel, D. (ed.) Tools & Notions for Program Construction: An Advanced Course, pp. 75–119. Cambridge University Press, Cambridge (1982)

    Google Scholar 

  10. Cousot, P., Cousot, R.: “A la burstall” intermittent assertions induction principles for proving inevitable ability properties of programs. Theor. Comput. Sci. 120(1), 123–155 (1993)

    Article  Google Scholar 

  11. Damm, W., Josko, B.: A sound and relatively * complete Hoare-logic for a language with higher type procedures. Acta Inf. 20, 59–101 (1983). https://doi.org/10.1007/BF00264295

    Article  MathSciNet  MATH  Google Scholar 

  12. Floyd, R.W.: Assigning meaning to programs. In: Schwartz, J. (ed.) Proceedings of a Symposium in Applied Mathematics, vol. 19, pp. 19–32. American Mathematical Society (1967)

    Google Scholar 

  13. Heizmann, M., Jones, N.D., Podelski, A.: Size-change termination and transition invariants. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 22–50. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15769-1_4

    Chapter  Google Scholar 

  14. Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12(10), 576–580 (1969). http://doi.acm.org/10.1145/363235.363259

    Article  Google Scholar 

  15. Katz, S., Manna, Z.: A closer look at termination. Acta Inf. 5, 333–352 (1975). https://doi.org/10.1007/BF00264565

    Article  MathSciNet  MATH  Google Scholar 

  16. Kleene, S.C.: Introduction to Meta-Mathematics. Elsevier North-Holland Pub. Co., Amsterdam (1952)

    Google Scholar 

  17. Lee, C.S., Jones, N.D., Ben-Amram, A.M.: The size-change principle for program termination. In: POPL, pp. 81–92. ACM (2001)

    Google Scholar 

  18. Leroy, X., Doligez, D., Frisch, A., Garrigue, J., Rémy, D., Vouillon, J.: The OCaml system, release 4.08, Documentation and user’s manual, February 2019. http://caml.inria.fr/pub/docs/manual-ocaml/, copyright \(\copyright \) 2013 Institut National de Recherche en Informatique et en Automatique

  19. Manna, Z., Ness, S., Vuillemin, J.: Inductive methods for proving properties of programs. Commun. ACM 16(8), 491–502 (1973)

    Article  MathSciNet  Google Scholar 

  20. Manna, Z., Pnueli, A.: Axiomatic approach to total correctness of programs. Acta Inf. 3, 243–263 (1974). https://doi.org/10.1007/BF00288637

    Article  MathSciNet  MATH  Google Scholar 

  21. Manna, Z., Vuillemin, J.: Fix point approach to the theory of computation. Commun. ACM 15(7), 528–536 (1972)

    Article  Google Scholar 

  22. Markowsky, G.: Chain-complete posets and directed sets with applications. Algebra Univ. 6(1), 53–68 (1976)

    Article  MathSciNet  Google Scholar 

  23. Park, D.: On the semantics of fair parallelism. In: Bjøorner, D. (ed.) Abstract Software Specifications. LNCS, vol. 86, pp. 504–526. Springer, Heidelberg (1980). https://doi.org/10.1007/3-540-10007-5_47

    Chapter  Google Scholar 

  24. Régis-Gianas, Y., Pottier, F.: A Hoare logic for call-by-value functional programs. In: Audebaud, P., Paulin-Mohring, C. (eds.) MPC 2008. LNCS, vol. 5133, pp. 305–335. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70594-9_17

    Chapter  Google Scholar 

  25. Schmidt, D.W.: Denotational Semantics: A Methodology for Language Development. William C. Brown Publishers, Dubuque (1988). http://people.cs.ksu.edu/~schmidt/text/DenSem-full-book.pdf

    Google Scholar 

  26. Scott, D.S.: Outline of a mathematical theory of computation. In: Proceedings of the Fourth Annual Princeton Conference on Information Sciences and Systems, pp. 169–176. Princeton University, March 1970

    Google Scholar 

  27. Scott, D.: The lattice of flow diagrams. In: Engeler, E. (ed.) Symposium on Semantics of Algorithmic Languages. LNM, vol. 188, pp. 311–366. Springer, Heidelberg (1971). https://doi.org/10.1007/BFb0059703

    Chapter  Google Scholar 

  28. Tarski, A.: A lattice theoretical fixpoint theorem and its applications. Pac. J. Math. 5, 285–310 (1955)

    Article  MathSciNet  Google Scholar 

  29. Turing, A.: Checking a large routine. In: Report of a Conference on High Speed Automatic Calculating Machines, University of Cambridge Mathematical Laboratory, Cambridge, England, pp. 67–69 (1949). http://www.turingarchive.org/browse.php/b/8

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Acknowledgements

I thank Francesco Ranzato for comments and suggestions on earlier versions of the paper. Work supported in part by NSF Grant CCF-1617717.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, USA

    Patrick Cousot

  2. IMDEA Software Institute, Madrid, Spain

    Patrick Cousot

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  1. Patrick Cousot
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Correspondence to Patrick Cousot .

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Editors and Affiliations

  1. University of Bologna, Bologna, Italy

    Maurizio Gabbrielli

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Cousot, P. (2020). On Fixpoint/Iteration/Variant Induction Principles for Proving Total Correctness of Programs with Denotational Semantics. In: Gabbrielli, M. (eds) Logic-Based Program Synthesis and Transformation. LOPSTR 2019. Lecture Notes in Computer Science(), vol 12042. Springer, Cham. https://doi.org/10.1007/978-3-030-45260-5_1

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  • DOI: https://doi.org/10.1007/978-3-030-45260-5_1

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  • Online ISBN: 978-3-030-45260-5

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