Weak Arithmetic Completeness of Object-Oriented First-Order Assertion Networks

  • Stijn de Gouw
  • Frank de Boer
  • Wolfgang Ahrendt
  • Richard Bubel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7741)

Abstract

We present a completeness proof of the inductive assertion method for object-oriented programs extended with auxiliary variables. The class of programs considered are assumed to compute over structures which include the standard interpretation of Presburger arithmetic. Further, the assertion language is first-order, i.e., quantification only ranges over basic types like that of the natural numbers, Boolean and Object.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahrendt, W., de Boer, F.S., Grabe, I.: Abstract Object Creation in Dynamic Logic. In: Cavalcanti, A., Dams, D. (eds.) FM 2009. LNCS, vol. 5850, pp. 612–627. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Apt, K.R., Bergstra, J.A., Meertens, L.G.L.T.: Recursive assertions are not enough - or are they? Theor. Comput. Sci. 8, 73–87 (1979)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Apt, K.R., de Boer, F.S., Olderog, E.-R., de Gouw, S.: Verification of object-oriented programs: A transformational approach. JCSS 78(3), 823–852 (2012)MATHGoogle Scholar
  4. 4.
    Beckert, B., Hähnle, R., Schmitt, P. (eds.): Verification of Object-Oriented Software. LNCS (LNAI), vol. 4334. Springer, Heidelberg (2007)Google Scholar
  5. 5.
    Cook, S.A.: Soundness and completeness of an axiom system for program verification. SIAM J. Comput. 7(1), 70–90 (1978)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    de Bakker, J., Meertens, L.: On the completeness of the inductive assertion method. Journal of Computer and System Sciences 11(3), 323–357 (1975)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Floyd, R.W.: Assigning meanings to programs. In: Schwartz, J.T. (ed.) Mathematical Aspects of Computer Science. Proc. of Symposia in Applied Mathematics, vol. 19, pp. 19–32. AMS (1967)Google Scholar
  8. 8.
    Gorelick, G.: A complete axiomatic system for proving assertions about recursive and non-recursive programs. Technical Report 75, Univ. of Toronto (1975)Google Scholar
  9. 9.
    Harel, D.: Arithmetical Completeness in Logics of Programs. In: Ausiello, G., Böhm, C. (eds.) ICALP 1978. LNCS, vol. 62, pp. 268–288. Springer, Heidelberg (1978)CrossRefGoogle Scholar
  10. 10.
    Lindström, P.: On extensions of elementary logic. Theoria 35(1), 1–11 (1969)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Manna, Z.: Mathematical Theory of Partial Correctness. In: Engeler, E. (ed.) Symposium on Semantics of Algorithmic Languages. Lecture Notes in Mathematics, vol. 188, pp. 252–269. Springer (1971)Google Scholar
  12. 12.
    McCarthy, J.: Towards a mathematical science of computation. In: IFIP, pp. 21–28. North-Holland (1962)Google Scholar
  13. 13.
    Olderog, E.-R.: On the notion of expressiveness and the rule of adaption. Theor. Comput. Sci. 24, 337–347 (1983)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Owicki, S.S., Gries, D.: An axiomatic proof technique for parallel programs i. Acta Inf. 6, 319–340 (1976)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Pierik, C.: Validation Techniques for Object-Oriented Proof Outlines. PhD thesis, Universiteit Utrecht (2006)Google Scholar
  16. 16.
    Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. Comptes Rendus du I congrs de Mathmaticiens des Pays Slaves, 92–101 (1929)Google Scholar
  17. 17.
    Suzuki, N., Jefferson, D.: Verification decidability of presburger array programs. J. ACM 27(1), 191–205 (1980)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Tucker, J., Zucker, J.: Program correctness over abstract data types, with error-state semantics. Elsevier Science Inc. (1988)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stijn de Gouw
    • 2
    • 3
  • Frank de Boer
    • 2
    • 3
  • Wolfgang Ahrendt
    • 1
  • Richard Bubel
    • 4
  1. 1.Chalmers UniversityGöteborgSweden
  2. 2.CWIAmsterdamThe Netherlands
  3. 3.Leiden UniversityThe Netherlands
  4. 4.Technische Universität DarmstadtGermany

Personalised recommendations