Correctness of programs over poor signatures

  • Hardi Hungar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 560)


A poor signature consists of one (unary) function symbol, a symbol for equality and a number of constant symbols. Due to the lack of internal structure, interpretations of poor signatures differ from interpretations of richer signatures in some respects. One of them is the completeness of Hoare-like systems relative to the first-order theory of the interpretation.

We demonstrate that Hoare-like proof systems for partial correctness are incomplete in any infinite (and not uniformly locally finite) interpretation of a poor signature, even if the systems are supplied with the first-order theory of the interpretation. The reason is that there are always relations which are definable by while-programs but not by formulas. This is shown by an application of a result used in model theory. We also answer the question which enrichments of the signature are necessary and sufficient to allow (nontrivial) infinite interpretations where Hoare-like systems are complete.


Function Symbol Proof System Boolean Formula Relation Symbol Constant Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    Apt, K.R. Ten years of Hoare's logic: A survey—part I. ACM TOPLAS 3 (1981) 431–483.CrossRefGoogle Scholar
  2. [BT1]
    Bergstra, J. A. and Tucker, J. V. Some natural structures which fail to possess a sound and decidable Hoare-like logic for their while-programs. TCS 17 (1982) 303–315.CrossRefGoogle Scholar
  3. [BT2]
    Bergstra, J. A. and Tucker, J. V. Hoare 's logic for programming languages with two data types. TCS 24 (1984) 215–221.Google Scholar
  4. [CGH]
    Clarke, E. M., German, S. M. and Halpern, J. Y. Effective axiomatizations of Hoare logics. JACM 30 (1983) 612–636.CrossRefGoogle Scholar
  5. [Co]
    Cook, S. A. Soundness and completeness of an axiom system for program verification. SIAM J. Comp. 7 (1978) 70–90.CrossRefGoogle Scholar
  6. [F]
    Friedman, H. Generalized Turing algorithms, and elementary recursion theory. in: Gandy and Yates (eds) Logic Colloquium '69, North-Holland, Amsterdam (1971) 361–390.Google Scholar
  7. [G]
    Gaifman, H. Finiteness is not a Σ 0-property. Isr. J. Math. 19 (1974) 359–368.Google Scholar
  8. [GeHa]
    German, S. M. and Halpern, J. Y. On the power of the hypothesis of expressiveness. IBM Res. Rep. RJ 4079 (45457) Yorktown Heights (1983).Google Scholar
  9. [GrHu]
    Grabowski, M. and Hungar, H. On the existence of effective Hoare logics. 3rd LiCS (1988) 428–435.Google Scholar
  10. [H]
    Harel, D. First-order dynamic logic. Springer LNCS 68 New York (1979).Google Scholar
  11. [HP]
    Harel, D. and Peleg, D. On static logics, dynamic logics, and complexity classes. Inf. and Contr. 60 (1984) 86–102.CrossRefGoogle Scholar
  12. [KU]
    Kfoury, A. J. and Urzyczyn, P. Necessary and sufficient conditions for the universality of programming formalisms. Acta Inf. 22 (1985) 347–377.CrossRefGoogle Scholar
  13. [Li]
    Lipton, R. J. A necessary and sufficient condition for the existence of Hoare logics. 18th FoCS (1977) 1–6.Google Scholar
  14. [LS]
    Loeckx, J. and Sieber, K. The foundation of program verification. 2nd ed., Wiley-Teubner (1987).Google Scholar
  15. [M]
    Marcus, L. Minimal models of theories of one function symbol. Isr. J. Math. 18 (1974) 117–131.Google Scholar
  16. [0]
    Olderog, E.-R. On the notion of expressiveness and the rule of adaptation. TCS 24 (1983) 337–347CrossRefGoogle Scholar
  17. [TU]
    Tiuryn, J. and Urzyczyn, P. Some relationships between logics of programs and complexity theory. TCS 60 (1988) 83–108.CrossRefGoogle Scholar
  18. [U]
    Urzyczyn, P. A necessary and sufficient condition in order that a Herbrand interpretation is expressive with respect to recursive programs. Inf. and Contr. 56 (1983) 212–219.CrossRefGoogle Scholar
  19. [W]
    Wand, M. A new incompleteness result for Hoare's system. JACM 25 (1978) 168–175.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Hardi Hungar
    • 1
  1. 1.Department of Computer ScienceUniversity OldenburgOldenburgGermany

Personalised recommendations