Logic of Programs 1979: Logic of Programs pp 23-101 | Cite as

PAL — Propositional algorithmic logic

  • Grażyna Mirkowska
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 125)


The aim of propositional algorithmic logic is to investigate the properties of program connectives. Complete axiomatic systems for deterministic as well as for nondeterministic interpretations of program variables are presented. They constitute basic sets of tools useful in the practice of proving the properties of program schemes. Propositional theories of data structures, e.g. the arithmetic of natural numbers and stacks, are constructed. This shows that in many aspects PAL is close to first-order algorithmic logic. Tautologies of PAL become tautologies of algorithmic logic after replacing program variables by programs and propositional variables by formulas. Another corollary to the completeness theorem asserts that it is possible to eliminate nondeterministic program variables and replace them by schemes with deterministic atoms.


Inductive Assumption Propositional Variable Program Variable Semantic Structure Program Scheme 
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. [1]
    Andreka H.,Nemeti I., Completeness of Floyd logic, Bulletin of section of logic, vol. 7 (1978)Google Scholar
  2. [2]
    Andreka H.,Nemeti I.,Sain I., Completeness problem in verification of programs and program schemes. Proc.MFCS 79 J.Becvar ed. in Lecture Notes on Computer Science vo.74Google Scholar
  3. [3]
    Banachowski L.,Kreczmar A.,Mirkowska G.,Rasiowa H.,Salwicki A., An introduction to algorithmic logic. Metamathematical investigations in the theory of programs, in Banach Center Publications vol.2 Mathematical Foundations of Computer Science (1977)Google Scholar
  4. [4]
    Berman F., A completeness technique for D-axiomatizable semantics, Proc.11th Ann.ACM Symp.on Thery of Computing Atlanta, Georgia May 1979Google Scholar
  5. [5]
    Chlebus B., On decidability of propositional algorithmic logic, Institute of Informatics Reports, University of Warsaw, 1979Google Scholar
  6. [6]
    Constable R., On the theory of programming logics, Proc. 9th Ann.ACM Symp. on Theory of Computing Boulder.Col. May 1977, pp. 269–285Google Scholar
  7. [7]
    Dahn B., Generalized Kripke-Models, l'Akademie Polonaise des Sci.Ser.Math. vol.XXI No 12(1973), pp. 1073–1077Google Scholar
  8. [8]
    Engeler E., Algorithmic properties of structures, Math.Systems Theory 1 (1967), pp. 183–195Google Scholar
  9. [9]
    Floyd R.W., Assigning Meanings to Programs, in Mathematical Aspects of Computer Science ed. J.T.Schwartz, 1967, pp.19–32Google Scholar
  10. [10]
    Fisher M.J.,Ladner R.E., Propositional modal logic of programs, Proc. 9th Ann.ACM Symp. on Theory of Computing. Boulder.Col., May 1977, pp.286–294Google Scholar
  11. [11]
    Grabowski M., The set of tautologies of zero-order algorithmic logic is decidable, Bull.Acad.Pol.Sci.Ser.Math. No 20 (1972) pp.575–582Google Scholar
  12. [12]
    Harel D.,Pratt V., Nondeterminism in logic of programs, Proc. 9th Ann.ACM Symp. on Theory of Computing, Boulder.Colorado, May 1977, pp.261–268Google Scholar
  13. [13]
    Hoare C.A., An axiomatic basis for computer programming, CACM 12, (1969), pp.576–580Google Scholar
  14. [14]
    Kozen D., A representation theorem for models of-free PDL, Report RC 7864, IBM Research, Yorktown Heights, New York, 1979Google Scholar
  15. [15]
    Kozen D., On the representation of dynamic algebras, Report RC 7898, IBM Research, Yorktown Heights, New York, 1979Google Scholar
  16. [16]
    Mirkowska G., On formalized systems of algorithmic logic, Bull. Acad.Pol.Sci.Ser.Math. 18 (1970) pp.499–505Google Scholar
  17. [17]
    Mirkowska G., Algorithmic logic with nondeterministic programs, Fundamenta InformaticaeGoogle Scholar
  18. [18]
    Mirkowska G., Multimodal logic, Technical report, University of Warszaw 1979Google Scholar
  19. [19]
    Mirkowska G., On algorithmic algebras, manuscript 1980Google Scholar
  20. [20]
    Mirkowska G., On propositional algorithmic theory related to arithmetic, manuscript 1980Google Scholar
  21. [21]
    Muldner T., Salwicki A., On algorithmic properties of concurent programs, manuscript 1979Google Scholar
  22. [22]
    Parikh R., A completeness result for PDL, Proc. MFCS '78, Lecture Notes in Computer Science, vol. 64, pp.403–416Google Scholar
  23. [23]
    Pratt V., Semantical Considerations of Floyd-Hoare logic, 17th IEEE Symposium on Foundations of Computer Science (1976), 109–121Google Scholar
  24. [24]
    Pratt V., Models of program logics, 20th IEEE Conference on Foundations of Computer Science, San Juan, PR,Oct.1979Google Scholar
  25. [25]
    Pratt V., Dynamic algebras and the nature of induction, preprint MIT /LCS/ TM-159, March 1980Google Scholar
  26. [26]
    Rasiowa H., Sikorski R., Mathematics of Metamathematics, PWN Warszaw 1963Google Scholar
  27. [27]
    Salwicki A., On algorithmic theory of stacks, ICS PAS report No 337, to appear in Fundamenta InformaticaeGoogle Scholar
  28. [28]
    Salwicki A., Formalized algorithmic languages, Bull.Acad.Pol. Sci.,Ser.Math. vol.18 No5, (1970), pp. 227–232Google Scholar
  29. [29]
    Segerberg K., A completeness theorem in the modal logic of programs, Notices of the AMS, 24,6, A-552, (1977)Google Scholar
  30. [30]
    Yanov J., On equivalence of operator schemes, Problems of Cybernetic 1, (1959), pp.1–100.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • Grażyna Mirkowska
    • 1
  1. 1.Institute of MathematicsUniversity of WarsawWarsawPoland

Personalised recommendations