Program schemata technique for propositional program logics: A 30-year history

Abstract

A survey is presented of the so-called program schemata technique for proving the decidability of propositional program logics. This method is based on the reduction to versions of the problem of relative totality for nondeterministic Yanov schemata.

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References

  1. 1.

    Ball, T., Podelski, A., and Rajamani, S.K., Boolean and Cartesian abstraction for model checking C programs, Microsoft Research, 2000, MSR-TR-2000-115. http://researchmicrosoftcom/apps/pubs/defaultaspx?idi821.

    Google Scholar 

  2. 2.

    Ben-Ari, M., Halpern, J.Y., and Pnueli, A., Deterministic propositional dynamic logic: Finite models, complexity, and completeness, J. Comput. Syst.Sci., 1982, vol. 25, no. 3, pp. 402–417.

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bradfield, J., and Stirling, C., Modal mu-calculi, in The Handbook of Modal Logic, Blackburn, P., van Benthem, J., and Wolter, F., Eds., Elsevier, 2006, pp. 721–756.

    Google Scholar 

  4. 4.

    Bruse, F., Friedmann, O., and Lange, M., Guarded transformation for the modal mu-calculus. http://arxiv. org/abs/1305.0648).

  5. 5.

    Buchi Weak Second Order Arithmetic and Finite Automata: The Collected Works of J. Richard Buchi, New York: Springer, 1990, pp. 398–424.

  6. 6.

    Chagrov, A. and Zakharyaschev, M., Modal Logic, Oxford: Oxford Univ. Press, 1997.

    Google Scholar 

  7. 7.

    Clarke, E.M., Grumberg, O., and Peled, D., Model Checking, MIT Press, 1999.

  8. 8.

    Cleaveland, R., Tableau-based model checking in the propositional mu-calculus, Acta Informatica, 1990, vol. 27, pp. 725–747.

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Emerson, E.A. and Jutla, C.J., The complexity of tree automata and logics of programs, SIAM J. Comput., 1999, vol. 29, pp. 132–158.

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Fischer, M., and Ladner, R., Propositional dynamic logic of regular programs, J. Comput. Syst. Sci., 1979, vol. 18, pp. 194–211.

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Harel, D., Kozen, D., and Tiuryn, J., Dynamic Logic, MIT Press, 2000.

    Google Scholar 

  12. 12.

    Kozen, D., Results on the propositional mu-calculus, Theor. Comput. Sci., 1983, vol. 27, pp. 333–354.

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Nepomniaschy, V.A., and Shilov, N.V., Non-deterministic program schemata and their relation to dynamic logic, Int. Conf. on Math. Logic and Its Applications, Plenum Press, 1987, pp. 137–147.

    Google Scholar 

  14. 14.

    Podlovchenko, R.I., A.A. Lyapunov and A.P. Ershov in the theory of program schemes and the development of its logic concepts, Lect. Notes Comput. Sci., 2001, vol. 2244, pp. 8–23.

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Rabin, M.O., Decidability of second order theories and automata on infinite trees, Trans. ASM, 1969, vol. 141, pp. 1–35.

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Rutledge, J.D., On Ianovs program schemata, JACM, 1964, vol. 11, no. 1, pp. 1–9.

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Rutledge, J.D., Program schemata as automata. Part I, J.Comput. Syst. Sci., 1973, vol. 7, no. 6, pp. 543–578.

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Shilov, N.V., Program schemata vs. automata for decidability of program logics, Theor. Comput. Sci., 1997, vol. 175, pp. 15–27.

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Shilov, N.V., An approach to design of automata-based axiomatization for propositional program and temporal logics (by example of linear temporal logic), in Logic, Computation, Hierarchies. Series: Ontos Mathematical Logic, Brattka, V., Diener, H., and Spreen, D., Eds., Ontos-Verlag/De Gruyter, 2014, vol. 4, pp. 297–324.

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Shilov, N.V., Program schemata technique to solve propositional program logics revised, Post-Proceedings of 10th Ershov Informatics Conference PSI-2015 (25–27 August 2015, Innopolis, Kazan, Russia). To appear in Springer Lect. Notes Comput. Sci., 2016, vol. 9609.

  21. 21.

    Stirling, C., Modal and temporal logics, in Handbook of Logic in Computer Science, Abramsky, S., Gabbay, D.M., and Maibaum, S.E., Eds., Oxford: Oxford Univ. Press, 1992, vol. 2, pp. 477–563.

    MathSciNet  Google Scholar 

  22. 22.

    Streett, R., Propositional dynamic logic of looping and converse is elementary decidable, Inf. Control, 1982, vol. 54, pp. 121–141.

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Tarski, A., A lattice-theoretical fixpoint theorem and its applications, Pac. J. Math., 1955, vol. 5, pp. 285–309.

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    The Description Logic Handbook: Theory, Implementation, Applications, Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., and Patel-Schneider, P.F., Eds., Cambridge: Cambridge Univ. Press, 2003.

  25. 25.

    Schlingloff, B.-H., On the expressive power of modal logics on trees, Lect. Notes Comput. Sci., 1992, vol. 620, pp. 441–451.

  26. 26.

    Valiev, M.K., Decision complexity of variants of propositional dynamic logic, Lect. Notes Comput. Sci., 1980, vol. 88, pp. 656–664.

    Article  MATH  Google Scholar 

  27. 27.

    Vardi, M.Y. and Wolper, P., Automata-theoretic techniques for modal logics of programs, J. Comput. Syst. Sci., 1986, vol. 32, no. 2, pp. 183–221.

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Walukiewicz, I., A complete deductive system for the mu-calculus, Proc. IEEE LICS’93, 1993, pp. 136–147.

    Google Scholar 

  29. 29.

    Walukiewicz, I., Completeness of Kozen’s axiomatisation of the propositional Mu-calculus, Information and Computation, 2000, vol. 157, pp. 142–182.

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Zakharov, V.A., An efficient and unified approach to the decidability of equivalence of propositional programs, Lect. Notes Comput. Sci., 1998, vol. 1443, pp. 247–258.

    MathSciNet  Article  Google Scholar 

  31. 31.

    Bodin, E.V., Shilov, N.V., and Ii, K.O., Programs logics made simple, in System Informatics: Collection of Scientific Works, Novosibirsk: Nauka, 2003, issue 8, pp. 206–249.

    Google Scholar 

  32. 32.

    Ershov, A.P., On Yanov’s operator schemata, in Problems of Cybernatics: Collection of Scientific Works, Moscow: Nauka, 1968, issue 20, pp. 181–200.

    Google Scholar 

  33. 33.

    Ershov, A.P., Introduction to Theoretical Programming (Talks on a Method): A Textbook, Moscow: Nauka, 1977. Engl. transl. Origins of Programming: Discourses on Methodology, New York: Springer, 1990.

    Google Scholar 

  34. 34.

    Karpov, Yu.G., Model Checking: Verification of Parallel and Distributed Program Schemata, St. Petersburg: BKhV-Peterburg, 2009.

    Google Scholar 

  35. 35.

    Kotov, V.E. and Sabel’fel’d, V.K., Theory of Program Schemata, Moscow: Nauka, 1991.

    Google Scholar 

  36. 36.

    Kfuri, A.D., Stolboushkin, A.P., and Uzhichin, P., Some open problems in the theory of program schemata and dynamical logics, Usp. Mat. Nauk, 1989, vol. 44. no. 1 (265), pp. 35–55.

    MathSciNet  Google Scholar 

  37. 37.

    Nepomnyashchii, V.A. and Ryakin, O.M., Applied Methods of Program Verification, Moscow: Radio i Svyaz’, 1988.

    Google Scholar 

  38. 38.

    Nepomnyashchii, V.A., and Shilov, N.V., Nondeterministic program schemata and their relation to dynamical logic, Kibernetika, 1988, no. 3, pp. 12–19.

    MathSciNet  Google Scholar 

  39. 39.

    Odintsov, S.P., Speranskii, S.O., and Drobyshevich, S.A., Introduction to Nonclassical Logics: A Textbook for Novosibirsk State University, Novosibirsk: RITs Novosibirsk. Gos. Univ., 2014.

    Google Scholar 

  40. 40.

    Podlovchenko, R.I., From Yanov schemata to the theory of program models, Mathematical Problems of Cybernetics, Yablonskii, S.V., Ed., Moscow: Nauka, Fizmatlit, 1998, issue 7, pp 281–302.

  41. 41.

    Shilov, N.V., Formalisms and tools for designing and maintaining ontology, System Informatics, Marchuk, A.G., Ed., Novosibirsk: Sib. Otd. Ross. Akad. Nauk, 2009, issue 11, pp. 10–48.

    MathSciNet  Google Scholar 

  42. 42.

    Shilov, N.V., Principles of Syntax, Semantics, Translation, and Verification of Programs, Novosibirsk: Novosibirsk. Gos. Univ., 2011.

    Google Scholar 

  43. 43.

    Shilov, N.V., Bernshtein, A.Yu., and Shilova, S.O., Application of nondeterministic monadic program schemata to the study of the properties of program logics with fixed points, International Conference Mal’tsev Meeting, Novosibirsk, November 11–15, 2013, Novosibirsk: Inst. Mat., Sib. Otd., Ross. Akad. Nauk, 2013, p. 58. http://wwwmathnscru/conference/malmeet/ 13/maltsev13pdf.

    Google Scholar 

  44. 44.

    Shilov, N.V., Shilova, S.O., and Bernshtein, A.Yu., Generalized totality of nondeterministic Yanov schemata and decidability of a program logic with fixed points, Modern Information Technologies and IT-Education: Collection of Selected Works from the IX International Research and Practice Conference, Moscow State University, November 14–16, 2013, Sukhomlin, V.A., Ed., Moscow: Mosk. Gos. Univ., 2014.

  45. 45.

    Yanov, Yu.I., On logical algorithmic schemata, in Problems of Cybernetics: Collection of Scientific Works, Moscow: Nauka, 1958, issue 1, pp. 75–127.

    Google Scholar 

  46. 46.

    Yanov, Yu.I., On local transformations of algorithmic schemata, in Problems of Cybernetics: Collection of Scientific Works, Moscow: Nauka, 1968, issue 20, pp. 201–216.

    Google Scholar 

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Correspondence to N. V. Shilov.

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Original Russian Text © N.V. Shilov, S.O. Shilova, A.Yu. Bernshtein, 2016, published in Programmirovanie, 2016, Vol. 42, No. 4.

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Shilov, N.V., Shilova, S.O. & Bernshtein, A.Y. Program schemata technique for propositional program logics: A 30-year history. Program Comput Soft 42, 239–256 (2016). https://doi.org/10.1134/S036176881604006X

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