# The taming of converse: Reasoning about two-way computations

Conference paper

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## Abstract

We consider variants of propositional dynamic logic (*PDL*) augmented with the *converse* construct. Intuitively, the converse *α*^{−} of a program *α* is a programs whose semantics is to run *α* backwards. While *PDL* consists of assertions about *weakest preconditions*, the *converse* construct enable us to make assertions about *strongest postconditions*. We investigate the interaction of *converse* with two constructs that deal with infinite computations: *loop* and *repeat*. We show that *converse - loop - PDL* is decidable in exponential time, and *converse - repeat - PDL* is decidable in nondeterministic exponential time.

## Keywords

Execution Sequence Hybrid Automaton Tree Automaton Loop Formula Propositional Dynamic Logic
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## References

- [BHP82]M. Ben-Ari, J.Y. Halpern, A. Pnueli, “Deterministic Propositional Dynamic Logic: Finite Models, Complexity, and Completeness”,
*J. Computer and System Science*, 25(1982), pp. 402–417.Google Scholar - [Bu62]J.R. Büchi, “On a Decision Method in Restricted Second Order Arithmetic”,
*Proc. Int'l Congr. Logic, Method and Phil. Sci. 1960*. Stanford University Press, 1962, pp. 1–12.Google Scholar - [Da84]R. Danecki, “
*Propositional Dynamic Logic with Strong Looping Predicate*”, 1984.Google Scholar - [dB80]J. de Bakker,
*Mathematical theory of program correctness*, Prentice hall, 1980.Google Scholar - [FL79]M.J. Fisher, R.E. Ladner, “Propositional Dynamic Logic of Regular Programs”,
*J. Computer and System Sciences*, 18(2), 1979, pp. 194–211.Google Scholar - [Ha83]J.Y. Halpern, private communication, 1983.Google Scholar
- [HS83a]D. Harel, R. Sherman, “Looping vs. Repeating in Dynamic Logic”,
*Information and Control*55(1982), pp. 175–192.Google Scholar - [HS83b]D. Harel, R. Sherman, “Propositional Dynamic Logic of Flowcharts”,
*Proc. Int. Conf. on Foundations of Computation Theory*, Lecture Notes in Computer Science, vol. 158, Springer-Verlag, Berlin, 1983, pp. 195–206.Google Scholar - [Pa80.Parikh, R.: A completeness result for PDL.
*Symp. on Math. Foundations of Computer Science, Zakopane, 1978*.Google Scholar - [Pr76]V.R. Pratt, “Semantical Considerations on Floyd-Hoare Logic”,
*Proc. 17th IEEE Symp. on Foundations of Computer Science*, Houston, October 1976, pp. 109–121.Google Scholar - [Pr79]V.R. Pratt, “Models of Program Logics”,
*Proc. 20th IEEE Symp. on Foundation of Computer Science*, San Juan, 1979, pp. 115–122.Google Scholar - [Pr80]V.R. Pratt, “A Near-Optimal Method for Reasoning about Action”,
*J. Computer and Systems Sciences*20(1980), pp. 231–254.Google Scholar - [Pr81]V.R. Pratt, “Using Graphs to understand PDL”,
*Proc. Workshop on Logics of Programs*, (D. Kozen, ed.), Yorktown-Heights, Lecture Notes in Computer Science, vol. 131, Springer-Verlag, Berlin, 1982, pp. 387–396.Google Scholar - [PS83]A. Pnueli, R. Sherman,
*“Propositional Dynamic Logic of Looping Flowcharts”*, Technical Report, Weizmann Institute, Rehovot, Israel, 1983.Google Scholar - [Ra70]M.O. Rabin, “Weakly Definable Relations and Special Automata”,
*Proc. Symp. Math. Logic and Foundations of Set Theory*(Y. Bar-Hillel, ed.), North-Holland, 1970, pp. 1–23.Google Scholar - [Sh84]R. Sherman,
*“Variants of Propositional Dynamic Logic,”*Ph.D. Dissertation, The Weizmann Inst. of Science, 1984.Google Scholar - [St80]R.S. Streett,
*“A Propositional Dynamic Logic for Reasoning about Program Divergence”*, M.Sc. Thesis, MIT, 1980.Google Scholar - [St82]R.S. Streett, “Propositional Dynamic Logic of Looping and Converse is elementarily decidable”,
*Information and Control*54(1982), pp. 121–141.Google Scholar - [VS85]M.Y. Vardi, L. Stockmeyer, “Improved Upper and Lower Bounds for Modal Logics of Programs”, To appear in
*Proc. 17th ACM Symp. on Theory of Computing*, Providence, May 1985.Google Scholar - [VW84]M. Y. Vardi, P. Wolper, “Automata Theoretic Techniques for Modal Logics of Programs”, IBM Research Report, October 1984. A preliminary version appeared in
*Proc. ACM Symp. on Theory of Computing*, Washington, April 1984, pp. 446–456.Google Scholar

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