Skip to main content
SpringerLink
Log in

Reachability Analysis for Some Models of Infinite-State Transition Systems

  • Conference paper
  • First Online:
CONCUR 2000 — Concurrency Theory (CONCUR 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1877))

Included in the following conference series:

Abstract

We introduce some new models of infinite-state transition systems. The basic model, called a (reversal-bounded) counter machine (CM), is a nondeterministic finite automaton augmented with finitely many reversal-bounded counters (i.e. each counter can be incremented or decremented by 1 and tested for zero, but the number of times it can change mode from nondecreasing to nonincreasing and vice-versa is bounded by a constant, independent of the computation). We extend a CM by augmenting it with some familiar data structures: (i) A pushdown counter machine (PCM) is a CM augmented with an unrestricted pushdown stack. (ii) A tape counter machine (TCM) is a CM augmented with a two-way read/write worktape that is restricted in that the number of times the head crosses the boundary between any two adjacent cells of the worktape is bounded by a constant, independent of the computation (thus, the worktape is finite-crossing). There is no bound on how long the head can remain on a cell. (iii) A queue counter machine (QCM) is a CM augmented with a queue that is restricted in that the number of alternations between non-deletion phase and non-insertion phase is bounded by a constant. A non-deletion (non-insertion) phase is a period consisting of insertions (deletions) and no-changes, i.e., the queue is idle. We show that emptiness, (binary, forward, and backward) reachability, nonsafety, and invariance for these machines are solvable. We also look at extensions of the models that allow the use of linear-relation tests among the counters and parameterized constants as “primitive” predicates. We investigate the conditions under which these problems are still solvable.

Supported in part by NSF grants IRI-9700370 and IIS-9817432.

Supported in part by NSF grant CCR-9970976.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Alur, C. Courcoibetis, and D. Dill. Model-checking in dense real time. Information & Computation, 104(1):2–34, 1993.

    Article  MATH  Google Scholar 

  2. R. Alur and D. Dill. Automata for modeling real-time systems. Theoretical Computer Science, 126(2):183–236, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Alur and T. A. Henzinger. A really temporal logic. J. ACM, 41(1):181–204, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. R. Burch, E. M. Clarke, K. L. McMillan, D. L. Dill, and L. H. Hwang. Symbolic model checking: 1020 states and beyond. Information & Computation, 98(2):142–170, June 1992.

    Google Scholar 

  5. A. Bouajjani, J. Esparza, and O. Maler. Reachability Analysis of Pushdown Automata: Application to Model-Checking. CONCUR 1997, pp. 135–150.

    Google Scholar 

  6. A. Bouajjani, R. Echahed and R. Robbana. “On the Automatic Verification of Systems with Continuous Variables and Unbounded Discrete Data Structures.” In Hybrid Systems II, LNCS 999, 1995.

    Google Scholar 

  7. B. Boigelot and P. Godefroid. “Symbolic verification of communication protocols with infinite state spaces using QDDs.” In Proc. Int. Conf. on Computer Aided Verification, 1996.

    Google Scholar 

  8. T. Bultan, R. Gerber, and W. Pugh. Model Checking Concurrent Systems with Unbounded Integer Variables: Symbolic Representations, Approximations, and Experimental Results. ACM Trans. on Programming Languages and Systems, 21(4):747–789, July 1999.

    Google Scholar 

  9. S. Bensalem, Y. Lakhnech, and S. Owre. Computing abstractions of infinite state systems compositionally and automatically. In Proc. 10th Int. Conf. on Computer Aided Verification, 1998.

    Google Scholar 

  10. B. Boigelot and P. Wolper, Symbolic verification with periodic sets, Proc. 6th Int. Conf. on Computer Aided Verification, 1994

    Google Scholar 

  11. H. Comon and Y. Jurski. Multiple counters automata, safety analysis and Presburger arithmetic. Proc. 10th Int. Conf. on Computer Aided Verification, 1998.

    Google Scholar 

  12. H. Comon and Y. Jurski. Timed Automata and the Theory of Real Numbers. Proc. CONCUR, 1999.

    Google Scholar 

  13. Z. Dang, O. H. Ibarra, T. Bultan, R. A. Kemmerer, and J. Su. Binary reachability analysis of discrete pushdown timed automata. To appear in Int. Conf. on Computer Aided Verification, 2000.

    Google Scholar 

  14. J. Dingel and T. Filkorn. Model checking for infinite state systems using data abstraction. Proc. 7th Int. Conf. on Computer Aided Verification, 1995.

    Google Scholar 

  15. D. Dams, R. Gerth, and O. Grumberg. Abstract interpretation of reactive systems. ACM Trans. on Programming Languages and Systems, 19(2):253–291, March 1997.

    Google Scholar 

  16. J. Esparza. Decidability of Model Checking for Infinite-State Concurrent Systems. Acta Informatica, 34(2):85–107, 1997.

    Article  MathSciNet  Google Scholar 

  17. A. Finkel and G. Sutre. Decidability of reachability problems for classes of two counters automata. STACS’2000, 346–357, Springer, 2000.

    Google Scholar 

  18. A. Finkel, B. Willems, and P. Wolper. A direct symbolic approach to model checking pushdown systems. INFINITY, 1997.

    Google Scholar 

  19. E. M. Gurari and O. H. Ibarra. The complexity of decision problems for finite-turn multicounter machines. JCSS, 22:220–229, 1981.

    MATH  MathSciNet  Google Scholar 

  20. S. A. Greibach. Checking automata and one-way stack languages. SDC Document TM 738/045/00, 1968.

    Google Scholar 

  21. T. A. Henzinger, X. Nicollin, J. Sifakis, and S. Yovine. Symbolic Model Checking for Real-time Systems. Information & Computation, 111(2):193–244, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  22. N. Halbwachs, P. Raymond, and Y. Proy. Verification of linear hybrid systems by means of convex approximations. In Proc. Int. Symposium on Static Analysis, B. LeCharlier ed., vol. 864, September 1994.

    Google Scholar 

  23. O. H. Ibarra. Reversal-bounded multicounter machines and their decision problems. J. ACM, Vol. 25, pp. 116–133, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  24. O. H. Ibarra and J. Su, A Technique for the Containment and Equivalence of Linear Constraint Queries. JCSS, 59(1):1–28, 1999.

    MATH  MathSciNet  Google Scholar 

  25. O. H. Ibarra, J. Su, and C. Bartzis, Counter Machines and the Safety and Disjointness Problems for Database Queries with Linear Constraints, to appear in Words, Sequences, Languages: Where Computer Science, Biology and Linguistics Meet, Kluwer, 2000.

    Google Scholar 

  26. O. H. Ibarra, J. Su, Z. Dang, T. Bultan, and R. Kemmerer, Counter Machines: Decidable Properties and Applications to Verification Problems, To appear in Proc. MFCS’2000.

    Google Scholar 

  27. Y. Matijasevic. Enumerable sets are Diophantine. Soviet Math. Dokl, Vol. 11, 1970, pp. 354–357.

    Google Scholar 

  28. M. Minsky. Recursive unsolvability of Post’s problem of Tag and other topics in the theory of Turing machines. Ann. of Math., 74:437–455, 1961.

    Article  MathSciNet  Google Scholar 

  29. E. Post. A variant of a recursively unsolvable problem. Bull. Am. Math. Soc., 52:264–268, 1946.

    Article  MATH  MathSciNet  Google Scholar 

  30. I. Walukiewicz. Pushdown processes: games and model checking. In Proc. Int. Conf. on Computer Aided Verification, 1996

    Google Scholar 

  31. P. Wolper and B. Boigelot. Verifying systems with infinite but regular state spaces. In Proc. Int. Conf. on Computer Aided Verification, 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of California, Santa Barbara, CA, 93106, USA

    Oscar H. Ibarra, Tevfik Bultan & Jianwen Su

Authors
  1. Oscar H. Ibarra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tevfik Bultan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianwen Su
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Pennsylvania State University, 220 Pond Laboratory, University Park, PA, 16802-6106, USA

    Catuscia Palamidessi

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ibarra, O.H., Bultan, T., Su, J. (2000). Reachability Analysis for Some Models of Infinite-State Transition Systems. In: Palamidessi, C. (eds) CONCUR 2000 — Concurrency Theory. CONCUR 2000. Lecture Notes in Computer Science, vol 1877. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44618-4_15

Download citation

  • DOI: https://doi.org/10.1007/3-540-44618-4_15

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67897-7

  • Online ISBN: 978-3-540-44618-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation