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Proof Pearl: A Formal Proof of Dally and Seitz’ Necessary and Sufficient Condition for Deadlock-Free Routing in Interconnection Networks

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Abstract

Avoiding deadlock is crucial to interconnection networks. In ’87, Dally and Seitz proposed a necessary and sufficient condition for deadlock-free routing. This condition states that a routing function is deadlock-free if and only if its channel dependency graph is acyclic. We formally define and prove a slightly different condition from which the original condition of Dally and Seitz can be derived. Dally and Seitz prove that a deadlock situation induces cyclic dependencies by reductio ad absurdum. In contrast we introduce the notion of a waiting graph from which we explicitly construct a cyclic dependency from a deadlock situation. Moreover, our proof is structured in such a way that it only depends on a small set of proof obligations associated to arbitrary routing functions and switching policies. Discharging these proof obligations is sufficient to instantiate our condition for deadlock-free routing on particular networks. Our condition and its proof have been formalized using the ACL2 theorem proving system.

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References

  1. Borrione, D., Helmy, A., Pierre, L., Schmaltz, J.: A formal approach to the verification of networks on chip. In: EURASIP Journal on Embedded Systems, 2009(Article ID 548324), 14 pp. doi:10.1155/2009/548324 (2009)

  2. Boyer, R.S., Strother Moore, J.: The addition of bounded quantification and partial functions to a computational logic and its theorem prover. J. Autom. Reason. 4(2), 117–172 (1988)

    Article  MATH  Google Scholar 

  3. Boyer, R.S., Strother Moore, J.: A Computation Logic Handbook. Academic Press (1988)

  4. Chen, R.C.: Deadlock prevention in message switched networks. In: ACM 74: Proceedings of the 1974 Annual Conference, pp. 306–310. ACM, New York, NY, USA (1974)

    Chapter  Google Scholar 

  5. Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press and McGraw Hill (1990)

  6. Dally, W.J., Seitz, C.L.: Deadlock-free message routing in multiprocessor interconnection networks. IEEE Trans. Comput. 36(5), 547–553 (1987)

    Article  MATH  Google Scholar 

  7. Dally, W.J., Towles, B.: Principles and Practices of Interconnection Networks. Morgan Kaufmann (2004)

  8. Duato, J.: A necessary and sufficient condition for deadlock-free adaptive routing in wormhole networks. IEEE Trans. Parallel Distrib. Syst. 6(10), 1055–1067 (1995)

    Article  Google Scholar 

  9. Duato, J., Yalamanchili, S., Ni, L.: Interconnection Networks: an Engineering Approach. IEEE Computer Society Press, Los Alamitos, CA, USA (1997)

    Google Scholar 

  10. Fleury, E., Fraigniaud, P.: A general theory for deadlock avoidance in wormhole-routed networks. IEEE Trans. Parallel Distrib. Syst. 9(7), 626–638 (1998)

    Article  Google Scholar 

  11. Kaufmann, M., Strother Moore, J.: An industrial strengh theorem prover of a logic based on common lisp. IEEE Trans. Softw. Eng. 23(4), 203–213 (1997)

    Article  Google Scholar 

  12. Kaufmann, M., Strother Moore, J.: Structured theory development for a mechanized logic. J. Autom. Reason. 26(2), 161–203 (1997)

    Article  Google Scholar 

  13. Kaufmann, M., Manolios, P., Strother Moore, J.: ACL2 Computer-Aided Reasoning: an Approach. Kluwer Academic Press (2000)

  14. Manolios, P., Strother Moore, J.: Partial functions in ACL2. J. Autom. Reason. 31(2), 107–127 (2003)

    Article  MATH  Google Scholar 

  15. Ni, L.M., Mckinley, P.K.: A survey of wormhole routing techniques in direct networks. IEEE Comput. 26, 62–76 (1993)

    Article  Google Scholar 

  16. Ray, S.: Quantification in tail-recursive function definitions. In: Manolios, P., Wilding, M. (eds.) Proceedings of the 6th International Workshop on the ACL2 Theorem Prover and its Applications (ACL2 2006). ACM International Conference Series, vol. 205, pp. 95–98. ACM, Seattle, WA (2006)

    Chapter  Google Scholar 

  17. Schmaltz, J., Borrione, D.: Towards a formal theory of on chip communications in the ACL2 logic. In: Proceedings of the Sixth International Workshop on the ACL2 Theorem Prover and its Applications, part of FloC’06, 14–15 August 2006. ACM, Seattle, WA (2006)

    Google Scholar 

  18. Schmaltz, J., Borrione, D.: A functional formalization of on chip communications. Form. Asp. Comput. 20, 241–258 (2008)

    Article  MATH  Google Scholar 

  19. Schwiebert, L., Jayasimha, D.N.: A universal proof technique for deadlock-free routing in interconnection networks. In: 7th Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 175–184 (1995)

  20. Verbeek, F., Schmaltz, J.: Formal validation of deadlock prevention in networks-on-chips. In: Ray, S., Russinoff, D. (eds.) Eighth International Workshop on the ACL2 Theorem Prover and its Application, pp. 135–145, 11–12 May 2009. Northeastern University, Boston MA, USA. ACM (2009)

  21. Verbeek, F., Schmaltz, J.: Formal specification of networks-on-chips: deadlock and evacuation. In: Proc. of Design, Automation, and Test in Europe (DATE’10), pp. 1701–1706 (2010)

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Authors and Affiliations

  1. Institute for Computing and Information Sciences, Radboud University Nijmegen, P.O. Box 9010, 6500, GL, Nijmegen, The Netherlands

    Freek Verbeek & Julien Schmaltz

  2. School of Computer Science, Open University of The Netherlands, P.O. Box 2960, 6401, DL, Heerlen, The Netherlands

    Freek Verbeek & Julien Schmaltz

Authors
  1. Freek Verbeek
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  2. Julien Schmaltz
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Correspondence to Julien Schmaltz.

Additional information

This research is supported by NWO/EW project Formal Validation of Deadlock Avoidance Mechanisms (FVDAM) under grant no. 612.064.811.

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Verbeek, F., Schmaltz, J. Proof Pearl: A Formal Proof of Dally and Seitz’ Necessary and Sufficient Condition for Deadlock-Free Routing in Interconnection Networks. J Autom Reasoning 48, 419–439 (2012). https://doi.org/10.1007/s10817-010-9206-x

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  • DOI: https://doi.org/10.1007/s10817-010-9206-x

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