From Geometric Semantics to Asynchronous Computability

  • Éric GoubaultEmail author
  • Samuel Mimram
  • Christine Tasson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9363)


We show that the protocol complex formalization of fault-tolerant protocols can be directly derived from a suitable semantics of the underlying synchronization and communication primitives, based on a geometrization of the state space. By constructing a one-to-one relationship between simplices of the protocol complex and (di)homotopy classes of (di)paths in the latter semantics, we describe a connection between these two geometric approaches to distributed computing: protocol complexes and directed algebraic topology. This is exemplified on atomic snapshot, iterated snapshot and layered immediate snapshot protocols, where a well-known combinatorial structure, interval orders, plays a key role. We believe that this correspondence between models will extend to proving impossibility results for much more intricate fault-tolerant distributed architectures.


Decision Task Simplicial Complex Local Memory Global Memory Interval Order 
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.
    Afek, Y., Attiya, H., Dolev, D., Gafni, E., Merritt, M., Shavit, N.: Atomic snapshots of shared memory. J. ACM 40(4), September 1993Google Scholar
  2. 2.
    Anderson, J.H.: Composite registers. In: Conference on Principles of Distributed Computing. ACM, New York (1993)Google Scholar
  3. 3.
    Biran, O., Moran, S., Zaks, S.: A combinatorial characterization of the distributed tasks which are solvable in the presence of one faulty processor. In: PoDC. ACM (1988)Google Scholar
  4. 4.
    Bonichon, R., Canet, G., Correnson, L., Goubault, E., Haucourt, E., Hirschowitz, M., Labbé, S., Mimram, S.: Rigorous evidence of freedom from concurrency faults in industrial control software. In: Flammini, F., Bologna, S., Vittorini, V. (eds.) SAFECOMP 2011. LNCS, vol. 6894, pp. 85–98. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  5. 5.
    Borowsky, E., Gafni, E.: Generalized FLP impossibility result for \(t\)-resilient synchronous computations. In: STOC (1993)Google Scholar
  6. 6.
    Dubut, J., Goubault, É., Goubault-Larrecq, J.: Natural homology. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 171–183. Springer, Heidelberg (2015) CrossRefGoogle Scholar
  7. 7.
    Fajstrup, L., Goubault, É., Haucourt, E., Mimram, S., Raussen, M.: Trace spaces: an efficient new technique for state-space reduction. In: Seidl, H. (ed.) Programming Languages and Systems. LNCS, vol. 7211, pp. 274–294. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  8. 8.
    Fajstrup, L., Goubault, É., Haucourt, E., Mimram, S., Raussen, M.: Directed Algebraic Topology and Concurrency. Springer (to be published) (2015)Google Scholar
  9. 9.
    Fajstrup, L., Goubault, É., Raußen, M.: Detecting deadlocks in concurrent systems. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 332–347. Springer, Heidelberg (1998) CrossRefGoogle Scholar
  10. 10.
    Fajstrup, L., Raussen, M., Goubault, É.: Algebraic topology and concurrency. TCS 357(1) (2006)Google Scholar
  11. 11.
    Fischer, M.J., Lynch, N.A., Paterson, M.S.: Impossibility of distributed consensus with one faulty process. Journal of the ACM (JACM) 32(2), 374–382 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fishburn, P.C.: Intransitive indifference with unequal indifference intervals. Journal of Mathematical Psychology 7(1), 144–149 (1970)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Gierz, G.: A Compendium of continuous lattices. Springer (1980)Google Scholar
  14. 14.
    Goubault, É.: Some geometric perspectives in concurrency theory. Homology, Homotopy and Appl. (2003)Google Scholar
  15. 15.
    Goubault, É., Haucourt, E.: A practical application of geometric semantics to static analysis of concurrent programs. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 503–517. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  16. 16.
    Goubault, É., Heindel, T., Mimram, S.: A geometric view of partial order reduction. MFPS, Electr. Notes. Theor. Comput. Sci. 298, (2013)Google Scholar
  17. 17.
    Goubault, É., Jensen, T.P.: Homology of higher-dimensional. In: Cleaveland, W.R. (ed.) CONCUR 1992. LNCS, vol. 630, pp. 254–268. Springer, Heidelberg (1992)Google Scholar
  18. 18.
    Goubault, É: The Geometry of Concurrency. Ph.D. dissertation, ENS (1995)Google Scholar
  19. 19.
    Goubault, É., Mimram, S., Tasson, C.: Iterated chromatic subdivisions are collapsible. Applied Categorical Structures (2014)Google Scholar
  20. 20.
    Grandis, M.: Directed Algebraic Topology: Models of Non-Reversible Worlds. New Mathematical Monographs, vol. 13. Cambridge University Press (2009)Google Scholar
  21. 21.
    Gunawardena, J.: Homotopy and concurrency. Bulletin of the EATCS 54, 184–193 (1994)zbMATHGoogle Scholar
  22. 22.
    Herlihy, M., Kozlov, D., Rajsbaum, S.: Distributed Computing Through Combinatorial Topology. Elsevier (2014)Google Scholar
  23. 23.
    Herlihy, M., Shavit, N.: The asynchronous computability theorem for \(t\)-resilient tasks. In: Proceedings of the Twenty-Fifth Annual ACM Aymposium on Theory of Computing, pp. 111–120. ACM (1993)Google Scholar
  24. 24.
    Herlihy, M., Shavit, N.: The topological structure of asynchronous computability. Journal of the ACM (JACM) 46(6), 858–923 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Kozlov, D.: Chromatic subdivision of a simplicial complex. Homology, Homotopy and Appl. 14 (2012)Google Scholar
  26. 26.
    Kozlov, D.: Topology of the view complex. arXiv preprint arXiv:1311.7283 (2013)
  27. 27.
    Loui, M.C., Abu-Amara, H.H.: Memory requirements for agreement among unreliable asynchronous processes. Advances in Computing Research 4 (1987)Google Scholar
  28. 28.
    Lynch, N.A.: Distributed algorithms. Morgan Kaufmann (1996)Google Scholar
  29. 29.
    Nachbin, L.: Topology and order. Van Nostrand, Van Nostrand mathematical studies (1965)zbMATHGoogle Scholar
  30. 30.
    Pratt, V.: Modeling concurrency with geometry. In: POPL. ACM Press (1991)Google Scholar
  31. 31.
    Saks, M.E., Zaharoglou, F.: Wait-free \(k\)-set agreement is impossible: the topology of public knowledge. In: STOC (1993)Google Scholar
  32. 32.
    van Glabbeek, R.: Bisimulation semantics for higher dimensional automata. Technical report, Stanford (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Éric Goubault
    • 1
    Email author
  • Samuel Mimram
    • 1
  • Christine Tasson
    • 2
  1. 1.LIXÉcole PolytechniquePalaiseauFrance
  2. 2.PPSUniversité ParisParisFrance

Personalised recommendations