Advertisement

Distributed Computing

, Volume 31, Issue 4, pp 289–316 | Cite as

Geometric and combinatorial views on asynchronous computability

  • Éric Goubault
  • Samuel Mimram
  • Christine Tasson
Article
  • 52 Downloads

Abstract

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.

Keywords

Fault-tolerant distributed computing Atomic snapshot protocol Protocol complex Directed algebraic topology Dihomotopy Interval order 

References

  1. 1.
    Afek, Y., Attiya, H., Dolev, D., Gafni, E., Merritt, M., Shavit, N.: Atomic snapshots of shared memory. J. ACM 40(4), 873–890 (1993)CrossRefzbMATHGoogle Scholar
  2. 2.
    Anderson, J.H.: Composite registers. In: Conference on Principles of Distributed Computing. ACM, New York (1993)Google Scholar
  3. 3.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  4. 4.
    Benavides, F., Rajsbaum, S.: The read/write protocol complex is collapsible. In: Latin American Symposium on Theoretical Informatics, pp. 179–191. Springer (2016)Google Scholar
  5. 5.
    Bezem, M., Klop, J.W., de Vrijer, R.: Term Rewriting Systems. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    Bonichon, R., Canet, G., Correnson, L., Goubault, É., Haucourt, E., Hirschowitz, M., Labbé, S., Mimram, S.: Rigorous evidence of freedom from concurrency faults in industrial control software. In: SAFECOMP (2011)Google Scholar
  8. 8.
    Borowsky, E., Gafni, E.: Generalized FLP impossibility result for \(t\)-resilient asynchronous computations. In: STOC (1993)Google Scholar
  9. 9.
    Castañeda, A., Rajsbaum, S., Raynal, M.: Specifying concurrent problems: beyond linearizability and up to tasks. In: International Symposium on Distributed Computing, pp. 420–435. Springer (2015)Google Scholar
  10. 10.
    Fajstrup, L., Goubault, É., Haucourt, E., Mimram, S., Raussen, M.: Trace spaces: an efficient new technique for state-space reduction. In: ESOP (2012)Google Scholar
  11. 11.
    Fajstrup, L., Goubault, É., Haucourt, E., Mimram, S., Raussen, M.: Directed Algebraic Topology and Concurrency. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar
  12. 12.
    Fajstrup, L., Goubault, É., Raussen, M.: Detecting deadlocks in concurrent systems. In: CONCUR, number 1466 in LNCS. Springer (1998)Google Scholar
  13. 13.
    Fajstrup, L., Raussen, M., Goubault, É.: Algebraic topology and concurrency. TCS 357(1), 241–278 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fischer, M.J., Lynch, N.A., Paterson, M.S.: Impossibility of distributed consensus with one faulty process. J. ACM (JACM) 32(2), 374–382 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fishburn, P.C.: Intransitive indifference with unequal indifference intervals. J. Math. Psychol. 7(1), 144–149 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gafni, E.: Snapshot for time: the one-shot case (2014). arXiv preprint arXiv:1408.3432
  17. 17.
    Gierz, G.: A Compendium of Continuous Lattices. Springer, Berlin (1980)CrossRefzbMATHGoogle Scholar
  18. 18.
    Goubault, É.: The Geometry of Concurrency. Ph.D. Dissertation, ENS (1995)Google Scholar
  19. 19.
    Goubault, É.: Some geometric perspectives in concurrency theory. Homol. Homotopy Appl. 5, 95–136 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Goubault, É., Haucourt, E.: A practical application of geometric semantics to static analysis of concurrent programs. In: CONCUR 2005. Springer (2005)Google Scholar
  21. 21.
    Goubault, É., Heindel, T., Mimram, S.: A geometric view of partial order reduction. MFPS Electr. Notes Theor. Comput. Sci. 298, 179–195 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Goubault, É., Jensen, T.P.: Homology of higher-dimensional automata. In: Proceedings of CONCUR (1992)Google Scholar
  23. 23.
    Goubault, É., Mimram, S., Tasson, C.: Iterated chromatic subdivisions are collapsible. Appl. Categ. Struct. 23, 777–818 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Grandis, M.: Directed Algebraic Topology: Models of Non-reversible Worlds. New Mathematical Monographs, vol. 13. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  25. 25.
    Gunawardena, J.: Homotopy and concurrency. Bull. EATCS 54, 184–193 (1994)zbMATHGoogle Scholar
  26. 26.
    Herlihy, M., Kozlov, D., Rajsbaum, S.: Distributed Computing Through Combinatorial Topology. Elsevier, Amsterdam (2014)zbMATHGoogle Scholar
  27. 27.
    Herlihy, M., Shavit, N.: The asynchronous computability theorem for \(t\)-resilient tasks. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pp. 111–120. ACM (1993)Google Scholar
  28. 28.
    Herlihy, M., Shavit, N.: The topological structure of asynchronous computability. J. ACM (JACM) 46(6), 858–923 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kozlov, D.: Chromatic subdivision of a simplicial complex. Homol. Homotopy Appl. 14(2), 197–209 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kozlov, D.: Topology of the view complex (2013). arXiv preprint arXiv:1311.7283
  31. 31.
    Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann, Burlington (1996)zbMATHGoogle Scholar
  32. 32.
    Nachbin, L.: Topology and Order. Van Nostrand Mathematical Studies. Van Nostrand, New York (1965)zbMATHGoogle Scholar
  33. 33.
    Pratt, V.: Modeling concurrency with geometry. In: POPL. ACM Press (1991)Google Scholar
  34. 34.
    Saks, M. E., Zaharoglou, F.: Wait-free \(k\)-set agreement is impossible: the topology of public knowledge. In: STOC (1993)Google Scholar
  35. 35.
    van Glabbeek, R.: Bisimulation semantics for higher dimensional automata. Technical Report, Stanford (1991)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LIX, École Polytechnique, CNRSUniversité Paris-SaclayPalaiseau CedexFrance
  2. 2.IRIF, Université Paris-Diderot, CNRS, Université Sorbonne Paris CitéParisFrance

Personalised recommendations