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Weak models of distributed computing, with connections to modal logic

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Abstract

This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree \(d\) receives messages through \(d\) input ports and sends messages through \(d\) output ports, both numbered with \(1,2,\ldots ,d\). In this work, \({\mathsf{VV}}_\mathsf{c}\) is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of \({\mathsf{VV}}_\mathsf{c}\):

  • \({\mathsf {VV}}\): Input port \(i\) and output port \(i\) are not necessarily connected to the same neighbour.

  • \(\mathsf{MV}\): Input ports are not numbered; algorithms receive a multiset of messages.

  • \(\mathsf{SV}\): Input ports are not numbered; algorithms receive a set of messages.

  • \(\mathsf{VB}\): Output ports are not numbered; algorithms send the same message to all output ports.

  • \(\mathsf{MB}\): Combination of \(\mathsf{MV}\) and \(\mathsf{VB}\).

  • \(\mathsf{SB}\): Combination of \(\mathsf{SV}\) and \(\mathsf{VB}\).

Now we have many trivial containment relations, such as \(\mathsf{SB}\subseteq \mathsf{MB}\subseteq \mathsf{VB}\subseteq {\mathsf {VV}}\subseteq {\mathsf{VV}}_\mathsf{c}\), but it is not obvious if, for example, either of \(\mathsf{VB}\subseteq \mathsf{SV}\) or \(\mathsf{SV}\subseteq \mathsf{VB}\) should hold. Nevertheless, it turns out that we can identify a linear order on these classes. We prove that \(\mathsf{SB}\subsetneq \mathsf{MB}= \mathsf{VB}\subsetneq \mathsf{SV}= \mathsf{MV}= {\mathsf {VV}}\subsetneq {\mathsf{VV}}_\mathsf{c}\). The same holds for the constant-time versions of these classes. We also show that the constant-time variants of these classes can be characterised by a corresponding modal logic. Hence the linear order identified in this work has direct implications in the study of the expressibility of modal logic. Conversely, one can use tools from modal logic to study these classes.

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References

  1. 1.

    Afek, Y., Alon, N., Bar-Joseph, Z., Cornejo, A., Haeupler, B., Kuhn, F.: Beeping a maximal independent set. In: Proceedings of 25th International Symposium on Distributed Computing (DISC 2011), Lecture Notes in Computer Science, vol. 6950, pp. 32–50. Springer (2011). doi:10.1007/978-3-642-24100-0_3

  2. 2.

    Angluin, D.: Local and global properties in networks of processors. In: Proceedings of 12th Annual ACM Symposium on Theory of Computing (STOC 1980), pp. 82–93. ACM Press (1980). doi:10.1145/800141.804655

  3. 3.

    Åstrand, M., Floréen, P., Polishchuk, V., Rybicki, J., Suomela, J., Uitto, J.: A local 2-approximation algorithm for the vertex cover problem. In: Proceedings of 23rd International Symposium on Distributed Computing (DISC 2009), Lecture Notes in Computer Science, vol. 5805, pp. 191–205. Springer (2009). doi:10.1007/978-3-642-04355-0_21

  4. 4.

    Åstrand, M., Polishchuk, V., Rybicki, J., Suomela, J., Uitto, J.: Local algorithms in (weakly) coloured graphs (2010). arXiv:1002.0125

  5. 5.

    Åstrand, M., Suomela, J.: Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks. In: Proceedings of 22nd Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2010), pp. 294–302. ACM Press (2010). doi:10.1145/1810479.1810533

  6. 6.

    Attiya, H., Snir, M., Warmuth, M.K.: Computing on an anonymous ring. J. ACM 35(4), 845–875 (1988). doi:10.1145/48014.48247

  7. 7.

    Benthem, J.v.: Modal correspondence theory. Ph.D. thesis, Instituut voor Logica en Grondslagenonderzoek van de Exacte Wetenschappen, Universiteit van Amsterdam (1977)

  8. 8.

    Blackburn, P., Benthem, J.v., Wolter, F. (eds.): Handbook of Modal Logic, Studies in Logic and Practical Reasoning, vol. 3. Elsevier, Amsterdam (2007)

  9. 9.

    Blackburn, P., Rijke, M.d., Venema, Y.: Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge, UK (2001)

  10. 10.

    Boldi, P., Shammah, S., Vigna, S., Codenotti, B., Gemmell, P., Simon, J.: Symmetry breaking in anonymous networks characterizations. In: Proceedings of 4th Israel Symposium on the Theory of Computing and Systems (ISTCS 1996), pp. 16–26. IEEE (1996)

  11. 11.

    Boldi, P., Vigna, S.: Computing vector functions on anonymous networks. In: Proceedings of the 4th Colloquium on Structural Information and Communication Complexity (SIROCCO 1997), pp. 201–214. Carleton Scientific (1997)

  12. 12.

    Boldi, P., Vigna, S.: Computing anonymously with arbitrary knowledge. In: Proceedings of 18th Annual ACM Symposium on Principles of Distributed Computing (PODC 1999), pp. 181–188. ACM Press (1999). doi:10.1145/301308.301355

  13. 13.

    Boldi, P., Vigna, S.: An effective characterization of computability in anonymous networks. In: Proceeedings of 15th International Symposium on Distributed Computing (DISC 2001), Lecture Notes in Computer Science, vol. 2180, pp. 33–47. Springer (2001). doi:10.1007/3-540-45414-4_3

  14. 14.

    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North-Holland, New York (1976)

  15. 15.

    Chalopin, J.: Algorithmique distribuée, calculs locaux et homomorphismes de graphes. Ph.D. thesis, LaBRI, Université Bordeaux 1 (2006)

  16. 16.

    Chalopin, J., Das, S., Santoro, N.: Groupings and pairings in anonymous networks. In: Proceedings of the 20th International Symposium on Distributed Computing (DISC 2006), Lecture Notes in Computer Science, vol. 4167, pp. 105–119. Springer (2006). doi:10.1007/11864219_8

  17. 17.

    Conradie, W.: Definability and changing perspectives: the beth property for three extensions of modal logic. Master’s thesis, Institute for Logic, Language and Computation, University of Amsterdam (2002)

  18. 18.

    Cornejo, A., Kuhn, F.: Deploying wireless networks with beeps. In: Proceedings of the 24th International Symposium on Distributed Computing (DISC 2010), Lecture Notes in Computer Science, vol. 6343, pp. 148–162. Springer (2010). doi:10.1007/978-3-642-15763-9_15

  19. 19.

    Czygrinow, A., Hańćkowiak, M., Krzywdziński, K., Szymańska, E., Wawrzyniak, W.: Brief announcement: distributed approximations for the semi-matching problem. In: Proceedings of 25th International Symposium on Distributed Computing (DISC 2011), Lecture Notes in Computer Science, vol. 6950, pp. 200–201. Springer (2011). doi:10.1007/978-3-642-24100-0_18

  20. 20.

    Czygrinow, A., Hańćkowiak, M., Wawrzyniak, W.: Fast distributed approximations in planar graphs. In: Proceedings of 22nd International Symposium on Distributed Computing (DISC 2008), Lecture Notes in Computer Science, vol. 5218, pp. 78–92. Springer (2008). doi:10.1007/978-3-540-87779-0_6

  21. 21.

    Diestel, R.: Graph Theory, 4th edn. Springer, Berlin (2010). http://diestel-graph-theory.com/

  22. 22.

    Diks, K., Kranakis, E., Malinowski, A., Pelc, A.: Anonymous wireless rings. Theor. Comput. Sci. 145(1–2), 95–109 (1995). doi:10.1016/0304-3975(94)00178-L

  23. 23.

    Emek, Y., Smula, J., Wattenhofer, R.: Stone age distributed computing (2012). arXiv:1202.1186

  24. 24.

    Fine, K.: In so many possible worlds. Notre Dame J. Formal Logic 13(4), 516–520 (1972). doi:10.1305/ndjfl/1093890715

  25. 25.

    Flocchini, P., Roncato, A., Santoro, N.: Computing on anonymous networks with sense of direction. Theor. Comput. Sci. 301(1–3), 355–379 (2003). doi:10.1016/S0304-3975(02)00592-3

  26. 26.

    Floréen, P., Hassinen, M., Kaasinen, J., Kaski, P., Musto, T., Suomela, J.: Local approximability of max–min and min–max linear programs. Theory Comput. Syst. 49(4), 672–697 (2011). doi:10.1007/s00224-010-9303-6

  27. 27.

    Floréen, P., Hassinen, M., Kaski, P., Suomela, J.: Local approximation algorithms for a class of 0/1 max–min linear programs (2008). arXiv:0806.0282

  28. 28.

    Floréen, P., Hassinen, M., Kaski, P., Suomela, J.: Tight local approximation results for max–min linear programs. In: Proceedings of the 4th International Workshop on Algorithmic Aspects of Wireless Sensor Networks (Algosensors 2008), Lecture Notes in Computer Science, vol. 5389, pp. 2–17. Springer (2008). doi:10.1007/978-3-540-92862-1_2. arXiv:0804.4815.

  29. 29.

    Floréen, P., Kaasinen, J., Kaski, P., Suomela, J.: An optimal local approximation algorithm for max-min linear programs. In: Proceedings of 21st Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2009), pp. 260–269. ACM Press (2009). doi:10.1145/1583991.1584058. arXiv:0809.1489.

  30. 30.

    Floréen, P., Kaski, P., Musto, T., Suomela, J.: Approximating max-min linear programs with local algorithms. In: Proceedings of 22nd IEEE International Parallel and Distributed Processing Symposium (IPDPS 2008). IEEE (2008). doi:10.1109/IPDPS.2008.4536235. arXiv:0710.1499.

  31. 31.

    Floréen, P., Kaski, P., Polishchuk, V., Suomela, J.: Almost stable matchings by truncating the Gale–Shapley algorithm. Algorithmica 58(1), 102–118 (2010). doi:10.1007/s00453-009-9353-9. arXiv:0812.4893

  32. 32.

    Göös, M., Hirvonen, J., Suomela, J.: Lower bounds for local approximation. In: Proceedings of 31st Annual ACM Symposium on Principles of Distributed Computing (PODC 2012), pp. 175–184. ACM Press (2012). doi:10.1145/2332432.2332465. arXiv:1201.6675.

  33. 33.

    Halpern, J.Y., Moses, Y.: Knowledge and common knowledge in a distributed environment. J. ACM 37(3), 549–587 (1990). doi:10.1145/79147.79161

  34. 34.

    Hella, L., Järvisalo, M., Kuusisto, A., Laurinharju, J., Lempiäinen, T., Luosto, K., Suomela, J., Virtema, J.: Weak models of distributed computing, with connections to modal logic. In: Proceedings of 31st Annual ACM Symposium on Principles of Distributed Computing (PODC 2012), pp. 185–194. ACM Press (2012). doi:10.1145/2332432.2332466. arXiv:1205.2051

  35. 35.

    Immerman, N.: Descriptive Complexity. Graduate Texts in Computer Science. Springer, Berlin (1999)

  36. 36.

    Kuhn, F.: The price of locality: exploring the complexity of distributed coordination primitives. Ph.D. thesis, ETH Zurich (2005)

  37. 37.

    Kuhn, F., Moscibroda, T., Wattenhofer, R.: The price of being near-sighted. In: Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 980–989. ACM Press (2006). doi:10.1145/1109557.1109666.

  38. 38.

    Kuhn, F., Wattenhofer, R.: On the complexity of distributed graph coloring. In: Proc. 25th Annual ACM Symposium on Principles of Distributed Computing (PODC 2006), pp. 7–15. ACM Press (2006). doi:10.1145/1146381.1146387.

  39. 39.

    Lenzen, C.: Synchronization and symmetry breaking in distributed systems. Ph.D. thesis, ETH Zurich (2011)

  40. 40.

    Lenzen, C., Oswald, Y.A., Wattenhofer, R.: What can be approximated locally? TIK Report 331, ETH Zurich, Computer Engineering and Networks Laboratory (2010). ftp://ftp.tik.ee.ethz.ch/pub/publications/TIK-Report-331.pdf

  41. 41.

    Lenzen, C., Wattenhofer, R.: Minimum dominating set approximation in graphs of bounded arboricity. In: Proceedings of 24th International Symposium on Distributed Computing (DISC 2010), Lecture Notes in Computer Science, vol. 6343, pp. 510–524. Springer (2010). doi:10.1007/978-3-642-15763-9_48

  42. 42.

    Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992). doi:10.1137/0221015

  43. 43.

    Mayer, A., Naor, M., Stockmeyer, L.: Local computations on static and dynamic graphs. In: Proceedings of 3rd Israel Symposium on the Theory of Computing and Systems (ISTCS 1995), pp. 268–278. IEEE (1995). doi:10.1109/ISTCS.1995.377023

  44. 44.

    Moran, S., Warmuth, M.K.: Gap theorems for distributed computation. SIAM J. Comput. 22(2), 379–394 (1993). doi:10.1137/0222028

  45. 45.

    Moscibroda, T.: Locality, scheduling, and selfishness: algorithmic foundations of highly decentralized networks. Ph.D. thesis, ETH Zurich (2006)

  46. 46.

    Naor, M., Stockmeyer, L.: What can be computed locally? SIAM J. Comput. 24(6), 1259–1277 (1995). doi:10.1137/S0097539793254571

  47. 47.

    Norris, N.: Classifying anonymous networks: when can two networks compute the same set of vector-valued functions? In: Proceedings of 1st Colloquium on Structural Information and Communication Complexity (SIROCCO 1994), pp. 83–98. Carleton University Press (1995)

  48. 48.

    Norris, N.: Computing functions on partially wireless networks. In: Proceedings 2nd Colloquium on Structural Information and Communication Complexity (SIROCCO 1995), pp. 53–64. Carleton University Press (1996)

  49. 49.

    Peleg, D.: Distributed computing: a locality-sensitive approach. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2000)

  50. 50.

    Petersen, J.: Die Theorie der regulären graphs. Acta Math. 15(1), 193–220 (1891). doi:10.1007/BF02392606

  51. 51.

    Polishchuk, V., Suomela, J.: A simple local 3-approximation algorithm for vertex cover. Inform. Process. Lett. 109(12), 642–645 (2009). doi:10.1016/j.ipl.2009.02.017. arXiv:0810.2175

  52. 52.

    de Rijke, M.: A note on graded modal logic. Stud. Log. 64(2), 271–283 (2000). doi:10.1023/A:1005245900406

  53. 53.

    Sangiorgi, D.: On the origins of bisimulation and coinduction. ACM Trans. Program. Lang. Syst. 31(4), Article 15 (2009). doi:10.1145/1516507.1516510

  54. 54.

    Suomela, J.: Distributed algorithms for edge dominating sets. In: Proceedings of 29th Annual ACM Symposium on Principles of Distributed Computing (PODC 2010), pp. 365–374. ACM Press (2010). doi:10.1145/1835698.1835783

  55. 55.

    Suomela, J.: Survey of local algorithms. ACM Comput. Surv. 45(2), 24:1–40 (2013). doi:10.1145/2431211.2431223. http://www.cs.helsinki.fi/local-survey/

  56. 56.

    Wiese, A., Kranakis, E.: Impact of locality on location aware unit disk graphs. Algorithms 1, 2–29 (2008). doi:10.3390/a1010002

  57. 57.

    Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55(3), 601–644 (1983). doi:10.1103/RevModPhys.55.601

  58. 58.

    Yamashita, M., Kameda, T.: Electing a leader when processor identity numbers are not distinct (extended abstract). In: Proceedings of 3rd International Workshop on Distributed Algorithms (WDAG 1989), Lecture Notes in Computer Science, vol. 392, pp. 303–314. Springer (1989). doi:10.1007/3-540-51687-5_52

  59. 59.

    Yamashita, M., Kameda, T.: Computing functions on asynchronous anonymous networks. Math. Syst. Theory 29(4), 331–356 (1996). doi:10.1007/BF01192691

  60. 60.

    Yamashita, M., Kameda, T.: Computing on anonymous networks: part I—characterizing the solvable cases. IEEE Trans. Parallel Distrib. Syst. 7(1), 69–89 (1996). doi:10.1109/71.481599

  61. 61.

    Yamashita, M., Kameda, T.: Computing on anonymous networks: part II—decision and membership problems. IEEE Trans. Parallel Distrib. Syst. 7(1), 90–96 (1996). doi:10.1109/71.481600

  62. 62.

    Yamashita, M., Kameda, T.: Leader election problem on networks in which processor identity numbers are not distinct. IEEE Trans. Parallel Distrib. Syst. 10(9), 878–887 (1999). doi:10.1109/71.798313

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Acknowledgments

This work is an extended and revised version of a preliminary conference report [34]. We thank anonymous reviewers for their helpful feedback, and Jérémie Chalopin, Mika Göös, and Joel Kaasinen for discussions and comments.

This work was supported in part by Academy of Finland (Grants 129761, 132380, 132812, and 252018), the research funds of University of Helsinki, and Finnish Cultural Foundation. Part of this work was conducted while Tuomo Lempiäinen was with the Department of Information and Computer Science at Aalto University.

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Correspondence to Jukka Suomela.

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Hella, L., Järvisalo, M., Kuusisto, A. et al. Weak models of distributed computing, with connections to modal logic. Distrib. Comput. 28, 31–53 (2015). https://doi.org/10.1007/s00446-013-0202-3

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Keywords

  • Distributed computing
  • Local algorithms
  • Modal logic
  • Models of computation