Distributed Computing

, Volume 28, Issue 1, pp 31–53 | Cite as

Weak models of distributed computing, with connections to modal logic

  • Lauri Hella
  • Matti Järvisalo
  • Antti Kuusisto
  • Juhana Laurinharju
  • Tuomo Lempiäinen
  • Kerkko Luosto
  • Jukka Suomela
  • Jonni Virtema
Article

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.

Keywords

Distributed computing Local algorithms Modal logic Models of computation 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Lauri Hella
    • 1
  • Matti Järvisalo
    • 2
  • Antti Kuusisto
    • 3
  • Juhana Laurinharju
    • 2
  • Tuomo Lempiäinen
    • 2
  • Kerkko Luosto
    • 1
  • Jukka Suomela
    • 2
  • Jonni Virtema
    • 1
  1. 1.School of Information SciencesUniversity of TampereTampereFinland
  2. 2.Department of Computer Science, Helsinki Institute for Information Technology HIITUniversity of HelsinkiHelsinkiFinland
  3. 3.Institute of Computer ScienceUniversity of WrocławWrocławPoland

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