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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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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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  1. School of Information Sciences, University of Tampere, Tampere, Finland

    Lauri Hella, Kerkko Luosto & Jonni Virtema

  2. Department of Computer Science, Helsinki Institute for Information Technology HIIT, University of Helsinki, Helsinki, Finland

    Matti Järvisalo, Juhana Laurinharju, Tuomo Lempiäinen & Jukka Suomela

  3. Institute of Computer Science, University of Wrocław, Wrocław, Poland

    Antti Kuusisto

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  1. Lauri Hella
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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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