Abstract
This paper contributes to exploring the connection between epistemic knowledge and communication complexity in distributed systems. We focus on Action Models, a well-known variant of dynamic epistemic logic, which allows to cleanly separate the state of knowledge of the processes and its update due to communication actions: Exactly like the set of possible global states, the possible actions are described by means of a Kripke model that specifies which communication actions are indistinguishable for which process. We first show that the number of connected components in the action model results in a lower bound for communication complexity. We then apply this result, in the restricted setting of a two processor system, for determining communication complexity lower bounds for solving a distributed computing problem \(\mathcal {P}\): We first determine some properties of the action model corresponding to any given protocol that solves \(\mathcal {P}\), and then use our action model communication complexity lower bounds. Finally, we demonstrate our approach by applying it to synchronous distributed function computation and to a simple instance of consensus in directed dynamic networks.
This work has been supported by the Austrian Science Fund FWF under the projects ADynNet (P28182) and RiSE/SHiNE (S11405).
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Notes
- 1.
Please observe the different fonts in our notation: in \(s\sim _a t\), \(\sim _a\) is taken from the epistemic model M, while in \(\texttt {s}\sim _a \texttt {t}\), \(\sim _a\) is from the action model \(\texttt {M}\).
- 2.
We note, however, that our findings do support this claim, as the action model is common a priori knowledge and clearly more complex in the private than in the public scenario, cp. Fig. 3.
- 3.
To be precise, this is only true if x is a preserved formula (as introduced in [8]), which requires x to be propositional or positive knowledge (but not \(x = \lnot K_a \phi \), for example). Thus we will also restrict ourselves to algorithms in which preconditions of actions only involve preserved formulas, which is essentially a non-restriction for distributed algorithms.
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Pfleger, D., Schmid, U. (2018). On Knowledge and Communication Complexity in Distributed Systems. In: Lotker, Z., Patt-Shamir, B. (eds) Structural Information and Communication Complexity. SIROCCO 2018. Lecture Notes in Computer Science(), vol 11085. Springer, Cham. https://doi.org/10.1007/978-3-030-01325-7_27
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