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The stability of biased threshold voting
Abstract
Imagine a network wherein nodes are arranged in a loop and each node is connected to receive data from the n nodes clockwise ahead of it. Let each node in the network carry a bit value which acts as a "vote"; this node votes 1 at time k if at least t of the n nodes connected to it vote 1 at time k $-$ 1, and it votes 0 otherwise. How must the nodes vote at time 0 if their votes are to all agree at some later time? This problem has potential application in fault-tolerant computing. For example, the nodes might additionally carry a numerical result, for which some variability is expected across the network, and the vote bit might be used to flag a suspected error in this result. To obtain approximate agreement across the network, the result carried by a node at time k could be taken as the average of the unflagged results carried at time k $-$ 1 by the n nodes connected to it. Of perhaps greater interest, however, is that here is a manageable instance of a much deeper problem. Dynamic systems, such as the weather, may be nearly or fully described by local mechanisms and yet exhibit global behavior which defies prediction. Such systems can typically be modeled as cellular automata. One fairly simple and well known cellular automaton, Conway's "Life", is rich enough to simulate Turing machines, which suggests how easily undecidable issues can arise from iterative interplay. The binary cellular automata investigated here are less lively. It will be shown that the initial vote assignments which eventually lead to all-1 or all-0 consensuses are respectively regular (in the language-theoretic sense that they are accepted by finite automata) when t $\le$ 3 and n $\ge$ 2t. From what is discovered, it seems plausible that this regularity is a general property, independent of the constraints just stated. Other results, some more specific, some more general, are also obtained.
Subject Area
Computer science
Recommended Citation
Wood, David Brandon, "The stability of biased threshold voting" (1993). ETD collection for University of Nebraska-Lincoln. AAI9416006.
https://digitalcommons.unl.edu/dissertations/AAI9416006