Computation on binary tree-networks

Abstract

In this paper, we show that, for any finite set X, the set A( X n ) of mappings from X n into itself is computable on a network N=( G, X) of size n, if and only if N contains an automaton which can directly receive informations from any automaton of the network. Then we prove that a non-bijective mapping F from {0,1} n into itself is computable on a binary tree-network N=( G, {0,1}) if and only if I(F) = Σ x|F −1(x)|/2≥2 n−p−1 , where p is the maximum number of pending vertices of a caterpillar of G, and ν denotes the greatest integer lower than or equal to ν.