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**1 - 1**of**1**### An Efficient Algorithm for Computing the Ith Letter of Phi^n(a)

"... Let \Sigma be a finite alphabet, and let ' : \Sigma ! \Sigma be a homomorphism, i.e., a mapping satisfying '(xy) = '(x)'(y) for all x; y 2 \Sigma . Let a 2 \Sigma, and let i 1, n 0 be integers. We give the first efficient algorithm for computing the ith letter of &ap ..."

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Let \Sigma be a finite alphabet, and let ' : \Sigma ! \Sigma be a homomorphism, i.e., a mapping satisfying '(xy) = '(x)'(y) for all x; y 2 \Sigma . Let a 2 \Sigma, and let i 1, n 0 be integers. We give the first efficient algorithm for computing the ith letter of ' n (a). Our algorithm runs in time polynomial in the size of the input, i.e., polynomial in log n, log i, and the description size of '. Our algorithm can be easily modified to give the distribution of letters in the prefix of length i of ' n (a). There are applications of our algorithm to computer graphics and biological modelling. If we consider finite-state transducers instead of homomorphisms, the corresponding problem is EXPTIME-hard. 1 Introduction Let \Sigma be a finite alphabet. A homomorphism is a map ' from \Sigma to \Sigma such that '(xy) = '(x)'(y) for all x; y 2 \Sigma . Let a 2 \Sigma; we define ' 0 (a) = a, and ' i (a) = '(' i\Gamma1 (a)) for i 1. For x 2 \Sigma , and a...