  This paper is devoted to the problem of finding characterizations for Cayley
automatic groups. The concept of Cayley automatic groups was recently
introduced by Kharlampovich, Khoussainov and Miasnikov. We address this problem
by introducing three numerical characteristics of Turing transducers: growth
functions, Folner functions and average length growth functions. These three
numerical characteristics are the analogs of growth functions, Folner functions
and drifts of simple random walks for Cayley graphs of groups. We study these
numerical characteristics for Turing transducers obtained from automatic
presentations of labeled directed graphs.
