The complexity and entropy of Turing machines

Addresses the relationship between dynamical systems theory and theoretical computer science, in particular the dynamical, information-theoretic and computational properties of systems that compute. These properties have been studied in cellular automata and the symbolic dynamics of maps over the unit interval, but have never been addressed in compact systems known to be capable of universal computation. Recent work is described in which the entropy, periodicity and regular language complexity of a large number of randomly generated Turing machines were calculated. The results are discussed in detail and compared with an identical analysis of a universal Turing machine. This comparison yields the first direct quantitative evidence that universal computation lies between ordered and chaotic behavior. The discussion concludes with a list of questions remaining to be answered about the phase-space portrait of computationally complex systems.<>