Abstract
We survey some work concerned with small universal Turing machines, cellular automata, tag systems, and other simple models of computation. For example it has been an open question for some time as to whether the smallest known universal Turing machines of Minsky, Rogozhin, Baiocchi and Kudlek are efficient (polynomial time) simulators of Turing machines. These are some of the most intuitively simple computational devices and previously the best known simulations were exponentially slow. We discuss recent work that shows that these machines are indeed efficient simulators. In addition, another related result shows that RuleĀ 110, a well-known elementary cellular automaton, is efficiently universal. We also discuss some old and new universal program size results, including the smallest known universal Turing machines. We finish the survey with results on generalised and restricted Turing machine models including machines with a periodic background on the tape (instead of a blank symbol), multiple tapes, multiple dimensions, and machines that never write to their tape. We then discuss some ideas for future work.
This paper is extended and updated fromĀ [110]. T.Ā Neary is supported by Science Foundation Ireland, Grant Number 09/RFP/CMS2212. D.Ā Woods is supported by National Science Foundation Grant 0832824, the Molecular Programming Project. We thank Astrid Haberleitner for her tireless work in translating a number of highly technical papers from German to English, and Beverley Henley for her support.
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Neary, T., Woods, D. (2012). The Complexity of Small Universal Turing Machines: A Survey. In: BielikovƔ, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., TurƔn, G. (eds) SOFSEM 2012: Theory and Practice of Computer Science. SOFSEM 2012. Lecture Notes in Computer Science, vol 7147. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27660-6_32
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