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Yurii Rogozhin’s Contributions to the Field of Small Universal Turing Machines

Woods D., Neary T.

Fundamenta Informaticae

2015

Vol. 138, nr 1/2

251--258

In the field of small universal Turing machines, Yurii Rogozhin holds a special prize: he was first to close off an infinite number of open questions by drawing a closed curve that separates the infinite set of Turing machines that are universal from a finite set of small machines for which we don’t yet know. Rogozhin did this by finding the smallest known universal Turing machines at the time, both in terms of number of states and number of symbols. This brief note summarises this and a few of Yurii’s other contributions to the field, including his work with Manfred Kudlek on small circular Post machines.

Small Semi-Weakly Universal Turing Machines

Woods D., Neary T.

Fundamenta Informaticae

2009

Vol. 91, nr 1

179-195

We present three small universal Turing machines that have 3 states and 7 symbols, 4 states and 5 symbols, and 2 states and 13 symbols, respectively. These machines are semi-weakly universal which means that on one side of the input they have an infinitely repeated word, and on the other side there is the usual infinitely repeated blank symbol. This work can be regarded as a continuation of early work by Watanabe on semi-weak machines. One of our machines has only 17 transition rules, making it the smallest known semi-weakly universal Turing machine. Interestingly, two of our machines are symmetric withWatanabe's 7-state and 3-symbol, and 5-state and 4-symbol machines, even though we use a different simulation technique.

Four Small Universal Turing Machines

Neary T., Woods D.

Fundamenta Informaticae

2009

Vol. 91, nr 1

123-144

We present universal Turing machines with state-symbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bi-tag system and are the smallest known single-tape universal Turing machines with 5, 4, 3 and 2-symbols, respectively. Our 5-symbolmachine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin. Also, all of the universalmachines we present here simulate Turing machines in polynomial time.