Small universal Turing machines
Theoretical Computer Science - Special issue on universal machines and computations
Computation: finite and infinite machines
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Small Semi-Weakly Universal Turing Machines
Fundamenta Informaticae - Machines, Computations and Universality, Part I
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We are interested by "small" Universal Turing Machines (in short: UTMs), in the framework of 2, 3 or 4 tape-symbols. In particular: - 2 tape-symbols. Apart from the old 24-states machine constructed by Rogozhin in 1982, we know two recent examples requiring 22 states, one due to Rogozhin and one to the author. - 3 tape-symbols. The best example we know, due to Rogozhin, requires 10 states. It uses a strategy quite hard to follow, in particular because even-length productions require a different treatment with respect to odd-length ones. - 4 tape-symbols. The best known machines require 7 states. Among them, the Rogozhin's one require only 26 commands; the Robinson's one, though requiring 27 commands, fournishes an easier way to recover the output when the TM halts. In particular, Robinson asked for a 7 × 4 UTM with only 26 commands and an easy treatment of the output. Here we will firstly construct a 7 × 4 UTM with an easy recover of the output which requires only 25 commands; then we will simulate such a machine by a (simple) 10 × 3 UT M and by a 19 × 2 UTM.