Three Small Universal Turing Machines

Abstract

We are interested by “small” Universal Turing Machines (in short: UTMs), in the framework of 2, 3 or 4 tape—symbols. In particular:

• 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.

• 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.

• 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 UTM and by a 19 × 2 UTM.