Abstract
A computation universal Turing machine, U, with 2 states, 4 letters, 1 head and 1 two-dimensional tape is constructed by a translation of a universal register-machine language into networks over some simple abstract automata and, finally, of such networks into U. As there exists no universal Turing machine with 2 states, 2 letters, 1 head and 1 two-dimensional tape only the 2-state, 3-letter case for such machines remains an open problem. An immediate consequence of the construction of U is the existence of a universal 2-state, 2-letter, 2-head, 1 two-dimensional tape Turing machine, giving a first sharp boundary of the necessary complexity of universal Turing machines.
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