A Note on Degrees of Self-Describing Turing Machines

Certain automata which build other automata are considered. Each such automaton contains a Turing machine. The properties of the Turing machine, of an offspring automaton, are completely determined by certain outputs of the Turing machine of the parent automaton. This makes it possible to speak of a parent Turing machine as describing its offspring Turing machine. I t is desirable to be able to conduct an algorithmic or partial algorithmic analysis of the properties of the descendants of an arbitrary Turing machine M given only a description of M. To this end, six types of Turing machines which describe certain effectively computable distortions of themselves (self-description being a special case) are defined. The Turing machines of two of these types are also universal Turing machines. For each of the six types of Turing machines, the problem of deciding whether or not a given Turing machine is of that type is considered. The degree of unsolvability of each of these decision problems is computed. The results of these computations are used to show that the corresponding decision problems have successively better partial algorithmic solutions which can be obtained algorithmically.