Decidability and Undecidability of the Halting Problem on Turing Machines, a Survey

The paper surveys the main results obtained for Turing machines about the frontier between a decidable halting problem and universality. The notion of decidability criterion is introduced. Techniques for decidability proofs and for contracting universal objects are sketchily explained. A new approach for finding very small universal Turing machines is considered in the last part of the paper.

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