Abstract
The problem of finding a Turing machine with undecidable halting problem whose program contains the smallest number of instructions is well known. Obviously, such a machine must satisfy the following condition: by deleting even a single instruction from its program, we get a machine with decidable halting problem. In this paper, Turing machines with undecidable halting problem satisfying this condition are called connected. We obtain a number of general properties of such machines and deduce their simplest corollaries concerning the minimal machine with undecidable halting problem.
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REFERENCES
A. I. Mal'tsev, Algorithms and Recursive Functions [in Russian], Nauka, Moscow, 1965.
L. M. Pavlotskaya, “Decidability of the halting problem for certain classes of Turing machines,” Mat. Zametki [Math. Notes], 13 (1973), no. 6, 899–909.
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Pavlotskaya, L.M. Turing Machines Connected to the Undecidability of the Halting Problem. Mathematical Notes 71, 667–675 (2002). https://doi.org/10.1023/A:1015840005656
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DOI: https://doi.org/10.1023/A:1015840005656