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From the construction, it is clear that (M') polynomially reducible to M. One can also convert a sym-universally halting Turing machine into a 1-tape sym-universally halting Turing machine: Let M be a deterministic sym-universally halting Turing machine recognizing a language X.
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For every recursive language X there exists a deterministic sym-universally halting Turing machine M with one tape recognizing X. Moreover if (M') is any deterministic Turing machine recognizing X then we can additionally assume that M polynomially reduces to (M').
Let (M') be a deterministic universally halting Turing machine with k tapes recognizing X.
Let ( MM _k) be a sym-universally halting Minsky machine.
Let ( MM _2) be a sym-universally halting 2-glass Minsky machine with (N+1) commands numbered (0,ldots,N).
Chess is a deterministic game.
This is a deterministic quantity.
In this paper, we use the fact that every Turing machine recognizing a recursive set is equivalent to a sym-universally halting Minsky machine, i.e., Minsky machine whose symmetrization halts on every non-accepted configuration (see Theorem 2.7).
The new Minsky machine accepts the same set of numbers and is still sym-universally halting with the same (up to the equivalence) time function.
We call a deterministic machine M sym-universally halting if ({mathrm {Sym}}(M)) does not have infinitely long reduced computations that start at a non-accepted configuration.
Moreover if c is not accepted by ( MM _3), then (c') is not accepted by M. Thus if M is sym-universally halting, then ( MM _3) is sym-universally halting.
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