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Also note that c cannot be an accepted configuration of ( MM _k) because otherwise z would be equal to (w(s_0)) in (S( MM _k)).
Let c be an accepted configuration of ( MM _k) such that (|c|=n) and the length of the computation connecting c with the accept configuration (s_0) of ( MM _k) is t(n).
The time function (T_M n)) of M is the minimal function such that every accepted configuration of length (le n) has an accepting computation of length (le T_M n)).
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It has limited the product design possibilities inside a limited number of accepted configurations.
We always assume that a machine halts on every accept configuration but it may halt on non-accept configurations too.
Recall that (s_0) is the accept configuration of ( MM _k), i.e., (s_0=(0;0,ldots,0)).
The problem of determining whether \(N\) accepts \(x\) can thus be reduced to that of checking whether there is a path in \(G_{N,x}\) from its initial configuration to an accepting configuration.
After the first k tapes of (M") form the accept configuration (s_{mathrm {acc}}(M')), the machine erases letters from the tape alphabet (A") on the history tape and halts, producing the accept configuration (s_{mathrm {acc}}(M")) of (M") (thus (s_{mathrm {acc}}(M")=s_0alpha _{k+1}q_{k+1}^0omega _{k+1})).
The accept configuration is (s_0=(0;0,ldots,0)) (the command number is 0, all glasses are empty) and input configurations have the form ((1 m,0,ldots,0)). Let us describe commands of Minsky machines more precisely.
For instance, given a deterministic Turing machine \(T\) which decides \(X\) in polynomial space, it may be shown that it is possible to construct a QBF-formula \ \phi_{T,x}\) of length polynomial in \(\lvert x\rvert\) which expresses using the reachability method that there is a path in the configuration graph for \(T\) from the initial configuration \(C_0 x)\) to an accepting configuration.
For the Turing machine we choose stop states (q_i^0) in each (Q_i), then a configuration w is accepted if there exists a computation starting with w and ending with a configuration where all state symbols are (q_i^0) and all tapes are empty (which is the accept configuration for the Turing machine).
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