68Q05-NondeterministicTuringMachine.tex

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\pmtitle{non-deterministic Turing machine}
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\begin{document}
The definition of a non-deterministic Turing machine is the same as the definition of a deterministic Turing machine except that $\delta$ is a relation,  not a function.  Hence, for any particular state and symbol, there may be multiple possible legal moves.

If $S\in\Gamma^+$ we say $T$ accepts $S$ if, when $S$ is the input, there is some finite sequence of legal moves such that $\delta$ is undefined on the state and symbol pair which results from the last move in the sequence and such that the final state is an element of $F$.  If $T$ does not accept $S$ then it rejects $S$.

An alternative definition of a non-deterministic Turing machine is as a deterministic Turing machine with an extra one-way, read-only tape, the guess tape.  Then we say $T$ accepts $S$ if there is any string $c(S)$ such that, when $c(S)$ is placed on the guess tape, $T$ accepts $S$.  We call $c(S)$ a \emph{certificate} for $S$, and otherwise that it rejects $S$.  In some cases the guess tape is allowed to be two-way; this generates different time and space complexity classes than the one-way case (the one-way case is equivalent to the original definition).
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