non-deterministic Turing machine

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 $S\in {\mathrm{\Gamma }}^{+}$ 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 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).