remember we are defining a 'computational method' and not an algorithm. what is a computational method naively?
a "procedure" that has all characteristics of an algorithm except that it possibly lacks finiteness, may be called a computational method.
simply put Q is a computational method.
Q = {all possible states of computations, I, Ω}
I = {all possible inputs}
Ω = {all possible outputs}
f = computational rule
f is a function from Q into itself.
f: Q ---> Q
[I] [Ω]
f should leave Ω pointwise fixed which means:
f(q) = q, ∀ q ∈ Ω
note it is not any different function but the same computationalrule just seperated to Ω</p>
Now a procedure will have a sequence. And obviously, a computational method must also have a sequence.
Hence,
Each input x in the set I defines a computational sequence x0, x1, x2, ..., as follows:
x0 = x and xk+1 = f(xk) for k ≥ 0.
How x0 = x?
Don't forget the input x is a sequence and so the initial input sequence would be x0.
As we are dealing with a sequence, and when we are concerning about 'k' states, the order and the position of elements in the sequence matters. And so, the computational rule f is such that the position or more precise word 'state' of the kth element would be k+1th state.
that way, we can seperately apply the function to each new state to get the state that follows.
if xk+1 is not in Ω, then it makes no sense by definition of a function. Hence the wording of Knuth.
The computational sequence is said to terminate in k steps if k is the smallest integer for which xk is in Ω and in this case, it is said to produce the output xk from x.
Thus this is the definition of a computational method. the computational rule is the algorithm.