Theories of abstract automata – Arbib

We have said that automata theory deals with the realization of partial functions F: X* —» Y* by some finitely specifiable substrate. Before we specify in more detail the forms (of which the Turing machine is one) of substrate which have figured most prominently in automata theory, it is useful to distinguish on-line machines from off-line machines. An on-line machine is one that may be thought of as processing data in an interactive situation—in processing a string it must yield a continual flow of outputs, processing each symbol completely (albeit in a way dependent on prior inputs) before it reads in the next symbol. This means that the corresponding function F: X* —> F* must have the following special property:

1 For each nonempty string u of X* there exists a function Fu: X* —» Y* such that for every nonempty v in X*
F(uv) = F(u) • Fu(v) that is, the input string u causes the machine to put out the string F(u) and to “change state” in such a way that it henceforth processes inputs according to a function Fu determined solely by F and u. We call a function sequential if it satisfies property 1.

From: Arbib, M.A., 1969,Theories of Abstract Automata, Prentice-Hall: Englewood Cliffs, N. J., 412 + xiii pages.