Discrete Transfinite Computation Models

In the past few years there has been a resurgence of interest in discrete models of computation that are allowed to act transfinitely. Such conceptual machines act in simple steps or stages, and have as a paradigm the standard Turing machine. This, during its progress moves one cell at a time, to the left or the right along an unbounded tape that it is reading, and subsequently alters symbols, changes states and moves on. This paradigm has been with us for 70 years, and for much of this chapter we shall consider variants of such a device. Our models will all be discrete acting computational digital models. We shall consider how Turing and other computing machines could possible behave when allowed to perform supertasks (by which we mean that they are allowed to complete an infinite sequence of tasks or operations). Such a machine is usually envisaged with an unbounded tape. If supertasks are allowed then naturally the whole of that tape comes into play, and we can imagine that tape already having some characteristic function written on it. The machine can then act on that tape and is then essentially a computer acting at a higher type, namely that of infinite sequences of 0, 1's, in other words of real numbers. Surprisingly, even if one restricts one's model to, say, a register machine