Transfinite machine models

Abstract

In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely . By ‘discrete’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original machine model. If we concentrate on this for a moment, the machine is considered to be running a program P perhaps on some natural number input n ∈ N and is calculating P (n). Normally we say this is a successful computation if the machine halts after a finite number of stages and we may read off some designated form of output: ‘P (n)↓’. However if the machine fails to halt after a finite time it may be exhibiting a variety of behaviours on its tape. Mathematically we may ask what happens ‘in the limit’ as the number of stages approaches ω. The machine may of course go haywire, and simply be rewriting a particular cell infinitely often, or else the Read/Write head may go ‘off to infinity’ as it moves inexorably down the tape. These kind of considerations are behind the notion of ‘computation in the limit’ which we consider below. Or, it may only rewrite finitely often to any cell on the tape, and leave something meaningful behind: an infinite string of 0, 1’s and thus an element of Cantor space 2. What kind of elements could be there? Considerations of what may lay on an output tape at an infinite stage first surface in the notion of ‘computation in the limit’ or ‘limit decidable’. Whilst the first publication on the matter seems to be two papers coincidentally appearing in the same year, 1965, as Martin Davis has commented, surely this was already known to Post?