Type 2 Turing Machines

A Type 2 Turing machine is a Turing machine with three tapes: An input tape, which is read-only; a working tape, which can be written to and read from; and, notably, an output tape, which is "append-only".

Real numbers

In this context, real numbers are represented as arbitrary infinite sequences of symbols. These sequences could for instance represent the digits of a real number. Such sequences need not be computable — this allowance is both important and philosophically unproblematic.^[4] Note that the programs that act on these sequences do need to be computable in a reasonable sense.

In the case of real numbers, the usual decimal or binary representations are not appropriate. Instead a signed digit representation first suggested by Brouwer often gets used: The number system is base 2, but the digits are ${\displaystyle {\overline {1}}}$ (representing ${\displaystyle -1}$), 0 and 1. In particular, this means ${\displaystyle 1/2}$ can be represented both as ${\displaystyle 0.1}$ and ${\displaystyle 1.{\overline {1}}}$.

To understand why decimal notation is inappropriate, consider the problem of computing ${\displaystyle z=x+y}$ where ${\displaystyle x=0.(3)}$ and ${\displaystyle y=0.(6)}$, and giving the result ${\displaystyle z}$ in decimal notation. The value of ${\displaystyle z}$ is either ${\displaystyle 0.(9)}$ or ${\displaystyle 1.(0)}$. If the latter result were given for instance, then a finite number ${\displaystyle n}$ of digits of ${\displaystyle x}$ would be read before choosing the digit ${\displaystyle 1}$ before the decimal point in ${\displaystyle z}$ — but then if the ${\displaystyle n+1}$th digit of ${\displaystyle x}$ were decreased to 2, then the result for ${\displaystyle z}$ would be wrong. Similarly, the former choice ${\displaystyle 0.(9)}$ for ${\displaystyle z}$ would be wrong sometimes. This is essentially the tablemaker's dilemma.

As well as signed digits, there are analogues of Cauchy sequences and Dedekind cuts that could in principle be used instead.

Computable functions

Computable functions are represented as programs on a Type 2 Turing machine. A program is considered total (in the sense of a total function as opposed to partial function) if it takes finite time to write any number of symbols on the output tape regardless of the input. A total programs run forever, generating increasingly more digits of the output.

Names

Results about computability associated with infinite sets often involve namings, which are maps between those sets and recursive representations of subsets thereof.

The issue of Type 1 versus Type 2 computability

Type 1 computability is the naive form of computable analysis in which one restricts the inputs to a machine to be computable numbers instead of arbitrary real numbers.

The difference between the two models lies in the fact that a program that is well-behaved over computable numbers (in the sense of being total) is not necessarily well-behaved over arbitrary real numbers. For instance, there are computable functions over the computable real numbers that map some bounded closed intervals to unbounded open intervals.^[5] These functions cannot be extended to arbitrary real numbers (without making them partial), as all computable functions ${\displaystyle \mathbb {R} \to \mathbb {R} }$ are continuous, and this would then violate the extreme value theorem. Since that sort of behaviour could be considered pathological, it is natural to insist that a function should only be considered total if it is total over all real numbers, not just the computable ones.

Realisability

In the event that one is unhappy with using Turing machines (on the grounds that they are low level and somewhat arbitrary), there is a realisability topos called the Kleene–Vesley topos in which one can reduce computable analysis to constructive analysis. This constructive analysis includes everything that is valid in the Brouwer school, and not just the Bishop school.^[6] Additionally, a theorem in this school of constructive analysis is that not all real numbers are computable, which is constructively non-equivalent to there exist uncomputable numbers. This school of constructive analysis is therefore in direct contradiction to schools of constructive analysis — such as Markov's — which claim that all functions are computable. It ultimately shows that while constructive existence implies computability, it is in fact unproblematic — even useful — to assert that not every function is computable.