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(ii) there are non-denumerably many functions f :N N,
• →
(iii) the Church-Turing thesis.
•
The first two propositions has the status of mathematical theorems as they
can be formulated within precisely defined formalisms. The third, cannot be
proved as it relates the intuitive notion of an effective method to the formal
models of computation.
TheenumerabilityoftheTuringmachinesfollowsfromtherequirementthat
everyTuringmachineprogrammustbestatedasafinitesetofinstructions,each
instruction being built from a finite number of tokens. Each programtherefore
is a finite string of tokens from a finite alphabet, and since such strings can be
enumerated,the setofTuringmachineprogramsisenumerable. Foracomplete
proof, an explicit numbering of the programs must be given, but that can be
done based on G¨odel numbering. Also, by explicitly giving an enumeration, a
particular non-computable function can be exhibited.
The non-enumerability of the functions f : N N can be proved using a
→
diagonal argument. The proof, though standard and well-known, will be given
here since it illustrates the diagonal method12 in a simple setting. First note
thatwe areconsideringallsuchfunctions, both partialandtotal. Suppose that
we are given an enumeration of all functions F = f . We can then define
n }∞n=0
{
a new function u, called the anti-diagonal function [23], where
1, if f (n) is undefined
u(n)= n
(cid:26)f n(n)+1, otherwise.
This is a well-defined total function. Note that questions of computability do
not enter at this stage. If the list F is complete, then the function u must be
oneofthe functions inthe list, sayf andthus u(x)=f (x) foreverynumber
m m
x. In particular u(m)=f (m) Using the definition of u we get
m
1, if f (m) is undefined
u(m)=f (m)= m
m (cid:26)f m(m)+1, otherwise.
This contradiction proves that the list F cannot be complete and the set of all
functions cannot be enumerated in any way.
12Themethodwasinvented byG.Cantor.
32
Computability enters when we ask the question whether the anti-diagonal
function can be computed or not. If the list F was compiled using Turing
machines, i.e. if the list is a list of all Turing computable functions, then the
proofshowsthatthereareTuringnon-computablefunctions. Thisargumentcan
be used on any computational model. If F is a list of all functions computable
withinacertainnamedmodel,thentherearefunctionsthatarenotcomputable
within this model. A priori, different models of computation could give rise to
different sets of computable functions. It is an empirical fact that this is not
the case.
It turns out that all computational models, claiming to capture the idea of
effective computability, that has been considered so far, can be proved to yield
the same set of computable functions.
Thenextquestionis,canthefunctionubecomputedeffectivelyatall,using
someintuitivemodel? Ifthatisthecase,thentheclassiccomputationalmodels
are to narrow. On the other hand, if the Church-Turingthesis is true, then the
function u is absolutely uncomputable.
Itisclearwhatthekeypointis. Ifweconsidertotalfunctions,i.e. functions
defined for all numbers, then the diagonal argument shows that any compu-
tational model that computes total functions is too narrow. In this case, the
clause taking care of the cases when the function is undefined, is not needed.
The anti-diagonal function can be defined, and it is easily proved that it can-
not occur in the list of functions. Therefore the model is incomplete. But in
this case, the anti-diagonal function is intuitively computable. This is often
phrased as saying that we can diagonalize out of any computational model for
total functions. This intuitive computation of the anti-diagonal function relies
on examining the list of functions and computing its values based on this list,
so it could also be seen as a meta-computation.
The questionofwhether itis possible todiagonalizeoutofthe modelornot
when partial functions are allowed, depends on whether there is any general
procedure to determine if a function is defined for a certain number or not. If
thereissuchaprocedure,itcouldbeusedtocomputetheanti-diagonalfunction.
Now,fortheTuringmachinemodel,afunctionisleftundefinedforacertain
argument in two cases. Either the machine stops in a non-standard configura-
tion,acasewhichcanbetakencareofbyproperprogramming. Orthemachine
neverstops. If there is a generaleffective method which is capable ofdetermin-
ing (in a finite amount of time) whether or not Turing computations halt, then
this method could be used to diagonalize out of the Turing model. This is the
halting problem. Itcan,infact,beshownthatthehaltingproblemisunsolvable
within the Turing model.
If one could devise a computational model, formal or intuitive, which were
able to solve the Turing halting problem, then that model would in some sense
be stronger than the Turing model. To date, there is no such model.
33
The halting problem
The general halting problem is the problem of designing an effective method,
intuitiveorwithintheTuringmachinemodel,todeterminewhetheraparticular
TuringmachineM willeverhaltwhenstartedtocomputewithinputdatam. If
n
acertaincomputationdoesnothalt,thismeansthatthecorrespondingfunction
is undefined. Therefore, the halting problem is closely related to the question
of computability.
The algorithm has access only to the Turing machine programs and the
input data on the tape. This makes sense, because it is of no use just to set
the machines running and wait to see if they will stop in a standard terminal
configuration. The machines, ’destined’ not to stop, will run forever, and the
answer cannot be obtained by waiting.
It can be proved that the halting problem is unsolvable within the Turing
machine model. The proof is non-trivial and technical, and we will just outline