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The Inversion of Functions Defined by Turing


J. McCarthy

Computer Science Department

Stanford University

Stanford, CA 95305



Consider the problem of designing a machine to solve well-defined intel-

lectual problems. We call a problem well-defined if there is a test which can

be applied to a proposed solution. In case the proposed solution is a solu-

tion, the test must confirm this in a finite number of steps. If the proposed

solution is not correct, we may either require that the test indicate this in a

finite number of steps or else allow it to go on indefinitely. Since any test may

be regarded as being performed by a Turing machine, this means that well-

defined intellectual problems may be regarded as those of inverting functions

and partial functions defined by Turing machines.

Let fm(n) be the partial function computed by the mth Turing machine.

It is not defined for a given value of n if the computation does not come to

an end. This paper deals with the problem of designing a Turing machine

which, when confronted by the number pair (m, r), computes as efficiently as

possible a function g(m, r) such that fm(g(m, r)) = r. Again, for particular

values of m and r no g(m, r) need exist. In fact, it has been shown that the

existence of g(m, r) is an undecidable question in that there does not exist a

Turing machine which will eventually come to a stop and print a 1 if g(m, r)

does not exist.

In spite of this, it is easy to show that a Turing machine exists which will

compute a g(m, r) if such exists. Essentially, it substitutes integers in fm(n)

until it comes to one such that f m(n) = r. It will therefore find g(m, r) if it

exists, but will never know enough to give up if g(m, r) does not exist. Since

the computation of fm(n) may not terminate for some n, it is necessary to

avoid getting stuck on such n’s Hence the machine calculates the numbers

m(n) in some order where f k

f k

m(n) is fm(n) if the computation of fm(n) ends

after k steps and is otherwise undefined

Our problem does not end once we have found this procedure for com-

puting g(m, r) because this procedure is extremely inefficient. It corresponds

to looking for a proof of a conjecture by checking in some order all possible

English essays

In order to do better than such a purely enumerative method, it is neces-

sary to use methods which take into account some of the structure of Turing

machines. But before discussing this, we must attempt to be somewhat (only

somewhat) more precise about what is meant by efficiency

The most obvious idea is to say that if T1 and T2 are two Turing machines

each computing a g(m, r), then for a particular m and r the more efficient one

is the one which carries out the computation in the fewest steps. However,

this won’t do since for any Turing machine there is another one which does k

steps of the original machine in one step However, the new machine has many

more different kinds of symbol than the old. It is probably also possible to

increase the speed by increasing the number of internal states, though this

is not so easy to show. (Shannon shows elsewhere in these studies that it

is possible to reduce the number of internal states to two at the cost of

increasing the number of symbols and reducing the speed.)

Hence we offer the following revised definition of the length of a compu-

tation performed by a Turing machine. For any universal Turing machine

there is a standard way of recoding on it the computation performed by the

given machine. Let this computation be recoded on a fixed universal Turing

machine and count the number of steps in the new computation. There are

certain difficulties here connected with the fact that the rate of computation

is limited if the tape of the universal machine is finite dimensional, and hence

the rate should probably be defined with respect to a machine whose tape

is infinite dimensional but each square of which has at most two states and

which has only two internal states. This requires a mild generalization of the

concept of Turing machines

The tape space is not required to be either homogeneous or isotropic We

hope to make these considerations precise in a later paper. For now, we only

remark that dimensionality is meant to be



log Vn

log n

where Vn is the number of squares accessible to a given one in n steps. It

will be made at least plausible that a machine with Q internal states and

S symbols should be considered as making about 1

2 log QS elementary steps

per step of computation and hence the number of steps in a computation

should be multiplied by this factor to get the length of the computation.

Having now an idea of what should be meant by the length (cid:96)(m, r, T )

of a particular computation of g(m, r) by the machine T , we can return to

the question of comparing two Turing machines. Of course, if (cid:96)(m, r, T1) ≤

(cid:96)(m, r, T2) for all m and r for which these numbers are finite, we should cer-

tainly say that T1 is more efficient than T2. (We only consider machines that

actually compute g(m, r) whenever it exists. Any machine can be modified

into such a machine by adding to it facilities for testing a conclusion and

having it spend a small fraction of its time trying the integers in order.)

However, it is not so easy to give a function which gives an over-all estimate

of the efficiency of a machine at computing g(m, r). The idea of assigning a

weight function p(m, r) and then calculating

p(m, r)(cid:96)(m, r, T )



does not work very well because (cid:96)(m, r, T ) is not bounded by any recursive

function of m and r. (Otherwise, a machine could be described for determin-

ing whether the computation of fm(n) terminates. It would simply carry out

some k(m, n) steps and conclude that if the computation had not terminated

by this time it was not going to.) There cannot be any machine which is as

fast as any other machine on any problem because there are rather simple

machines whose only procedure is to guess a constant which are fast when

g(m, r) happens to equal that constant.

We now return to the question of how to design machines which make use

of information concerning the structure of Turing machines. The straight-

forward approach would be to try to develop a formalized theory of Turing

machines in which length of a computation is defined and then try to get

a decision procedure for this formal theory. This is known to be a hopeless

task. Systems much simpler than Turing machine theory have been shown to

have unsolvable decision procedures. So, we look for a way of evading these


Before discussing how this may be done, it is worthwhile to bring up some

more enumerative procedures. First, let fk(x, y) be the function of two vari-

ables computed by the kth Turing machine. Our procedure consists in trying

the numbers fk(m, r) in order (again diagonalizing to avoid computations

which don’t end.) This is based on the plausible idea that, in searching for

the solution to a problem, the given data should be taken into account.

The next complication which suggests itself is to revise the order in which

recursive functions are considered. One way is to consider ffk((cid:96))(m, r) diag-

onalized on k and (cid:96). This is based on the idea that the best procedure is

more likely to be recursively simple, rather than merely to have a low number

in the ordering. More generally, the enumeration of partial recursive func-

tions should give an early place to compositions of the functions which have

already appeared.

This process of elaborating the schemes by which numbers are tested can

be carried much further. Intuitively it seems that each successive complica-

tion improves the performance on difficult problems, but imposes a delay in

getting started

The difficulty with the afore-mentioned methods and their elaborations

is that they have no concept of partial success. It is a great advantage in

conducting a search to be able to know when one is close. This suggests first

of all that a function fk(x, y) be tried out first on simpler problems known

to have simple solutions before very many steps of the computation fk(m, r)

are carried out.

At this point it becomes unclear how to proceed further. What has been

done is only a semiformal description of a few of the processes common to

scientific reasoning and we have no guarantee of being exhaustive. Of course,

exhaustiveness in examining for usefulness can be attained for any effectively

enumerable class of objects simply by going through the enumeration. How-

ever, enumerative methods are always inefficient. What is needed is a general

procedure which ensures that all relevant concepts which can be computed

with are examined and that the irrelevant are eliminated as rapidly as possi-

ble. Here we remark that a property is useful mainly when it permits a new

enumeration of the objects possessing it and not merely a test which can be

applied. With the enumeration the objects not possessing the property need

not even be examined Of course, it is then necessary to be able to express

all other relevant properties as functions of the new enumeration.

In order to get around the fact that all formal systems which are anywhere

near adequate for describing recursive function theory are incomplete, we

avoid restriction to any one of them by introducing the notion of a formal

theory (not for the first time, of course).

For our purposes a formal theory is a decidable predicate V (p, t1, . . . , tk →

t) which is to be regarded as meaning that p is a proof that the statements

t1, . . . , tk imply the statement t in the theory We do not require that state-

ments have negatives or disjunctions, although theories with these and other

properties of propositional calculi will presumably belong to the most useful


An interpretation of a formal theory is a computable function i(t) map-

ping a class of statements of the theory into statements of other theories

or concrete propositions. The only kind of concrete proposition which we

shall mention for now is P (m, n, r, k) meaning fm(n) = r in a computation

of length ≤ k. Such a proposition is of course verifiable

The theories and interpretations of them can be enumerated. A function

of their G¨odel numbers is called a status function.

(To be regarded as a

current estimate of their validity and relevance, etc.) An action scheme

is a computation rule which computes from a status function (its G¨odel

number) a new status function, perhaps gives an estimate of g(m, r) if it has

determined this, and computes a new action scheme.

REMARK 1. A status function would consist primarily of estimates of

the validity of the separate theories and would place theories which had

not been examined in an “unknown” category. However, an action scheme

might modify the status of a whole class of theories simultaneously in some

systematic way.

REMARK 2. The reader should note that a formal equivalence could

probably be established between formal theories and certain action schemes

Thus, knowledge can be interpreted as a predisposition to act in certain ways

under given conditions, and an action scheme can be interpreted as a belief

that certain action is appropriate under these conditions. This equivalence

should not be made because a simple theory may have a very complicated

interpretation as an action scheme and conversely, and in this inversion prob-

lem the simplicity of an object is one of its most important properties

This suggests an Alexandrian solution to a knotty question. Perhaps, a

machine should be regarded as thinking if and only if its behavior can most

concisely be described by statements, including those of the form “It now

believes Pythagoras’ theorem.” This would not be the case for a phonograph

record of a proof of Pythagoras1 theorem. Such statements will probably be

especially applicable to predictions of the future behavior under a variety of


REMARK 3. In determining a procedure which has action schemes, the-

ories, etc., we are imposing our ideas of the structure of the problem on the

machine. The point is to do this so that the procedures we offer are general

(will eventually solve every solvable problem) and also are improvable by the

methods built into the machine.

REMARK 4. Except for the concrete proposition none of the statements

of a formal theory have necessary interpretations which are stateable in ad-

vance. However, if the machine were working well, an observer of its internal

mechanism might come to the conclusion that a certain statement of a the-

ory is being used as though it were, say (x)fm(x) = 0, etc. An important

merit of a particular theory might be that in it the verification of a proof

at the correlate of a certain concrete proposition is much shorter than the

computation of the concrete proposition, i.e., shorter than k in P (m, n, r, k).

REMARK 5. The relation of certain action schemes to certain theories

might warrant our regarding certain statements in them as being normative,

i.e., of the form “the next axiom scheme should be no. m” or “theory k should

be dropped from consideration.”

REMARK 6. Not every worthwhile problem is well-defined in the sense of

this paper. In particular, if there exist more or less satisfactory answers with

no way of deciding whether an answer already obtained can be improved on

a reasonable time, the problem is not well-defined.

/@steam.stanford.edu:/u/jmc/s96/inversion.tex: begun 1996

May 24, latexed 1996

Jun 1 at 9:44 a.m.