PROGRAMS WITH COMMON SENSE

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305

jmc@cs.stanford.edu

http://www-formal.stanford.edu/jmc/

Introduction

Interesting work is being done in programming computers to solve problems

which require a high degree of intelligence in humans. However, certain

elementary verbal reasoning processes so simple that they can be carried

out by any non-feeble minded human have yet to be simulated by machine

programs.

This paper will discuss programs to manipulate in a suitable formal lan-

guage (most likely a part of the predicate calculus) common instrumental

statements. The basic program will draw immediate conclusions from a list

of premises. These conclusions will be either declarative or imperative sen-

tences. When an imperative sentence is deduced the program takes a cor-

responding action. These actions may include printing sentences, moving

sentences on lists, and reinitiating the basic deduction process on these lists.

Facilities will be provided for communication with humans in the system

via manual intervention and display devices connected to the computer.

The advice taker is a proposed program for solving problems by manip-

ulating sentences in formal languages. The main difference between it and

other programs or proposed programs for manipulating formal languages (the

Logic Theory Machine of Newell, Simon and Shaw and the Geometry Pro-

gram of Gelernter) is that in the previous programs the formal system was

the subject matter but the heuristics were all embodied in the program. In

this program the procedures will be described as much as possible in the

language itself and, in particular, the heuristics are all so described.

The main advantages we expect the advice taker to have is that its behav-

ior will be improvable merely by making statements to it, telling it about its

symbolic environment and what is wanted from it. To make these statements

will require little if any knowledge of the program or the previous knowledge

of the advice taker. One will be able to assume that the advice taker will

have available to it a fairly wide class of immediate logical consequences of

anything it is told and its previous knowledge. This property is expected to

have much in common with what makes us describe certain humans as hav-

ing common sense. We shall therefore say that a program has common sense

if it automatically deduces for itself a sufficiently wide class of immediate

consequences of anything it is told and what it already knows.

The design of this system will be a joint project with Marvin Minsky, but

Minsky is not to be held responsible for the views expressed here.1

Before describing the advice taker in any detail, I would like to describe

more fully our motivation for proceeding in this direction. Our ultimate

objective is to make programs that learn from their experience as effectively

as humans do. It may not be realized how far we are presently from this

objective. It is not hard to make machines learn from experience to make

simple changes in their behavior of a kind which has been anticipated by

the programmer. For example, Samuel has included in his checker program

facilities for improving the weights the machine assigns to various factors in

evaluating positions. He has also included a scheme whereby the machine

remembers games it has played previously and deviates from its previous

play when it finds a position which it previously lost. Suppose, however, that

we wanted an improvement in behavior corresponding, say, to the discovery

by the machine of the principle of the opposition in checkers. No present or

presently proposed schemes are capable of discovering phenomena as abstract

as this.

If one wants a machine to be able to discover an abstraction, it seems

most likely that the machine must be able to represent this abstraction in

11996: This was wishful thinking. Minsky’s approach to AI was quite different.

some relatively simple way.

There is one known way of making a machine capable of learning arbi-

trary behavior; thus to anticipate every kind of behavior. This is to make it

possible for the machine to simulate arbitrary behaviors and try them out.

These behaviors may be represented either by nerve nets (Minsky 1956),

by Turing machines (McCarthy 1956), or by calculator programs (Friedberg

1958). The difficulty is two-fold. First, in any of these representations the

density of interesting behaviors is incredibly low. Second, and even more

important, small interesting changes in behavior expressed at a high level of

abstraction do not have simple representations. It is as though the human

genetic structure were represented by a set of blue-prints. Then a mutation

would usually result in a wart or a failure of parts to meet, or even an un-

grammatical blue-print which could not be translated into an animal at all.

It is very difficult to see how the genetic representation scheme manages to

be general enough to represent the great variety of animals observed and yet

be such that so many interesting changes in the organism are represented by

small genetic changes. The problem of how such a representation controls the

development of a fertilized egg into a mature animal is even more difficult.

In our opinion, a system which is to evolve intelligence of human order

should have at least the following features:

1. All behaviors must be representable in the system. Therefore, the system should either be able to construct arbitrary automata or to

   program in some general purpose programming language.

2. Interesting changes in behavior must be expressible in a simple way.
3. All aspects of behavior except the most routine must be improvable. In particular, the improving mechanism should be improvable.
4. The machine must have or evolve concepts of partial success because on difficult problems decisive successes or failures come too infrequently.
5. The system must be able to create subroutines which can be included in procedures as units. The learning of subroutines is complicated by

   the fact that the effect of a subroutine is not usually good or bad in

   itself. Therefore, the mechanism that selects subroutines should have

   concepts of interesting or powerful subroutine whose application may

   be good under suitable conditions.

   Of the 5 points mentioned above, our work concentrates mainly on the

   second. We base ourselves on the idea that: In order for a program to be

   capable of learning something it must first be capable of being told it. In fact,

   in the early versions we shall concentrate entirely on this point and attempt

   to achieve a system which can be told to make a specific improvement in its

   behavior with no more knowledge of its internal structure or previous knowl-

   edge than is required in order to instruct a human. Once this is achieved, we

   may be able to tell the advice taker how to learn from experience.

   The main distinction between the way one programs a computer and

   modifies the program and the way one instructs a human or will instruct the

   advice taker is this: A machine is instructed mainly in the form of a sequence

   of imperative sentences; while a human is instructed mainly in declarative

   sentences describing the situation in which action is required together with

   a few imperatives that say what is wanted. We shall list the advantages of

   of the two methods of instruction.

   Advantages of Imperative Sentences

6. A procedure described in imperatives is already laid out and is carried out faster.
7. One starts with a machine in a basic state and does not assume previous knowledge on the part of the machine.

   Advantages of Declarative Sentences

8. Advantage can be taken of previous knowledge.
9. Declarative sentences have logical consequences and it can be arranged that the machine will have available sufficiently simple logical conse-

   quences of what it is told and what it previously knew.

10. The meaning of declaratives is much less dependent on their order than is the case with imperatives. This makes it easier to have after-

    thoughts.

11. The effect of a declarative is less dependent on the previous state of the system so that less knowledge of this state is required on the part

    of the instructor.

    The only way we know of expressing abstractions (such as the previous

    example of the opposition in checkers) is in language. That is why we have

    decided to program a system which reasons verbally.

    2 The Construction of the Advice Taker

    The advice taker system has the following main features:

12. There is a method of representing expressions in the computer. These expressions are defined recursively as follows: A class of entities called terms

    is defined and a term is an expression. A sequence of expressions is an ex-

    pression. These expressions are represented in the machine by list structures

    (Newell and Simon 1957).

13. Certain of these expressions may be regarded as declarative sentences in a certain logical system which will be analogous to a universal Post canon-

    ical system. The particular system chosen will depend on programming

    considerations but will probably have a single rule of inference which will

    combine substitution for variables with modus ponens. The purpose of the

    combination is to avoid choking the machine with special cases of general

    propositions already deduced.

14. There is an immediate deduction routine which when given a set of premises will deduce a set of immediate conclusions. Initially, the immediate

    deduction routine will simply write down all one-step consequences of the

    premises. Later, this may be elaborated so that the routine will produce

    some other conclusions which may be of interest. However, this routine

    will not use semantic heuristics; i.e., heuristics which depend on the subject

    matter under discussion.

    The intelligence, if any, of the advice taker will not be embodied in the

    immediate deduction routine. This intelligence will be embodied in the pro-

    cedures which choose the lists of premises to which the immediate deduction

    routine is to be applied. Of course, the program should never attempt to ap-

    ply the immediate deduction routine simultaneously to the list of everything

    it knows. This would make the deduction routine take too long.

15. Not all expressions are interpreted by the system as declarative sen- tences.Some are the names of entities of various kinds. Certain formulas

    represent objects. For our purposes, an entity is an object if we have some-

    thing to say about it other than the things which may be deduced from the

    form of its name. For example, to most people, the number 3812 is not an

    object: they have nothing to say about it except what can be deduced from

    its structure. On the other hand, to most Americans the number 1776 is an

    object because they have filed somewhere the fact that it represents the year

    when the American Revolution started. In the advice taker each object has

    a property list in which are listed the specific things we have to say about

    it. Some things which can be deduced from the name of the object may be

    included in the property list anyhow if the deduction was actually carried

    out and was difficult enough so that the system does not want to carry it out

    again.

16. Entities other than declarative sentences which can be represented by formulas in the system are individuals, functions, and programs.
17. The program is intended to operate cyclically as follows. The immedi- ate deduction routine is applied to a list of premises and a list of individuals.

    Some of the conclusions have the form of imperative sentences. These are

    obeyed. Included in the set of imperatives which may be obeyed is the routine

    which deduces and obeys.

    We shall illustrate the way the advice taker is supposed to act by means

    of an example. Assume that I am seated at my desk at home and I wish to go

    to the airport. My car is at my home also. The solution of the problem is to

    walk to the car and drive the car to the airport. First, we shall give a formal

    statement of the premises the advice taker uses to draw the conclusions. Then

    we shall discuss the heuristics which cause the advice taker to assemble these

    premises from the totality of facts it has available. The premises come in

    groups, and we shall explain the interpretation of each group.

18. First, we have a predicate “at”. “at(x, y)” is a formalization of

    “xisaty”. Under this heading we have the premises

    at(I, desk)

    at(desk, home)

    at(car, home)

    at(home, county)

    at(airport, county)

    (1)

    (2)

    (3)

    (4)

    (5)

    (6)

    (7)

    (8)

    (10)

    (11)

    (12)

    (13)

    (14)

    We shall need the fact that the relation“at” is transitive which might be

    written directly as

    at(x, y), at(y, z) → at(x, z)

    or alternatively we might instead use the more abstract premises

    transitive(at)

    transitive(u) → (u(x, y), u(y, z) → u(x, z))

    from which (6) can be deduced.

19. There are two rules concerning the feasibility of walking and driving. walkable(x), at(y, x), at(z, x), at(I, y) → can(go(y, z, walking))

    (9)

    drivable(x), at(y, x), at(z, x), at(car, y), at(I, car) → can(go(y, z, driving))

    There are also two specific facts

    walkable(home)

    drivable(county)

20. Next we have a rule concerned with the properties of going. did(go(x, y, z)) → at(I, y)
21. The problem itself is posed by the premise: want(at(I, airport))
22. The above are all the premises concerned with the particular problem. The last group of premises are common to almost all problems of this sort.

    They are:

    (x → can(y)), (did(y) → z) → canachult(x, y, z)

    (15)

    The predicate “canachult(x, y, z)” means that in a situation to which x ap-

    plies, the action y can be performed and ultimately brings about a situation

    to which z applies. A sort of transitivity is described by

    canachult(x, y, z), canachult(z, u, v) → canachult(x, prog(y, u), v).

    (16)

    Here prog(u, v) is the program of first carrying out u and then v. (Some

    kind of identification of a single action u with the one step program prog(u)

    is obviously required, but the details of how this will fit into the formalism

    have not yet been worked out).

    The final premise is the one which causes action to be taken.

    x, canachult(x, prog(y, z), w), want(w) → do(y)

    (17)

    The argument the advice taker must produce in order to solve the problem

    deduces the following propositions in more or less the following order:

23. at(I, desk) → can(go(desk, car, walking))
24. at(I, car) → can(go(home, airport, driving))
25. did(go(desk, car, walking)) → at(I, car)
26. did(go(home, airport, driving)) → at(I, airport)
27. canachult(at(I, desk), go(desk, car, walking), at(I, car))
28. canachult(at(I, car), go(home, airport, driving), at(I, airport))
29. canachult(at(I, desk), prog(go(desk, car, walking), go(home, airport, driving)) → at(I, airport))
30. do(go(desk, car, walking)) The deduction of the last proposition initiates action.

    The above proposed reasoning raises two major questions of heuristic.

    The first is that of how the 17 premises are collected, and the second is

    that of how the deduction proceeds once they are found. We cannot give

    complete answers to either question in the present paper; they are obviously

    not completely separate since some of the deductions might be made before

    some of the premises are collected. Let us first consider the question of where

    the 17 premises came from.

    First of all, we assert that except for the 14th premise want(at(I, airport)),

    which sets the goal, and the 1st premise at(I, desk), which we shall get from

    a routine which answers the question “whereamI (cid:48)(cid:48), all the premises can rea-

    sonably be expected to be specifically present in the memory of a machine

    which has competence of human order in finding its way around. That is,

    none of them are so specific to the problem at hand that assuming their pres-

    ence in memory constitutes an anticipation of this particular problem or of a

    class of problems narrower than those which any human can expect to have

    previously solved. We must impose this requirement if we are to be able to

    say that the advice taker exhibits common sense.

    On the other hand, while we may reasonably assume that the premises

    are in memory, we still have to describe how they are assembled into a

    list by themselves to which the deduction routine may be applied. Tenta-

    tively, we expect the advice taker to proceed as follows: initially, the sentence

    “want(at(I, airport))” is on a certain list L, called the main list, all by itself.

    The program begins with an observation routine which looks at the main list

    and puts certain statements about the contents of this list on a list called

    “observations of the main list”. We shall not specify at present what all the

    possible outputs of this observation routine are but merely say that in this

    case it will observe that “the only statement on L has the form (cid:48)want(u(x))(cid:48).”

    (We write this out in English because we have not yet settled on a formalism

    for representing statements of this kind). The “deduce and obey” routine

    is then applied to the combination of the “observations of the main list”

    list, and a list called the “standing orders list”. This list is rather small

    and is never changed, or at least is only changed in major changes of the

    advice taker. The contents of the “standing orders” list has not been worked

    out, but what must be deduced is the extraction of certain statements from

    property lists. Namely, the program first looks at “want(at(I, airport))” and

    attempts to copy the statements on its property list. Let us assume that it

    fails in this attempt because “want(at(I, airport))” does not have the sta-

    tus of an object and hence has no property list. (One might expect that if

    the problem of going to the airport has arisen before, “want(at(I, airport))”

    would be an object, but this might depend on whether there were routines

    for generalizing previous experience that would allow something of general

    use to be filed under that heading). Next in order of increasing generality the

    machine would see if anything were filed under “want(at(I, x))” which would

    deal with the general problem of getting somewhere. One would expect that

    premises (6), (or (7) and (8)), (9), (10), (13), would be so filed. There would

    also be the formula

    want(at(I, x)) → do(observe(whereamI))

    which would give us premise (1). There would also be a reference to the next

    higher level of abstraction in the goal statement which would cause a look at

    the property list of “want(x)”. This would give us (15), (16), and (17).

    We shall not try to follow the solution further except to remark that on

    the property list of “want(at(I, x))” there would be a rule that starts with

    the premises “at(I, y)” and “want(I, x)” and has as conclusion a search for

    the property list of “go(y, x, z)”. This would presumably fail, and then there

    would have to be heuristics that would initiate a search for a y such that

    “at(I, y)” and “at(airport, y)”. This would be done by looking on the prop-

    erty lists of the origin and the destination and working up. Then premise

    (10) would be found which has as one of its premises at(I, car). A repeti-

    tion of the above would find premise (9), which would complete the set of

    premises since the other “at” premises would have been found as by-products

    of previous searches.

    We hope that the presence of the heuristic rules mentioned on the prop-

    erty lists where we have put them will seem plausible to the reader. It should

    be noticed that on the higher level of abstraction many of the statements are

    of the stimulus-response form. One might conjecture that division in man

    between conscious and unconscious thought occurs at the boundary between

    stimulus-response heuristics which do not have to be reasoned about but only

    obeyed, and the others which have to serve as premises in deductions.

    We hope to formalize the heuristics in another paper before we start

    programming the system.

    3 References

    Friedberg, R. (1958). A Learning Machine, Part I IBM Journal of Research

    and Development 2, No. 1.

    McCarthy, John (1956). The Inversion of Functions Defined by Turing Ma-

    chines, in Automata Studies, Annals of Mathematical Study No. 34, Prince-

    ton, pp. 177–181.

    Minsky, M.L. (1956). Heuristic Aspects of the Artificial Intellegence Prob-

    lem. Lincoln Laboratory Report,pp.34–55.

    Newell, A., Shaw, J. C. and Simon, H.A.(1957). Empirical Explorations of

    the Logic Theory Machine. A case Study in Heuristic. Proceedings of the

    Western Joint Computer Conference, published by the Institute of Radio

    Engineers, New York, 1957, pp. 218–230.

    4 DISCUSSION OF THE PAPER

    PROF. Y. BAR-HILLEL: Dr. McCarthy’s paper belongs in the Journal of

    Half-Baked Ideas, the creation of which was recently proposed by Dr. I. J.

    Good. Dr. McCarthy will probably be the first to admit this. Before he goes

    on to bake his ideas fully, it might be well to give him some advice and raise

    some objections. He himself mentions some possible objections, but I do not

    think that he treats them with the full consideration they deserve; there are

    others he does not mention.

    For lack of time, I shall not go into the first part of his paper, although

    I think that it contains a lot of highly unclear philosophical, or pseudo-

    philosophical assumptions. I shall rather spend my time in commenting on

    the example he works out in his paper at some length. Before I start, let

    me voice my protest against the general assumption of Dr. McCarthy —

    slightly caricatured — that a machine, if only its program is specified with

    a sufficient degree of carelessness, will be able to carry out satisfactory even

    rather difficult tasks.

    Consider the assumption that the relation he designates by at is transitive

    (page 7). However, since he takes both “at(I, desk)” and “at(desk, home)”

    as premises, I presume – though this is never made quite clear – that at means

    something like being-a-physical-part or in-the-immediate-spatial-neighborhood-of. But then the relation is clearly not transitive. If A is in the immediate

    spatial neighborhood of B and B in the immediate spatial neighborhood of C

    then A need not be in the immediate spatial neighborhood of C. Otherwise,

    everything would turn out to be in the immediate spatial neighborhood of

    everything, which is surely not Dr. McCarthy’s intention. Of course, start-

    ing from false premises, one can still arrive at right conclusions. We do such

    things quite often, and a machine could do it. But it would probably be bad

    advice to allow a machine to do such things consistently.

    Many of the other 23 steps in Dr. McCarthy’s argument are equally or

    more questionable, but I don’t think we should spend our time showing this in

    detail. My major question is the following: On page 9 McCarthy states that

    a machine which has a competence of human order in finding its way around

    will have almost all the premises of the argument stored in its memory. I am

    at a complete loss to understand the point of this remark. If Dr. McCarthy

    wants to say no more than that a machine, in order to behave like a human

    being, must have the knowledge of a human being, then this is surely not a

    very important remark to make. But if not, what was the intention of this

    remark?

    The decisive question how a machine, even assuming that it will have

    somehow countless millions of facts stored in its memory, will be able to pick

    out those facts which will serve as premises for its deduction is promised to

    receive its treatment in another paper, which is quite right for a half-baked

    idea.

    It sounds rather incredible that the machine could have arrived at its

    conclusion — which, in plain English, is “Walk from your desk to your car!”

    — by sound deduction. This conclusion surely could not possibly follow from

    the premise in any serious sense. Might it not be occasionally cheaper to call

    a taxi and have it take you over to the airport? Couldn’t you decide to cancel

    your flight or to do a hundred other things? I don’t think it would be wise

    to develop a programming language so powerful as to make a machine arrive

    at the conclusion Dr. McCarthy apparently intends it to make.

    Let me also point out that in the example the time factor has never been

    mentioned, probably for the sake of simplicity. But clearly this factor is here

    so important that it could not possibly be disregarded without distorting the

    whole argument. Does not the solution depend, among thousands of other

    things, also upon the time of my being at my desk, the time at which I have

    to be at the airport, the distance from the airport, the speed of my car, etc.

    To make the argument deductively sound, its complexity will have to

    be increased by many orders of magnitude. So long as this is not realized,

    any discussions of machines able to perform the deductive — and inductive!

    — operations necessary for treating problems of the kind brought forward

    by Dr. McCarthy is totally pointless. The gap between Dr. McCarthy’s

    general programme (with which I have little quarrel, after discounting its

    “philosophical” features) and its execution even in such a simple case as the

    one discussed seems to me so enormous that much more has to be done to

    persuade me that even the first step in bridging this gap has already been

    taken.

    DR. O. G. SELFRIDGE: I have a question which I think applies to this. It

    seems to me in much of that work, the old absolutist Prof. Bar-Hillel has

    really put his finger on something; he is really worried about the deduction

    actually made. He seemed really to worry that that system is not consistent,

    and he made a remark that conclusions should not be drawn from false

    premises.

    In my experience those are the only conclusions that have ever

    been drawn. I have never yet heard of someone drawing correct conclusions

    from correct premises. I mean this seriously. This, I think is Dr. Minsky’s

    point this morning. What this leads to is that the notion of deductive logic

    being something sitting there sacred which you can borrow for particularly

    sacred uses and producing inviolable results is a lot of nonsense. Deductive

    logic is inferrred as much as anything else. Most women have never inferred

    it, but they get along pretty well, marrying happy husbands, raising happy

    children, without ever using deductive logic at all. My feeling is that my

    criticism of Dr. McCarthy is the other way. He assumes deductive logic,

    whereas in fact that is something to be concocted.

    This is another important point which I think Prof. Bar-Hillel ignores in

    this, the criticism of the programme should not be as to whether it is logically

    consistent, but only will he be able to wave it around saying “this in fact

    works the way I want it”. Dr. McCarthy would be the first to admit that his

    proramme is not now working, so it has to be changed. Then can you make

    the changes in the programme to make it work? That has nothing to do with

    logic. Can he amend it in such a way that it includes the logic as well as the

    little details of the programme? Can he manage in such a way that it works

    the way he does? He said at the begining of his talk that when he makes an

    arbitrary change in the programme it will not work usually, evidence, to me

    at least, that small changes in his programme will not obviously make the

    programme work and might even improve it. His next point is whether he

    can make small changes that in fact make it work. That is what we do not

    know yet.

    PROF. Y. BAR-HILLEL: May I ask whether you could thrash this out with

    Dr. McCarthy? It was my impression that Dr. McCarthy’s advice taker

    was meant to be able, among other things, to arrive at a certain conclusion

    from appropriate premises by faultless deductive reasoning. If this is still his

    programme, then I think your defence is totally beside the point.

    DR. O. G. SELFRIDGE: I am not defending his programme, I am only

    defending him.

    DR. J. McCARTHY: Are you using the word ‘programme’ in the technical

    sense of a bunch of cards or in the sense of a project that you get money for?

    PROF. Y. BAR-HILLEL: When I uttered my doubts that a machine working

    under the programme outlined by Dr. McCarthy would be able to do what

    he expects it to do, I was using ‘programme’ in the technical sense.

    DR. O. G. SELFRIDGE: In that case your criticisms are not so much philo-

    sophical as technical.

    PROF. Y. BAR-HILLEL: They are purely technical. I said that I shall not

    make any philosophical criticisms, for lack of time.

    DR. O. G. SELFRIDGE: A technical objection does not make ideas half-

    baked.

    PROF. Y. BAR-HILLEL: A deductive argument, where you have first to

    find out what are the relevant premises, is something which many humans

    are not always able to carry out successfully. I do not see the slightest reason

    to believe that at present machines should be able to perform things that

    humans find trouble in doing. I do not think there could possibly exist a

    programme which would, given any problem, divide all facts in the universe

    into those which are and those which are not relevant for that problem.

    Developing such a programme seems to me by 1010 orders of magnitude

    more difficult than, say, the Newell-Simon problem of developing a heuristic

    for deduction in the propositional calculus. This cavalier way of jumping

    over orders of magnitude only tends to becloud the issue and throw doubt

    on ways of thinking for which I have a great deal of respect. By developing

    a powerful programme language you may have paved the way for the first

    step in solving problems of the kind treated in your example, but the claim

    of being well on the way towards their solution is a gross exaggeration. This

    was the major point of my objections.

    DR. MCCARTHY (in reply): Prof. Bar-Hillel has correctly observed that

    my paper is based on unstated philosophical assumptions although what

    he means by “pseudo-philosophical” is unclear. Whenever we program a

    computer to learn from experience we build into the program a sort of epis-

    temology. It might be argued that this epistemology should be made explicit

    before one writes the programme, but epistemology is in a foggier state than

    computer programming even in the present half-baked state of the latter.

    I hope that once we have succeeded in making computer programs reason

    about the world, we will be able to reformulate epistemology as a branch of

    applied mathematics no more mysterious or controversial than physics.

    On re-reading my paper I can’t see how Prof. Bar-Hillel could see in it a

    proposal to specify a computer program carelessly. Since other people have

    proposed this as a device for achieving “creativity”, I can only conclude that

    he has some other paper in mind.

    In his criticism of my use of the symbol “at”, Prof. Bar-Hillel seems to

    have misunderstood the intent of the example. First of all, I was not trying

    to formalize the sentence form, A is at B, as it is used in English. “at” merely

    was intended to serve as a convenient mnemonic for the relation between a

    place and a sub-place. Second, I was not proposing a practical problem for

    the program to solve but rather an example intended to allow us to think

    about the kinds of reasoning involved and how a machine may be made to

    perform them.

    Prof. Bar-Hillel’s major point concerns my statement that the premises

    listed could be assumed to be in memory. The intention of this statement is

    to explain why I have not included formalizations of statements like, “it is

    possible to drive from my home to the airport” among my premises. If there

    were n known places in the county there would be

    such sentences and, since we are quite sure that we do not have each of them

    in our memories, it would be cheating to allow the machine to start with

    them.

    The rest of Prof. Bar-Hillel’s criticisms concern ways in which the model

    mentioned does not reflect the real world; I have already explained that

    this was not my intention. He is certainly right that the complexity of the

    model will have to be increased for it to deal with practical problems. What

    we disagree on is my contention that the conceptual difficulties arise at the

    present level of complexity and that solving them will allow us to increase

    the complexity of the model easily.

    With regard to the discussion between Prof. Bar-Hillel and Oliver Self-

    ridge — the logic is intended to be faultless although its premises cannot be

    guaranteed. The intended conclusion is “do(go(desk, car, walking))”—not,

    of course, “at(I, airport)”. The model oversimplifies but is not intended to

    oversimplify to the extent of allowing one to deduce one’s way to the airport.

    n(n − 1)