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FREE WILL—EVEN FOR ROBOTS

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305, USA

jmc@cs.stanford.edu

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

2000 Feb 14, 4:12 p.m.

I can, but I won’t.1

Abstract

Human free will is a product of evolution and contributes to the

success of the human animal. Useful robots will also require free will

of a similar kind, and we will have to design it into them.

Free will is not an all-or-nothing thing. Some agents have more

free will, or free will of different kinds, than others, and we will try to

analyze this phenomenon. Our objectives are primarily technological,

i.e. to study what aspects of free will can make robots more useful,

and we will not try to console those who find determinism distressing.

We distinguish between having choices and being conscious of these

choices; both are important, even for robots, and consciousness of

choices requires more structure in the agent than just having choices

and is important for robots. Consciousness of free will is therefore not

just an epiphenomenon of structure serving other purposes.

Free will does not require a very complex system. Young children

and rather simple computer systems can represent internally ‘I can,

but I won’t’ and behave accordingly.

Naturally I hope this detailed design stance (Dennett 1978) will

help understand human free will. It takes the compatibilist philosoph-

ical position.

There may be some readers interested in what the paper says about

human free will and who are put off by logical formulas. The formulas

are not important for the arguments about human free will; they are

present for people contemplating AI systems using mathematical logic.

They can skip the formulas, but the coherence of what remains is not

absolutely guaranteed.

Introduction—two aspects of free will

Free will, both in humans and in computer programs has two aspects—the

external aspect and the introspective aspect.

The external aspect is the set of results that an agent P can achieve, i.e.

what it can do in a situation s,

P oss(P, s) = {x|Can(P, x, s)}.

(1)

Thus in the present situation, I can find my drink. In one sense I can climb

on the roof of my house and jump off. In another sense I can’t. (The different

senses of can will be discussed in Section 3.1). In a certain position, a chess

program can checkmate its opponent and can also move into a position lead-

ing to the opponent giving checkmate. What is x in Can(P, x, s)? In English

it usually has the grammatical form of an action, but in the interesting cases

it is not an elementary action like those treated in situation calculus. Thus

we have ‘I can go to Australia’, ‘I can make a million dollars’, ‘I can get a

new house’. Often the what is to be achieved is a fluent, e.g. the state of

having a new house.

In the most important case, P oss(P, s) depends only on the causal posi-

tion of P in the world and not on the internal structure of P .

The introspective aspect involves the agent P ’s knowledge of P oss(P, s),

i.e. its knowledge of what it can achieve. Here is where the human sense of

free will comes in. It depends on P having an internal structure that allows

certain aspects of its current state to be interpreted as expressing knowledge.

I know I can find my drink. In a simple chess position I would know I could

give checkmate in three, because the chess problem column in the newspaper

said so, although I mightn’t yet have been able to figure out how.

Some present computer programs, e.g. chess programs, have an extensive

P oss(P, s). However, their knowledge of P oss(P, s) as a set is very limited.

Indeed it is too limited for optimal functionality, and robots’ knowledge of

their possibilities need to be made more like that of humans. For example, a

robot may conclude that in the present situation it has too limited a set of

possibilities. It may then undertake to ensure that in future similar situations

it will have more choices.

1.1 Preliminary philosophical remarks

Consider a machine, e.g. a computer program, that is entirely deterministic,

i.e. is completely specified and contains no random element. A major ques-

tion for philosophers is whether a human is deterministic in the above sense.

If the answer is yes, then we must either regard the human as having no free

will or regard free will as compatible with determinism. Some philosophers,

called compatibilists, e.g. Daniel Dennett (Dennett 1984), take this view, and

regard a person to have free will if his actions are determined by his internal

decision processes even if these processes themselves are deterministic.2 My

view is compatibilist, but I don’t need to take a position on determinism

itself.

AI depends on a compatibilist view, but having taken it, there is a lot to

be learned about the specific forms of free will that can be designed. That

is the subject of this article.

I don’t discuss the aspects of free will related to assigning credit or blame

for actions according to whether they were done freely. More generally, the

considerations of this article are orthogonal to many studied by philosophers,

but I think they apply to human free will nevertheless.

Specifically, East Germany did not deny its citizens the kind of free will

that some hope to establish via quantum mechanics or chaos theory. It did

deny its citizens choices in the sense discussed in this article.

Logical AI has some further philosophical presuppositions. These are

discussed in (McCarthy 1999b).

Informal discussion

There are different kinds and levels of free will. An automobile has none,

a chess program has a minimal kind of free will, and a human has a lot.

Human-level AI systems, i.e. those that match or exceed human intelligence

will need a lot more than present chess programs, and most likely will need

almost as much as a human possesses, even to be useful servants.

Consider chess programs. What kinds of free will do they have and can

they have? A usual chess program, given a position, generates a list of moves

It then goes down the list and tries the moves

available in the position.

successively getting a score for each move.

It chooses the move with the

highest score (or perhaps the first move considered good enough to achieve

a certain objective.)

That the program considers alternatives is our reason for ascribing to it a

little free will, whereas we ascribe none to the automobile. How is the chess

program’s free will limited, and what more could we ask? Could further free

will help make it a more effective program?

A human doesn’t usually consider his choices sequentially, scoring each

and comparing only the scores. The human compares the consequences of

the different choices in detail. Would it help a chess program to do that?

Human chess players do it.

Beyond that is considering the set Legals(p) of legal moves in position p

as an object. A human considers his set of choices and doesn’t just consider

each choice individually. A chess position is called ‘cramped’ if there are few

non-disastrous moves, and it is considered useful to cramp the opponent’s

position even if one hasn’t other reasons for considering the position bad for

the opponent. Very likely, a program that could play as well as Deep Blue

but doing 10−6 as much computation would need a more elaborate choice

structure, i.e. more free will. For example, one fluent of chess positions, e.g.

having an open file for a rook, can be regarded as giving a better position

than another without assigning numerical values to positions.

3 The finite automaton model of free will and

can

This section treats P oss(P, s) for finite automata. Finite automata raise the

question of what an agent can do in a sharp form. However, they are not

a useful representation of an agent’s introspective knowledge of what it can

do.

To the extent that a person or machine can achieve any of different goals,

that person or machine has free will. Our ideas on this show up most sharply

considering systems of interacting discrete finite automata. These are as

deterministic as you can get, which is why I chose them to illustrate free

will.

The material of this section revises that in (McCarthy and Hayes 1969),

section 2.4 entitled ‘The automaton representation and the notion of can’.

Let S be a system of interacting discrete finite automata such as that

shown in figure 1.

Figure 1: System S.

Each box represents a subautomaton and each line represents a signal.

Time takes on integer values and the dynamic behavior of the whole automa-

ton is given by the equations:

a1(t + 1) = A1(a1(t), s2(t))

a2(t + 1) = A2(a2(t), s1(t), s3(t), s10(t))

a3(t + 1) = A3(a3(t), s4(t), s5(t), s6(t), s8(t))

a4(t + 1) = A4(a4(t), s7(t))

(2)

s2(t) = S2(a2(t))

s3(t) = S3(a1(t))

s4(t) = S4(a2(t))

s5(t) = S5(a1(t))

s7(t) = S7(a3(t))

s8(t) = S8(a4(t))

s9(t) = S9(a4(t))

s10(t) = S10(a4(t))

(3)

The interpretation of these equations is that the state of any subautoma-

ton at time t + 1 is determined by its state at time t and by the signals

received at time t. The value of a particular signal at time t is determined by

the state at time t of the automaton from which it comes. Signals without a

source subautomaton represent inputs from the outside and signals without

a destination represent outputs.

Finite automata are the simplest examples of systems that interact over

time. They are completely deterministic; if we know the initial states of all

the automata and if we know the inputs as a function of time, the behavior

of the system is completely determined by equations (2) and (3) for all future

time.

The automaton representation consists in regarding the world as a system

of interacting subautomata. For example, we might regard each person in the

room as a subautomaton and the environment as consisting of one or more

additional subautomata. As we shall see, this representation has many of the

qualitative properties of interactions among things and persons. However, if

we take the representation too seriously and attempt to represent particular

interesting systems as systems of interacting automata, we encounter the

following difficulties:

  1. The number of states required in the subautomata is very large, for ex- ample 21010

    , if we try to represent a person’s knowledge. Automata this large

    have to be represented by systems of equations or by computer programs, or

    in some other way that does not involve mentioning states individually. In

    Section 4 we’ll represent them partially, by sentences of logic.

  2. Geometric information is hard to represent. Consider, for example, the location of a multi-jointed object such as a person or a matter of even

    more difficulty—the shape of a lump of clay.

  3. The system of fixed interconnections is inadequate. Since a person may handle any object in the room, an adequate automaton representation

    would require signal lines connecting him with every object.

  4. The most serious objection, however, is that (in the terminology of (McCarthy and Hayes 1969)) the automaton representation is epistemologi-

    cally inadequate. Namely, we do not ever know a person well enough to list

    his internal states. The kind of information we do have about him needs to

    be expressed in some other way.

    Nevertheless, we may use the automaton representation for concepts of

    can, causes, useful kinds of counterfactual statements (‘If another car had

    come over the hill when you passed just now, there would have been a head-on

    collision’). See (Costello and McCarthy 1999).

    Figure 2: Another system S.

    Figure 3: System S1.

    Let us consider the notion of can. Let S be a system of subautomata

    without external inputs such as that of figure 2. Let p be one of the subau-

    tomata, and suppose that there are m signal lines coming out of p. What p

    can do is defined in terms of a new system S p, which is obtained from the

    system S by disconnecting the m signal lines coming from p and replacing

    them by m external input lines to the system. In figure 2, subautomaton 1

    has one output, and in the system S1 (figure 3) this is replaced by an external

    input. The new system S p always has the same set of states as the system S.

    Now let π be a condition on the state such as, ‘a2 is even’ or ‘a2 = a3’. (In

    the applications π may be a condition like ‘The box is under the bananas’.)

    We shall write

    can(p, π, s)

    which is read, ‘The subautomaton p can bring about the condition π in the

    situation s’ if there is a sequence of outputs from the automaton S p that

    will eventually put S into a state a(cid:48) that satisfies π(a(cid:48)). In other words, in

    determining what p can achieve, we consider the effects of sequences of its

    actions, quite apart from the conditions that determine what it actually will

    do.

    Here’s an example based on figure 2. In order to write formulas conve-

    niently, we use natural numberss for the values of the states of the subau-

    tomata and the signals.

    a1(t + 1) = a1(t) + s2(t)

    a2(t + 1) = a2(t) + s1(t) + 2s3(t)

    a3(t + 1) = if a3(t) = 0 then 0 else a3(t) + 1

    s1(t) = if a1(t) = 0 then 2 else 1

    s2(t) = 1

    s3(t) = if a3(t) = 0 then 0 else 1.

    (4)

    Consider the initial state of S to be one in which all the subautomata are

    in state 0. We have the following propositions:

  5. Subautomaton 2 will never be in state 1. [It starts in state 0 and goes to state 2 at time 1. After that it can never decrease.]
  6. Subautomaton 1 can put Subautomaton 2 in state 1 but won’t. [If

    Subautomaton 1 emitted 1 at time 0 instead of 2, Subautomaton 2 would go

    to state 1.]

  7. Subautomaton 3 cannot put Subautomaton 2 in state 1. [The output from Subautomaton 1 suffices to put Subautomaton 2 in state 1 at time 1,

    after which it can never decrease.]

    We claim that this notion of can is, to a first approximation, the appro-

    priate one for a robot to use internally in deciding what to do by reasoning.

    We also claim that it corresponds in many cases to the common sense notion

    of can used in everyday speech.

    In the first place, suppose we have a computer program that decides what

    to do by reasoning. Then its output is determined by the decisions it makes

    in the reasoning process. It does not know (has not computed) in advance

    what it will do, and, therefore, it is appropriate that it considers that it can

    do anything that can be achieved by some sequence of its outputs. Common-

    sense reasoning seems to operate in the same way.

    The above rather simple notion of can requires some elaboration, both to

    represent adequately the commonsense notion and for practical purposes in

    the reasoning program.

    First, suppose that the system of automata admits external inputs. There

    are two ways of defining can in this case. One way is to assert can(p, π, s)

    if p can achieve π regardless of what signals appear on the external inputs.

    Thus, we require the existence of a sequence of outputs of p that achieves

    the goal regardless of the sequence of external inputs to the system. Note

    that, in this definition of can, we are not requiring that p have any way of

    knowing what the external inputs were. An alternative definition requires

    the outputs to depend on the inputs of p. This is equivalent to saying that p

    can achieve a goal, provided the goal would be achieved for arbitrary inputs

    by some automaton put in place of p. With either of these definitions can

    becomes a function of the place of the subautomaton in the system rather

    than of the subautomaton itself. Both of these treatments are likely to be

    useful, and so we shall call the first concept cana and the second canb.

    3.1 Representing a person by a system of subautomata

    The idea that what a person can do depends on his position rather than

    on his characteristics is somewhat counter-intuitive. This impression can be

    mitigated as follows: Imagine the person to be made up of several subau-

    tomata; the output of the outer subautomaton is the motion of the joints. If

    we break the connection to the world at that point we can answer questions

    like, ‘Can he fit through a given hole?’ We shall get some counter-intuitive

    answers, however, such as that he can run at top speed for an hour or can

    jump over a building, since these are sequences of motions of his joints that

    would achieve these results.

    The next step, however, is to consider a subautomaton that receives the

    nerve impulses from the spinal cord and transmits them to the muscles. If

    we break at the input to this automaton, we shall no longer say that he can

    jump over a building or run long at top speed since the limitations of the

    muscles will be taken into account. We shall, however, say that he can ride

    a unicycle since appropriate nerve signals would achieve this result.

    The notion of can corresponding to the intuitive notion in the largest

    number of cases might be obtained by hypothesizing an organ of will, which

    makes decisions to do things and transmits these decisions to the main part

    of the brain that tries to carry them out and contains all the knowledge of

    particular facts.3

    If we make the break at this point we shall be able to

    say that so-and-so cannot dial the President’s secret and private telephone

    number because he does not know it, even though if the question were asked

    could he dial that particular number, the answer would be yes. However,

    even this break would not give the statement, ‘I cannot go without saying

    goodbye, because this would hurt the child’s feelings’.

    On the basis of these examples, one might try to postulate a sequence

    of narrower and narrower notions of can terminating in a notion according

    to which a person can do only what he actually does. This extreme notion

    would then be superfluous. Actually, one should not look for a single best

    notion of can; each of the above-mentioned notions is useful and is actually

    used in some circumstances. Sometimes, more than one notion is used in a

    single sentence, when two different levels of constraint are mentioned.

    Nondeterministic systems as approximations to deterministic systems are

    discussed in (McCarthy 1999a). For now we’ll settle for an example involving

    a chess program. It can be reasoned about at various levels. Superhuman

    Martians can compute what it will do by looking at the initial electronic

    state and following the electronics. Someone with less computational power

    can interpret the program on another computer knowing the program and

    the position and determine the move that will be made. A mere human chess

    player may be reduced to saying that certain moves are excluded as obviously

    disastrous but be unable to decide which of (say) two moves the program will

    make. The chess player’s model is a nondeterministic approximation to the

    program.

    3.2 Causality

    Besides its use in explicating the notion of can, the automaton representation

    of the world is very suited for illustrating notions of causality. For, we may

    say that subautomaton p caused the condition π in state s, if changing the

    output of p would prevent π. In fact the whole idea of a system of interacting

    automata is mainly a formalization of the commonsense notion of causality.

    The automaton representation can be used to explicate certain counter-

    factual conditional sentences. For example, we have the sentence, ‘If another

    car had come over the hill when you just passed, there would have been a

    head-on collision’. We can imagine an automaton representation in which

    whether a car came over the hill is one of the outputs of a traffic subautoma-

    ton. (Costello and McCarthy 1999) discusses useful counterfactuals, like the

    above that are imbedded in a description of a situation and have conse-

    quences. One use is that they permit learning from an experience you didn’t

    quite have and would rather not have.

    3.3 Good analyses into subautomata

    In the foregoing we have taken the representation of the situation as a system

    of interacting subautomata for granted. Indeed if you want to take them for

    granted you can skip this section.

    However, a given overall automaton system might be represented as a

    system of interacting subautomata in a number of ways, and different rep-

    resentations might yield different results about what a given subautomaton

    can achieve, what would have happened if some subautomaton had acted

    differently, or what caused what. Indeed, in a different representation, the

    same or corresponding subautomata might not be identifiable. Therefore,

    these notions depend on the representation chosen.

    For example, suppose a pair of Martians observe the situation in a room.

    One Martian analyzes it as a collection of interacting people as we do, but

    the second Martian groups all the heads together into one subautomaton

    and all the bodies into another.4 How is the first Martian to convince the

    second that his representation is to be preferred? Roughly speaking, he

    would argue that the interaction between the heads and bodies of the same

    person is closer than the interaction between the different heads, and so

    more of an analysis has been achieved from ‘the primordial muddle’ with the

    conventional representation. He will be especially convincing when he points

    out that when the meeting is over the heads will stop interacting with each

    other, but will continue to interact with their respective bodies.

    We can express this kind of argument formally in terms of automata as

    follows: Suppose we have an autonomous automaton A, i.e. an automaton

    without inputs, and let it have k states. Further, let m and n be two integers

    such that mn ≥ k. Now label k points of an m-by-n array with the states

    of A. This can be done in (cid:16)mn

    k (cid:17)! ways. For each of these ways we have a

    representation of the automaton A as a system of an m-state automaton B

    interacting with an n-state automaton C. Namely, corresponding to each

    row of the array we have a state of B and to each column a state of C.

    The signals are in 1–1 correspondence with the states themselves; thus each

    subautomaton has just as many values of its output as it has states.

    Now it may happen that two of these signals are equivalent in their effect

    on the other subautomaton, and we use this equivalence relation to form

    equivalence classes of signals. We may then regard the equivalence classes as

    the signals themselves. Suppose then that there are now r signals from B to C

    and s signals from C to B. We ask how small r and s can be taken in general

    compared to m and n. The answer may be obtained by counting the number

    of inequivalent automata with k states and comparing it with the number

    of systems of two automata with m and n states respectively and r and s

    signals going in the respective directions. The result is not worth working

    out in detail, but tells us that only a few of the k state automata admit such

    a decomposition with r and s small compared to m and n. Therefore, if

    an automaton happens to admit such a decomposition it is very unusual for

    it to admit a second such decomposition that is not equivalent to the first

    with respect to some renaming of states. Applying this argument to the real

    world, we may say that it is overwhelmingly probable that our customary

    decomposition of the world automaton into separate people and things has

    a unique, objective and usually preferred status. Therefore, the notions of

    can, of causality, and of counterfactual associated with this decomposition

    also have a preferred status.

    These considerations are similar to those used by Shannon, (Shannon 1938)

    to find lower bounds on the number of relay contacts required on the average

    to realize a boolean function.

    An automaton can do various things. However, the automaton model

    proposed so far does not involve consciousness of the choices available. This

    requires that the automata be given a mental structure in which facts are

    represented by sentences. This is better done in a more sophisticated model

    than finite automata. We start on it in the next section.

    4 Formalism for introspective free will

    The previous section concerned only external free will, and it isn’t convenient

    to represent knowledge by the states of subautomata of a reasoning automa-

    ton. (McCarthy 1979) has a more extensive formalization of knowing what

    and knowing that.

    The situation calculus, (McCarthy and Hayes 1969) and (Shanahan 1997),

    offers a better formalism for a robot to represent facts about its own possi-

    bilities.

    4.1 A minimal example of introspective free will

    The following statement by Suppes (Suppes 1994) provides a good excuse for

    beginning with a very simple example of introspective free will.

    There are, it seems to me, two central principles that should gov-

    ern our account of free will. The first is that small causes can

    produce large effects. The second is that random phenomena are

    maximally complex, and it is complexity that is phenomenologi-

    cally in many human actions that are not constrained but satisfy

    ordinary human notions of being free actions.[my emphasis]

    I don’t agree that complexity is essentially involved so here’s a minimal

    example that expresses, ‘I can, but I won’t’.

    Because the agent is reasoning about its own actions, as is common in sit-

    uation calculus formalization, the agent is not explicitly represented. Making

    the agent explicit offers no difficulties.

    If an action a is possible in a situation s, then the situation Result(a, s)

    that results from performing the action is achievable.

    P ossible(a, s) → Can(Result(a, s), s)

    If a situation Result(a, s) is achievable and every other situation that is

    achievable is less good, then the action a should be done.

    Can(Result(a, s), s) ∧ (∀s(cid:48))(Can(s(cid:48), s) → s(cid:48) <good Result(a, s))

    → Should(a, s)

    Here <good means ‘not so good as’.

    done.

    Actions leading to situations inferior to what can be achieved won’t be

    Can(Result(a, s), s) ∧ Result(a(cid:48), s) <good Result(a, s)

    → ¬W illDo(a(cid:48), s)

    This is reasonably close to formalizing ‘It can, but it won’t’ except for

    not taking into account the distinction between ‘but’ and ‘and’. As truth

    functions, ‘but’ and ‘and’ are equivalent. Uttering ‘p but q’ is a different

    speech act from uttering ‘p and q’, but this article is not the place to discuss

    the difference.

    (5)

    (6)

    (7)

    4.2 Representing more about an agent’s capability

    Here are some examples of introspective free will and some considerations.

    They need to be represented in logic so that a robot could use them to learn

    from its past and plan its future.

  8. Did I make the wrong decision just now? Can I reverse it?
  9. ‘Yesterday I could have made my reservation and got a cheap fare.’
  10. ‘Next year I can apply to any university in the country. I don’t need to make up my mind now.’.
  11. ‘If I haven’t studied calculus, I will be unable to take differential equa- tions.’
  12. ‘If I learn to program computers, I will have more choice of occupation.’
  13. ‘It is better to have an increased set of choices.’
  14. ‘I am not allowed to harm human beings.’ Asimov imagined his three laws of robotics, of which this is one, as built into his imaginary positronicbrains. In his numerous science fiction stories, the robots treated them

    as though engraved on tablets and requiring interpretation. This is

    necessary, because the robots did have to imagine their choices and

    their consequences.

  15. Some of a person’s behavior is controlled by reflexes and other auto- matic mechanisms. We rightly regard reflexive actions as not being

    deliberate and are always trying to get better control of them.

    • The coach helps the baseball player analyze how he swings at the

    ball and helps him improve the reflexive actions involved.

    • I’m a sucker for knight forks and for redheads and need to think

    more in such chess and social situations.

  16. In the introduction I wrote P oss(P, s) = {x|can(P, x, s)}.

    (8)

    What kind of an entity is x? In situation calculus, the simplest operand

    of can is a situation as treated above, but also we can consider an action

    itself or a propositional fluent. A propositional fluent p is a predicate

    taking a situation argument, and an agent can reason that it can (or

    cannot) bring about a future situation in which p holds.

    (McCarthy 1996) includes an extensive discussion of what consciousness,

    including consciousness of self, will be required for robots.

    5 Conclusions

  17. Human level AI requires the ability of the agent to reason about its past, present, future and hypothetical choices.
  18. What an agent can do is determined by its environment rather than by its internal structure.

    knowing about them.

  19. Having choices is usefully distinguished from the higher capability of
  20. What people can do and know about what they can do is similar to what robots can do and know.

    AI needs a more developed formal theory of free will, i.e. the structures

    of choice a robot can have and what it can usefully know about them.

    6 Acknowledgments

    I thank Eyal Amir, Daniel Dennett, and Patrick Suppes for useful discussions.

    This work was supported by the DARPA High Performance Knowledge

    Bases program and by the AFOSR New World Vistas program under AFOSR

    grant AF F49620-97-1-0207.

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    7 Notes

    1Sarah McCarthy, at age 4, personal communication.

    2Some people ask whether making the system probabilistic or quantum

    mechanical or classical chaotic makes a difference in the matter of free will.

    I agree with those who say it doesn’t.

    3The idea of an organ of will cannot be given a precise definition, which

    has caused philosophers and psychologists to denounce as senseless ideas

    that separate will from intellect. However, it may be a useful approximate

    concept in the sense of (McCarthy 1999a). It presumably won’t correspond

    to a specific part of the brain.

    4An inhabitant of momentum space might regard the Fourier components

    of the distribution of matter as the separate interacting subautomata.

    5http://www-formal.stanford.edu/jmc/counterfactuals.html

    6http://www-formal.stanford.edu/jmc/consciousness.html

    7http://www-formal.stanford.edu/jmc/approximate.html

    8http://www-formal.stanford.edu/jmc/phil2.html

    9http://www-formal.stanford.edu/jmc/mcchay69.html