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Towards a Mathematical Science of

Computation

J. McCarthy

Computer Science Department

Stanford University

Stanford, CA 95305

jmc@cs.stanford.edu

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

May 14, 1:20 p.m.

Abstract

Introduction

In this paper I shall discuss the prospects for a mathematical science of

computation. In a mathematical science, it is possible to deduce from the

basic assumptions, the important properties of the entities treated by the

science. Thus, from Newton’s law of gravitation and his laws of motion, one

can deduce that the planetary orbits obey Kepler’s laws.

What are the entities with which the science of computation deals?

What kinds of facts about these entities would we like to derive?

What are the basic assumptions from which we should start?

What important results have already been obtained? How can the math-

ematical science help in the solution of practical problems?

I would like to propose some partial answers to these questions. These

partial answers suggest some problems for future work. First I shall give

some very sketchy general answers to the questions. Then I shall present

some recent results on three specific questions. Finally, I shall try to draw

some conclusions about practical applications and problems for future work.

This paper should be considered together with my earlier paper [McC63].

The main results of the present paper are new but there is some overlap so

that this paper would be self-contained. However some of the topics in this

paper such as conditional expressions, recursive definition of functions, and

proof by recursion induction are considered in much greater detail in the

earlier paper.

2 What Are The Entities With Which Com-

puter Science Deals?

These are problems, procedures, data spaces, programs representing proce-

dures in particular programming languages, and computers.

Problems and procedures are often confused. A problem is defined by the

criterion which determines whether a proposed solution is accepted. One can

understand a problem completely without having any method of solution.

Procedures are usually built up from elementary procedures. What these

elementary procedures may be, and how more complex procedures are con-

structed from them, is one of the first topics in computer science. This

subject is not hard to understand since there is a precise notion of a com-

putable function to guide us, and computability relative to a given collection

of initial functions is easy to define.

Procedures operate on members of certain data spaces and produce mem-

bers of other data spaces, using in general still other data spaces as interme-

diates. A number of operations are known for constructing new data spaces

from simpler ones, but there is as yet no general theory of representable data

spaces comparable to the theory of computable functions. Some results are

given in [McC63].

Programs are symbolic expressions representing procedures. The same

procedure may be represented by different programs in different program-

ming languages. We shall discuss the problem of defining a programming

language semantically by stating what procedures the programs represent.

As for the syntax of programming languages, the rules which allow us to

determine whether an expression belongs to the language have been formal-

ized, but the parts of the syntax which relate closely to the semantics have

not been so well studied. The problem of translating procedures from one

programming language to another has been much studied, and we shall try

to give a definition of the correctness of the translation.

Computers are finite automata. From our point of view, a computer is

defined by the effect of executing a program with given input on the state

of its memory and on its outputs. Computer science must study the various

ways elements of data spaces are represented in the memory of the computer

and how procedures are represented by computer programs. From this point

of view, most of the current work on automata theory is beside the point.

3 What Kinds of Facts About Problems, Pro-

cedures, data Spaces, Programs, And Com-

puters Would We Like to Derive?

Primarily. we would like to be able to prove that given procedures solve

given problems. However, proving this may involve proving a whole host of

other kinds of statement such as:

  1. Two procedures are equivalent, i.e. compute the same function.
  2. A number of computable functions satisfy a certain relationship, such as an algebraic identity or a formula of the functional calculus.
  3. A certain procedure terminates for certain initial data, or for all initial data.
  4. A certain translation procedure correctly translates procedures between one programming language and another.
  5. One procedure is more efficient than another equivalent procedure in the sense of taking fewer steps or requiring less memory.
  6. A certain transformation of programs preserves the function expressed but increases the efficiency.
  7. A certain class of problems is unsolvable by any procedure, or requires procedures of a certain type for its solution.

    4 What Are The Axioms And Rules of Infer-

    ence of A Mathematical Science of Compu-

    tation?

    Ideally we would like a mathematical theory in which every true statement

    about procedures would have a proof, and preferably a proof that is easy to

    find, not too long, and easy to check. In 1931, G¨odel proved a result, one

    of whose immediate consequences is that there is no complete mathemati-

    cal theory of computation. Given any mathematical theory of computation

    there are true statements expressible in it which do not have proofs. Never-

    theless, we can hope for a theory which is adequate for practical purposes,

    like proving that compilers work; the unprovable statements tend to be of a

    rather sweeping character, such as that the system itself is consistent.

    It is almost possible to take over one of the systems of elementary number

    theory such as that given in Mostowski’s book ”Sentences Undecidable in

    Formalized Arithmetic” since the content of a theory of computation is quite

    similar. Unfortunately, this and similar systems were designed to make it easy

    to prove meta-theorems about the system, rather than to prove theorems in

    the system. As a result, the integers are given such a special role that the

    proofs of quite easy statements about simple procedures would be extremely

    long.

    Therefore it is necessary to construct a new, though similar, theory in

    which neither the integers nor any other domain, (e.g. strings of symbols)

    are given a special role. Some partial results in this direction are described

    in this paper. Namely, an integer-free formalism for describing computations

    has been developed and shown to be adequate in the cases where it can be

    compared with other formalisms. Some methods of proof have been devel-

    oped, but there is still a gap when it comes to methods of proving that a

    procedure terminates. The theory also requires extension in order to treat

    the properties of data spaces.

    5 What Important Results Have Been Ob-

    tained Relevant to A Mathematical Science

    of Computation?

    In 1936 the notion of a computable function was clarified by Turing, and

    he showed the existence of universal computers that, with an appropriate

    program, could compute anything computed by any other computer. All our

    stored program computers, when provided with unlimited auxiliary storage,

    are universal in Turing’s sense. In some subconscious sense even the sales

    departments of computer manufacturers are aware of this, and they do not

    advertise magic instructions that cannot be simulated on competitors ma-

    chines, but only that their machines are faster, cheaper, have more memory,

    or are easier to program.

    The second major result was the existence of classes of unsolvable prob-

    lems. This keeps all but the most ignorant of us out of certain Quixotic

    enterprises such as trying to invent a debugging procedure that can infallibly

    tell if a program being examined will get into a loop.

    Later in this paper we shall discuss the relevance of the results of math-

    ematical logic on creative sets to the problem of whether it is possible for a

    machine to be as intelligent as a human. In my opinion it is very important

    to build a firmer bridge between logic and recursive function theory on the

    one side, and the practice of computation on the other.

    Much of the work on the theory of finite automata has been motivated

    by the hope of applying it to computation. I think this hope is mostly in

    vain because the fact of finiteness is used to show that the automaton will

    eventually repeat a state. However, anyone who waits for an IBM 7090 to

    repeat a state, solely because it is a finite automaton, is in for a very long

    wait.

    6 How Can A Mathematical Science of Com-

    putation Help in The Solution of Practical

    Problems?

    Naturally, the most important applications of a science cannot be foreseen

    when it is just beginning. However, the following applications can be fore-

    seen.

  8. At present, programming languages are constructed in a very unsys- tematic way. A number of proposed features are invented, and then we argue

    about whether each feature is worth its cost. A better understanding of the

    structure of computations and of data spaces will make it easier to see what

    features are really desirable.

  9. It should be possible almost to eliminate debugging. Debugging is the testing of a program on cases one hopes are typical, until it seems to work.

    This hope is frequently vain.

    Instead of debugging a program, one should prove that it meets its speci-

    fications, and this proof should be checked by a computer program. For this

    to be possible, formal systems are required in which it is easy to write proofs.

    There is a good prospect of doing this, because we can require the computer

    to do much more work in checking each step than a human is willing to do.

    Therefore, the steps can be bigger than with present formal systems. The

    prospects for this are discussed in [McC62].

    7 Using Conditional Expressions to Define

    Functions Recursively

    In ordinary mathematical language, there are certain tools for defining new

    functions in terms of old ones. These tools are composition and the identifi-

    cation of variables. As it happens, these tools are inadequate computable in

    terms of old ones. It is then customary to define all functions that can rea-

    sonably be regarded as to give a verbal definition. For example, the function

    |x| is usually defined in words.

    If we add a single formal tool, namely conditional expressions to our

    mathematical notation, and if we allow conditional expressions to be used

    recursively in a manner that I shall describe, we can define, in a completely

    formal way, all functions that can reasonably be regarded as computable in

    terms of an initial set. We will use the ALGOL 60 notation for conditional

    expressions in this paper.

    We shall use conditional expressions in the simple form

    where p is a propositional expression whose value is true or false. The value

    of the conditional expression is a if p has the value true and b if p has the

    if p then a else b

    value false. When conditional expressions are used, the stock of predicates

    one has available for constructing p’s is just as important as the stock of

    ordinary functions to be used in forming the a’s and b’s by composition. The

    following are examples of functions defined by conditional expressions:

  10. |x| = if x < 0 then − x else x
  11. n! = if n = 0 then 1 else n × (n − 1)!
  12. n! = g(n, 1) where

    g(n, s) = if n = 0 then s else g(n − 1, n × s)

  13. n! = f (n, 0, 1) where

    f (n, m, p) = if m = n then p else f (n, m + 1, (m + 1) × p)

  14. n− = pred (n, 0) where

    pred (n, m) = if m(cid:48) = n then m else pred (n, m(cid:48))

  15. m + n = if n = 0 then m else m(cid:48) + n− The first of these is the only non-recursive definition in the group. Next,

    we have three different procedures for computing n!; they can be shown to

    be equivalent by the methods to be described in this paper. Then we define

    the predecessor function n− for positive integers (3− = 2) in terms of the

    successor function n(cid:48)(2(cid:48) = 3) Finally, we define addition in terms of the

    successor, the predecessor and equality. In all of the definitions, except for

    the first, the domain of the variables is taken to be the set of non-negative

    integers.

    As an example of the use of these definitions, we shall show how to com-

    pute 2! by the second definition of n!. We have

    2! = g(2, 1)

    = if 2 = 0 then 1 else g(2 − 1, 2 × 1)

    = g(1, 2)

    = if 1 = 0 then 2 else g(1 − 1, 1 × 2)

    = g(0, 2)

    = if 0 = 0 then 2 else g(0 − 1, 0 × 2)

    = 2

    Note that if we attempted to compute n! for any n but a non-negative

    integer the successive reductions would not terminate. In such cases we say

    that the computation does not converge. Some recursive functions converge

    for all values of their arguments, some converge for some values of the argu-

    ments, and some never converge. Functions that always converge are called

    total functions, and the others are called partial functions. One of the ear-

    liest major results of recursive function theory is that there is no formalism

    that gives all the computable total functions and no partial functions.

    We have proved [McC63] that the above method of defining computable

    functions, starting from just the successor function n(cid:48) and equality, yields all

    the computable functions of integers. This leads us to assert that we have the

    complete set of tools for defining functions which are computable in terms of

    given base functions.

    If we examine the next to last line of our computation of 2! we see that

    we cannot simply regard the conditional expression

    if p then a else b

    as a function of the three quantities p, a, and b.

    If we did so, we would

    be obliged to compute g(−1, 0) before evaluating the conditional expression,

    and this computation would not converge. What must happen is that when

    p is true we take a as the value of the conditional expression without even

    looking at b.

    Any reference to recursive functions in the rest of this paper refers to

    functions defined by the above methods.

    8 Proving Statements About Recursive Func-

    tions

    In the previous section we presented a formalism for describing functions

    which are computable in terms of given base functions. We would like to

    have a mathematical theory strong enough to admit proofs of those facts

    about computing procedures that one ordinarily needs to show that computer

    programs meet their specifications. In [McC63] we showed that our formalism

    for expressing computable functions, was strong enough so that all the partial

    recursive functions of integers could be obtained from the successor function

    and equality. Now we would like a theory strong enough so that the addition

    of some form of Peano’s axioms would allow the proof of all the theorems of

    one of the forms of elementary number theory.

    We do not yet have such a theory. The difficulty is to keep the axioms

    and rules of inference of the theory free of the integers or other special do-

    main, just as we have succeeded in doing with the formalism for constructing

    functions.

    We shall now list the tools we have so far for proving relations between

    recursive functions. They are discussed in more detail in [McC63].

  16. Substitution of expressions for free variables in equations and other relations.
  17. Replacement of a sub-expression in a relation by an expression which has been proved equal to the sub-expression. (This is known in elementary

    mathematics as substitution of equals for equals.) When we are dealing with

    conditional expressions, a stronger form of this rule than the usual one is

    valid. Suppose we have an expression of the form

    if p then a else (if q then b else c)

    and we wish to replace b by d. Under these circumstances we do not have to

    prove the equation b = d in general, but only the weaker statement

    ∼ p ∧ q ⊃ b = d

    This is because b affects the value of the conditional expression only in case

    ∼ p ∧ q is true.

  18. Conditional expressions satisfy a number of identities, and a complete theory of conditional expressions, very similar to propositional calculus, is

    thoroughly treated in [McC63]. We shall list a few of the identities taken as

    axioms here.

  19. (if true then a else b) = a
  20. (if false then a else b) = b
  21. if p then (if q then a else b) else (if q then C else d) = if q then (if p then a else c) else (if p then b else d)
  22. f (if p then a else b) = if p then f (a) else f (b)
  23. Finally, we have a rule called recursion induction for making arguments that in the usual formulations are made by mathematical induction. This

    rule may be regarded as taking one of the theorems of recursive function

    theory and giving it the status of a rule of inference. Recursion induction

    may be described as follows:

    Suppose we have a recursion equation defining a function f , namely

    f (x1, . . . , xn) = ε{f, x1, . . . , xn}

    (1)

    where the right hand side will be a conditional expression in all non-trivial

    cases, and suppose that

  24. the calculation of f (x1, . . . , xn) according to this rule converges for a certain set A of n-tuples,
  25. the functions g(x1, . . . , xn) and h(x1, . . . , xn) each satisfy equation (1) when g or h is substituted for f . Then we may conclude that g(x1, . . . , xn) =h(x1, . . . , xn) for all n-tuples (x1, . . . , xn) in the set A.

    As an example of the use of recursion induction we shall prove that the

    function g defined by

    g(n, s) = if n = 0 then s else g(n − 1, n × s)

    satisfies g(n, s) = n! × s given the usual facts about n!. We shall take for

    granted the fact that g(n, s) converges for all non-negative integral n and s.

    Then we need only show that n! × s satisfies the equation for g(n, s). We

    have

    n! × s = if n = 0 then n! × s else n! × s

    if n = 0 then s else (n − 1)! × (n × s)

    and this has the same form as the equation satisfied by g(n, s). The steps in

    this derivation are fairly obvious but also follow from the above-mentioned

    axioms and rules of inference. In particular, the extended rule of replacement

    is used. A number of additional examples of proof by recursion induction

    are given in [McC63], and it has been used to prove some fairly complicated

    relations between computable functions.

    9 Recursive Functions, Flow Charts, And Al-

    golic Programs

    In this section 1 want to establish a relation between the use of recursive

    functions to define computations, the flow chart notation, and programs ex-

    pressed as sequences of ALGOL-type assignment statements, together with

    conditional go to’s. The latter we shall call Algolic programs with the idea

    of later extending the notation and methods of proof to cover more of the

    language of ALGOL. Remember that our purpose here is not to create yet

    another programming language, but to give tools for proving facts about

    programs.

    In order to regard programs as recursive functions, we shall define the

    state vector ξ of a program at a given time, to be the set of current assign-

    ments of values to the variables of the program. In the case of a machine

    language program, the state vector is the set of current contents of those reg-

    isters whose contents change during the course of execution of the program.

    When a block of program having a single entrance and exit is executed,

    the effect of this execution on the state vector may be described by a function

    ξ (cid:48) = r(ξ) giving the new state vector in terms of the old. Consider a program

    described by the flow chart of fig. 1.

    r

    s

    t

    f

    p

    g

    q

    Fig. 1 Flow chart for equations shown below.

    r(ξ) = s(f (ξ))

    s(ξ) = if p(ξ) then r(ξ) else t(g(ξ))

    t(ξ) = if q1(ξ) then r(ξ) else if q2(ξ) ξ else t(g(ξ))

    The two inner computation blocks are described by functions f (ξ) and

    g(ξ), telling how these blocks affect the state vector. The decision ovals

    are described by predicates p(ξ), q1(ξ), and q2(ξ) that give the conditions for

    taking the various paths. We wish to describe the function r(ξ) that gives the

    effect of the whole block, in terms of the functions f and g and the predicates

    p, q1, and q2. This is done with the help of two auxiliary functions s and t.

    s(ξ) describes the change in ξ between the point labelled s in the chart and

    the final exit from the block; t is defined similarly with respect to the point

    labelled t. We can read the following equations directly from the flow chart:

    r(ξ) = s(f (ξ))

    s(ξ) = if p(ξ) then r(ξ) else t(g(ξ))

    t(ξ) = if q1(ξ) then r(ξ) else if q2(ξ) then ξ else t(g(ξ))

    In fact, these equations contain exactly the same information as does the

    flow chart.

    Consider the function mentioned earlier, given by

    g(n, s) = if n = 0 then s else g(n − 1, n × s)

    It corresponds, as we leave the reader to convince himself, to the Algolic

    program

    if n = 0 then go to b;

    s := n × s;

    n := n − 1;

    go to a;

    a :

    b :

    Its flow chart is given in fig. 2.

    This flow chart is described by the function fac(ξ) and the equation

    fac(ξ) = if p(ξ) then ξ else fac(g(f (ξ)))

    where p(ξ) corresponds to the condition n = 0, f (ξ) to the statement n :=

    n − 1, and g(ξ) to the statement s := n × s.

    n = 0?

    s := n*s

    n := n - 1

    Fig. 2 Flow chart for fac(ξ) as described by the following equations.

    f ac(ξ) = if p(ξ) then ξ else f ac(g(f (ξ)))

    where

    and so

    p(ξ)c(n, ξ) = 0,

    f (ξ) = a(s, c(n, ξ)c(s, ξ), ξ),

    g(ξ) = a(n, c(n, ξ) − 1, ξ)

    f ac(ξ) = a(n, 0, a(s, c(n, ξ)! × c(s, ξ))).

    We shall regard the state vector as having many components, but the only

    components affected by the present program are the s-component and the

    n-component. In order to compute explicitly with the state vectors, we intro-

    duce the function c(var , ξ) which denotes the value assigned to the variable

    var in the state vector ξ, and we also introduce the function a(var , val , ξ)

    which denotes the state vector that results when the value assigned to var in

    ξ is changed to val , and the values of the other variables are left unchanged.

    The predicates and functions of state vectors mentioned above then be-

    come:

    p(ξ) = (c(n, ξ) = 0)

    g(ξ) = a(n, c(n, ξ) − 1, ξ)

    f (ξ) = a(s, c(n, ξ) × c(s, ξ), ξ)

    We can prove by recursion induction that

    fac(ξ) = a(n, 0, a(s, c(n, ξ)! × c(s, ξ), ξ),

    but in order to do so we must use the following facts about the basic state

    vector functions:

  26. c(u, a(v, α, ξ)) = if u = v then α else c(u, ξ)
  27. a(v, c(v, ξ), ξ) = ξ
  28. a(u, α, a(v, β, ξ)) = if u = v then a(u, α, ξ) else a(v, β, a(u, α, ξ)) The proof parallels the previous proof that

    g(n, s) = n! × s,

    but the formulae are longer.

    While all flow charts correspond immediately to recursive functions of

    the state vector, the converse is not the case. The translation from recursive

    function to flow chart and hence to Algolic program is immediate, only if

    the recursion equations are in iterative form. Suppose we have a recursion

    equation

    r(x1, . . . , xn) = E{r, x1, . . . , xn, f1, . . . , fm}

    where E{r, x1, = ldots, xn, f1, . . . , fm} is a conditional expression defining the

    function r in terms of the functions f1, . . . , fm. E is said to be in iterative

    form if r never occurs as the argument of a function but only in terms of the

    main conditional expression in the form

    . . . then r(. . .).

    In the examples, all the recursive definitions except

    n! = if n = 0 then 1 else n × (n − 1)!

    In that one, (n − 1)! occurs as a term of a prod-

    are in interative form.

    uct. Non-iterative recursive functions translate into Algolic programs that

    use themselves recursively as procedures. Numerical calculations can usually

    be transformed into iterative form, but computations with symbolic expres-

    sions usually cannot, unless quantities playing the role of push-down lists are

    introduced explicitly.

    10 Recursion Induction on Algolic Programs

    In this section we extend the principle of recursion induction so that it can

    be applied directly to Algolic programs without first transforming them to

    recursive functions. Consider the Algolic program

    We want to prove it equivalent to the program

    if n = 0 then go to b;

    s := n × s;

    n := n − 1;

    go to a;

    a :

    b :

    s := n! × s

    n := 0

    a :

    b :

    Before giving this proof we shall describe the principle of recursion induction

    as it applies to a simple class of flow charts. Suppose we have a block of

    program represented by fig. 3a. Suppose we have proved that this block is

    equivalent to the flow chart of fig. 3b where the shaded block is the original

    block again. This corresponds to the idea of a function satisfying a functional

    equation. We may now conclude that the block of fig. 3a is equivalent to the

    flow chart of fig. 3c for those state vectors for which the program does not

    get stuck in a loop in fig. 3c.

    (a)

    f

    (b)

    p

    (c)

    p

    f

    g

    g

    Fig. 3 Recursion induction applied to flow charts.

    This can be seen as follows. Suppose that, for a particular input, the

    program of fig. 3c goes around the loop n times before it comes out. Consider

    the flow chart that results when the whole of fig. 3b is substituted for the

    shaded (or yellow) block, and then substituted into this figure again, etc. for

    a total of n substitutions. The state vector in going through this flow chart

    will undergo exactly the same tests and changes as it does in going through

    fig. 3c, and therefore will come out in n steps without ever going through the

    shaded (or yellow) block. Therefore, it must come out with the same value

    as in fig. 3c. Therefore, for any vectors for which the computation of fig. 3c

    converges, fig. 3a is equivalent to fig. 3c.

    Thus, in order to use this principle to prove the equivalence of two blocks

    we prove that they satisfy the same relation, of type 3a-3b, and that the

    corresponding 3c converges.

    We shall apply this method to showing that the two programs mentioned

    above are equivalent. The first program may be transformed as follows:

    a :

    if n = 0 then go to b; s := n × s; n := n − 1;

    go to a : b :

    a :

    if n = 0 then go to b; s := n × s; n := n − 1;

    a1 : if n = 0 then go to b; s := n × s; n := n − 1;

    go to a1; b :

    The operation used to transform the program is simply to write separately

    the first execution of a loop.

    In the case of the second program we go:

    a : s := n! × s; n := 0; b :

    a :

    if n = 0 then go to c; s := n! × s; n := 0;

    go to b; c : s := n! × s; n := 0; b :

    a :

    if n = 0 then go to c; s := (n − 1)! × (n × s); n := 0;

    go to b; c :; b :

    from

    to

    from

    to

    to

    to

    a :

    if n = 0 then go to b; s := n × s; n := n − 1; s :=

    = n! × s; n := 0; b :

    The operations used here are: first, the introduction of a spurious branch,

    after which the same action is performed regardless of which way the branch

    goes; second, a simplification based on the fact that if the branch goes one

    way then n = 0, together with rewriting one of the right-hand sides of an as-

    signment statement; and third, elimination of a label corresponding to a null

    statement, and what might be called an anti-simplification of a sequence of

    assignment statements. We have not yet worked out a set of formal rules jus-

    tifying these steps, but the rules can be obtained from the rules given above

    for substitution, replacement and manipulation of conditional expressions.

    The result of these transformations is to show that the two programs each

    satisfy a relation of the form: program is equivalent to

    a : if n = 0 then go to b; s := n × s; n := 0; program.

    The program corresponding to fig. 3c in this case, is precisely the first

    program which we shall assume always converges. This completes the proof.

    11 The Description of Programming LanguagesProgramming languages must be described syntactically and semantically.

    Here is the distinction. The syntactic description includes:

  29. A description of the morphology, namely what symbolic expressions represent grammatical programs.
  30. The rules for the analysis of a program into parts and its synthesis from the parts. Thus we must be able to recognize a term representing a

    sum and to extract from it the terms representing the summands.

    The semantic description of the language must tell what the programs

    mean. The meaning of a program is its effect on the state vector in the case

    of a machine independent language, and its effect on the contents of memory

    in the case of a machine language program.

    12 Abstract Syntax of Programming LanguagesThe Backus normal form that is used in the ALGOL report, describes the

    morphology of ALGOL programs in a synthetic manner. Namely, it describes

    how the various kinds of program are built up from their parts. This would

    be better for translating into ALGOL than it is for the more usual problem

    of translating from ALGOL. The form of syntax we shall now describe differs

    from the Backus normal form in two ways. First, it is analytic rather than

    synthetic; it tells how to take a program apart, rather than how to put it

    together. Second, it is abstract in that it is independent of the notation used

    to represent, say sums, but only affirms that they can be recognized and

    taken apart.

    Consider a fragment of the ALGOL arithmetic terms, including constants

    and variables and sums and products. As a further simplification we shall

    consider sums and products as having exactly two arguments, although the

    general case is not much more difficult. Suppose we have a term t. From the

    point of view of translating it we must first determine whether it represents

    a constant, a variable, a sum, or a product. Therefore we postulate the

    existence of four predicates: isconst(t), isvar (t), issum(t) and isprod (t). We

    shall suppose that each term satisfies exactly one of these predicates.

    Consider the terms that are variables. A translator will have to be able

    to tell whether two symbols are occurrences of the same variable, so we need

    a predicate of equality on the space of variables. If the variables will have to

    be sorted an ordering relation is also required.

    Consider the sums. Our translator must be able to pick out the sum-

    mands, and therefore we postulate the existence of functions addend (t) and

    augend (t) defined on those terms that satisfy issum(t). Similarly, we have

    the functions mplier (t) and mpcand (t) defined on the products. Since the

    analysis of a term must terminate, the recursively defined predicate

    term(t) = isvar (t) ∨ isconst(t)issum(t) ∧ term

    (addend (t)) ∧ term(augend (t)) ∨ isprod (t) ∧ term

    (mplier (t)) ∧ term(mpcand (t))

    must converge and have the value true for all terms.

    The predicates and functions whose existence and relations define the

    syntax, are precisely those needed to translate from the language, or to define

    the semantics. That is why we need not care whether sums are represented

    by a + b, or +ab, or (PLUS A B), or even by G¨odel numbers 7a11b.

    It is useful to consider languages which have both an analytic and a syn-

    thetic syntax satisfying certain relations. Continuing our example of terms,

    the synthetic syntax is defined by two functions mksum(t, u) and mkprod (t, u)

    which, when given two terms t and u, yield their sum and product respec-

    tively. We shall call the syntax regular if the analytic and the synthetic

    syntax are related by the plausible relations:

  31. issum(mksum(t, u)) and isprod (mkprod (t, u))
  32. addend (mksum(t, u)) = t; mplier (mkprod (t, u)) = t augend (mksum(t, u)) = u, mpcand (mkprod (t, u)) = u
  33. issum(t) ⊃ mksum(addend (t), augend (t)) = t and isprod (t) ⊃ mkprod (mplier (t), mpcand (t)) = t

    Once the abstract syntax of a language has been decided, then one can

    choose the domain of symbolic expressions to be used. Then one can define

    the syntactic functions explicitly and satisfy one self, preferably by proving

    it, that they satisfy the proper relations. If both the analytic and synthetic

    functions are given, we do not have the difficult and sometimes unsolvable

    analysis problems that arise when languages are described synthetically only.

    In ALGOL the relations between the analytic and the synthetic functions

    are not quite regular, namely the relations hold only up to an equivalence.

    Thus, redundant parentheses are allowed, etc.

    13 Semantics

    The analytic syntactic functions can be used to define the semantics of a

    programming language.

    In order to define the meaning of the arithmetic

    terms described above, we need two more functions of a semantic nature,

    one analytic and one synthetic.

    If the term t is a constant then val (t) is

    the number which t denotes. If α is a number mkconst(α) is the symbolic

    expression denoting this number. We have the obvious relations

  34. val (mkconst(α)) = α
  35. isconst(mkconst(α))
  36. isconst(t) ⊃ mkconst(val (t)) = t Now we can define the meaning of terms by saying that the value of a

    term for a state vector ξ is given by

    value(t, ξ) = if isvar (t) then c(t, ξ) else if isconst(t)

    then val (t) else if issum(t) then value (addend (t), ξ) +

    +value(augend (t), ξ) else if isprod (t) then value(mplier (t),

    ξ) × value(mpcand (t), ξ)

    We can go farther and describe the meaning of a program in a program-

    ming language as follows: The meaning of a program is defined by its effect

    on the state vector. Admittedly, this definition will have to be elaborated to

    take input and output into account.

    In the case of ALGOL we should have a function

    ξ (cid:48) = algol (π, ξ),

    which gives the value ξ (cid:48) of the state vector after the ALGOL program π has

    stopped, given that it was started at its beginning and that the state vector

    was initially ξ. We expect to publish elsewhere a recursive description of the

    meaning of a small subset of ALGOL.

    Translation rules can also be described formally. Namely,

  37. A machine is described by a function x(cid:48) = machine(p, x)

    giving the effect of operating the machine program p on a machine vector x.

  38. An invertible representation x = rep(ξ) of a state vector as a machine
  39. A function p = trans(π) translating source programs into machine vector is given.

    programs is given.

    The correctness of the translation is defined by the equation

    rep(algol (π, ξ)) = machine(trans(π), rep(ξ)).

    It should be noted that trans(π) is an abstract function and not a machine

    program. In order to describe compilers, we need another abstract invertible

    function, u = repp(π), which gives a representation of the source program in

    the machine memory ready to be translated. A compiler is then a machine

    program such that trans(π) = machine(compiler , repp(π)).

    14 The Limitations of Machines And Men as

    Problem-Solvers

    Can a man make a computer program that is as intelligent as he is? The

    question has two parts. First, we ask whether it is possible in principle, in

    view of the mathematical results on undecidability and incompleteness. The

    second part is a question of the state of the programming art as it concerns

    artificial intelligence. Even if the answer to the first question is “no”, one

    can still try to go as far as possible in solving problems by machine.

    My guess is that there is no such limitation in principle. However, a

    complete discussion involves the deeper parts of recursive function theory,

    and the necessary mathematics has not all been developed.

    Consider the problem of deciding whether a sentence of the lower pred-

    icate calculus is a theorem. Many problems of actual mathematical or sci-

    entific interest can be formulated in this form, and this problem has been

    proved equivalent to many other problems including problem of determining

    whether a program on a computer with unlimited memory will ever stop. Ac-

    cording to Post, this is equivalent to deciding whether an integer is a member

    of what Post calls a creative set. It was proved by Myhill that all creative

    sets are equivalent in a quite strong sense, so that there is really only one

    class of problems at this level of unsolvability.

    Concerning this problem the following facts are known:

    1) There is a procedure which will do the following: If the number is in

    the creative set, the procedure will say “yes”, and if the number is not in the

    creative set the procedure will not say “yes”, it may say “no” or it may run

    on indefinitely.

    2) There is no procedure which will always say “yes” when the answer

    is “yes”, and will always say “no” when the answer is “no” If a procedure

    has property (1) it must sometimes run on indefinitely. Thus there is no

    procedure which can always decide definitely whether a number is a member

    of a creative set, or equivalently, whether a sentence of the lower predicate

    calculus is a theorem, or whether an ALGOL program with given input will

    ever stop. This is the sense in which these problems are unsolvable.

    3) Now we come to Post’s surprising result. We might try to do as well

    as possible. Namely, we can try to find a procedure that always says “yes”

    when the answer is “yes”, and never says “yes” when the answer is “no”, and

    which says “no” for as large a class of the negative cases as possible, thus

    narrowing down the set of cases where the procedure goes on indefinitely as

    much as we can. Post showed that one cannot even do as well as possible.

    Namely, Post gave a procedure which, when given any other partial decision

    procedure, would give a better one. The better procedure decides all the

    cases the first procedure decided, and an infinite class of new ones. At first

    sight this seems to suggest that Emil Post was more intelligent than any

    possible machine, although Post himself modestly refrained from drawing this

    conclusion. Whatever program you propose, Post can do better. However,

    this is unfair to the programs. Namely, Post’s improvement procedure is

    itself mechanical and can be carried out by machine, so that the machine can

    improve its own procedure or can even improve Post, if given a description

    of Post’s own methods of trying to decide sentences of the lower predicate

    calculus. It is like a contest to see who can name the largest number. The

    fact that I can add one to any number you give, does not prove that I am

    better than you are.

    4) However, the situation is more complicated than this. Obviously, the

    improvement process may be carried out any finite number of times and a

    little thought shows that the process can be carried out an infinite number of

    times. Namely, let p be the original procedure, and let Post’s improvement

    procedure be called P , then P np represents the original procedure improved

    n times. We can define P ωp as a procedure that applies p for a while, then

    PD for a while, then p again, then Pp, then P 2p, then p, then Pp, then P 2p,

    then P 3p, etc. This procedure will decide any problem that any P n will

    decide, and so may justifiably be called P ωp. However, P ωp is itself subject

    to Post’s original improvement process and the result may be called P ω+1p.

    How far can this go? The answer is technical. Namely, given any recursive

    transfinite ordinal α, one can define P αp. A recursive ordinal is a recursive

    ordering of the integers that is a well-ordering in the sense that any subset of

    the integers has a least member in the sense of the ordering. Thus, we have

    a contest in trying to name the largest recursive ordinal. Here we seem to be

    stuck, because the limit of the recursive ordinals is not recursive. However,

    this does not exclude the possibility that there is a different improvement

    process Q, such that Qp is better than P αp for any recursive ordinal α.

    5) There is yet another complication. Suppose someone names what he

    claims is a large recursive ordinal. We, or our program, can name a larger

    one by adding one, but how do we know that the procedure that he claims is

    a recursive well-ordering, really is? He says he has proved it in some system

    of arithmetic. In the first place we may not be confident that his system of

    arithmetic is correct or even consistent. But even if the system is correct,

    by G¨odel’s theorem it is incomplete. In particular, there are recursive or-

    dinals that cannot be proved to be so in that system. Rosenbloom, in his

    book “Elements of Mathematical Logic” drew the conclusion that man was

    in principle superior to machines because, given any formal system of arith-

    metic, he could give a stronger system containing true and provable sentences

    that could not be proved in the original system. What Rosenbloom missed,

    presumably for sentimental reasons, is that the improvement procedure is

    itself mechanical, and subject to iteration through the recursive ordinals, so

    that we are back in the large ordinal race. Again we face the complication

    of proving that our large ordinals are really ordinals.

    6) These considerations have little practical significance in their present

    form. Namely, the original improvement process is such that the improved

    process is likely to be too slow to get results in a reasonable time, whether

    carried out by man or by machine. It may be possible, however, to put this

    hierarchy in some usable form.

    In conclusion, let me re-assert that the question of whether there are

    limitations in principle of what problems man can make machines solve for

    him as compared to his own ability to solve problems, really is a technical

    question in recursive function theory.

    References

    [McC62] John McCarthy. Checking mathematical proofs by computer.

    In Proceedings Symposium on Recursive Function Theory (1961).

    American Mathematical Society, 1962.

    [McC63] J. McCarthy. A basis for a mathematical theory of computation.1 In

    P. Braffort and D. Hirschberg, editors, Computer Programming and

    Formal Systems, pages 33–70. North-Holland, Amsterdam, 1963.

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

    May 1, latexed 1996

    May 14 at 1:20 p.m.

    1http://www-formal.stanford.edu/jmc/basis/basis.html