Church's Thesis Without TearsAuthor(s): Fred RichmanReviewed work(s):Source: The Journal of Symbolic Logic, Vol. 48, No. 3 (Sep., 1983), pp. 797-803Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2273473 .Accessed: 27/07/2012 10:51

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 48, Number 3, Sept. 1983

CHURCH'S THESIS WITHOUT TEARS

FRED RICHMAN

?1. Introduction. The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computa- tion, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of com- putable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.

The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase "you can find" implies "by a finite calculation." In particular if P is some property, then the statement "for each m there is n such that P(m, n)" means that we can construct a (computable) function 0 such that P(m, 0(m)) for all m. Church's thesis has a dif- ferent flavor in an environment like this where the notion of a computable function is primitive.

One point of such a treatment of Church's thesis is to make available to Bishop- style constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable func- tions other than recursive functions makes it quite implausible that a Bishop- style constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

Another point of our treatment is to gain the benefits of economy and generality that accompany an axiomatic development. Economy is achieved by bypassing the technically involved theory of recursive functions. Generality flows from the fact that the set of partial recursive functions is not the only model for the axiom CPF of ?2. In this respect our approach can be viewed as axiomatic recursion theory

Received October 24, 1981 revised January 20, 1982. ? 1983, Association for Symbolic Logic

0022-4812/83/4803-0031/$01.70

797

798 FRED RICHMAN

in the spirit of Wagner and Strong [FRDM]; indeed CPF implies that the partial functions form an wo-BRFT as defined in [FRDM].

We are particularly interested in applications to constructive analysis. A real number is a Cauchy sequence of rational numbers, which is essentially a function from the positive integers to the positive integers. Thus CPF has implications in the theory of real numbers. Except for the last section on the recursion and iso- morphism theorems, it is these implications that we are most concerned with.

?2. Partial functions. Let N denote the set of positive integers. A partialfunction algorithm from N to Nis a function A from N x N to N U { I } such that if A(x, n)

I, then A(x, n + 1) = A(x, n). We think of A(x, n) as being the output after n steps of the algorithm A applied to the input x. The partial function f associated with A is defined on

domf = {xeN :A(x,n) I for some n}

by setting f(x) = A(x, n) for any n such that A(x, n) : 1. Two partial functions f and g are equal if domf = dom g andf(x) = g(x) whenever x e domf. A partial function that is defined on all of N is called total. The difference between f and A, which are essentially the same object, is in how we look at them, and the equality relation to which they are subject.

The central feature of the Church-Markov-Turing theory of computability is the following:

CPF. The set of partial functions is countable. That is, there is a sequence fi, f2, . . . of partial functions so that given a partial

function g you can find a positive integer m such that g = ft. Note that this does not mean that the set of partial function algorithms is countable. In fact partial function algorithms are in 1-1 correspondence with functions, and a simple diagonal augument shows that given any sequence of functions, we can construct a function different from any in the given sequence. On the other hand we do have a sequence A1, A2, . . . of partial function algorithms that define the partial functions fl, f2,

.... We shall assume that these sequences are fixed throughout the remainder of the paper.

If a partial function f is defined at x, then f is not undefined at x, that is, it is false that A(x, n) = I for all n The converse statement that if it is false that A(x, n) = I for all n, then there exists n such that A(x, n) : I is an instance of Markov's principle. As neither Bishop-style constructivists nor intuitionists accept Markov's principle, we will maintain a distinction between the statements "f is defined at x" and "f is not undefined at x". The distinction rests upon the demand that there be some sort of prior bound on the length of a computation, and not just an assurance that it will eventually terminate (compare [RGRS, *10, p. 51]).

?3. The halting problem. The problem of determining whether or not fm(x) is defined is called the halting problem. To solve the halting problem means to con- struct a function g: N x N -+ {O, 1 } such thatfm(x) is defined if and only if g(m, x) = 1. A solution to the halting problem would be a special instance of Bishop's limited principle of omniscience (LPO) which states that given any function h: N -+

{0, 1} we can find n e N such that h(n) = 0 implies h(x) = 0 for all x in N. If the

CHURCH S THESIS WITHOUT TEARS 799

existence of a certain construction implies LPO, Bishop considers it essentially impossible that the construction can be effected; this kind of argument is known as a Brouwerian counterexample. It is easily seen that LPO is equivalent to the state- ment that the interval [- 1, 1] is equal to the union [- 1, 0) U {O} U (0, 1]. The main construction in demonstrating that the halting problem is unsolvable is the following:

LEMMA 3. 1. Given a total function g from N to N, there is a positive integer m so that fm(m) is defined exactly when g(m) 0.

PROOF. Define A(x, n) = I if g(x) ? 0, and A(x, n) = 0 if g(x) = 0. Clearly A(x, n) defines a partial functionfm of the required form. El

THEOREM 3.2. There is no total function gfrom N to N such that g(m) : 0 exactly when fm(m) is defined, or such that g(m) : 0 exactly when fm(m) is not undefined. O

A subset S of N is detachable (recursive) if there is a function g from N to {0, 1} such that x e S if and only if g(x) = 1. The set K = {m: fm(m) is defined} is count- able (recursively enumerable) because we can enumerate the elements of K by set- ting an = m if Am(m, n) : I, and an = Z where fm is the constant function 0 otherwise. Enumerating the amn gives us an enumeration of K. On the other hand neither K nor K' = Im fm(m) is not undefined} is detachable from N as that would contradict Theorem 3.2.

LEMMA 3.3. Let K = {m: fm(m) is defined} and L = {m e K: fm(m) > O}. If or is a total function from N to {0, 1}, then there exists a number n in K such that o(n) =

o if and only if n e L. Thus if S is any detachable set, then S n K ? L. PROOF. Choose n such that 1 -a f=r. E The lesser limited principle of omniscience (LLPO) of [BSH2] states that the

interval [- 1, 1] is equal to [- 1, 0] U [0, 1]. Clearly LPO implies LLPO. Like LPO, this weaker principle is used for Brouwerian counterexamples and is false if we assume CPF (see [SHNN, p. 108]). Before proving this we recall the notion of an operation [BSH 1].

By an operation a: [- 1, , 1 } we mean a computation that, for each x in [- 1, 1], produces a(x) = + 1. An operation need not be a function in that we might have x = y but a(x) :A a(y). The point is that x and y may be different as sequences of rationals yet equal as real numbers. Thus a exhibits [- 1, 1] as the (not necessarily disjoint) union of the two subsets a-l(- 1) and ca-(1). If Ix - a(x)l < 1 for all x, then a1-(- 1) c [- 1, 0] and a1-(l) c [0, 1]. The exis- tence of such an a is what it means to say that [-1, 1] [-1, 0] U [0, 1]; but such an a cannot exist in light of the following.

THEOREM 3.4. If a: [-1, 1] {-1, 1} is an operation, then Ix - a(x)I > I for some x in [-1, 1].

PROOF. For each m in N define a real number am in [-1, 1] as follows. Let

amn = am(n-l) if n > l and Am(m, n - 1) I I;

otherwise, let

amn= 0 if Am(m, n)= 1, = 1/n if Am(m, n) > O0

800 FRED RICHMAN

For each m the sequence amn is Cauchy; let am be its limit. If m e K = {m :fm(m) is defined}, then am = 0 and am > 0 if and only if

m eL = {m e K: fm(m) > O}. The function v: N-+ {O, 1} defined by v(m) =

(a(am) + 1)/2 is total. By Lemma 3.3 we can find m in K such that v(m) = 0 if and only if m e L. Setx = am. E

?4. Continuity and intermediate values. There are some positive consequences of

CPF. The following consequence of the unsolvability of the halting problem shows that functions on the reals must have some continuity properties (see [ABER, Theorem 6.3]).

LEMMA 4.1. Let S be a closed set of real numbers and an a sequence in S converg- ing to c. There is no operation g on S such that gx = 1 if x - an for some n, and gx =(0 if x = c.

PROOF. Define 0(m, n) to be 1 if Am(m, n) = l, and b(m, n) = 0 otherwise. Then for each m the series of real numbers

a1 + E 0(m, n) (a,+, - an) neN

converges to a real number rm E S. Iffm(m) is defined, then rm = an for some n so g(rm) = 1, while if fm(m) is undefined, then rm = c so g(rm) = 0. Thus g(rm)

if and only if fm(m) is not undefined. This contradicts Theorem 3.2. E

THEOREM 4.2. Every integer valuedfunction on the real numbers is constant. PROOF. It suffices to show that any function g taking the real numbers to {0, 1 }

is constant. If g were not constant, then by interval halving we could construct Cauchy sequences {an} and {bn} of real numbers such that g(an) = 1 and g(bn) = 0

for all n, and an -bn converges to 0. Let c be the limit of these Cauchy sequences and suppose, say, that g(c) = 0. This contradicts Lemma 4.1. E

Next we consider a weak form of continuity for arbitrary functions which was

proved by Markov in 1958 (see [ABER, Theorem 7. 1]). THEOREM 4.3 (WEAK CONTINUITY). Let S be a closed set of real numbers and an

a sequence in S converging to c. Iff: S -+ R is a function and If(an) - f(c) > ? for all n, then 5 < 0.

PROOF. Suppose 3 > 0. Define an operation g: S -+ {O, 1} such that if g(x) = 0, then If(x)- f(c)I < 3, while if g(x) = 1, then If(x) - f(c)I > 0. Then g(c) - 0

and g(an) = 1, contradicting Lemma 3.1. LI The Russian constructivists went on to prove that every function on the reals is

continuous [CEIT], [ABER, Appendix 3]. However Beeson [BEES] has shown that

this proof makes essential use of Markov's principle. The intermediate value theorem for uniformly continuous functions does not

admit a constructive proof [BSH1, Problem 12, p. 59]. However an approximate intermediate value theorem is available [BSH1, Problem 11, p. 59] and if the

function is required to be nonconstant in every open interval, then the intermediate value theorem can be proved [BSH1, Problem 13, p. 59]. In fact pointwise con-

tinuity is all that is needed for these last two results. If we assume CPF, we do

not even need this [ABER, Theorem 7.6]. THEOREM 4.4. Let f be a function on an interval [a, b] such that f(a) < t < f(b).

For each e > 0 there is c e [a, b] such that If (c) - t I < e. Iff is nonconstant on each

CHURCH S THESIS WITHOUT TEARS 801

nonempty subinterval of [a, b], then we can construct c so that f(c) = t. PROOF. By interval halving we can construct sequences a, < a2 < < b2 < b1

such that

(1)f(an) < t < f(bn) or (2) lf(aj) - tI < 6/3 and an = bn5

and a. -b, goes to 0. If f is nonconstant on every nonempty subinterval we can always achieve (1) by approximate interval halving, so the sequences are indepen- dent of 6. Let c be the limit of the sequence a,. Suppose lf(c) - t I > 6/2. We may assume thatf(c) > t + e/2. Then If(c) - f(an)l ? 6/6 for all n, contradicting Theo- rem 3.2. Thus lf(c) - tI < e. D

?5. Specker's sequence. A remarkable construction of Specker [SPEC] is an ascending sequence (a") of rational numbers in the unit interval so that for every real number r the sequence an is eventually bounded away from r.

THEOREM 5.1. There exists an ascending sequence of rational numbers a. in the Cantor set such that for any real number r there is v in N and s > 0 satisfying Ir - a,, I a if n ? v.

PROOF. Let a"(m) denote the mth digit in the ternary expansion of a.. Define a. by

a"(m) = 2 if Am(m, n) = O and m < n,

= 0 otherwise.

Let b(i) be the ith ternary digit of some number in the Cantor set C. Then there is a positive integer m such that b(i) = fm(i) with fm total. Thus for n sufficiently large we have

b(m) = fm(m) = 2 -a(m)

whence if r is in C the theorem is true. To verify the theorem for general r define be C so that for each n either

d(r, C) > 1/(n + 1) and bn+1 = b, or d(r, b,+1) < 1/n.

Then the b. form a Cauchy sequence converging to a number b in C. Thus there exist v and s such that lb - anI ? s if n ? v. If d(r, b) < s we are in good shape; otherwise we can find n so that d(r, C) > 1/(n + 1) and we are in even better shape. D

Specker's sequence is an example of a bounded ascending sequence of real numbers with no supremum. It is an easy exercise to use it to give counterexamples to the Heine-Borel theorem and to the theorem that every continuous function on the closed unit interval is uniformly continuous.

?6. The isomorphism and recursion theorems. There are many possible ways of enumerating the partial functions. Indeed if 0 maps N onto N. then f0(),f0(2), . . . is another enumeration. That all enumerations are essentially the same is the con- tent of the isomorphism theorem [RGRS, p. 191], [MAYO, p. 114], [FRDM, Theorem 2.5]. First we show that every partial function appears infinitely often in the sequencef1,f2,

802 FRED RICHMAN

LEMMA 6.1. Let f be a partial function and J a finite subset of N such that f, = f for each j in J. Then there exists i 0 J such that fj = f.

PROOF. There is a total function 0 such thatfo(m) = f if fm(m) is defined, otherwise

f(m) is everywhere undefined. By Lemma 3.1 there is m such that 0(m) 0 J if and only if fm(m) is defined. If 0(m) 0 J, then fm(m) is defined, so f()- = f. In this case we can set i = 0(m). If 0(m) e J, then fm(m) is undefined, so f = fo(m) is everywhere undefined. Thus it remains to prove the lemma in casef is everywhere undefined.

Supposef is everywhere undefined. There is a total function 0 such thatfo(m) 1

iffm(m) is defined; otherwise f(m) is everywhere undefined. By Lemma 3.1 there is m such that 0(m) e J if and only iffm(m) is defined. If 0(m) e J, thenfm(m) is defined so fo(m) is everywhere defined-a contradiction. Hence 0(m) 0 J and fm) is every- where undefined, as desired. L

THEOREM 6.2. If g1, g2, . . . is an enumeration of the partial functions, then there is a one-to-one function Ofrom N onto N such that gi = fo() for each i in N.

PROOF. By Lemma 6.1 we can construct a strictly increasing function A from N to N such that g, = fA() for each i in N. Interchanging the roles of f and g we can also construct a strictly increasing function ,u from N to N such that fi = g,,(i) for each i in N. We define 0 and 0-1 inductively by

0(m) = n if m = 0-1(n) for n < m,

= A(m) otherwise.

0-1(m) = n ifm = 0(n) for n < m,

= ,(m) otherwise. E

We close with the recursion theorem, another example of how the major theorems of computability theory can be deduced from CPF (compare [FRDM, Theorem 1.6]).

THEOREM 6.3. Let 0: N -- N be a total function. Then there exists n such that

fn = fq(n)-

PROOF. Let A(x) = f,(x). Then there is a total function g such that fg(x) = AX

whenever A(x) is defined. Choose m such that 0(g(x)) = fm(x) and set n = g(m). Then

fn = fg(m) f(m) = f(g(m)) = fo(f) LI

REFERENCES

[ABER] 0. ABERTH, Computabhe analysis, McGraw-Hill, New York, 1980.

[BEES] M. BEESON, The non-derivability in intuitionistic formal systems of theorems on the con- tinuity of effective operations, this JOURNAL, vol. 40 (1975), pp. 321-346.

[BSHI] E. BISHOP, Foundations of constructive analysis, McGraw-Hill, New York, 1967.

[BSH2] , Schizophrenia in contemporary mathematics, American Mathematical Society Colloquium Lectures, Missoula, Montana, 1973.

[CEIT] G.S. CEITIN, Algorithmic operators in constructive metric spaces, American Mathematical Society Translations, vol. 64 (1967), pp. 1-80.

[FRDM] H. FRIEDMAN, Axiomatic recursion theory, Logic Colloquiim 69 (R. Gandy and M. Yates, Editors), North-Holland, Amsterdam, 1971, pp. 113-137.

CHURCH S THESIS WITHOUT TEARS 803

[MARK] A.A. MARKOV, On constructive functions, American Mathematical Society Transla- tions, vol. 29 (1963), pp. 163-195.

[MAYO] M. MACHTEY and P. YOUNG, An introduction to the general theory of algorithms, North-Holland, Amsterdam, 1978.

[RGRS] H. ROGERS, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.

[SHNN] N.A. SANIN, Constructive real numbers and function spaces, American Mathematical Society Translations, vol. 21 (1968).

[SPEC] E. SPECKER, Nicht konstruktiv beweisbare Sdtze der Analysis, this JOURNAL, VOI. 14 (1949), pp. 145-158.

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