1.7. Comment.Church’s thesis will be given a precise formulation in the next section: the basic functions and the elementary operations will be given exp- licitly. The exact mathematical theory of computability begins at that point.
But it seemed important to indicate first the general significance of the discovery that such families of functions and operations exist at all and can even be given explicitly, a result that is far from obvious.
This is an experimental fact, one of the most important discovered by logic.
In the next section we discuss evidence of its value and usefulness. Now we merely note that this fact is related to the finiteness of the basic logical and set- theoretic principles of mathematics (implicit, for example, in L1Set), but is not identical to this finiteness.
The domain of definition D(h) is also defined by recursion:
x1, . . . , xn,1 ∈D(h)⇔ x1, . . . , xn ∈D(f),
x1, . . . , xn, k+ 1 ∈D(h)⇔ x1, . . . , xn, k ∈D(h), and x1, . . . , xn, k, h(x1, . . . , xn, k) ∈D(g) fork1.
(d)The µ-operator. This operation associates to a partial function f from (Z+)n+1 to Z+ the partial function h from (Z+)n to Z+ that is defined as follows:
D(h) =(
x1, . . . , xn|∃xn+11, f(x1, . . . , xn, xn+1) = 1 and x1, . . . , xn, k ∈D(f) for allkxn+1)
; h(x1, . . . , xn) = min(
xn+1|f(x1, . . . , xn, xn+1) = 1) .
The general role of µ is to introduce “implicitly defined” functions, as is often done in many areas of mathematics. Three remarks about the definition of µ should be made at this point. First, we obviously chose the minimal y withf(x1, . . . , xn, y) = 1 in order to ensure that the functionhis single-valued.
The second observation is that at first glance, it might seem that the domain of definition ofhis artificially narrow. If, for example, we havef(x1, . . . , xn,2) = 1 andf(x1, . . . , xn,1) is not defined, then we have takenh(x1, . . . , xn) to be unde- fined, rather than equal to 2. This is done because we want to preserve intuitive semicomputability in going fromftoh, as will be discussed in somewhat greater detail below (see 2.7(a)).
Finally, we note that all the operations before µ, if applied to everywhere defined functions, give an everywhere defined function. This is obviously not the case for µ. Thus, µ is the only one of the operations that causes partial functions to arise unavoidably.
2.4. Definition.
(a) A sequence of partial functions f1, . . . , fN is called a partial recursive (respectively primitive recursive) description of the function fN =f if
f1 belongs to the family of basic functions;
fi, i2, either belongs to the family of basic functions, or else is obtained by applying one of the elementary operations (respecti- vely one of the elementary operations other thanµ) to certain of the functionsf1, . . . , fi−1.
(b) A functionf is called partial recursive (respectively primitive recursive) if it admits a partial recursive (respectively primitive recursive) description.
(The analogy with the definition of a deduction in a formal language immedi- ately catches our attention, and can sometimes be of use.)
2.5. Church’s Thesis(usual form)
(a) A functionf is semicomputable if and only if it is partial recursive.
(b) A function f is computable if and only if both f and χD(f) are partial recursive.
Remark on terminology. Everywhere defined partial recursive functions are also calledgeneral recursivefunctions. If the domain of definition is either clear or not essential in a given context, we simply use the term “recursive.” (Note that every primitive recursive function is general recursive.)
2.6.Use of Church’s thesis.Before discussing in detail the arguments supporting Church’s thesis, we indicate how it is used in practice in mathematics. Two basic applications are especially evident in the literature.
(a) Church’s thesis used for a definition of algorithmic undecidability.
Suppose we have a countable sequence of mathematical “problems”
P1, P2, . . . . Further, suppose that each problem has a “yes” or “no” an- swer, and that the conditions in Pn are written out “effectively” as a function of n. Such a sequence P = (Pn) is called a “mass problem.” We associate to such a problem a function f fromZ+ to Z+
D(f) = {i∈Z+|Pi has “yes” for an answer}; f(i) = 1, ifi∈D(f).
A mass problem P is called algorithmically decidable if the functions f and χD(f)are partial recursive. Otherwise,P is calledalgorithmically undecidable.
We also distinguish the case in which onlyχD(f)is not partial recursive from the case in which evenf is not partial recursive. The second type of undecidability is worse than the first; we saw examples of this in§1. Finally, a whole hierarchy of “degrees of undecidability” can be rigorously defined and investigated.
A well-known example of a mass problem is theproblem of word identities in groups.LetGbe a finitely defined group, and leta1, . . . , ar∈Gbe elements.
A “reduced word” in a1, . . . , ar is an expression of the formaεi1i· · ·aεik
k, where k 1, εj = ±1, and εj = εj+1 whenever ij = ij+1. We number all the reduced words and ask the question Pn: “Does the nth word represent the unit element of the group G?” The “mass problem” (Pn) turns out to be algorithmically decidable for certain groups G and elements a1, . . . , an and algorithmically undecidable for others (Novikov, Boone, Higman). The func- tion f in this case is always partial recursive, but χD(f) is not always (see Chapter VIII).
For another example of an undecidable problem, this one connected with Diophantine equations, see Chapter VI.
(b) Church’s thesis as a heuristic principle. The intuitive notion of
“semicomputability” at first seems broader than the notion of “partial recur- siveness,” and many problems concerning partial recursive functions become much easier if we replace the conditions in the problems by informal ideas and allow such ideas to be used to solve the problems. For example, the formula
e = lim(1 + 1/n)n and the Euclidean algorithm make it intuitively clear that the functions f, g:Z+→Z+ given by
f(n) = thenth digit in the decimal expansion of e, g(n) = thenth prime number
are computable, but the verification that they are recursive requires rather painstaking constructions.
Church’s thesis allows us to solve such problems in two stages: (1) finding an informal solution using any intuitive algorithms we need, and (2) formalizing the solution. The second stage presupposes a certain proficiency in finding a partial recursive description for a wide variety of semicomputable functions, and Church’s thesis assures us that such a description exists.
As proofs of recursiveness become more and more numerous in the literature, it becomes increasingly common to go through only the first stage of the solu- tion; a striking example of this is Hartley Rogers’ book Theory of Recursive Functions and Effective Computability (McGraw-Hill, New York, 1967). We shall also take such liberties toward the end of this book. All the same, there is a certain danger in this practice. It is possible that the habit of increasingly using informal arguments delayed the discovery of such a fundamental fact as the result that recursively enumerable sets and Diophantine sets coincide.
2.7. Arguments in support of Church’s thesis
(a) First of all, the basic functions clearly must be computable, no mat- ter how we precisely define the notion of computability. Furthermore, when the elementary operations are applied to semicomputable functions, they again give a semicomputable function. A program to semicompute the latter function can easily be put together from the programs that semicompute the original func- tions. We shall consider only the case of the µ-operator in detail, leaving the simple construction of the other three programs to the reader.
In the notation of 2.3(d), let f be a semicomputable function from (Z+)n+1 to Z+. In order to computeh(x1, . . . , xn), we go through the vectors x1, x2, . . . , xn,1, x1, . . . , xn,2, . . .in the order of increasing last coordinate, and compute the values off at these vectors. If x1, . . . , xn ∈D(h), wherehis obtained fromf by applying theµ-operator, then the program forfsuccessively computes
f(x1, . . . , xn,1), . . . , f(x1, . . . , xn, y−1),
and finally f(x1, . . . , xn, y) = 1. The least suchy, if it exists, must be given as output; it will be the value ofhat the point x1, . . . , xn. On the other hand, if it turns out that one of the valuesf(x1, . . . , xn, k) (before we reachf = 1) is not defined, then either the program that semicomputes f will work infinitely long, or else it will give an answer not in Z+, which must then be given as output. But then, by definition,his not defined at the point x1, . . . , xn, and the behavior of the program for h still agrees with the definition of h being semicomputable.
From all this we conclude thatpartial recursivefunctionsare semicomputable.
However, the stronger part of Church’s thesis is the converse: semicomputable functions are partial recursive. (The definition of computability in terms of semicomputability is simply taken from §1 without any changes.) As has been said, this result is an experimental fact. The experimental evidence for it is divided into several classes, which we consider in (b)–(d) below.
(b) In the literature we find a huge collection of recursive descriptions of various computable and semicomputable functions. See, for example, R´ozsa P´eter, Recursive Functions(Academic Press, New York, 1967). We shall give part of this list in the next section. We also find certain techniques for composing recursive descriptions that are applicable to entire classes of (semi)computable functions. Every time an author has tried to find a partial recursive description of a (semi)computable function, he has met with success.
(c) Turing proposed a mathematical characterization of an abstract com- puter, and gave strong arguments to the effect that this computer is universal, i.e., it can (semi)compute any (semi)computable function. His arguments came from a detailed analysis of the characteristic features of determinate computa- tional processes. (We again recall that we have not at all concerned ourselves with formalizing computational processes, but only with the results of such pro- cesses.) It turned out that the class of functions that are semicomputable by Turing machines exactly coincides with the class of partial recursive functions.
(d) Church, Post, Markov, Kolmogorov, Uspenskiˇı, and others have pro- posed other deterministic schemes for processing information of a general (not necessarily number-theoretic) character. In all cases it has turned out that if the sets of input and output are numbered in a suitable “effective” way, these meth- ods lead to a class of maps from Z+ to Z+ that coincides with some subclass of the partial recursive functions.
For further discussion of Church’s thesis, we refer the reader to the literature;
see, in particular, S. Kleene, Introduction to Metamathematics(Van Nostrand, New York–Toronto, 1952).