Computable Analysis by S. Mazur; A. Grzegorczyk; H. RasiowaReview by: R. L. GoodsteinThe Journal of Symbolic Logic, Vol. 36, No. 1 (Mar., 1971), pp. 148-150Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271528 .

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148 REVIEWS

turns out that sHAn is not closed under 3a, a result expected from the analogy with the semi- hyperarithmetic (I'l) sets. This result leads to a brief study of the class of sets expressible in the form 3a R(-, a) with R in sHAn. These are related to sHAn sets much as E is related to UI1.

PETER G. HINMAN

ROBERT I. SOARE. Recursion theory and Dedekind cuts. Transactions of the American Mathematical Society, vol. 140 (1969), pp. 271-294.

ROBERT I. SOARE. Cohesive sets and recursively enumerable Dedekind cuts. Pacific journal of mathematics, vol. 31 (1969), pp. 215-231.

These papers present a new approach in the study of reducibilities, and should be read in the above order. As in recursive analysis, a Dedekind section in the rationals can be associated with each subset A of the natural numbers N. To this section one can assign a new subset L(A) of N. The dense linear ordering of the rationals affects the recursive structure of L(A) and allows sim- pler proofs of results due to Jockusch and Young. A complete picture of the mutual reducibility of A and L(A) is given: They must lie in the same Turing degree (Theorem 1.2), but can be in different truth-table degrees (Theorem 2.3); A is truth-table but not necessarily bounded truth- table reducible to L(A). If A is r.e., so is L(A) but the converse does not hold. Let us confine ourselves to non-recursive r.e. cuts. Such a cut cannot be: many-one complete (Corollary 1.10), creative (Corollary 1.11), quasi-creative (Corollary 1.13), a semicylinder or splinter without being a cylinder (Lemmas 3.4, 3.6). We can find cuts: in every truth-table degree (Theorem 2.4), that are not cylinders (Theorem 3.5), in every Turing degree that are semicreative (Theorem 4.4). Like Jockusch's semirecursive sets (Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420-436) to which they are related, cuts satisfy such strange results as: If L(A) is many-one reducible to L(B), then L(B) is Turing reducible to L(A) (Theorem 1.9). However "Turing reducible" cannot be replaced by "truth-table reducible" in this result (Theorem 2.8).

The second paper examines just how sparse an infinite set A can be, but yet with L(A) r.e. It gives a r.e. cut L(A) where: (1) A is dominant in the sense that the functionf(n), enumerating the elements of A in order of magnitude without repetition, increases faster than any recursive function; and (2) A contains an infinite retraceable subset. It also gives a cohesive set C with r.e. L(C). We note that C is hyperhyperimmune, whereas A is not. Though more emphasis on the intuitive geometrical arguments behind some of the proofs would have been welcome, the papers are clearly written and contain many ideas and results not mentioned in this review.

BRIAN H. MAYOH

S. MAZUR. Computable analysis, edited by A. Grzegorczyk and H. Rasiowa, Rozprawy matematyczne no. 33, Pan'stwowe Wydawnictwo Naukowe, Warsaw 1963, 111 pp.

Though published in 1963, these lecture notes are based on lectures which Mazur gave on computable analysis at the Institute of Mathematics in Warsaw in 1949-1950. Mazur's lectures included material on recursive real numbers which he and Banach had obtained between 1936 and 1939 as well as later work by Mazur himself; this earlier work, Sur lesfonctions calculables, was never published except for a mention in the Annales de la Societe Polonaise de Mathe- matique, vol. 16 (1937), page 223.

In the present paper, primitive recursive functions are called "recursive," and general recur- sive functions are called " computable," and a series of parallel results are obtained for recursive and computable real numbers and recursive and computable functions of real numbers. The classes of natural numbers, recursive functions, and computable functions are denoted by 21, R, C, respectively. A real number a is said to be recursive if there are functionsf, g e R such that

ja-f(n) - g~(n)~ a -~ ~ ~ + | n + 1|n + 1

for all n e 21. The class of recursive real numbers is denoted by A1. It is shown that various definitions of 2 are equivalent. For example, a is a recursive real number if there are functions f g e R such that If(n)/(g(n) + 1) - lal I < 1/(n + 1) for all n e 2. It is proved that N is a field.

The class of real numbers which have expansions in the scale of k is denoted by Mk. Specifi- cally, a e N2 if there is an fe R such that la = I =o f(n)/kn, 0 C f(n + 1) < k, for all

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REVIEWS 149

n e 21. The class of real numbers definable by recursive cuts is denoted by N3, so that a e 213 if there is an f e R such that tat < p/(q + 1) if and only if f(p, q) = 0 for all p, q E 21. The class A* is the class of real numbers such that there is an f e R for which Jai = En' of(n)/n!, 0 _ f(n + 1) < n + 1, for all n e 21. Mazur proves that 213 = (but the proof is non-constructive since it proceeds by cases, distinguishing the cases when a recursive real is rational and when it is irrational), and that 213 C Ad' c 21. As is well known, Specker proved (XV 67) that 1R3 #

2 1.

The corresponding definition of the class X of computable reals is that a is a computable real number if there aref, g e C such that

I n + 1 I n + 1

for all n e 2. Parallel to 2, 2Z3 the classes Wk, ( are defined, and it is proved that 21 is a proper subset of (E. R. M. Robinson, in his reference to Specker's paper in XVI 280(2), pointed out that X = 02, but his proof is non-constructive.

The second section of the paper, concerning recursive and computable sequences, introduces three notions of recursive and computable sequences which are distinct even in the case of com- putability. A sequence {ak} of real numbers is said to be recursive (computable) if there are functions A, g e R (C) such that

|a f(k- n)<jikX n)f | for all k, n e 21.

Let the class of recursive (computable) sequences be denoted by p (y). Further let pq (y3) denote the class of all sequences {al} of real numbers for which there exist f, g e R (C) such that at = Z. o(f(k, i)/qt) - g(k) and let p3 (y3) denote the class of sequences (ak} such that there exists f e R (C) such that akt < (p - q)/(r + 1) if and only if f(k, p, q, r) = 0 for all k, p, q, r e 2. It is shown that y3 a yq and that pq a p, y3 a Y, y3 a y. The second section ends with a discus- sion of recursive (computable) convergence. A sequence {ak} is said to be recursively (comput- ably) convergent to a if there is an h e R (C) such that lak- al < 1/(m + 1) for all k > h(m). It is proved that if {ak) e y, and if {ak} is monotonic and tends to a e XY then {a,<} is coM- putably convergent to a; if the condition that {ak} is monotonic is dropped then there exists a strictly increasing f E C such that {af(k)) is computably convergent to a. Amongst other results in this section there is a (non-constructive) proof that any root of a polynomial with recursive (computable) coefficients is recursive (computable).

The next section introduces the notion of recursive (computable) functional (different from Kleene's recursive functionals). A functional (D defined over C (R) is computable (recursive) if for a computable (recursive) sequence {f( the sequence of values an = D<gf> is computable (recursive). Computable functionals are continuous in the sense that if {fij -*f then {PD<f>} D<f>, which implies that a computable functional depends only on a finite number of values of the argument function. For recursive functionals only a weaker result holds.

The last section studies recursive and computable real functions. A real function, defined on a set of recursive (computable) numbers, is said to be recursive (computable) if it transforms any recursive (computable) sequence into a recursive (computable) sequence. The main results of this section are (1) that the four arithmetical operations do not lead out of the class of computable (recursive) functions, but 1/x is not recursive on the set of all positive recursive numbers; (2) the functions sgn x, [x] are not computable (even on (0, 1)), but sgn x is comput- able (but not recursive) on the set of computable (recursive) numbers different from zero; (3) a power series with recursive (computable) sequence of coefficients, and with recursive (comput- able) center, on the set of all recursive (computable) numbers in the interval of convergence, is not necessarily recursive (computable), but is recursive (computable) on the recursive (com- putable) reals in any closed interval within the interval of convergence; (4) a computable function on the set of all computable numbers in an interval is continuous and has the Darboux property. Finally, the relation between the author's notion of a recursive function (f E 2) and the corresponding notion introduced by Specker (_f S 6) is compared, and it is shown that if f e 2M then f e 6 if and only if f is recursive and recursively uniformly continuous; specifi-

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150 REVIEWS

cally, iff e m on the set 'g3 of all non-negative recursive numbers, and f is recursively uniformly continuous on A3, then the only uniformly continuous extensions off on 93 belongs to 6S.

It is important to point out that this paper presents a study of recursive and computable numbers and functions by entirely classical methods, and that it can make no claim to pre- senting a system of constructive analysis, that is a study of constructive objects by constructive methods. R. L. GOODSTEIN

MICHAEL 0. RABIN. A simple method for undecidability proofs and some applications. Logic, methodology and philosophy of science, Proceedings of the 1964 International Congress, edited by Yehoshua Bar-Hillel, Studies in logic and the foundations of mathematics, North- Holland Publishing Company, Amsterdam 1965, pp. 58-68.

This paper gives a useful method of showing a theory undecidable by using formulas of the theory to represent the universe and the non-logical constants of an undecidable theory T in such a way that every model of T is obtained from some model of the original theory. An advantage of the method is that T need not be finitely axiomatizable or essentially undecidable. The method is presented in several precisely formulated variations and then is used to obtain elegant proofs of some previously known as well as some new undecidability results. The theory T1 of an irreflexive, symmetric, binary relation is shown undecidable by interpreting in it the theory of an arbitrary binary relation. T1 is then in turn interpreted in the theory of atomistic distributive lattices to obtain a simple proof of the undecidability of that theory. An improve- ment of Tarski's undecidability result for the theory of groups is obtained by using the method to show that the theory of all groups which are the free product of two free groups with an amalgamated subgroup is undecidable. Finally, a more involved application of the method is used to obtain the new result that the theory of finite commutative rings is undecidable.

WILLIAM HANF

PHILIP K. HOOPER. The undecidability of the Turing machine immortality problem. The journal of symbolic logic, vol. 31 (1966), pp. 219-234.

The Turing machine immortality problem can be stated as follows: Give an algorithm which for any Turing machine decides whether or not it has an immortal generalized instantaneous description.

A generalized instantaneous description of a Turing machine is a triple consisting of an infinitely inscribed tape, a scanned square, and the state of the Turing machine. Hooper's reference to generalized instantaneous descriptions simply as "instantaneous descriptions " is somewhat confusing, since usually in the literature an instantaneous description has only finitely many non-empty squares on the tape. The same problem with (finite) instantaneous descriptions is sometimes referred to as the uniform halting problem, and its undecidability also follows from Hooper's paper. (The undecidability of the uniform halting problem can, how- ever, be proved in ways essentially simpler than Hooper's.) A generalized instantaneous description is said to be immortal if and only if starting with it the Turing machine never halts.

The major result of the paper is that the Turing machine immortality problem is recursively undecidable. The proof is complicated and extremely ingenious, and the author does quite an admirable job in explaining it. This reviewer feels that a simpler proof of the same theorem may well exist. The essence of Hooper's proof is the following.

There is an algorithm which associates with every 2-symbol Turing machine T operating on a semi-infinite tape a Turing machine T such that T has an immortal generalized instantaneous description if and only if the instantaneous description of T, consisting of the blank tape with the first square scanned in the (specified) initial state of T, is immortal. Since there can be no algorithm which decides for an arbitrary 2-symbol Turing machine whether or not the instan- taneous description <blank tape, first square scanned, initial state> is immortal, the unde- cidability of the immortality problem follows.

There is one misleading point in the proof. In the third paragraph of Part IV, the right search is done over x + 4 squares not including the square (which contains a 0) in which the search starts. Thus at the end of an unsuccessful search, we have on the tape O0[Oo01 where the first 0 is the point at which the machine got into the right (bounded) search routine 3x.

A number of supplementary results are also discussed. GABOR T. HERMAN

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