A Theory Without Recursive Models by A. GrzegorczykReview by: Thomas FrayneThe Journal of Symbolic Logic, Vol. 28, No. 1 (Mar., 1963), pp. 102-103Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271343 .

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102 REVIEWS

two questions, whether addition is definable in SC, and whether SC is decidable. The author does not discuss this matter in detail, but from his results (including a detailed description of the decision method for SC) and from what is said in the Robinson paper, one can infer that the answers to the questions are, respectively, "no" and "yes'.

Further results are: (1) that the restriction of predicate variables to ultimately periodic predicates does not alter the truth-value of a formula without free variables; and (2) that the first-order theory is decidable whose domain of discourse is the set of non-negative real (or, alternatively, rational) numbers with addition and the two constant predicates, "is an integral power of 2" and "is an integer."

The second paper closes with three unsolved problems: First, if Ax(2x + 1) and Ax(2x + 2) are used instead of successor, is SC still decidable? Second, if the domain of individuals of SC is enlarged to the ordinals less than wt2 and the ordering on these ordinals is added as a primitive, is the system still decidable? And third, is there an effective manner of determining whether or not there exists a finite automaton satisfying a given condition expressed in SC?

Exposition and technical matters: The author's style in both papers, together with an inherent subtlety in the subject matter, makes for difficult reading. Too often is part of a proof omitted which the author expects the reader to fill in for himself. The rule should be that such an omission must be something that the reader can fill in by a mere computation; this rule is violated frequently in both papers, considerable ingenuity being required on the part of the reader. Another fault is the tendency to disguise as a remark or corollary an important theorem whose difficult proof is omitted.

There is a minor technical defect in the proof of theorem 2 of the first paper (p. 76). The clause U[r(t)] in the line marked (1) does not work for special predicates r. U is a special output and will therefore be stable under the null input transition; but there is no guarantee that the r-states are null after time x, and so special r-predicates can be guaranteed to represent the states only for a finite amount of time. The reviewer suggests replacing this clause by U[r(x)]. Another more complicated change would then have to be made in the line marked (2).

In the second paper, the last paragraph of the proof of the crucial and difficult lemma 9 is obscure to the reviewer, who had to finish the proof to convince himself of the validity of the lemma. The reviewer also finds the connection between the proof and the statement of lemma 4 to be obscure; it would appear that the second and third occurrences of R(i, x, y) in the statement should be changed to R(i, z, y), with a corresponding change in the second line of the proof. The proof of lemma 8 has an error in that s as recursively defined does not always satisfy the condition stated. (When pi(O) -T, P2(0) F and, for all t, pl(t') p2(t') F, this condition, which is a biconditional, has its right side true and left side false.) However, if si is defined by s1(t) = s(t) p2(t) then the condition is satisfied with s1 in place of s. (The reviewer acknowledges the aid of his student, Miss Amy Eliasoff, in this matter.)

Errata, first paper: p. 72, line 15, 'in' for 'im'; p. 73, line 11, insert 'A W(Y)' im- mediately to the left of '- '; p. 74, line 5, 'E' for 'G'; p. 74, line 9, 'lemma 5' for 'lemma 4'; p. 75, last line, 'Ul' for 'U'; p. 76, line 7, insert 'w' after '-'; p. 76, line 15, '15' for 'U5'; same correction on line 16; p. 90, line 8, 'u' for 'y'; p. 91, something has been dropped from the sentence ending on line 16.

Second paper: p. 6, line 11, all occurrences of 'i' should be bold-face; p. 4, lines 22 through 28, all occurrences of 'r' should be bold-face.

ROBERT McNAUGHTON

A. GRZEGORCZYK. A theory without recursive models. Bulletin de l'Academie Polonaise des Sciences, 'S6rie des sciences math6matiques, astronomiques et physi- ques, vol. 10 (1962), pp. 63-69.

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REVIEWS 103

Four finitely axiomatizable first order theories F C F1 C F2 C F3, in a language with two individual constants 0 and S, and one binary operation symbol 1, are presented. The theory F is essentially undecidable but has a primative recursive model; F1 has no primitive recursive models but has a general recursive model; F2 has no general recursive model but has an arithmetical w-standard model; F3 has no arith- metical w-standard model.

The theory F has as axioms: 0 - Six, Six = Sly x = y, x = v Vy x _ Sly, V/{tI0 = x vAu[l(Slu) = g (flu)]}. F1 has in addition: AxyVzAu zlu = xi(ylu), an axiom calling for the existence of q such that qlw = 0 or 1, and qlw = I if and only if w is a square, and an axiom calling for the existence of g so that for every natural number x, glx = fix - hix. THOMAS FRAYNE

S. JASKOWSKI. Example of a class of systems of ordinary differential equations having no decision method for existence problems. Bulletin de l'Acaddmie Polo- naise des Sciences, classe troisibme, vol. 2 (1954), pp. 155-157.

S. JASKOWSKI. Primer kiassa sistim obyknovlnnyh diffire'ncial'nyh uravnlnij, nd imJ4t6go algorifma razrdgimosti dld problem o sutstvovanii. Russian version of the preceding. B6lltdn' Pol'skoj Akadgmii Nauk, Otd. 3, vol. 2 (1954), pp. 153-155.

The author formulates the result of J. Pepis (III 160) in a sentence stating that for a special relation R the problem, whether or not (P) for any A e 1 there exists k e 9 such that R(A, A), is undecidable because equivalent to the decision problem of the functional calculus. Then the author gives a system of ten ordinary differential equations with a certain initial condition (1 1), and reduces the following problem (D) to the problem (P). Hence he concludes that the problem, whether or not (D) for any 0 of the interval (-1, 1), the system of equations (1) -(10) possesses a real solution over the interval (-1, 1), satisfying the initial condition (11), is undecidable.

A. GRZEGORCZYK

W. W. TAIT. Nested recursion. Mathematische Annalen, Bd. 143 (1961), S. 236-250.

Nach Ergebnissen des Referenten fuhrt bei der Anordnung "<" der naturlichen Zahlen weder die Wertverlaufsrekursion - wobei zur Definition eines Funktions- wertes qp(xi, ..., xm, y + 1) nicht gerade der auf der unmittelbar vorangehenden Stelle angenommene Wert qo(xl, ... , xm, y), sondern an beliebigen vorherigen Stellen angenommenen Werte von p benutzt werden -, noch die eingeschachtelte Rekursion - wobei, in der Definition von (xl, ..., xmI y + 1), aus gegebenen Funktionen und aus in Bezug auf die Rekursionsvariable y friuheren q-Werten durch Substitutionen aufgebauten Terme fur die Parameter xj., . .. , xz gesetzt werden - von der KMasse der primitiv-rekursiven Funktionen heraus. (Fiir die letztere Tatsache ergibt sich in vorliegender Arbeit auch ein neuer Beweis.) Es wurde nichts neues bedeuten, wenn bei der eingeschachtelten Rekursion die genannte Ersetzung der Variablen durch Terme auch fur die Rekursionsvariable y erfolgen wUrde, denn dann hatte man wegen der Forderung, dass dabei nur in Bezug auf y friihere P-Werte auftreten duirfen, mit einem eingeschachtelten Wertverlaufsrekursion zu tun, die auch dann mit Hilfe der Wertverlaufsfunktion

v

P(Xl, *. Am' Y) = I pT(xIP...xAi) i-O

aufgel6st werden k6nnte, welche samtliche Vorganger von q(xi, . M ) Xm V + 1) zusammenfasst.

Hier bedeutet eine solche Zusammenfassung keine Schwierigkeit; aber bei anderen Wohlordnungen < der natiurlichen Zahlen, auf welche vorliegende Arbeit den Begriff der eingeschachtelten Rekursion verallgemeinert, wobei eine Zahl unendlich viele

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