Recursive Arithmetic of Skolem by H. A. PogorzelskiReview by: R. L. GoodsteinThe Journal of Symbolic Logic, Vol. 29, No. 2 (Jun., 1964), pp. 101-102Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2270426 .

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REVIEWS 101

There is given, in this paper, the realization of a two-valued propositional calculus by means of the pneumatic logical units. A realization of two-valued propositional functions such as negation, identity, implication, conjunction, disjunction, equivalence, stroke function, and symmetric difference, is shown.

The pneumatic delay-line is introduced and pneumatic logical circuits containing delay-lines are considered.

Some applications of pneumatic logical circuits are also given. ANDRZEJ ROWICKI

BURTON DREBEN, A. S. KAHR, and HAO WANG. Classification of AEA formulas by letter atoms. Bulletin of the American Mathematical Society, vol. 68 (1962), pp. 528-532.

This paper is concerned with the decision problem with regard to satisfiability of various subclasses of the class G of all closed formulas of the form AxEuyM(x, u, y) whose quantifier-free matrices M(x, u, y) contain only binary functional symbols. Nine pairs of individual variables can occur in the atomic parts of the matrices M(x, u, y). Consider any three of the four pairs xy, yx, uy, yu. By XXVII 225(1), if H is any subclass of G (and, in particular, G itself) containing the subclass of all formulas in whose atomic parts just these three pairs occur, then H is a reduction class and is thus unsolvable. By taking any two of the above four pairs and combining them with the other five pairs, a subclass of G is specified. Thus six subclasses of G are obtained which the authors denote by J = {xy, uy}, J* = {yx, yu}, L = {xy, yx}, L* = {uy, yu}, Q = {xy, yu}, Q* = {yx, uy}.

The main result of the paper states that J (and J*) is solvable. More specifically, a given formula AxEuAyM(x, u, y) of J is satisfiable if and only if the conjunction of numerical instances M(O, 1, 0), ..M(E, E + 1, E) is truth-functionally consistent, where E = 26 and 6 is the number of disjuncts in the full disjunctive normal form of M(x, u, y). This result is of some interest because J is an infinite class of formulas which includes formulas satisfiable only in infinite domains.

In a final paragraph it is remarked that L and L* are solvable and include no for- mulas satisfiable only in infinite domains. The general status of Q and Q* is left open.

F. C. OGLESBY

JOYCE FRIEDMAN. A semi-decision procedure for the functional calculus. Journal of the Association for Computing Machinery, vol. 10 (1963), pp. 1-24.

JOYCE FRIEDMAN. A computer program for a solvable case of the decision problem. Ibid., pp. 348-356.

In these two papers a decision procedure is provided for formulas in Skolem normal form with prefix (3yj) ... (3ym) (z1) ... (Zn) and matrix M such that either: (1) every elementary part contains at least one of the variables zi or contains only one variable or contains both yi and Y2 and no other variables; or else (2) every elementary part contains at least one of the variables zi or contains only one variable or contains all the variables y1 . More generally, a semi-decision procedure is provided for the general Skolem normal form, so that corresponding to a matrix M a method is given for constructing a matrix M* such that M j M* and such that (3y') ... (3ym) (zr) ...

(zn)M is provable if and only if (3y') ... (3Ym) (zi)... (z.)M* is provable. These methods have application to problems of theorem-proving on computers. An error in Rule 7 of the first of these two papers is corrected in the second paper, and the proof of Theorem 4 of the first paper is revised accordingly. FREDERIC B. FITCH

H. A. POGORZELSKI. Recursive arithmetic of Skolem. Mathematica Scan- dinavica, vol. 11 (1962), PP. 33-36.

H. A. POGORZELSKI. Recursive arithmetic of Skolem II. Ibid., pp. 156-160. These two papers present formalisations in recursive arithmetic of theorems on

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102 REVIEWS

unique representation by Mycielski numbers and their generalizations. An Ml(My- cielski)-number is a number which is not a power of a smaller number. The first twelve Ml-numbers are

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17.

Since there is a prime (and therefore an Ml-number) between n and 2n for any n > 2, the nth Mi-number is less than 2n, so that the sequence of Mi-numbers is primitive recursive.

Defining an exponential chain [xm] by the recursion

[Xi] = Xi

[Xn+l] = [Xn+l, [Xn]]

where [x, y] stands for xythe author proves that the representation of a number by an exponential chain of Ml-numbers is unique, i.e., that if Xn and Yn are sequences of Ml-numbers then

[Xm] = [Yn]

if and only if m = n and xi = yj for all i < m. By means of the (doubly recursive) Ackermann functions Sk(X, y), defined by

4i(x, y) = x, k+i(x, 0) = 1, 5k+i(x, y + 1) = Sik(x, Sk+i(x, y)), the author gener- alises the concept of an Ml-number to an Mk-number, where x is an Mk-number if there are no numbers y, z less than x such that x = Sk(Y, z). Thus for instance 9 32

is not an Mi-number, but is an M2-number (whereas 33 is not an M2-number). Defining a k-chain Xkxr by the recursion

it~ ~ ~ ~~~X .X. xr = Xi

Xk+lX = k(X+l , Xkxr)

the author proves that if xr and y, are sequences of Mk-numbers then

Xk X = Xky

if and only if m = n and xr = yr for all r ? m. R. L. GOODSTEIN

J. P. CLEAVE. Creative functions. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, vol. 7 (1961), pp. 205-212.

The notions of a creative set and of one-one reducibility were introduced by E. L. Post (X 18); he proved the existence of a creative set to which every recursively enumerable (r.e.) set is one-one reducible. Implicit in Post's theory is the concept of (recursive) isomorphism; the sets a and fi are isomorphic, if there exists a recursive permutation which maps a onto fl. Consider the following theorems: (A) there exists a creative set, (B) every r.e. set is one-one reducible to every creative set, (C) any two creative sets are isomorphic. Theorems (B) and (C) are due to J. Myhill (XXII 73). R. M. Smullyan (Theory of formal systems, Princeton 1961) generalized the notions of creativity, one-one reducibility, and isomorphism to ordered pairs of disjoint r.e. sets and proved the analogues of (A), (B), and (C) for such ordered pairs. This process of generalization has reached a natural end in the present paper by J. P. Cleave. Here these three notions are introduced for (finite and infinite) sequences of mutually disjoint r.e. sets. The author then proves the analogues of (A), (B), and (C) for such sequences. A more explicit summary follows.

Positive integers (numbers) are denoted by lower case Latin letters and collections of numbers (sets) by lower case Greek letters; e stands for the set of all numbers. The sequence xi, . . ., Xk or x(1), . .. , x(k) of variables is abbreviated by Xk . For any set a we write a for the complement of a with respect to e. Let the domain of a partial

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