(1.2) Cylindrification: If F:Nr N8 Is in (R, Then So Is N X F N' + N1 + 8 Defined by F(Y,X) = (Y',X') Iff Fx = X' and Y = Y' for X E NT, X' E N8, Y,Y' E N

ITERATION AND RECURSION BY SAMUEL EILENBERG AND CALVIN C. ELGOT

COLUMBIA UNIVERSITY, NEW YORK, NEW YORK, AND IBM WATSON RESEARCH CENTER, YORKTOWN HEIGHTS, NEW YORK Communicated August 1, 1968 Introduction.-The theory of recursive functions and the accompanying theory of recursively enumerable sets were developed parallel to the theory of Turing (and related) machines. While in the theory of machines the iterative character of the procedures is fairly clear, the same cannot be said for the theory of recursive functions. It is noted here that when proper attention is paid to the phenomenon of iteration, a simple and highly algebraicized description of the class of recursive partial functions and recursively enumerable sets may be obtained. 1. Recursive Partial Functions.-Let N be the monoid of nonnegative in- tegers. We shall denote by N' the r-fold product N X ... X N, with NO inter- preted as the monoid consisting of 0 alone. Let aR be a class of partial functions f:N`-* N' r > O. s > 0 given for various indices r, s. Such a class will be called distinguished if it is closed with respect to the operations (1.1)-(1.4) listed below and if it contains the partial functions listed in (1.5)-(1.9). (1.1) Composition: If f:NT-.. N8 and g:N8 - Nt are in iR, then sowis gf: N r Nt. (1.2) Cylindrification: If f:Nr N8 is in (R, then so is N X f N' + N1 + 8 defined by f(y,x) = (y',x') iff fx = x' and y = y' for x e NT, x' e N8, y,y' e N. (1.3) Union: If fg NT -- N8 are in 61 and have disjoint domains of definition, then the partial function {fx, if fx defined f UfUg g:Nr -*. NsN, (f UUg~x) =gx, ifgxdefined is in 61. (1.4) Iteration: If fg:N NT are in 61 and have disjoint domains, then the partial function gf* = g U gf U gfl U ... is in 61. (1.5) Induced functions: For any function f:{ 1,...,s} -- {1,... ,r}, the induced function f#: Nr -- N8 given by

A#xi, . O) = (Xfl, * Xf-') is in 61. (1.6) Unit: The function U:NO -° N defined by UO = 0 is in 61. (1.7) Successor: The function S:N -- N given by Sx = x + 1 is in 61. 378 Downloaded by guest on September 29, 2021 VOL. 61, 1968 MA THEMA TICS: EILENBERG AND ELGOT 379

(1.8) Antiunit: The partial function U-1: N -- NO given by U-10 = 0, U-1x undefined otherwise, is in (R. (1.9) Predecessor: The partial function S-1:N -) N given by S-x = x - 1 if x > 0 and S-10 undefined is in 61. THEOREM 1. The smallest distinguished class (Ro coincides with the class of recursive partial functions. 2. Recursively Enumerable Sets.-We next consider a sequence 8 = {Sr} r = 0,1,... where each Sr is a class of subsets of NT. Such a sequence is called distinguished if the following conditions hold. (2.1) If X,Y e SrT thenX n Y, X U Y E Sr. (2.2) If X 1e S, then N X X 6 Sr+l. (2.3) If X e S2r is the graph of a relation R: Nr-- NT, then the set X* which is the graph of the relation

R* = I U R U R2 U ... R' U ... is in S2r- (2.4) IfX ES,. andf:{ 1,.. .,s} - {1. . ..,r} isafunction, thenf#X eS8. (2.5) {0} e S,. (2.6) {(n,n + 1), neN} ES2. THEOREM 2. The class of recursively enumerable sets is the smallest distinguished sequence S. We recall that a set X C NT is called recursively enumerable if it is the image (or equivalently, the domain of definition) of a recursive partial function. (3) Generalizations: Let B be a set of (total) functions N -0 N, let 61(B) be the least distinguished class of partial functions containing B, and let S(B) be the least distinguished sequence of sets containing the graphs of the functions in B. THEOREM 3. The sets in S(B) are the images of partial functions in (R(B). They are also the domains of definition of the partial functions in 61(B). THEOREM 4. A partial function f: N' -- N8 is in (R(B) if it is of the form f = hg where g:N -- NT, h:N -- N8 are (total) functions in 61(B). The monoid N is the free monoid with 1 as base. One could replace it by the free monoid W on any finite set of letters 2 = { a }. Besides the shift to multi- plicative notation entailing the replacement of 0 by 1, conditions (1.7) and (1.9) have to be modified as follows: (1.7') Successors: For each a e 2, the left successor function L: W W defined by Lw = aw is in 61. (1.9') Predecessors: For each a e 2, the partial function L-': WW defined by L,(acw) = w, L4 undefined otherwise, is in (R. Axiom (2.6) also has to be modified accordingly. With these changes the results stated above hold. Downloaded by guest on September 29, 2021