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A Note on Orders of Enumeration Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1972 A Note on Orders of Enumeration John Helm Albert Meyer Paul Young Report Number: 72-050 Helm, John; Meyer, Albert; and Young, Paul, "A Note on Orders of Enumeration" (1972). Department of Computer Science Technical Reports. Paper 420. https://docs.lib.purdue.edu/cstech/420 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. ON ORDERS OF ENUMERATIONS AND TRANSLATIONS by 1 2 3 John Helm , A lbert Meyer , § Paul Young January 1972 CSD TR 50 PURDUE UNIVERSITY A basic result of intui t ive recursion theory is that a set (of natural numbers) is decidable (recursive) iffit can be effectively enumerated in its natural order (of increasing magni tude). The chief theorems of this paper give si mpl e , but very basic facts relating to enumerations in natural order of magni tude and the extent to which translat ions must preserve the orders of the sets being enumera t ed. Under very general conditions for what constitutes a programm ing system for enumerating the recursively enumerable (r.e.) se ts , we prove in Theorem 1 that not every recursive set is "best" enumerated in its natural order , and we later show that under these same general condi t ions , in every programm ing syst em , every recursive set is enumerable in its natural order. We accomplish the latter result by extending Rogers ' Isomorphism Theorem to a result which asserts that under these same general condi t ions , every programm ing system can be effectively translated into any other in a manner which preserves the order of the sets being enumera t ed, (Theorems 4 and S) . In fac t , we show that for every transl a tor, t , there is an order preserving pre transl a tor, p , such that t is order preserving modulo p; I . e . t o p is an order preserving translator. In addi t ion, the restriction of the original programm ing system to the recursive set-(range p}is a standard programm ing system on which t is order preserving. Along the way we establish the existence of 2 sets best enumerated in their na tura l order and , for every r . e . se t , the existence of bad orders for enumerating the se t . A companion paper , "Notes on difficulties of enumerat ions ," contains some observations relat ing difficul ty of enumera t ions , difficul ty of comput a t ions , difficul ty of decision probl ems , one-one and many-one reduc ibi l i t i es , and classifications of the r . e . se ts . All proofs in this paper aTe fairly straightforward. NOTATION DEFINITIONS We let denote a canonical enumeration of all finite sets (of natural numbers). Given x , we can effectively list D ^ and know when the listing is compl e t ed. Except for the i ndex i ngs^x W ^ of the r . e . sets which we are about to describe and the notat ion w" described l a t er, our not a t ion < general ly follows [7]. x , y> is (an encoding of) the ordered pa ir of inte- gers x and y . Defini t ion, ([8]). An enumerat ion technique is a total recursive function E(x,y) such that for every (r.e.) set W there exists an e such p that W = U D _ , . We call e an index of W and denote U n_ , . by W , the n E(e ,n) n E(e,nj e superscript E- being suppressed whenever there is no danger of ambigui ty. D - U d F ' (e,n) m ^ n E O ,m) ' An enumeration technique is called standard if the sj- theorem is sa t isfi ed, i . e . , if there exists a total recursive function S such that for all e and x , W , . = {y| < y,x > eW }. We assume without further ment ion that Sc (e,xj e all enumeration techniques considered in this paper are standard. The reader should have no troubl e convincing hi mse lf that all of the usual models for enumerating r . e . sets obviously yield standard enumerat ion t echniques . E . g . , we m ight let D g ^ denote the set of integers enumerated ei ther in exactly n steps of Turing machine e or in no more than n steps of Turing machine e . C l early , each enumeration technique E part i a l ly orders each r . e . set / e in a natural way which can be extended to a total ordering; E E that is, we may define x < ^y if x precedes y in the enumeration of W , and f i in the case that x and y appear at the same stage of the enumerat ion we assume that x and y are arbitrarily ordered by some convention uniform in e: 4 Defini t ion . For each enumera t ion t echnique E , x < y is some fixed r . e . e t ernary relat ion of x , e , and y such that for each integer e , E (i) <© total ly orders W 0 , and (ii) fp > o 5 x , . - D _ , , 6 D c i ED„,, - - [x <1 y]. c p E E C l early , if the binary re l a t ions < e and <e o are ident i ca l , e e o o We shal l general ly be working w i th t wo arbi trary , but fixed enumerat ion t echniques E and E . We abbrevi a t e N ^ and to W and W ° respec t ive ly and g E E o si m i l arly < and < ° by < and < . Defini t ion , ft8]). For each enumera t ion t echnique E , we define > 1 A . (n) = (uy> 11 D , - J If A . (n) exi s t s , W ? is the set consist ing j * tF f u ,yj of the first n e l ements of VL in the order < y This is si m i lar to Defini t ion , ([1]). For each st andard indexing of the part i a l recursive func t ions , a Blum measure is a r . e . sequence of part i a l recursive funct ions such that for all i , doma in = doma in 4>. and the t ernary re l a t ion <&j(x) <_ y is dec idabl e (under the convent ion that it is fa lse if ^ (x) is undefined) . I ORDERS OF ENUMERATIONS We now prove tha t any infini t e , coinfini t e , recursive set has a recursive superset whose na tura l order is not the best order for enumera t ing the se t . The original proof , [2], was by priori ty argument , but the proof given here is more in the spiri t of [3], and seems to bear evidence for the c l a i m made in [31 that "the exist ence of al most everywhere compl ex zero-one va lued func t ions is often sufficient to establish the existence of inherently compl ex comput a t ions" Although the proof we are about to give is largely calculat ional , the reader will be well-advised to bear in m ind that the calculations are merely formal validation of the following intuitive argument: Let R be any infinite and coinfinit® recursive se t , and let f be a 0-1 valued function which is much harder to compute than R is to enumerate in increasing order. Let r enumerate TT in increasing order and let S = RU(r(n) | f(n) = 0> . Since R£S , S B can be enumerated almost as easily as R , ([JQ]). " t if S can be enumerated easily in natural order, then S-R can also be enumerated easily in natural order. But if this were possibl e , the following algorithm would give an easy way to compute f(n): enumerate the fiTSt n+1 elements of S-R; if r(n) appears set f(n) = 0 , otherwise f(n) = 1 . We now formalize this in: Theorem 1. Let R be an infinite and coinfinite recursive set . L e t ^ ) be any effective operator carrying total recursive functions to total recur- sive functions. Then theTe exists a recursive set S such that RCS , and some index e for S such that for any j if W^ = S and < . is the increasing order 4 for S , thentfJcA ) (n) <_ A . (n) for all but finitely many n . ® J 1 Proof. Let r enumerate R in 1-1 increasing order. Define r by - 1 - 1 - 1 r ( y ) - max{z |r(z) £ y). (Thus r r ( y ) <_ y and r (y- l ) 1 y O Define a total recursive function l by = R u { ? C n ) ( l = 0 } V i ) i * i ° ' and define a very large upper bound , t , on the difficulty of enumerating these supersets of R by t(z) « M X < g( A ) ( y " V - l ) = z h m )|T 6 Thus for any i , for almost all y , (a . e . , y) ) C y ) (a) ( j K ( i ) - Define = undefined if has fewer than r(y)+l elements if FOr) c otherw ise . Define = C n ) f A T + f I s ( i ) i i ( C n ) D I Y S (i , n , y) . Q .= 0 otherw ise , and S(n,y) = max{S (i ,n,y),S(n,y-l),S(n-l ,y)}.
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