Zeilschr. f. math. Lugik und Gl’undlagen d . Math. Bd. 31, S. 495 - 501 (1985)

LIKEAR ORDER TYPES O F NONRECURSIVE PRESENTABILITY

by DEV KUMAR ROY in Miami, Florida (U.S.A.)

0. Recursively enumerable (r.e.) presented linear orders were first looked a t in the tradition of effective studies of algebraic objects initiated by METAKIDES and NERODE in [4] and followed up by many other recursion theorists. A partial bibliography ap- pears in [7]. An r.e. presented linear order arises when the rationals are condensed modulo a Z1 congruence relation. From quite another direction, ROSENSTEIN in his comprehensive book [6] has a chapter on recursive linear orders in which he develops many interesting results on recursiveness in the setting of linear orders. With regard to order types, ROSENSTEIN has the folloning hierarchy (based on work of MANASTER and ROSEXSTEIN, and FEIXER [l]) :

IT0-types c I7,-types c . . . c I7,,-types c D,,+,-types c . . .; note that t is a II,, order type iff t is a the order types realized by r.c. presented linear orders live in this hierarchy.

What makes the question more interesting is that r.e. presented linear orders are qimply one instance of looking a t a universal structure modulo a Z1 congruence rela- tion. R.e. presented vector spaces, for instance. mod out V , by a L’, congruence. What happens when the congruence is of higher complexity? It is easy to see [7] that there are presentations of arbitrary r.e. degree within the simplest order types. In this paper the congruence is allowed to be L‘,, , I7,, etc. and it is seen that the base level inclusions generalize in an obvious way, locating the general order types in the hierarchy men- tioned above.

The technique of condensing the rationals by relations in the arithmetical hierarchy, introduced in this paper, when combined with the location of the corresponding types, turns out to be a useful tool in the construction of specific order types. This provides a nice blend of the spirit of collapsing intervals in Q with the notion of choosing isomorphic sets of rationals, where the isomorphism is not even required to bc recursive.

Though basic definitions are repeated here, the motivation behind r.e. presented linear orders appears in [7]. Recursion theoretic notation and terminology arc. as in [SJ. The representation techniques used here were pioneered by ROSENSTETN . chapter 16 of [6] is highly recommended. Many interesting results on recursive and not-so-recursive linear orders appear there and in LERMAN’S and ROSENSTErN’s paper [2], [31.

I . The follon ing is afi effectively equivalent verhion of the definition of r.e. presented linear order. Let N - [ O . I , 2. . , .). Q the rationals, q the order typp of t h o rationals and 0~ the ordei’ type of N Let X = { {a , /J) E Q x Q ’ IL 5 0) Lclt f lw a fixed pt fwt ivp

1 - 1 correspondence from N onto X . W , is the ith I’ e. subset of N , and f ( W , ) is de- noted by X , . X , indicates which rationals are to be identified to sl point; this is for-

order type. A natural question is where

496 DET KPMAR R O Y

mally achieved by constructing the transitive closure R, of the set

[ (a , b ) E X: c 5 a 5 b 5 d for some (c, ( 1 ) E X,} u ( ( a . 4): q E Q ) .

Defin i t ion 1.1. Given R,, let [ q ] = (.r E Q : ( q . x) 01' ( x . q ) E R,]. Let Q/R, =

= { [ q ] : q E Q}. Define < on Q/R, by

Lal] < Lq2] iff (Vz E [al]) ( V y E [q2 I) (.r < y as rationals).

Then (QjR, ~ <) is an r.e. presented litiear order.

Note that the ordering < on Q/R, is the one inherited fiom Q: also that I?, is re-

Def in i t ion 1.2. A linear order type z is recursive if there is a recursive subsrt of Q of type z. z iz; Zn (IT,) if there is a Zn (ITn) subset of Q of type z. z is an r.e. prrsentPd type if it is realized 1)y an r.e. presented linear order.

cursively mumerable.

Propohitions 1.3 through 1.5 are proved in [6].

P ropos i t i on 1.3. z is a recursivr type iff there i v m i r . p . subset of Q of type z.

Propoai t ion 1.4. Iti general, every &+,-type i s also a IIn- typr .

Propos i t i on 1.5. For every n 2 0% the class of nn-typPcT is properly iiiclzidrd in the

Defin i t ion 1.6. The linear order (Q/R, , <> ih recur5ively pwsented if X - R, i h r.e. An order type z is a recursively preseiited type if it i.: realized by a recursively pre- sented linear order.

class of IIn+,-types.

Them are examples of r.e. presented linear orders, in [ T I . which are creative and therefort. very far indeed from being recursively presented. This of course does not mean that their types are not recursive types. Theorem 1.9 below will prove, hou ever. that there are indeed r.e. presented types ~ h i c l i are not recursivc.

Theorem 1.7. A type z is recursive iff it i s recursively presented.

Proof. Suppose z is recursively presented, and realized by (Q/S. <) (with S re- cursive). Let q o , q l , . . . be an effective enumeration of Q. Let Eo = { q o ) . Given En = I l Q l o 1 41, > ' ' . > !I,,). let

En if there is a E 5 m with ( q l k . qnt l ) or (qnt l , q , , ) E S: En u {qn+ l } otherwise.

En+1 =

Then E = U { E n : n E N} is an r.e. subset of Q isomorphic to (Q/S, <); therefore by 1.3, T is recursive.

Now suppose z is a recursive type. Let z be realized by the r.e. set of rationals g(N), where g is recursive and 1 - 1. A map h : g(N) --t Q and a subset R of X will be con- structed so that (Q/R. <) is r.e. presented and q t+ [h(q)] establishes an isomorphism.

S tage 0 : Find g(0). Set h(g(0)) = g(0 ) . Put ( g ( 0 ) - 1, g(0) + 1) into R. Begin an enumeration to put the closure of this interval into R.

Stage n + 1 : Find g(n + 1). One of the following is true:

Case 1 : g(n + 1) < g(k) for all k 5 n.

Case 2 : g(n + 1) > g ( E ) for all k 5 n.

LINEAR ORDEK TYPES OF NONRECURSIW PRESENTABILITY 49 i

C'ase 3 : g ( i ) < g(n + 1) < g ( i ) , and there are no g(k ) , k 5 n, between g ( i ) and g ( r i + 1 ) or between g(v + 1) and g ( j ) .

Let [ y ] ; be the least rational in [qI by stage 1 2 , [y]> the largest. If superscript 71 does not appear. then [qJn l . [qlM are least and greatest at' the time that they are referred to.

In case 1 . define h(g(u + 1)) = [Min(hg(k): k s In case 2. drfine h(y(n + 1)) = [Max{hg(k): k 5 n)]> + 1,

In case 3. define h(g(n + 1)) = 1/2{[hg(i)l:, + [hg(j)1t7}

- 1,

( 'outinue the construction of R as fullows:

Put (N - 1. x) into K . where A = [Min(hg(i): i 5 n + 1 ) I,,, Plit (6. j3 + 1) into R. where j3 = [Max(hg(i): i 5 TL + l}J,

For all g ( i ) such that there is a g(j) greater than it ( i . j 5 n + 1), Ict g ( i ' ) be the least such g(j).

Put (hg(i) . 1/2([hg(i)], + [hg( i ' )J , , J ) into K .

C'ontinue enumerations to achieve t,he closure of the subset of R enumerated thus far. I t is clear that there is always room for new h(g(i)) , because the rationals are dense. Also at the end of time, (QIR. <) is an r.e. presented linear order isomorphic to g(N). Finally, the presentation is recursive because event'ually every rational niust be in some [hg(i)l: and q , E [hg( i , )J is R-equivalent to q2 E [hg(i2)] iff i , = i,.

Thus the flo-types are all recursively presented. Furthermore, there are r.e. presented types that are not recursively presented, i.e., the I7,-types are properly contained in the r.e. presented types. This fact appears later, as Theorem 1.9. First, it will be estab- lished t.hat the r.e. presented types are all IFl.

Theorem 1.8. Every r.e. presented order type is U , . Proof . Let the type z be realized by (QIR, <). The follouing procedure chooses

a n, set of representatives. The construction is similar to that in Theorem 1.7, except that now an oracle for R is used. Similar to the earlier construction,

E" if there is a k 5 n with qk - qfli l (mod R) ( En u {q,,+l} otherwise. Efl+1 =

Then E = U(E": n E N ) has type z. E is a 17, subset of Q 1)cwuse given arbitrary q E Q, n such that q = qn can be found effectively. and q E E iff n = 0 or (Vk - I n - 1 ) (qk -c q,, mod R); and the last condition is 17,.

Thus the r.e. presented types split into two classes. Those among thtm that are reciirsively presented are no. and the rest' are I7, - fl,. And both classes are non empty.

Theorem 1.9. There is an r.e. presented type z which is not f f c , .

Proof . Let B be a Z3 - 17, subset. of N. Let z be the type

7 + 3 + a, + 3 + n, + 3 + N~ f . . . , where

7 if n # B 7 + 2 + 7 if n E B . .x, =

498 D E V K U M A R R O Y

It will be shown that z is r.e. presented. Also that if .the order t8ype z is no then S = {n: L Y ~ = q + 2 + q> must be IT,. Since S = B, this will prove by contradic- tion that z cannot, be Do.

The construction of r.e. presented (QIR, <) realizing z proceeds as folloiJx : Pa,rti- tion Q into

( - 00, 0) w [0,4) u [4, 8 ) w [8, 12) u u [4k , 4 ( k + 1)) u . . . Each [ 4 k , 4 ( k + 1)) is further part,itioned as follows:

( 4 k ) U (4k , 4k + 1 ) W [4k + 1, 4 k + 21 \J (4k + 2.4k + 3) U [4k + 3, 4k + 1).

The bars above indicate that the intervals ( 4 k , 4k + I ) , [4k + 1,4k + 21 are identified to points via R. [4k + 3 , 4 k + 4) will be used to produce R ~ . It is clear that the other identifications produce the 3-point markers between successive 'Y, . Finally. each inter- val [4k + 3, 5 k ) is again partit>ioiicd int,o co many intervals as follo\vs:

[ 4 ~ + 3 , 4 k + 3 + ip) U . . . w [4k + 3 + (1 - L I P ) , ~ r l : + 3 + (1 - 1/2'71+1)) W . . .

Son-, k E B * (32) (Qy) (32) P(s , y, 2 , k ) ,

where P is a recursive predicate. Assume also t,hat each k E B has a unique witness r : it is easy to see that the usual definition of a Z, predicate leads to this one, by a change in the recursive predicate at the end. For each k , do the following in the interval 14k + 3, 4k + 4); and dovetail all the procedui*es. Recall that) [4E + 3, 4k + 4) lras been partitioned into o-many intervals; index these in order by x = 0, 1, 2 , . . . Son- for fixed k and r. do thc following within the specified snbint8erval. (Call the s u b interval [ y , d ) . )

1

Set y = 0, z = 0, Increase A = y z by 1

t

Identify (via R) y with m = (A + 6)/2

Set A = m , z = 0 Increase y by 1

If k E B , the unique subinterval correhponding to the witness r will be identified to a point. This, with the left endpoint of the next subinterval. provides the 2 of q + 2 + 71. If k I$ B. every suhinterval will contribute 1 + 77: A& thrn is 17.

LINEAR ORDER TYPES OF NONRECURSIVE PRESENTABILITY 499

R. suitably closed under transitivity etc. is r.e. To see this, note tha t any member- ship question about. R can event.ually be narrowed down to whether or not some ideii- tification is made within a subinterval corresponding to fixed k. x. And t,liis identi- ficat,ion is made iff (Vy' 2 y) (32) P(x , y', z , k ) . This completes the Construction. Lemma 1.10 completes the proof.

L e m m a 1.10. If z = 7 + 3 + :yo + 3 + + . . . is CIo,. where ( V n ) (n,, = 11 or A,, = 11 + 3 + 11). then the set S = ( N : fi,, = 7 + 3 + q } is I7,.

Proof . Let A be a recursive subset of Q realizing z. Intuitively, n E S iff thew is a gap between the ,zth and (?L + 1)th 5-point markers. Identifying natui*al numbers with corresponding rat,ionals by some fixed cffeet,ivct corrcxspondence. and lett.ing %ci

denote the power of the ith prime in the unique factorization of u, and abbreviating .I' E =1 A y E A A .I' < y A i (3;) ( 2 E B A :c < z < y) by .cGy, then

'w E S iff (Vu Vzi Vw) { [ ( V i 5 ti, + 1) ( T L ~ G z ~ ~ G ~ P ~ ) A (VJi 5 n ) ( w i < 7 ( i + l )

A 1 (3R: 37J) (ZGy A TJ < Tho)

A (v i 5 ) L ) l ( 3 c -+ (3x 3y) (wn < L A y < ' i ln+ l A d y ) ) .

3 2 ) (.uG'yGz A wi < x A z < z r i + , ) . l

G is TI, , and a Tarski-ICurato\vski reduction shows that S is IT,.

definition. 2. Looking ahcad to presentations of higlicr complexity, one makes the following

D e f i n i t i o n 2.1. Let R S be a En (U,,) set which is transit'ive. contains the diag- onal of Q2. and is coiivcx, i.e.: ( a , b ) E K implies ( c , d ) E R whenever a =< c 2 d 5 6 ; let LyjR = :.c: (.c. q ) or (y. z) E K ) : tJhen ( [ a ] : y E Q} with the inherited ordering is a S,, (17,) pre.se,ited linear order, denoted by (QIR, <). z is a Z,, (L?,,) pre.sented type if it is realized by a L',, (n,) presented linear order.

Theorem 1.8 generalizrs at once to show t,hat evciy Z;, (n,) presentcd t.ype is 17, (Z,,). On the other hand, starting with an oraclc for a I7,, set of rationals S. and an effective enumeration of Q, the construct,ion for the latt,er half of Theorem 1.7 generalizes to give a congruence R such that (QjR, <) realizes t'he typc of S. R is recursive in S. and therefore the, type of ( Q / H . <) is :I,,+, presented. Using tlw fol- lowing a I)usive 1)ii t o 11 vion s not a t,ion :

Z,, for [z: t is a S,,-typt>} ~ QjZ,, for. I T : z is a Zll presentcd type: etc..

the following ensiics:

Q ' f f n + i s xnti = nn. In fact. QjU,,, I = n,,. because

Q/ I,,+l Q/17,,+l (because c v r y ,1,,+, set is n,,+,) s n,, s Q/&, .

Furthermow. Thcorcm 1.9 also generalizes: Start with a su1)st.t S of N which is L',,,, hilt, not Uning an cxactly siniilai. construction, oht~ain t>lie t>ype z = 7 + 3 + + lo + 3 + i1 + . . . \\.itti

if i $ S

' '1 + 2 -+ 1 1 . if i E S : I t i , \ . =

3.p

500 DEV KUlMAR ROT

z is then in Q/Zn+,. Again it follows t'hat z is not in fin. Putting all this together, one obtains :

. . . Q / Z n E Q / n n + i = nn = Q / , J n + l c Q/Zn+l s Q/nn+2 = [ I n + , . . , This places the general presented types among si:l)orders of Q of general complexity.

3. Remarks and examples:

The inclusion ITn c Q/,Yn+l s nn+, leaves open the question: arc there Un+, typw which are not Zn+, presented'? If this is tmc, there would be a class of order types of strictly intermediate complexity bet'ween fin and Un+, , which \voulcl 1)e tr~iljr interest- ing. Whet,her or not this is true, tlic easy inclusions Z7,, s Q/,Y,i+l nn+, . and the const8ruction of Theoreni 1.9 provitle an easier \Yay of establishing that I7, c TIf,+ than thc earlier proofs.

(londcnsat,ion by congruencr's in the arit'lmiet ical hierai.cIiy, thc fact' that evcq- En+, t)ype is lIn. and tlie fact tmhat every n,+, prcsented type is Uf, , are useful tools. In constructing an order type of lowest possible complexit'y, one can try a 1it)tle less hard by seeking Z,+, instead of Z7,. or I / , + , prestmt(d instead of IIn. This u.orks in thc other direction as \veil.

As an example. finite condensations of recursive linear orders are considered. ROSEKSTEIK [6] sho~vs that tlie finit'e condensation of a rccursive linear order niust he n2 ~ and constructs a recursive linear order n-hich has nonrecursive finite conden- sat,ion. He asks if the co~idcneat~ion in his proof is Dl or strict,ly U 2 : the latt,er would serve t o show that, U2 was tlie best possible result. However. the mcthods dcvclopccl her(. will be used to show that the type in question is 17, presented, hence ZZ, . This piece of bad news 1eavc.s open the qneshion of whvther finite condensat'ions of recursive linear orders have to he 11,.

Defin i t , ion 3.1. Ginan a linear order ( F , <) with field F and linear ordering <. and L, y E F wit11 .c < y. identify ,r and y iff the int,erval [x, y l is finite. The resulting condensation is t'lie f inite condematioii of ( F , <).

Propos i t ion 3 . 2 . If z i s a rrcztrsine type theti the firiite condensation of a l i w a r order renlizitbg z has Z2 presented typr .

Proof . Gvtting carried away by presentations. the following simple proof is pres- ented: Let t be realized by the recursively present'ed linear order (QIR, <). Let g be an effective correspondence from N ont'o Q. Define S s X by

( a . b ) E S - a 5 b A (312) [ ( V i 5 lh(,u) - 2 ) ( y ( n i ) < g ( . r z i + , ) )

A (g ("0 ) - R a ) /' ( 8 ( ' 2 1 n ( n ) - l ) - R b , A (vp) (a, < 4 < b ---t (3i < lh (n) ) (q N~ g('l?,i)))].

Since x - y iff ( x , y) or (y, a) E R and R is rccursive, S is Z2. Closing S under tran- sitivity, etc. as required by definition 2.1 yields a Z2 presented linear order which is t>lie finitc condensation of ( Q I K , <).

N, there is a, recursive l inear order Theore in 3.3 (ROSEKSTEIN). Given a Z3 set C ujhich has f ini te conde?isatio~ of type z = q + a, + q + a , + q + a 2 + . . ., where

3 if 11 E c a n = { 2 if n fic.

LINEAR ORDER TYPES OF NONRECURSIVE PRESENTABILITY 501

This is ROSEXSTEIK’S example of a nonrecursivc finite condensation, in the case that C is Z; but not 17,. If z were to be 17, but not IF,, the finite condensations of recursive linear orders nould be exactly located. But that is not thc case.

Propos i t ion 3 4. z as in T?iroreni 3.3 is 17, Proof . Partition Q into

- ( - co, 0 ) u [O, 1 ) u [l} u (1, 21 u . . . u (3k + 2 , 3k + 3 ) u

u [3k + 3 . 3 k + 4) u (3k + 4} u (3k + 4 , 3 k + 51 u . . . As before, the bars indicate identification to a point. The working intervals will be [O. 1). [ 3 , 4 ) . [6. 7 ) , etc.: these will correspond to TL = 0, 1. 2 , . . . Let the Z3 set C be defined by

71 E c 0 (32) (Vy) (32) P(d.. y. z . ‘12)

ith P recursive. Within each working interval la, 0) choose a recursive increasing xequence ao, a,, n 2 . . . . converging to @, a i th cr < a0. Identify [axo.@) under the rongruence S being constiucted. iff (Vy 3) Y(.co, y, z , I L ) . On performing suitable closure operations, S is indeed 17,. The details are omitted, but note that nontrivial questions about membership in S can be referred to the existence of some ocx0 with r0 bounded, and therefore to the formula (32, 2) (Vy) (32) P(zo. y, z , n) . Thus (Q/A’. <) is n2 presented. It also realizes z. therefore z is III, .

One last observation: if in fact Z,,,, presented types are strictly intermediate be- tueen III, and ll,,, , -types, then Proposition 3.2 cuts down the complexity of finite condensations of recursive linear orders to something below 17, .

References

[ l ] FEINER, L., Degrees of non-recarsive presentability. Proc. Amer. Math. Sor. 3H (1973), 621 -624. r2] LERMAN, M., On recursive linear orderings. In: Proc. Logic Tear 1979-80: The Univ. of Con-

necticut (31. LERMAN, *J. SCHMERL, 1%. SOARE editors), Springer Lecture Notes in Mathematics

131 LERBIAN, &I.. and J. G . ROSENSTEIN, On recursive linear orderinps 11. In: Proc. Patras Con-

[4] METAKIDES, G . , and 9. NERODE, Recursively enumerable vector spaces. Annals Math. Logic 11

[ 5 ] ROGERS, JR.. H.. Theory of Recursive Functions and Effective (‘omputability. MrGraw Hill

[a] KOSENSTEIS. .J. Q., Linear Orderings. Academic Press, New York - London 1982. [7] ROY. D. K.. H.e. presented linear orders. J. Symbolic Logic 1 A (1983), 369-376.

S59 (1981), pp. 132 - 142.

ference 1980, Sorth-Holland Publ. Comp., Amsterdam -New York 1981, pp. 123- 136.

(1977), 147-171.

Sew York 1Yti7.

Dev Iiumar Roy Florida International University Dept. of Mathematics Tamiami Campus Miami, Florida 33199 U.S.A.

(Eingegangen am 10. Februar 1984)