Non-obtainable Continuous Functionals by Dag Normann, Oslo - 79

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1 Non-obtinble Continuous Functionls by Dg Normnn, Oslo - 79 Abstrct For ech k > 3 we construct continuous functionl 6 of type k+l with recursive ssocite such tht 6 is not Kleene-computble in ny continuous functionl of type < k. Introduction. The countble or continuous functionls were first defined independently by Kleene 05] nd Kreisel [6] Kleene's countble functionls is sub-clss of the totl functionls while Kreisel's continuous functionls re equivlence-clsses of functions f : IN ~ jn. In this pper we will regrd the countble functionls s type-structure < Ct (k) >k {,) where ech ~ Ct(k+l) is totl mp ~ : Ct(k)~ w. This is equivlent to Kreisel s definition nd it ws lso used in e.g. Bergstr [] nd Gndy - Hylnd [3]. We will work with fixed k > 3. We let n, m, k, i, j etc. denote nturl numbers, f, g, h,, B, y will denote elements of Ct(l), F will denote n element of Ct(k-),, ~ will denote elements of Ct(k) nd ~ will denote n element of Ct(k+l). We let, T, TI, o denote finite sequences which we without mentioning will identify with their sequence-numbers. (n-) will denote the n'th coordinte of when 0 < n ; lh(). We use the stndrd nottion f(n) = <f(o),.,f(n-)> nd cr(n) = <(O),...,(n-)> whenever n < lh(). Kleene [5] showed tht the clss of countble functionls

2 - 2- is closed under Sl- S9 (Kleene [4]), nd he showed tht ll computble functionls re recursive, i.e. hve recursive ssocites. Lter Tit showed tht the converse is not true. The fnfunctionl ~ is recursive but not computble in ny f. ~ is functionl working on two rguments G E Ct(2) nd f. If we let Cf = {g ; Vn g(n) : f(n)} = g (n) ~ G(g ) = G(g )) 2 2 Tit never published his result, but sufficient rguments re given in e.g. Gndy- Hylnd [3], Fenstd [2] nd Normnn [8] follows Lter Gndy defined new functionl r in Ct(3) s where Gndy showed tht r (* denotes conctention ) is recursive nd Hylnd showed tht r is not computble in ~ nd ny f. The proof is bsed on some mteril in Bergstr (] nd cn be found in Gndy- Hylnd [3] nd Normnn [8]. The following problem still remin@ open: "Are ll continuous functionls computble in n element of Ct(3)?" we solve this problem by constructing recursive ~ In this pper E Ct(k+l) for ll k ~ 3 such tht A is not computble in ny ~ E Ct(k). 2. Conventions nd preliminries. conventions: From now on we will use the following nottion nd

3 - 3 - Let Bn denote the set of functionls in Ct(n) with n ssocite extending. We will then hve B = {f. f(lh(o)) = o} 0 When we use the letters nd T we will lwys ssume tht Bk- 'I 0 ' k- ' B 'I 0 0 Lenun Bk- k- s+l.,. k-2 k-2 c B ~ vo,s(-r(o) = 3'T B 0 ~ B 'IT " o('it) = s+l - T b If Bk- Bk- k- then. k- k- c u 0 u B 3<n B 0 c B - T Tn - T c If Bk- :. Bk- u u Bk- then there is n extension - T Tn of such tht Bk- n (Bk- u u Bk-) = 0. T Tn ) Both this nd the next lemm re elementry nd we will not prove them here. Lemm 2 k- _j_ Let I = {; B 0 r 0}. There is primitive recursive fmily {F 0 } 0 EI in Ct(k-) such tht i F 0 E Bk~l b ii F = F ~ T Bk-l (J -r"(j r... contins just F which is constnt. iii If o<t nd Bk- rt. k- then k- B FT f. T - B. There is primitive recursive dense fmily { Ct(k-2) such tht the reltion ~ E recursive. k-2 B o ~i}i~!n in is primitive For ech F we let F(~.) The following result

4 - 4 - ws essentilly first proved in Normnn [7]. Lter S. Dvornickov simplified the proof. His proof is given in Normnn [9]. Lemm 3 Let H = {hp; FE Ct(k-)}. Then b If A is then there is primitive recursive R such tht E A~ Vh E H 3n R(~(n),fi(n),n) Definition Let G E Ct(n), n>2. We cll semi-ssocite for G if vm G E Bn ~(m) In proving the properties of ~ nd r mentioned bove we mke use of the following observtion: If G E Ct(2) then computtion {e}(g) depends only on G restricted to countble set, nmely -sc(g) = {f f is computble in G}.; So if is semi-ssocite for G securing ll f E -sc(g) then there is n n such tht {e}(g) is uniquely determined by This ws proved in [ 3] Our next lemm gives higher type version of this observtion. Lemm 4 Let E Ct(k), {e}( ) ~ s by Sl - S9. Then there is fk_ 2 -set A c H such tht if (F) is used in subcomputtion of {e}( ) then hf EA.

5 - 5 - Proof c will be Let be n ssocite for >. Then the following ~et ~k-zc.) C = { <d,,,,g,t> ech f.' g. l. J re ssocites for functionls G., T. l. J of type < k-2 nd {d}(~, >,)~t is subcomputtion of {e}( >)} From C it is esy to construct A s we wnt. Lemm 5 Proof Let {e}( >) ~ s. Let. be semi-ssocite for > such tht whenever >(F) is used in subcomputtion of {e}(~) then. secures ll ssocites for F. Then there is n n such tht vw E Bk c {e}(w)+ {e}(w) ~ s ) (n) The stndrd proof used when. is n ssocite will work in this cse too. Remrk Lemms 4 nd 5 my esily be proved for list of rguments insted of just for >. 3. The construction The strtegy now is s follows. We construct recursively compct set K such tht i All S E K re semi-ssocites for ko.

6 - 6 - ii No B E K is n ssocite. iii If A c H is Il then there is B E K such tht -,..,k-2 if hf E A then B secures ll ssocites for F. 2. For ech we construct sequence 0 m uniformly primitive recursive in such tht lim will be the principl m-.oo m ssocite for. 3. We show tht if 6K() = ~n vm~n VSEK ( S(m) +! ) then 6K hs recursive ssocite. 4. If V ( 6K() = {e}(cp,~) ) then by lemm 4 nd lemm 5 there Remrk will be B ~ K such tht 6K(k0) is determined by finite prt S(n) s it seems. of B We will show tht this is s bsurd - 4 give the min ide behind the construction. In order to crry through the technicl rguments we must choose both K nd o! with some cre nd define 6K in slightly different wy. From now on let I(,h) be the following reltion I &, h) ~' 3 B E I k _ z () ( B C::: H A h E B ) Then I is nd by lemm 3.b there is primitive recursive reltion R such tht For ech cr let

7 - 7 - ={ o ( (cs) -) i + if cr(o) = o if cr(cs) >- 0 i E {"', 2} where ( ) nd ( ) 2 re the two projection mps of the stndrd piring opertor <, > For ech cr we let hcr be the lrgest sequence such tht hcr(i) = s if 3o(cr(o) = s+l A ~i E B~- 2 ) If B~-l contins more thn one element then h C is finite sequence uniformly recursive in Define cr. if 3n R((n),ficr (n),fi 0 otherwise contins just one element or if 2 (n),n) P is uniformly recursive in nd P is semi-ssocite for ko. Lemm 6 If A c H then there is n E {0,} ln such tht if hf E A then secures ll ssocites for F. b P is not n ssocite. c nd then Proof Let E { 0, } :W be such tht A is rk_2 CCJ) Let B = {h : h f A} where h (n) = (h(n){ Then B c H is rk_ 2 () so r (,h ) for ll h E A. Let h 2 (n) = (h(n)) 2

8 - 8 - Then for h E A 3n R((n),fi (n), fi (n),n) 2 Let f3 be n ssocite for F ' hp E A Let h = h F. Then h = lim hccm)) nd h = lim h(s(m)) It follows tht m-.oo 2 Jll.+oo 2 for some m P (S(m)) =.. rk_ 2C)} b Let be given. Let c = U{B ~ H. B is c c H nd c is ~k-2. /I:(,h ). Choose h E H such tht 2 Then So there is n h E H ' C nd then vnir((n),fi (n), fi (n),n) Let h = hf h = h nd 2 2 F 2 let B be n ssocite for F = <F,F >. It is cler tht 2 F cnnot be constnt ( since otherwise so k- BS(n) will lwys contin more thn one element. ( If k- B 0 contins just one element, tht element is constnt ). c It follows tht P will not secure B This is trivil from the following monotonity property: which gin follows trivilly from the definition of h. ( Use lemm l. ) "[' This ends the proof of lemm 6. h cr nd Let K = { P E { o } n~} k : ' Then Kk is compct nd contins only semi-ssocites for ko none of which re ssocites. We will now show tht from such compct sets K we my construct interesting functionls of type k+l.

9 - 9 - Definition Let ~ E Ct(k). Let o! be the sequence of length m defined s follows. For o<m let s+l if 3T<m (o*t A Bk-l c Bk-l) T - 0 A VT<m (Bk-l c Bk-l ~ ~(F ) = s) T 0 T 0 otherwise Lemm 7 lim o~ is the principl ssocite for ~. m Jll-+00 The proof is stndrd. i The proof. Lemm 8 Let K be compct set of semi-ssocites for type k function~ls such tht K contins no ssocites. Then the functionl Proof ~K(~) = ~n vm~n VSEK 3o<m (S(o)=O A o!ccr)>o ) is well-defined nd hs n ssocite recursive in K {<n~tt, {TT ',TTk } = {S(n) : SEK}} n i.e. in Let be n ssocite for ~. It is sufficient to show tht ~K(~) is uniformly recursive in, K. For ech 3 ' if s is semi-ssocite nd Vo ( (o) > 0 ~ S (cr) > 0 ) then 3 is n ssocite. So Since vs E K 3cr (S(o)=O A (cr)>o ) K is compct we my choose these o's mong finite set

10 - 0- {cr,,crk}. Choose m so lrge tht ll these sequences hve proper extensions <m. Then Recursively in, K we my pick m to be the lest such m. 0 We then know tht is the lest n<m - 0 such tht We my find this n uniformly recursive in K, m 0 This shows tht ~K is recursive in K nd ends the proof of lemm 8. Let ~k = ~K. Then ~k E Ct(k+l) hs recursive k ssocite. Lemm 9 ~k is not Kleene-computble in ny ~ E Ct(k). Proof Assume tht the lemm is flse. Then there is ~ E Ct(k) nd n e such tht By lemm 4 there is AcH such tht whenever is used in subcomputtion of {e}(ko,~) then HF E A. By lemm 6. there is P E Kk securing ll ssocites for F whenever hp E A. By lemm 5 there is n n such tht whenever k E Bp (n) then

11 - - We defined for E Ct (k) but we cn use the sme definition for ll defined on ll F Let if k- B contins just oneelement (F ) = 0 +l or if 3T<n (F E Bk-l A P (T)=l) T otherwise By lemm 2.ii we see tht 0 is well-defined. Moreover, if k V E BP(n) ( (F)=s) then 0 (F)=s, so ll finite prts of Clim my be extended to elements in k Bp (n) vm>n v (n<<m:::o o 0 () < P () ) - m - b If <n nd P ()=O then there is n m such tht o Proof m>m "* o 0 ()=O - o m For ech m, we hve tht om 0 () is either 0 or (F )+ 0 nd ( F ) is either 0 or +l. 0 If o 0 ()=+2 m then But (F 3 <m ( ~ A Bk-l c Bk-l A (F )= (F )=+l) 0 0 )F+l when F so this 0 tht OmO() E {0,} for ll. is impossible. Assume tht n<<m nd o 0 ()=l. Since - m o 0 ()>O "* o()= (F )+ m m for ll, we must hve It follows If this is becuse contins just one element we hve

12 - 2 - constructed P If Bk-l in such wy tht P ()=l. contins more thn one element we must hve 3T<n (F E Bk-l A P (T)=l ) T Then T< nd by lemm 2.iii we must hve But then by lemm 6.c. cf> So o 0 ()=l ~ P m ()=l b If P ()=O then p () =. k- B k- k- B c B T contins more thn one element. If Bk-l c u {Bk-l : T<n A P (T)=l} then by lemm l.b. T 3T<n (P (T)=l A Bk-l c Bk-l). T But by lemm 6.c. P ()=l so this is impossible. So : T<n A p (T)=l} By lemm l.c. there re extensions nd 2 of such tht ~ + 2 T<TI A p (T)=l} : 0 nd Then cp (F ) = + o cf> tht om 0 ()=O. nd cp (F ) = + o 2 2 For This ends the proof of the clim. m> 2 we see By the clim we hve 3m >n Vm>m V< m (omo() < P ()). Choose m > mx{~k(ko),m 0 }. Let V<m cp(f ) = cf> (F ). o k E Bp (n) be such tht As we remrked fter the definition of o this is possible. cf> Then s=o o m m nd V<m o() () So ~k(cf>)~m. But since m -

13 - 3 - E B~ (n) we hve ~k( ) = ~k(ko). This is contrdiction nd. the lemm is proved. Theorem We hve now showed For ech k > 2 there is recursive functionl ~ in Ct(k+l) such tht ~ is not computble in ny functionl in Ct(k). Proof For k=2 we my use the fn-functionl while for k~3 we hve showed tht ~k is n exmple. 5. References. J. Bergstr, Computbility nd continuity in finite types, Disserttion, Utrecht J. E. Fenstd, Generl Recursion Theory, Springer Verlg (979) 3. R. 0. Gndy nd J. M. E. Hylnd, Computble nd recursively countble functions of higher type, in R. 0. Gndy nd J. M. E. Hylnd (eds) Logic Colloquium , North Hollnd S. C. Kleene, Recursive Functionls nd Quntifiers of Finite Types I, Trns. Am. Mth. Soc. 9 (959) -52; nd II 08 (963) S. C. Kleene, Countble functionls, in A. Heyting (ed): Constructivity in Mthemtics North Hollnd G. Kreisel, Interprettion of Anlysis by mens of Constructive Functionls of Finite Types, in A. Heyting (ed): Constructivity in Mthemtics 0-28 North Hollnd 959

14 D. Normnn, Countble functionls nd the nlytic hierrchy, Oslo Preprint no 7, D. Normnn, Recursion on the countble functionls, Lecture Note, in preprtion. 9. D. Normnn, Countble functionls nd the projective hierrchy, in preprtion.