Refinment of the "up to a constant" ordering using contructive co-immunity and alike. Application to the Min/Max hierarchy of Kolmogorov complexities

Reﬁnmen t of the “up to a consta nt” ordering using con tructiv e co-immun ity and alik e. Application to the M in/ M ax hierarc h y of Kolmogoro v complexiti es Marie Ferbus-Zanda LIAF A, Universit ´ e P aris 7 2, pl. Jussieu 75251 P aris Cedex 05 F rance ferbus@log ique.jussieu.fr Serg e Grigorieff LIAF A, Universit ´ e P aris 7 2, pl. Jussieu 75251 P aris Cedex 05 F rance seg@liafa. jussieu.fr No v em b er 16, 2021 Con ten ts 1 In tro duction 2 1.1 Comparing total functions N → N . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Second order Kolmogoro v complexity . . . . . . . . . . . . . . . . . . . . . . 4 1.3 A strong hierarc hy theorem for Kolmogoro v complexities . . . . . . . . . . . 5 2 P artial computable functionals and oracular recursion theory 5 2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Some classical results from recursion theory . . . . . . . . . . . . . . . . . . 6 2.3 P artial comput able functionals . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Uniform relativization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Acceptable en umerations of some sub classes of ∆ 0 2 . . . . . . . . . . . . . . 8 2.6 The min and max operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Coimmunit y and density 11 3.1 Constructive co immunity and constru ctive density . . . . . . . . . . . . . . 11 3.2 Uniform constructive density . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Constructive ( C , D )- density . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 The OftLess relations and the ≪ orderings 15 4.1 Relations OftLess C , D F , O ftLess C , D F ↑ on maps N → N . . . . . . . . . . . . . 15 4.2 Monotonicit y versus recursive low er b oun d . . . . . . . . . . . . . . . . . . . 16 4.3 T ransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 Orderings ≪ C , D F , ≪ C , D F ↑ on maps N → N . . . . . . . . . . . . . . . . . . . 19 4.5 Left comp osition an d ≪ C , D F . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1 5 F unctional Kolmogorov complexity 21 5.1 Kolmogoro v complexity of a fun ctional . . . . . . . . . . . . . . . . . . . . . 21 5.2 F u nctional inv ariance theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6 The M in/ M ax hierarch y of Kolmogoro v complexitie s 23 6.1 M in/ M ax K olmogorov complex ities . . . . . . . . . . . . . . . . . . . . . . 23 6.2 F u nctional M in/ M a x Kolmogorov complexities . . . . . . . . . . . . . . . . 24 7 F unctional versus oracular 25 8 Reﬁning the oracular Min/Max hierarc hy with the ≪ , ≪ ↑ orderings 26 8.1 Barzdins’ theorem in a u niform setting . . . . . . . . . . . . . . . . . . . . . 26 8.2 Comparing K and K max ` a la Barzdins . . . . . . . . . . . . . . . . . . . . 29 8.3 Comparing K and K min ` a la Barzdins . . . . . . . . . . . . . . . . . . . . . 33 8.4 Comparing K min and K max ` a la Barzdins . . . . . . . . . . . . . . . . . . 36 8.5 Syntactical complexit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8.6 The hierarc hy th eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Abstract W e introduce orde r ings ≪ C , D F betw een to tal functions f , g : N → N which reﬁne the p oint wise “up to a constant” ordering ≤ ct and also insure that f ( x ) is often muc h less than g ( x ). With such ≪ C , D F ’s, we prove a strong hiera rch y theorem fo r Kolmog orov complexities ob- tained with jump ora cles and/or M ax or M in of partia l recursive func- tions. W e introduce a notion of se c ond or der conditional Ko lmo gorov com- plexity which y ields a uniform bound for the “up to a constant ” com- parisons in volv ed in the hierarch y theorem. 1 In tro duc tion 1.1 Comparing total functions N → N Notation 1.1. Equalit y , inequ alit y and str ict inequalit y up to a constant b et ween total fu nctions I → N , where I is an y set, are denoted as follo ws: f ≤ ct g ⇔ ∃ c ∈ N ∀ x ∈ I f ( x ) ≤ g ( x ) + c f = ct g ⇔ f ≤ ct g ∧ g ≤ ct f ⇔ ∃ c ∈ N ∀ x ∈ I | f ( x ) − g ( x ) | ≤ c f < ct g ⇔ f ≤ ct g ∧ ¬ ( g ≤ ct f ) ⇔ f ≤ ct g ∧ ∀ c ∈ N ∃ x ∈ I g ( x ) > f ( x ) + c T otal f unctions f , g : N → N can b e compared in diverse wa ys. T he sim- plest one is p oin t wise comparison via the partial ordering relation ∀ x f ( x ) < g ( x ). In case fun ctions are considered up to an additiv e constan t, for in- stance with Kolmogoro v complexit y , p oin t wise comparison has to b e re- placed by the ≤ ct preordering or th e < ct ordering. 2 Observe that the < ct ordering is an inﬁn ite in tersection: f < ct g ⇔ f ≤ ct g ∧ ∀ c ∈ N f < io g − c where < io (io stands for “inﬁn itely often”) is the non transitive relation f < io g ⇔ { x : f ( x ) < g ( x ) } is in ﬁ nite Relation < io can b e muc h reﬁned via lo calization: instead of merely de- manding { x : f ( x ) < g ( x ) } to b e inﬁn ite, one can ask it to h av e inﬁ nite in tersection with every inﬁnite set in a family C of sets. In case C is the family of all subsets of N , this giv es the relation { x : f ( x ) < g ( x ) } is coﬁn ite whic h is a partial ordering relation. In case C is the family of r.e. sets, this is related to the idea of coimm un it y . An instance of s u c h a relation app ears in a classical result ab out Kolmogoro v complexit y K , du e to Barzdins (cf. [9 ] T h m.2.7.1 iii, p.167, or Zvonkin & Levin, [17] p.92.), whic h states that, f or any total recur siv e f u nction φ which tends to + ∞ , the set { x : K ( x ) < φ ( x ) } meets every inﬁn ite r.e. set. In practice, for simp le classes C , an inﬁ nite s ubset of X ∩ { x : f ( x ) < g ( x ) } , for X inﬁnite in C , can alwa ys b e f ou n d in a n ot to o complex class D . Whic h leads to consider the relation OftLess C , D suc h that f OftLess C , D g ⇔ ∀ X ∈ C ∃ Y ∈ D ( X is inﬁ n ite ⇒ Y is inﬁnite ∧ Y ⊆ { x : f ( x ) < g ( x ) } ) If C = D then th is relation is transitiv e, hence is a strict partial ordering. Ho w ev er, in case C 6 = D , transitivit y may fail (for instance, a coun terexample is obtained via Lemma 8.10). The key observ ation for the pap er is as f ollo ws: F or any C , D , the r ela tion f ≤ ct g ∧ ∀ c ( f OftLess C , D g − c ) is tr ansitive, henc e is a p artial strict or dering r eﬁning < ct . In other wor ds, c onsidering OftLess C , D up to any c onstant and mixing it with ≤ ct always le ads to an or d ering. If F is a family of total fu nctions N → N wh ic h tend to + ∞ and F is closed b y translations (i.e. φ ∈ F imp lies max(0 , φ − c ) ∈ F ), then the ab o ve observ atio n also applies to the relation f ≤ ct g ∧ ∀ φ ∈ F f OftLess C , D φ ◦ g , i.e. the relation f ≤ ct g ∧ ∀ φ ∈ F ∀ X ∈ C ∃ Y ∈ D ( X is inﬁn ite ⇒ Y is inﬁ nite ∧ Y ⊆ { x : f ( x ) < g ( x ) } ) whic h is also a partial strict ord er in g reﬁnin g the ordering < ct . Enric hin g this r elation with the requirement that a co de for an inﬁnite subset Y of X ∩ { x : f ( x ) < φ ( g ( x )) } can b e eﬀectiv ely compu ted from co des for φ and X , w e get the relation Of tLess C , D F whic h is the main concern of th is 3 pap er. In § 2 we review some needed elemen ts of oracular computabilit y . This is done in terms of partial computable functionals so as to get uniformity in the oracle. In § 3 we recall Xiang Li’s notion of constructiv e immunit y and introdu ce the related notions of ( C , D )-densit y and constructiv e densit y . In § 4 w e in tro d uce the relation OftLess C , D F and its v arian t OftLess C , D F ↑ (where only total monotone increasing fu nctions in F are considered) and pro ve that their intersect ions ≪ C , D F and ≪ C , D F ↑ with ≤ ct are strict orderings reﬁning the ordering < ct . 1.2 Second order Kolmogoro v complexit y In relation with the partial compu table functional approac h to oracular com- putabilit y (cf. § 2), w e dev elop in § 5 a functional version K ( x || A ) of Kol- mo go r ov c ompl exity. This amoun ts to a simp le, seemingly unnoticed, fact: Or acular Kolmo gor ov c o mplexity K A c an b e obtaine d by instantiating to A the se c ond or der p ar ameter of a variant of c onditiona l Kolmo g or o v c omp lex- ity in which the c ondition i s a set of inte gers r ather than an inte ger. The or a cle is thus viewe d as a se c ond or der c ondition al p ar ameter. The usual p ro of of the in v ariance theorem go es through. This s econd-ord er conditional complexit y allo ws for a un if orm choic e of oracular K olmogoro v complexities (this is d etailed in § 7) since, for any A , K ( x || A ) = ct K A ( x ) i.e. ∀ A ∃ c ∀ x | K A ( x ) − K ( x || A ) | ≤ c . A t ypical b eneﬁt of the functional version of K is as follo ws. Usu al prop er- ties with K in vol ving equalit y or inequalit y “up to a constan t” go through oracles. Let c A b e the in vol ved constan t for the oracle A version. F or a single equality or inequality inv olving K A , it ma y b e p ossible to mo dify K A (b y an ad d itiv e constan t) so that c A = 0. But this is no more p ossible for sev eral equalities or inequalities since the needed mo diﬁcations of K A ma y – a p riori – b e in compatible. Th u s , for a system of equalities or inequalities, th er e is n o a p riori A - computable b oun d of the inv olv ed constan t c A for the oracle A v ersion. Ho w ev er, in case (wh ich is also usual) su c h prop erties also go th r ough the functional v ersion, the constan t b ound inv olv ed in the functional v ersion is v alid for any oracle. In other words, wher e as the or acular version a priori al lows no A -c omputable b ound of the c onstant, the functional version do es al low a c onstant b ound. This fact is applied in § 8.6 to get sharp er results. 4 In § 6 we recall the v arian ts K max , K min of Kolmogoro v complexit y intro- duced in our p ap er [5] and we extend them to functional v ersions. Th e p re- cise relation b et w een suc h fun ctional v ersions and the oracular K max , K min is detailed in § 7. 1.3 A st rong hierarc hy t heorem for Kolmogoro v c o mplexi- ties In § 8.1 w e pro ve of a v ersion of Barzdins’ result cited in § 1.1 (cf. also § 3.1) with as m uch eﬀectivit y as p ossible whic h in vo lves an ordering relation in- tro duced in § 4 and can b e stated as K ≪ Σ 0 1 , Σ 0 1 PR log. Also, the fun ctional v ersions of Kolmogoro v complexit y and the functional approac h to oracular computabilit y allo w to get a fu nctional ve rs ion of this result, hence to get eﬀectivit y relativ e to th e oracle. W e extend this result in § 8.2, 8.3, 8.4 and pro ve that K, K max , K min can b e compared via th e ab o ve OftLess and ≪ relations, with more complex classes C , D , n amely C = Σ 0 1 ∪ Π 0 1 and D = ∃ <φ (Σ 0 1 ∧ Π 0 1 ) or the v aria nts in wh ic h Π 0 1 is constrained w ith a “recursive ly b ounded growth” condition (cf. Def.8.7). Also, the class F can b e extended to M in P R , i.e. th e class of inﬁma of partial recursive sequences of functions. The ab o v e class D is a sub class of ∆ 0 2 whic h can b e obtained via b ounded existen tial quant iﬁcation o ver b oolean com binations of Σ 0 1 relations. In § 8.5, w e sho w that such synta ctical complexities naturally app ear when compar- ing K, K max , K min . Finally , in § 8.6 we pro ve th e main application of the ≪ C , D F and ≪ C , D F ↑ order- ings, whic h is a strong h ierarc hy theorem for the Kolmogoro v complexities K, K max , K min and their oracular v ersions u sing the successiv e jumps. 2 P artial computable functionals and oracu lar re- cursion theory 2.1 Notations Notation 2.1. 1. [ B asic sets] X , Y denote pro ducts of non emp t y ﬁn ite families of spaces of the form N or Z or Σ ∗ where Σ is some ﬁnite alphab et. 2. [Partial r e cu rsive functions] Let A ⊆ N . W e denote P R X → Y (resp. P R X → Y ,A ) th e family of partial recursiv e (resp. A -recursiv e) functions b e- t w een basic sets X and Y . 3. [Bije ctions b etwe en b asic sp ac es] F or any basic spaces X , Y and Z w e ﬁx s ome p articular total recursiv e bijection from X × Y to Z and denote h x , y i X × Y , Z , or simply h x , y i , the image in Z of the pair ( x , y ). 5 2.2 Some classical results from recursion theory W e shall use the follo wing classical results fr om computabilit y theory (cf. Odifredd i’s b o ok [11] p.372–3 74, 288–2 92, or Sho enﬁeld’s b o ok [15]). Prop osition 2.2. 1. (Post’s Th eorem, 1948 [13]) A set is Σ 0 n +1 (r esp. ∆ 0 n +1 ) if and only if it is r e cu rsively enumer a ble (r.e.) (r esp. r e cursive) in or acle ∅ ( n ) . 2. (P ost, 1944 [12]) F or any or acle A , e v ery i nﬁnite A -r.e. set X c ontains an inﬁnite set Y which is r e cursive in A . M or e over, one c an r e cu rsively go fr om an r.e. c o de f or X to r.e . c o d es for suc h a Y and its c omplement . In p a rticular, every inﬁnite Σ 0 n +1 set X c ontains an inﬁnite ∆ 0 n +1 subset Y . Also , one c a n r e cursively go fr om a Σ 0 n +1 -c o de for X to Σ 0 n +1 -c o des for such a Y and its c omplement. 3. R e c al l that an A -r.e. set X ⊂ N is maximal if it is c oinﬁnite and for any A -r.e. set Y ⊇ X e ither N \ Y is ﬁnite or Y \ X is ﬁnite. (F riedb erg, 1958 [6]) Ther e exists maximal A -r.e . sets. Remark 2.3. 1. Since ev ery Π 0 n set is Σ 0 n +1 , p oint 2 of the ab o v e prop osition yields that ev ery inﬁnite Π 0 n set con tains an inﬁnite ∆ 0 n +1 subset. This cannot b e impro ved: the complemen t of an y maximal recursivel y enumerable set is an inﬁnite Π 0 1 set wh ic h d o es not con tain any inﬁn ite recursive set. 2. Any total function ψ with graph in Σ 0 n is in fact ∅ ( n − 1) -recursiv e and h as graph in ∆ 0 n since y 6 = ψ ( x ) ⇔ ∃ z 6 = y z = ψ ( x ). 2.3 P artial computable functionals Def.2.4 is classical, cf. Roge rs [14] p.361, or Odifredd i [11] p.178. Deﬁnition 2.4. A (partial) functional F : X × P ( N ) → Y is partial com- putable if there exists an oracle T ur ing machine M suc h that, giv en A ∈ P ( N ) as oracle and x ∈ X as in put, - M halts and accepts if and only if F ( A, x ) is deﬁ n ed, - if M halts and accepts then its output is F ( A, x ). The family of partial computable functionals X × P ( N ) → Y i s den oted P C X × P ( N ) → Y . The notion of acceptable enumeration of partial r ecursiv e fun ctions (cf. Rogers [14] E x . 2.10 p.41, or Odifrr eddi [11], p.215) extends to functionals. Deﬁnition 2.5. W e denote X , Y , Z some basic sets (cf. Notation 2.1). 1. An enumeration (Φ i ) i ∈ N of partial computable f unctionals X × P ( N ) → Y is ac c ept able if 6 i. ( i, x , A ) 7→ Φ i ( x , A ) is a p artial computable functional. ii. Ev ery partial compu table fu n ctional X × P ( N ) → Y is enumerated: ∀ Ψ ∈ P C X × P ( N ) → Y ∃ i Φ i = Ψ iii. th e parametrization (also called s-m-n) prop ert y h olds : for every basic set Z , there exists a total recursive fu nction s Z X : N × Z → N such that ∀ i ∀ z ∈ Z ∀ x ∈ X ∀ A ⊆ N Φ i ( h z , x i , A ) = Φ s Z X ( i, z ) ( x , A ) where h z , x i is the image of the p air ( z , x ) b y s ome ﬁ xed total recur siv e bijection Z × X → X (cf. Notation 2.1). 2. An en umeration ( W i ) i ∈ N of Σ 0 1 subsets of X × P ( N ) is ac c eptable if ther e exists an ac c ept able en um eration (Φ i ) i ∈ N of p artial r ecursiv e fun ctionals such that W i is th e domain of Φ i . In p articular, ( W i ) i ∈ N is Σ 0 1 as a subset of N × X × P ( N ). Prop osition 2.6. Ther e exists an ac c eptable enu mer a tion of p ar tial c om- putable functionals X × P ( N ) → Y . 2.4 Uniform relativization When dealing with oracles A , it is often p ossible to get resu lts inv olving r e cursive transfer fun ctions rather than A -recursiv e on es. T o do so, we m ust consider enumerations of A -r.e. sets and p artial A -recursiv e fu nctions wh ic h are obtained f rom enumerations of p artial c omputable functionals by ﬁxing the second order argumen t A . Such en umerations will b e called uni f orm enumer atio ns . This amoun ts to consider r elativ e computabilit y as a concept dep enden t on the prior n otion of partial computable f u nctional, though , historica lly , relativ e computabilit y came ﬁrst, cf. Hinman’s b o ok [7 ] 5.15 p.68. Prop osition 2.7. L et (Φ i ) i ∈ N b e an ac c ep table enumer ation of p artial c omputable fu nctionals X × P ( N ) → Y . F or A ⊆ N , deﬁne ϕ A i : X → Y and W A i ⊆ X fr om Φ i and W i by ﬁxing the se c ond or der ar gument as fol lows: ϕ A i ( x ) = Φ i ( x , A ) W A i = domain ( ϕ A i ) = { x : ( x , A ) ∈ W i } Then the se quenc es ( ϕ A i ) i ∈ N and ( W A i ) i ∈ N ar e ac c eptable enumer ations of the family P R X → Y ,A ) of p artial A -r e cursive func tions X → Y a nd that of A -r.e. sub sets of X . Such ac c epta ble enumer ations ar e c al le d uniform enumerations . 7 Rogers’ theorem (cf. Odifred di [11] p.219) extends to p artial computable funti onals, hence to uniform enumerations. Theorem 2.8. 1. (Rogers’ theorem) If (Ψ i ) i ∈ N and (Φ i ) i ∈ N ar e b oth ac c eptable enumer- ations of p artial c omp utable functionals X × P ( N ) → Y , then ther e exists some r e c ursive bij e ction θ : N → N such that Ψ i = Φ θ ( i ) for al l i ∈ N . 2. If ( ψ A i ) i ∈ N and ( ϕ A i ) i ∈ N ar e u niform enumer ations of p artial A -r e cursive functions then ther e exists some recursiv e bije ction θ : N → N such that ψ A i = ϕ A θ ( i ) for al l i ∈ N . Uniform enumerations allo w for eﬀ e ctive (as opp osed to A -eﬀectiv e) clo- sure results for a lot of op erations on partial A -recursiv e functions and A -r.e. sets wh ic h corresp ond to closure p rop erties of partial computable function- als admitting sets and partial functions as arguments, cf. Hinman [7] § I I.2, I I.4. 2.5 Acceptable en umerations of some sub classes of ∆ 0 2 Comparison of K and K min , K max in the hierarc hy theorem 8.14 inv olv es particular ∆ 0 2 sets describ ed in Def.2.12 b elo w. F irst, w e ﬁx a notion of b ound ed quan tiﬁcation p ertinent for our app lications. Deﬁnition 2.9. 1. W e consid er on eac h basic set a norm suc h th at - || x || = | x | if x ∈ N or Z , - || x || = l eng th ( x ) if x ∈ Σ ∗ where Σ is a ﬁnite alph ab et, - || ( x 1 , ..., x k ) || = max( || x 1 || , ..., || x 1 || ). 2. S u pp ose µ : N → N is a total fun ction (resp. µ : N × P ( N ) → N is a total fu nctional) whic h is monotone increasing (resp. with resp ect to its ﬁrst argum ent). Let X is a b asic set. F or R ⊆ X × ( { 0 , 1 } ∗ ) m and R ⊆ X × ( { 0 , 1 } ∗ ) m × P ( N ), w e let ∃ ≤ µ R = { x : ∃ ~ u ( | u 1 | , . . . , | u m | ≤ µ ( || x || ) ∧ R ( ~ u , x )) } ∃ ≤ µ R = { ( x , A ) : ∃ ~ u ( | u 1 | , . . . , | u m | ≤ µ ( || x || ) ∧ R ( ~ u , x , A )) } If C ⊆ P ( X ) (resp . C ⊆ P ( X × P ( N )), w e denote ∃ ≤ µ C the sub class of subsets of X (resp. X × P ( N )) consisting of all sets ∃ ≤ µ R where R is in C . Note 2.10. In view of app licatio n s to Kolmogoro v complexit y , w e c ho ose b ound ed qu an tiﬁcations o ve r binary words (where the b ound applies to the length). Of course, going from µ to 2 µ , we can redu ce to b ound ed quan tiﬁ- cations o ve r N . As is well k n o wn, b ounded quant iﬁcation do es not increase syntactic al complexit y of ∆ 0 2 sets. 8 Prop osition 2.11. If µ : N → N has Σ 0 2 gr ap h then ∃ ≤ µ ∆ 0 2 ⊆ ∆ 0 2 , b e it for r elat ions in X or in X × P ( N ) . Pr o of. In case µ ( x ) = x this is just th e commutati on of a b oun ded quantiﬁ- cation with an unb ounded one. In general, we ha ve ∃ ~ u ( ~ u ≤ µ ( || x || ) ∧ R ( x , ~ u )) ⇔ ∃ y ( y = µ ( || x || ) ∧ ∃ ~ u ( ~ u ≤ y ∧ R ( x , ~ u ))) ⇔ ∀ y ( y = µ ( || x || ) ⇒ ∃ ~ u ( ~ u ≤ y ∧ R ( x , ~ u ))) whic h are r esp ectiv ely ∃ (Σ 0 2 ∧ ∆ 0 2 ), h ence Σ 0 2 , and ∀ (Π 0 2 ∨ ∆ 0 2 ), h ence Π 0 2 . Deﬁnition 2.12. Let C b e a syn tactical class among Σ 0 1 , Π 0 1 , Σ 0 1 ∨ Π 0 1 , ∃ ≤ µ (Σ 0 1 ∧ Π 0 1 ) 1. L et C [ X ] b e the family of subsets of X whic h are C -deﬁn ab le. An accept- able enumeration ( W C [ X ] i ) i ∈ N of C [ X ] is an en u m eration obtained from accept- able en umerations ( W X × ( { 0 , 1 } ∗ ) m i ) i ∈ N of r.e. su bsets of the X × ( { 0 , 1 } ∗ ) m ’s as follo ws: W Π 0 1 [ X ] i = X \ W X i W Σ 0 1 ∧ Π 0 1 [ X ] i = W X j ∩ ( X \ W X k ) where i = h j, k i W Σ 0 1 ∨ Π 0 1 [ X ] i = W X j ∪ ( X \ W X k ) where i = h j, k i W ∃ ≤ µ (Σ 0 1 ∧ Π 0 1 )[ X ] i = ∃ ≤ µ W (Σ 0 1 ∧ Π 0 1 )[ X × ( { 0 , 1 } ∗ ) m ] j where i = h j, m i 2. Let C [ X × P ( N )] b e the family of su bsets of X × P ( N ) whic h are C - deﬁnable. An acceptable en umeration ( W C [ X ] i ) i ∈ N of C [ X × P ( N )] is deﬁned similarly from acceptable en umerations ( W X × ( { 0 , 1 } ∗ ) m × P ( N ) i ) i ∈ N of Σ 0 1 sub- sets of the X × ( { 0 , 1 } ∗ ) m ’s. 3. Let A ⊆ N and C A b e th e A -oracle s y ntactic al class asso ciated to C . An enumeration ( W C A [ X ] i ) i ∈ N of C A [ X ] is uniform if it is obtained from an acceptable enumeration ( W C [ X × P ( N )] i ) i ∈ N of C [ X × P ( N )] b y ﬁxing the second order argument. I .e. W C A [ X ] i = { x ∈ X : ( x , A ) ∈ W C [ X × P ( N )] i } . 2.6 The min and max op erators The follo wing d eﬁnitions and r esults collect material fr om [5, 4 ]. Deﬁnition 2.13. Let X b e s ome basic s et. W e denote min and max the op erators which map partial functions ϕ : X × N → N and p artial fu n ctionals Φ : X × P ( N ) × N → N on to partial functions min ϕ, max ϕ : D → N and 9 functionals m in Φ , max Φ : D → N suc h that (min ϕ )( x ) = min { ϕ ( x , t ) : t ∈ N ∧ ϕ ( x , t ) is deﬁn ed } (max ϕ )( x ) = max { ϕ ( x , t ) : t ∈ N ∧ ϕ ( x , t ) is deﬁned } (min Φ)( x , A ) = min { Φ( x , A, t ) : t ∈ N ∧ φ ( x , t ) is deﬁned } (max Φ)( x , A ) = max { Φ( x , A, t ) : t ∈ N ∧ φ ( x , t ) is deﬁ n ed } with the con ve ntion that min ∅ and max ∅ and the max of an in ﬁnite set are undeﬁn ed. 2. W e let M in X → N P R = { min ϕ : ϕ ∈ P R X × N → N } M in X → N P R A = { min ϕ : ϕ ∈ P R X × N → N ,A } M in X × P ( N ) → N P C = { min Φ : Φ ∈ P C P ( N ) × X × N → N } The classes M ax X → N P R , M ax X → N P R A and M ax X × P ( N ) → N P C are d eﬁned similarly from the max op erator. Note 2.14. 1. S imple examples of functions in M in P R are Kolmogoro v complexities K and H . Examples of fun ctions in M ax P R are th e Busy Bea v er function an d the (partial) function giving the cardin al of W n (if ﬁnite). 2. The fu nctional K ( || ), deﬁn ed in § 5, is in M in P ( N ) × N → N P C . Let’s men tion an easy resu lt as concerns the synt actical complexit y of these fu nctions. Prop osition 2.15. Any function in M in X → N P R ∪ M ax X → N P R has Σ 0 1 ∧ Π 0 1 gr ap h. The r esult extends to functionals and also r elativizes. Pr o of. Observ e th at y = (min ϕ )( x ) can b e written ( ∃ t y = ϕ ( x , t )) ∧ ( ∀ t ∀ s ( ϕ ( x , t ) con v erges in s steps ⇒ y ≤ ϕ ( x , t ))) Idem for y = (max ϕ )( x ) with ≥ in place of ≤ . W e shall use the follo wing straigh tforw ard corollary of th e ab o v e Prop o- sition. Prop osition 2.16. Al l functions in M in P R and M ax P R ar e p artial r e cu r- sive in ∅ ′ . An enumeration theorem holds for the families in tro du ced in Def.2.1 3 . 10 Prop osition 2.17. Ther e exists an ac c eptable enumer ation ( φ i ) i ∈ N of M in X → N P R (wher e ac c eptable me ans that the analo gs of c onditions i–iii of D e f.2.5 hold. In p articular, the function ( i, x ) 7→ φ i ( x ) is itself in M in N × X → N P R ). Idem with the class M ax X → N P R and the functional classes M in X × P ( N ) → N P C and M in X × P ( N ) → N P C . The follo wing simp le result ab out M in P R and M ax P R will b e useful. Prop osition 2.18. 1. If φ ∈ M in X → N P R and f : Y → X is in P R Y → X then φ ◦ f ∈ M in Y → N P R . Idem with M ax P R in plac e of M in P R . 2. If ψ ∈ M in N → N P R is monoto ne incr e asing and φ ∈ M in X → N P R then ψ ◦ φ ∈ M in X → N P R . Pr o of. 1. Let φ ( x ) = min t ϕ ( x , t ) wh ere ϕ is partial recursive. Th en φ ( f ( y )) = min t ϕ ( f ( y ) , t ) is in M in Y → N P R since ϕ ( f ( y ) , t ) is in P R Y → N . 2. Let φ ( x ) = min t ϕ ( x , t ) and ψ ( x ) = min u θ ( x, u ) where ϕ, θ are partial recursiv e. Since ψ is monotone in creasing, letting ( π 1 , π 2 ) : N → N 2 b e the in ve rs e of Can tor bijection, we ha ve ψ ( φ ( x )) = ψ (min t ϕ ( x , t )) = min t ( ψ ( ϕ ( x , t ))) = min t (min u θ ( ϕ ( x , t ) , u )) = min v θ ( ϕ ( x , π 1 ( v )) , π 2 ( v )) whic h is in M in X → N P R since θ ( ϕ ( x , π 1 ( v )) , π 2 ( v )) is partial recursiv e. 3 Coimm unit y and densit y 3.1 Constructiv e coimm unit y and constructiv e densit y A classical result ab out Kolmogoro v complexit y K , due to Barzdins (cf. [9] Thm.2.7.1 iii, p.167, or Zv onkin & L evin, [17] p.92.), states that if ϕ is total recursiv e and tends to + ∞ then { x : K ( x ) < ϕ ( x ) } is an r.e. set which meets ev ery inﬁnite r.e. set, i.e. { x : K ( x ) < ϕ ( x ) } ∩ W i is an inﬁnite r .e. set whenever W i is inﬁn ite. (The case ϕ is monotone increasing is d ue to Kolmogoro v, cf. [17] p.90, or [9] Thm .2.3.1 iii, p.119–1 20). In p articular, K has no total recursive unboun ded lo w er b ound . In § 8 w e extend in v arious w a ys this result to sets w h ic h are no more r.e. sets and in v olv e Kolmogoro v complexities K min or K max . W e also consider eﬀectiv eness of such prop erties in a sense related to the notion of constructiv e imm un ity , ﬁrst consider ed in Xiang Li, 1983 [10] (cf. Odifreddi’s b o ok [11] p.267). 11 Deﬁnition 3.1. Let ( W i ) i ∈ N b e an acceptable en um eration of recursive ly en umerable subsets of some basic set X . 1i. (Dekk er, 1958). A set X ⊆ X is immune if it is inﬁnite and con tains no inﬁnite r.e. set. 1ii. (Xiang Li, 1983 [10]). A set X ⊆ X is c o nstructively immune if it is inﬁnite an d ther e exists some partial recursive f unction ϕ : N → X s uc h that ∀ i ( W i is inﬁnite ⇒ ϕ ( i ) is deﬁne d ∧ ϕ ( i ) ∈ W i \ X ) 2i. A set Z ⊆ X is Σ 0 1 -dense if it conta ins an inﬁnite r.e. subset of any inﬁnite r.e. set included in X . 2ii. A set Z ⊆ X is constru ctiv ely Σ 0 1 -dense if ther e exists some total recursiv e function λ suc h that ∀ i ( W i is inﬁnite ⇒ W λ ( i ) is an inﬁnite subset of Z ∩ W i ) Note 3.2. Rog ers’Th m.2.8 ins u res th at the ab ov e notion of constru ctiv e imm un ity and Σ 0 1 -densit y do not dep end on the c hosen enumeration of r.e. sets. Prop osition 3.3. Z ⊆ X is c onstructively immune if and only if it is inﬁnite and its c omplement i s c on structively Σ 0 1 -dense. Pr o of. ⇐ . Let ϕ ( i ) b e the p oin t whic h app ears ﬁrst in th e enumeratio n of W λ ( i ) (of course, ϕ ( i ) is undeﬁ n ed in case W λ ( i ) is emp t y). ⇒ . Deﬁne a partial recursiv e function µ ( i, n ) whic h satisﬁes: - µ ( i, 0) = ϕ ( i ) - µ ( i, n + 1) = ϕ ( i n ) where i n is suc h that W i n = W i \ { µ ( i, m ) : m ≤ n } Using the parametrizatio n theorem, let λ b e total recursiv e so that W λ ( i ) = { µ ( i, m ) : m ∈ N } . If W i is inﬁ nite then all µ ( i, m )’s are d eﬁned and distinct and b elong to W i ∩ Z . T h us , W λ ( i ) is an inﬁnite su bset of W i ∩ Z . In case Z is r.e., constructiv e Σ 0 1 -densit y amoun ts to say that Z ∩ W i is inﬁnite wh enev er W i is inﬁn ite. Barzdin’s result give s an instance of a constructiv ely Σ 0 1 -dense r.e. set. Other examples are maximal r .e. sets. Prop osition 3.4. An y maximal r.e. set Z is c onstr uctiv ely Σ 0 1 -dense. Pr o of. Let Z ⊆ X b e r.e. where X is some b asic set. W e p ro ve that f or eve ry inﬁnite r.e. set W i ⊆ X the in tersection Z ∩ W i is also in ﬁnite. In fact, su pp ose Z ∩ W i is ﬁnite. Th en W i \ Z is an inﬁn ite r.e. set d isj oin t from Z . Th u s , Z ′ = Z ∪ W i is an r.e. set co ntaining Z suc h that the 12 diﬀerence Z ′ \ Z = W i \ Z is inﬁn ite. Since Z is maximal this imp lies that Z ′ is coﬁnite. Th us, X \ Z = ( X \ Z ′ ) ∪ ( W i \ ( Z ∩ W i )) = A ∪ ( W i \ B ) where A, B are ﬁnite sets. Hence X \ Z is r .e. and, consequent ly Z is recursiv e. A con tradiction. 3.2 Uniform constructive densit y In order to d eal with Kolmogoro v complexities K ∅ ′ , K ∅ ′′ , . . . an d their M in/ M ax v ersions, we shall consider constructiv e densit y for Σ 0 n sets. This will b e d one through r elativiza tion of Σ 0 1 -densit y w ith r esp ect to jump oracle ∅ ( n − 1) . There is t w o natural w a ys to relativize Σ 0 1 -densit y to an oracle A : ( ∗ ) C on s ider the W A i ’s and ask for λ A -recursive . ( ∗∗ ) Consider the W A i ’s and ask for λ recursive. The second w a y , wh ic h is the stron ger one, will b e the one p ertinen t for applications to Kolmogoro v complexities. Of course, to deal with ( ∗∗ ), we m ust consider u niform en umerations of A -r.e. sets and partial A -recursive functions (cf. Prop.2.7), i.e. w e ha ve to consider the notion of constru ctiv e densit y with fun ctionals. This will, in fact, giv e a str ong version of ( ∗∗ ) in which λ is a total r e cursive function which do es not dep end on A . Deﬁnition 3.5. 1. Let Z ⊆ X × P ( N ). F o r A ⊆ N , let’s denote Z A = { x ∈ X : ( x , A ) ∈ Z } . Consider an acc eptable enumeration ( W i ) i ∈ N of Σ 0 1 subsets of X × P ( N ) (cf. Def.2.5) and let W A i = { x : ( x , A ) ∈ W i } . Z is constru ctiv ely Σ 0 1 -dense if there exists some total r e cursive function λ suc h that, for all i ∈ N and all A ∈ P ( N ), ( ∗ ) ∀ i ∈ N ∀ A ∈ P ( N ) ( W A i is inﬁnite ⇒ W A λ ( i ) is an inﬁnite subset of W A i ∩ Z A ) 2. Z ⊆ X is constru ctiv ely uniformly Σ 0 ,A 1 -dense if there exists some con- structiv ely Σ 0 1 -dense set Z ⊆ X × P ( N ) suc h that Z = Z A . In p articular, there exists some total r e c u rsive function λ s u c h that ( ∗∗ ) ∀ i ∈ N ( W A i is inﬁnite ⇒ W A λ ( i ) is an i nﬁnite subset of W A i When A = ∅ ( n − 1) w e shall also sa y that Z is constructiv ely uniformly Σ 0 n - dense. Note 3.6. Thm.2.8 insures that the ab o v e notion of constr u ctiv e un iform Σ 0 ,A 1 -densit y do es not dep end on the chosen enumeration of A -r.e. sets, as long as it is uniform, cf. P rop.2.7. 13 Remark 3.7. Using P oint 2 of Prop.2.2, one can supp ose that if W A i is inﬁnite then W A λ ( i ) is A -recursiv e and an A -r.e. co de for its complemen t is giv en b y another total recursiv e function λ ′ . Note 3.8. In the vein of what we mentio ned at the start of § 3.1, if ϕ : N → N is total A -recursiv e and tends to + ∞ then Lemma.8.1 insures that { x : K A ( x ) < ϕ ( x ) } is an A -r.e. set whic h is uniform ly constructiv ely Σ 0 ,A 1 - dense. In case ϕ ( x ) < ct log( x ), this set is coinﬁnite sin ce it excludes int egers with incompr essib le binary represen tations. Remark 3.9. Imm un it y can also b e relativized according to the d iﬀeren t p olicies ( ∗ ) and ( ∗∗ ). Also, Prop.3.3 admits straigh tforwa rd extensions to the fun ctional setting an d the u n iform relativized one. Finally , let’s observ e that Pr op.3.4 r elativize s in the uniform sense. Prop osition 3.10. Any maximal A -r.e. set Z i s uniformly c onstructively Σ 0 ,A 1 -dense. Pr o of. Let Z = { x : ( x , A ) ∈ Z } w here Z ⊆ X × P ( N ) is Σ 0 1 . There is a total recursiv e fun ction θ su c h that, for all A and i , Z ∩ W i = W θ ( i ) . In particular, Z ∩ W A i = W A θ ( i ) and the argument of Prop.3.4 goes through. 3.3 Constructiv e ( C , D ) -densit y Comparison of K and K min , K max in the h ierarc hy theorem 8.14 leads to a particular v ersion of constru ctiv e d ensit y applied to Σ 0 1 and to Π 0 1 sets and in vo lving su b classes ∃ ≤ µ (Σ 0 1 ∧ Π 0 1 ) of ∆ 0 2 sets d escrib ed in Def.2.9 b elo w. W e now in tro d uce some cen tral notions of this pap er. Deﬁnition 3.11. Let X b e a b asic set. 1i. Let S , T b e families of subsets of X . A set Z ⊆ X is ( S , T )-den s e if for ev ery inﬁnite set X ∈ S the inte rsection Z ∩ X con tains an inﬁn ite subset Y wh ic h is in T . ii. Let S , T b e families of subsets of X × P ( N ). A set Z ⊆ X × P ( N ) is ( S , T )-dense if f or eve ry X ∈ S there exists Y ∈ T s uc h that, for eve ry A , letting X A = { x : ( x , A ) ∈ X } , X A is inﬁn ite ⇒ Y A is in ﬁnite and included in X A ∩ Z A 2. Let C , D b e syn tactical classes as in Def.2.12. i. Z is constructive ly ( C , D )-dense if it is ( C [ X ] , D [ X ])-dens e in the sense of 1i ab o v e and, moreov er, a D -co de for Y can b e recur siv ely obtained from a C -cod e for X . I n other words, there exists some total recursiv e fun ction λ : N → N su c h that, for all i W C [ X ] i is in ﬁnite ⇒ W D [ X ] λ ( i ) is in ﬁnite and includ ed in W C [ X ] i ∩ Z 14 ii. A set Z ⊆ X × P ( N ) is constructiv ely ( C , D )-dense if it is ( C [ X ] , D [ X ])- dense in the sense of 1ii ab o ve and, moreo v er, an D -co de for Y can b e recursiv ely obtained from a C -co d e for X . In other w ords, there exists some total recurs ive function λ : N → N such that, for all i ( W A i ) C [ X × P ( N )] is in ﬁnite ⇒ ( W A λ ( i ) ) D [ X × P ( N )] is inﬁn ite and includ ed in ( W A i ) C [ X × P ( N )] ∩ Z A ) Note 3.12. 1. Clearly , (constru ctiv e) (Σ 0 1 , Σ 0 1 )-densit y is exactly (constructiv e) Σ 0 1 - densit y in the sense of Def.3.5. 2. See Lemmas 8.6, 8.8 for examples of constructiv e (Σ 0 1 , ∃ ≤ µ (Σ 0 1 ∧ Π 0 1 ))- densit y and (Π 0 1 , ∃ ≤ µ (Σ 0 1 ∧ Π 0 1 ))-densit y . Let’s state a s im p le result ab out ( C , D )-densit y . Prop osition 3.13. 1. The family of (c onstructively) ( C , D ) -dense subsets of X (r esp. X × P ( N ) ) is sup erset close d. 2. L et Z 1 , Z 2 ⊆ X . If Z 1 is (c onstructively) ( C , D ) -dense and Z 2 is (c on- structively) ( D , E ) -dense then Z 1 ∩ Z 2 is (c onstructively) ( C , E ) -dense. Idem for Z 1 , Z 2 ⊆ X × P ( N ) . Pr o of. P oin t 1 is obvio u s . As for p oint 2, let X b e an inﬁnite set in C [ X ]. Using ( C , D )-dens ity of Z 1 w e (recursiv ely) get (a co de for) an inﬁnite X 1 ⊆ X ∩ Z 1 in D . Then, u sing ( D , E )-densit y of Z 2 , w e (recursively) get (a co de for) an inﬁnite X 2 ⊆ X 1 ∩ Z 2 ⊆ X ∩ ( Z 1 ∩ Z 2 ) in E . F or Z 1 , Z 2 , X ⊆ X × P ( N ), ﬁx the second order argument A and argue similarly with Z A 1 , Z A 2 , X A . 4 The OftLess relations and the ≪ orderings In this § we in tro du ce the cen tral n otions of this p ap er to compare the gro wth of total functions f , g : N → N . 4.1 Relations OftLess C , D F , OftLess C , D F ↑ on maps N → N Deﬁnition 4.1. Let C , D b e synt actical classes (cf. Def.2. 12 ) and F b e a coun table family of fun ctions N → N and ( φ i ) i ∈ N b e a (non necessarily injectiv e) enumeration of F (in § 8, F will b e P R or M in P R , cf. Def.2.13). W e let f OftLess C , D F g (resp . f O f tLess C , D F ↑ g ) b e the relation b et we en total fun ctions f , g : N → N d eﬁned by the follo wing conditions: 15 i. F or ev ery total (resp. and monotone increasing) function φ : N → N in F whic h tends to + ∞ , the set { x : f ( x ) < φ ( g ( x )) } is constructive ly ( C , D )-dense. ii. The constru ctive ( C , D )-densit y in condition i is u niform in φ : There exists some total r ecur siv e λ : N 2 → N suc h that, for all i, j , φ i is total (resp. and monotone increasing) and tends to + ∞ ∧ W C j is inﬁn ite ⇒ W D λ ( i,j ) is an inﬁnite su bset of W C j ∩ { x : f ( x ) < φ i ( g ( x )) } Remark 4.2. 1. The notation OftLess stresses the f act that f is often m uch smaller than g : consider functions φ which are m uc h smaller than th e ident ity function, e.g. max(0 , z − c ), ⌊ z /c ⌋ , ⌊ log( z ) ⌋ , log ∗ ( z ),. . . 2. OftLess C , D F carries the con ten ts, reformulat ed in terms of unif orm con- structiv e ( C , D )-den s it y , of Barzdins result cited ab o ve, and that of adequate v arian ts th at we shall pro ve ab out K max and K min (cf. Lemmas 8.1 , 8.6, 8.8). 3. Su pp ose F con tains all trans lation fu nctions z 7→ max(0 , z − c ). If f OftLess C , D F ↑ g (a fortiori if f OftLess C , D F g ) then g is necessarily un- b ound ed. Else, if c is a b oun d for g , consider φ ( z ) = max(0 , z − c ) to get a con tradiction. 4. OftLess C , D F ↑ is an extension of OftLess C , D F whic h has muc h b etter prop erties (cf. Thm.4.4). 4.2 Monotonicit y versu s rec ursive low er b ound In case F = P R , th e monotonicit y condition can b e put in another equiv a- len t form. Prop osition 4.3. R elatio n f OftLess C , D P R ↑ g holds if and only if c on ditions i, ii in Def .4.1 hold for every total functions φ, φ i : N → N which r e cursively tend to + ∞ , i.e. ther e ar e r e cursive gr o wth mo d ulus ξ , ξ i : N → N such that ∀ N ∀ n ≥ ξ ( N ) φ ( n ) ≥ N , ∀ N ∀ n ≥ ξ i ( N ) φ i ( n ) ≥ N Pr o of. ⇒ . If φ is total r ecursiv e and m on otone increasing and tend s to + ∞ then it tends r ecur siv ely to + ∞ : a p ossible recursiv e gro wth mo dulu s is ξ ( N ) = least x su c h that φ ( x ) ≥ N ⇐ . O b serv e that an y total φ ∈ P R whic h tends recursiv ely to + ∞ has a total recurs ive minorant ψ which also tend s to + ∞ , namely ψ (0) = ϕ (0) , ψ ( N + 1) = ϕ ( ξ (1 + ψ ( N ))) where ξ is a recursive gro wth mo du lus of ϕ . Of course, if tru e for ψ , conditions i, ii are also true for φ . 16 4.3 T ransitivit y It is clear that if C 6 = D th en OftLess C , D F and O f tLess C , D F ↑ ma y n ot b e transitiv e, h ence may n ot b e orderin gs. Ho wev er, we ha ve the follo wing result. Theorem 4.4 (T ransitivit y theorem) . 1. L et B , C , D b e syntactic al classes and F , G b e c ountable classes of func- tions c ontaining the identity function I d : N → N . Then, i. e OftLess B , C F f OftLess C , D G g = ⇒ e OftLess B , D G g ii. e OftLess B , C F ↑ f OftLess C , D G ↑ g = ⇒    e OftLess B , D G ↑ g e OftLess B , D F ↑ g In c ase F is r e cursively close d by ne gative tr anslation of the output, i.e. φ ∈ F ⇒ ∀ c max(0 , φ − c ) ∈ F and ther e exists a total r e cursive function θ : N 2 → N su ch that max(0 , φ i − c ) = φ θ ( i,c ) then iii. e ≤ ct f OftLess C , D F g = ⇒ e OftLess C , D F g In c ase F is r e cursively close d by ne gative tr anslation of the output and also by ne gative tr ansl ation of the input, i.e. φ ∈ F ⇒ ∀ c x 7→ φ (max(0 , x − c )) ∈ F and ther e exists a total r e cursive function ζ : N 2 → N such that, for al l x , φ i (max(0 , x − c )) = φ ζ ( i,c ) ( x ) then iv . e ≤ ct f OftLess C , D F ↑ g ≤ ct h = ⇒ e OftLess C , D F ↑ h 2. C ase C = D . R elations OftLess C , C F and OftLess C , C F ↑ ar e strict or d erings. Pr o of. 1i. S upp ose e OftLess B , C F f OftLess C , D G g . Ob serv e that { x : e ( x ) < f ( x ) } ∩ { x : f ( x ) < φ ( g ( x )) } ⊆ { x : e ( x ) < φ ( g ( x )) } Since I d ∈ F , the sets on the left are r esp ectiv ely constructiv ely ( B , C )- dense and ( C , D )-dense, uniformly in φ . Applying Prop.3.13, we see that { x : e ( x ) < φ ( g ( x )) } is constructiv ely ( B , D )-den se, wh ence e Of tLess B , D G g . 1ii. T he ab o v e argum en t also giv es e OftLess B , D G ↑ g . T o get e OftLess B , D F ↑ g , argue as ab o v e and observ e that { x : e ( x ) < φ ( f ( x )) } ∩ { x : f ( x ) < g ( x ) } ⊆ { x : e ( x ) < φ ( g ( x )) } 17 whenev er φ is monotone increasing. 1iii. Let c b e suc h that e ( x ) ≤ f ( x ) + c for all x . If φ ∈ F is total and tends to + ∞ , so is its negativ e outpu t translation b φ c ( z ) = max { 0 , φ ( z ) − c } Supp ose f ( x ) < b φ c ( g ( x )). Th en b φ c ( g ( x )) > 0 so that b φ c ( g ( x )) = φ ( g ( x )) − c f ( x ) < φ ( g ( x )) − c e ( x ) ≤ f ( x ) + c < φ ( g ( x )) This prov es th e follo wing inclusion { x : f ( x ) < b φ c ( g ( x )) } ⊆ { x : e ( x ) < φ ( g ( x ) } (1) Relation f OftLess C , D F g insures that { x : f ( x ) < b φ c ( g ( x )) } is constructiv ely ( C , D )-dense. Inclusion (1) implies that the same is true with { x : e ( x ) < φ ( g ( x )) } . S ince a co de for b φ c is recursively obtained from a co de for φ , this pro ves e Of tLess C , D F g . 1iv. Let c b e no w s u c h that e ( x ) ≤ f ( x ) + c and g ( x ) ≤ h ( x ) + c for all x . If φ ∈ F is total, monotone increasing and tends to + ∞ , so is its negativ e input and output translation e φ c ( z ) = max { 0 , φ (max(0 , z − c )) − c } Supp ose f ( x ) < e φ c ( g ( x )). Th en e φ c ( g ( x )) > 0 so that e φ c ( g ( x )) = φ (max (0 , g ( x ) − c ) − c f ( x ) < φ (max(0 , g ( x ) − c ) − c e ( x ) ≤ f ( x ) + c < φ (max(0 , g ( x ) − c ) No w, g ( x ) ≤ h ( x ) + c and φ is monotone increasing, hence e ( x ) < φ (max(0 , g ( x ) − c ) ≤ φ (max(0 , h ( x )) = φ ( h ( x )) This prov es in clusion { x : f ( x ) < e φ c ( g ( x )) } ⊆ { x : e ( x ) < φ ( h ( x ) } (2) Relation f OftLess C , D F ↑ g insures that { x : f ( x ) < e φ c ( g ( x )) } is constructiv ely ( D , E )-dense. Inclusion (2) implies that the same is true with { x : e ( x ) < φ ( h ( x )) } . Since a co de for e φ c is recursiv ely obtained fr om a code for φ , this pro ves e Of tLess C , D F ↑ h . 2. T ransitivit y of OftLess C , C F and O ftLess C , C F ↑ is an obvio us consequence of 1i–ii. As for irreﬂexivit y , arguing with φ = I d (which is in F ), w e s ee that f ( x ) < φ ( f ( x )) is imp ossible, so that f Of tLess C , C F f and f OftLess D , C F ↑ f are alw a ys false. Thus, OftLess C , C F and OftLess C , C F ↑ are strict orderings. 18 4.4 Orderings ≪ C , D F , ≪ C , D F ↑ on maps N → N P oin ts iii-iv of the ab o v e theorem sh o w that taking intersect ion with th e “up to a constan t” ord ering ≤ ct transforms the relations OftLess C , D F and OftLess C , D F ↑ in to strict ord erings ≪ C , D F and ≪ C , D F ↑ . Deﬁnition 4.5. ≪ C , D F and ≪ C , D F ↑ are the intersectio ns of the OftLess C , D F and OftLess C , D F ↑ relations with the ≤ ct ordering on total maps N → N . Theorem 4.6 ( ≪ C , D F and ≪ C , D F ↑ are strict orderings) . L et A , B , C , D b e syntactic al classes and let F , G b e c ountable classes of func- tions N → N which c ontain I d and which ar e r e cursively close d by output and input tr anslation (cf. Thm.4.4) r elative to some enumer ations of F , G . Then i. e ≪ B , C F f ≪ C , D G g ⇒ e ≪ B , D G g ii. e ≪ B , C F ↑ f ≪ C , D G ↑ g ⇒    e ≪ B , D F ↑ g e ≪ B , D G ↑ g iii. e ≤ ct f ≪ C , D G g ⇒ e ≪ C , D G g iv . e ≤ ct f ≪ C , D G ↑ g ≤ ct h ⇒ e ≪ C , D G ↑ h In p articular, pr op erties iii and i v c an b e applie d with ≤ ct r epla c e d by ≪ A , B F or ≪ A , B F ↑ . R elat ions ≪ C , D F and ≪ C , D F ↑ ar e strict or derings such that v . f ≪ C , D F g ⇒ f ≪ C , D F ↑ g ⇒ f < ct g Pr o of. Conditions i–iv are s tr aigh tforw ard consequ en ces of the similar con- ditions in Thm.4.4. Condition i–ii yields transitivity of ≪ C , D F and ≪ C , D F ↑ . Implication f ≪ C , D F g ⇒ f ≪ C , D F ↑ g is trivial. Let’s p ro v e that ≪ C , D F re- ﬁnes < ct (and not merely ≤ ct ). Supp ose f ≪ C , D F g. Then f ≤ ct g . Also, lett ing ψ ( z ) = max (0 , z − c ), w e see that { x : f ( x ) < ψ ( g ( x ) } = { x : f ( x ) < g ( x ) − c } is inﬁnite, hence the condition ∀ x g ( x ) ≤ f ( x ) + c is imp ossible, whatev er b e c . T h us , f < ct g . The ab o v e th eorem sh o ws that comp osition of the orderin gs ≪ C , D F and ≪ C , D F ↑ is remark ably ﬂexible. In particular, 19 Corollary 4.7. If 1 ≤ i ≤ n and 1 ≤ j < k ≤ m ≤ n then f 0 ≪ D 1 , E 1 F 1 ↑ . . . ≪ D n , E n F n ↑ f n ⇒ f 0 ≪ D i , E i F i ↑ f n f 0 ≪ C 1 , C 2 F 2 ↑ . . . ≪ C n − 1 , C n F n ↑ f n ⇒ f 0 ≪ C j , C m F k ↑ f n 4.5 Left comp osition and ≪ C , D F Def.4.1, 4.5 compare total fun ctions f , g : N → N via the asso ciated sets { x : f ( x ) < φ ( g ( x )) } for φ ∈ F . O ne could also compare f , g via the sets { x : φ ( f ( x )) < g ( x ) } for φ ∈ F . Similar prop erties could b e deriv ed. Though we shall not use it in the sequel, there is a prop ert y of ≪ C , D F and ≪ C , D F ↑ whic h is inte resting on its o wn and give s an alternativ e deﬁn ition of ≪ C , D F and ≪ C , D F ↑ where the inequalit y f ( x ) < φ ( g ( x )) gets a sym m etric form ψ ( f ( x )) < φ ( g ( x )) in v olving functions ψ , φ on b oth sides of the inequalit y . W e prov e it in case F is P R or M in P R . Prop osition 4.8. L et C , D b e syntactic al classes and F b e P R or M in P R . L et ψ : N → N b e a total r e cursive function. Then f ≪ C , D F g ⇒ ψ ◦ f ≪ C , D F g f ≪ C , D F ↑ g ⇒ ψ ◦ f ≪ C , D F ↑ g Mor e over, the c ons tructive density aﬀer ent to the r elations ψ ◦ f ≪ C , D F g and ψ ◦ f ≪ C , D F ↑ g is uniform in ψ . Pr o of. 1. Let φ : N → N b e a total fun ction in F whic h tend to + ∞ . W e pro ve that { x : ψ ( f ( x )) < φ ( g ( x )) } is constructiv ely ( C , D )-dense. Set ψ ′ ( z ) = max( z , max { ψ ( u ) : u ≤ z } ). Then ψ ′ ≥ I d is total recursiv e, monotone increasing and u n b ounded. Sin ce ψ ≤ ψ ′ , we ha v e { x : ψ ′ ( f ( x )) < φ ( g ( x )) } ⊆ { x : ψ ( f ( x )) < φ ( g ( x )) } (3) 2. Deﬁne α, ζ : N → N as follo ws: α ( z ) = largest u su c h that ψ ′ ( u ) ≤ φ ( z ) ζ ( z ) = smallest s su c h th at ψ ′ is constan t on [ s, α ( z )] Since φ ( z ) and ψ ′ ( z ) tend to + ∞ so do α ( z ) and ζ . Also, ∀ u < ζ ( z ) ψ ′ ( u ) < ψ ′ ( ζ ( z )) = ψ ′ ( α ( z )) ≤ φ ( z ) (4) Finally , α and ζ are in F . If F = P R , th is is trivial. If F = M in P R and φ ( x ) = min t φ t ( x ) then observe that α ( x ) = min t α t ( x ) and ζ ( x ) = min t ζ t ( x ) (wher e α t , ζ t are deﬁned from ψ , φ t as are α, ζ from ψ , φ ). 20 3. Condition (4 ) app lied to z = g ( x ) , u = f ( x ) insu res f ( x ) < ζ ( g ( x )) ⇒ ψ ′ ( f ( x )) < φ ( g ( x )) whence { x : f ( x ) < ζ ( g ( x )) } ⊆ { x : ψ ′ ( f ( x )) < φ ( g ( x )) } (5) Condition f OftLess C , D F g applied to ζ insures that { x : f ( x ) < ζ ( g ( x )) } is constructiv ely ( C , D )-dense. Usin g inclusions (5) and (3), we see that so is { x : ψ ( f ( x )) < φ ( g ( x )) } . 4. In case f OftLess C , D F ↑ g , then φ is monotone increasing. Since ψ is also monotone increasing, so are α, ζ and we get ψ ◦ f OftLess C , D F g . 5. Finally , observ e that all the constru ction is un iform in ψ and φ . 5 F unctional Kolmogoro v complexit y The purp ose of this section is to reconsider the oracular v ersion of Kol- mogoro v complexit y . W e shall view the oracle as a p arameter in a second order v arian t of conditional Kolmogoro v complexit y . 5.1 Kolmogoro v complexit y of a functional Deﬁnition 5.1. Let X b e a b asic set. The Kolmogoro v complexit y K F : X × P ( N ) → N asso ciated to a partial functional F : { 0 , 1 } ∗ × P ( N ) → X is deﬁned as follo ws: K F ( x || A ) = smallest | p | su c h that ( F ( p , A ) = x ) Note 5.2. 1. F orgetting the A , w e get the classical notion K F ( x ) with F : { 0 , 1 } ∗ → X . F reezing th e A also leads to the classical oracular notion. Th is is the con- ten ts of the next obvious prop osition and of T hm.7.1 b elo w. 2. The double bar || is used so as to get no confusion with usual conditional Kolmogoro v complexit y where the condition is a ﬁ rst-order ob ject. 3. The abov e deﬁnition can ob viously b e extended to cond itional Kol- mogoro v complexit y K F ( x | y || A ) wh ere F : { 0 , 1 } ∗ × Y × P ( N ) → X . Prop osition 5.3. L et F b e as in Def.5.1. F or A ∈ P ( N ) , denote F A : { 0 , 1 } ∗ → X the function su c h that F A ( p ) = F ( p , A ) . Then, for al l x ∈ X , K F A ( x ) = K F ( x || A ) 21 5.2 F unctional inv ariance theorem The u sual pro of of the inv a riance theorem (Kolmogoro v, 1965 [8 ]) extends easily when considering partial computable fu nctionals { 0 , 1 } ∗ × P ( N ) → N in place of partial r ecursiv e functions { 0 , 1 } ∗ → N , leading to what w e call functional Kolmo go r ov c omplexity and denote K ( x || A ). Theorem 5.4 (F unctional In v ariance Theorem) . 1. L et F b e the family of p ar tial c omputable functionals { 0 , 1 } ∗ × P ( N ) → X . When F varies in F , ther e is a le ast K F up to an additive c onsta nt: ∃ F ∈ F ∀ G ∈ F K F ≤ ct K G Such an F i s said to b e optimal in F . We let K ( || ) b e K F wher e F is some ﬁxe d optima l f u nctional. 2. L et ( F k ) k ∈ N b e a p artial c omputable enu mer ation of P C { 0 , 1 } ∗ × P ( N ) → X . L et U : { 0 , 1 } ∗ × P ( N ) → X b e such that U (0 k 1 p , A ) = F k ( p , A ) U (0 k ) = 0 Then U is optimal in P C { 0 , 1 } ∗ × P ( N ) → X . Pr o of. It clearly suﬃ ces to prov e P oin t 2. T h e usual pr o of of th e classical in v ariance theorem giv es indeed th e functional version stated ab o v e. K F k ( x || A ) = min {| p | : F k ( p , A ) = x } = min {| p | : U (0 k 1 p , A ) = x } = min {| 0 k 1 p | − k − 1 : U (0 k 1 p , A ) = x } ≥ min {| q | − k − 1 : U ( q , A ) = x } = min {| q | : U ( q , A ) = x } − k − 1 = K U ( x || A ) − k − 1 Whence K U ≤ K F k + k + 1 and therefore K U ≤ ct K F k . Remark 5.5. 1. Ob viously , K F ( x || A ) d o es dep end on A . F or example, if x ∈ N is incompressible then K F ( x || ∅ ) = ct log( x ) whereas K F ( x || { x } ) = ct 0. The conte nts of the functional inv a riance theorem is that, for some F ’s (the optimal ones) the num b er max {K F ( x || A ) − K G ( x || A ) : x ∈ N , A ∈ P ( N ) } is ﬁnite for any giv en G . 2. F or th e fu nctional inv ariance theorem, we only h av e to supp ose the en umeration ( F k ) k ∈ N to b e p artial computable as a fun ctional N × { 0 , 1 } ∗ × P ( N ) → X . There is no need that it b e acc eptable (cf. Def.2.5). 22 As for the usual Kolmogoro v complexit y , c ompu table ap p ro ximation from ab o v e is p ossible. Prop osition 5.6. Ther e exists a total c omputable functional ( x , t, A ) ∈ X × P ( N ) × N 7→ K t ( x || A ) which is de cr e asing with r esp e ct to t and such that, for al l x , A , K ( x || A ) = min {K t ( x || A ) : t ∈ N } Pr o of. Letting K = K U where K ∈ P C { 0 , 1 } ∗ → N , set B ( x , t, A ) = {| p | : | p | ≤ t ∧ U ( p , A ) = x ∧ U ( p , A ) halts in ≤ t steps } T ( x , A ) = smallest t suc h that B ( x , t, A ) 6 = ∅ K t ( x || A ) = smallest | p | ∈ B ( x , t, A ) ∪ B ( x , T ( x , A ) , A ) 6 The M in/ M ax hierarc h y of Kolmogoro v com- plexities Inﬁnite computations in relation with K olmogoro v complexit y we re ﬁrst considered in Chaitin, 1976 [3] and S olo v a y , 1977 [16]. Bec her & Daicz & Chaitin, 2001 [1], int ro duced a v arian t H ∞ of th e p reﬁx v ersion of Kol- mogoro v complexit y b y allo wing programs leading to p ossibly inﬁnite com- putations but ﬁnite output (i.e. remo v e the sole halting condition). This v arian t satisﬁes H ∅ ′ < ct H ∞ < ct H (cf. [1, 2]). In [5], 20 04, w e in tro duced a mac h in e-free d eﬁnition K max of the usu al (n on preﬁx) Kolmogoro v v ersion K ∞ , together with a dual v ersion K min . The pro of in [2] of the abov e inequalitie s extends easily to the K setting for K max . Ho wev er, a diﬀerent argumen t is required in ord er to get the K min v ersion (cf. [5]). 6.1 M in/ M ax Kolmogoro v complexities The follo wing deﬁnitions and th eorems collec ts material from [5]. Th e clas- sical wa y to d eﬁ ne Kolmogoro v complexit y extends directly to these classes. Theorem 6.1 ( M in/ M ax Inv ariance theorem) . 1. L et F b e M in { 0 , 1 } ∗ → N P R or M ax { 0 , 1 } ∗ → N P R (cf. De f.2.13). When φ varies i n F ther e is a le ast K φ , up to an additive c onstant (cf. Notation 2.1): ∃ φ ∈ M in { 0 , 1 } ∗ → N P R ∀ ψ ∈ M in { 0 , 1 } ∗ → N P R K φ ≤ ct K ψ ∃ φ ∈ M ax { 0 , 1 } ∗ → N P R ∀ ψ ∈ M ax { 0 , 1 } ∗ → N P R A K φ ≤ ct K ψ 23 Such φ ’s ar e said to optimal in F . We let - K min denote K φ wher e φ is any function optimal in M in { 0 , 1 } ∗ → N P R , - K max denote K φ wher e φ is any function optimal in M ax { 0 , 1 } ∗ → N P R . 2. Supp ose ( φ k ) k ∈ N is an enumer ation of M in { 0 , 1 } ∗ → N P R such that the function ( k , p ) 7→ φ k ( p ) is in M in N ×{ 0 , 1 } ∗ → N P R . L et U min b e such that U min (0 k 1 p ) = φ k ( p ) U min (0 k ) = φ k ( λ ) Then U min is optimal in M in { 0 , 1 } ∗ → N P R . Idem with M ax { 0 , 1 } ∗ → N P R . 3. R elativizing to an or acle A ⊆ N , one similarly deﬁnes K A min and K A max and the analo g of Point 2 also holds. Remark 6.2 ([5]) . There exists optimal functions for M ax { 0 , 1 } ∗ → N P R of the form max f w here f : { 0 , 1 } ∗ × N → N is total r e c ursive . This is false for M in { 0 , 1 } ∗ → N P R . Relativizing to the s u ccessiv e ju mps oracle s, we get an inﬁnite family of Kolmogoro v complexities for which holds a h ierarch y theorem. Theorem 6.3 (The M in/ M ax Kolmogoro v hierarc hy , [5]) . log > ct K > ct K min K max > ct K ∅ ′ > ct K ∅ ′ min K ∅ ′ max > ct K ∅ ′′ > ct K ∅ ′′ min K ∅ ′′ max > ct K ∅ ′′′ ... Strict inequalities K > ct K max > ct K ∅ ′ > ct K ∅ ′ max > ct K ∅ ′′ w ere ﬁrst pro ved by Bec her & Chaitin, 2001–2002 [1] (for the p reﬁx v arian ts). The main application of the ≪ C , D F and ≪ C , D F ↑ orderings in tro du ced in § 4 is a strong imp ro v ement of th is hierarc hy theorem, cf. Thm.8.14. Finally , we shall n eed the follo wing resu lt (cf. [5], or [1] as concerns K max ). Theorem 6.4. K, K min , K max ar e r e cursive in ∅ ′ . 6.2 F unctional M in/ M ax Kolmogoro v complexities The Inv ariance T heorems f or M ax P R and M in P R (cf. Thm.6.1) admit functional v ersions, the p ro ofs of which are exactly the same as that in Thm.5.4. Theorem 6.5 ( M in/ M ax F unctional In v ariance Th eorem) . 1. When F : { 0 , 1 } ∗ × P ( N ) → N varies over M in { 0 , 1 } ∗ × P ( N ) → N P C or over M ax { 0 , 1 } ∗ × P ( N ) → N P C , ther e is a le ast K F up to an additive c onstant: ∃ F ∈ M in P ( N ) ×{ 0 , 1 } ∗ → N P C ∀ G ∈ M in P ( N ) ×{ 0 , 1 } ∗ → N P C K F ≤ ct K G ∃ F ∈ M ax P ( N ) ×{ 0 , 1 } ∗ → N P C ∀ G ∈ M ax P ( N ) ×{ 0 , 1 } ∗ → N P C K F ≤ ct K G 24 Such an F is said to b e optimal in M in P ( N ) ×{ 0 , 1 } ∗ → N P C or in M ax P ( N ) ×{ 0 , 1 } ∗ → N P C . We let K min ( || ) = K F and K max ( || ) = K F b e some ﬁxe d such optimal functionals. 2. L et ( F k ) k ∈ N b e an enumer ation of M in P ( N ) ×{ 0 , 1 } ∗ → N P C which is itself in M in N × P ( N ) ×{ 0 , 1 } ∗ → N P C . L et U min b e such that U min (0 k 1 p , A ) = F k ( p , A ) U min (0 k ) = 0 Then U min is optimal in M in { 0 , 1 } ∗ → N P C . One deﬁnes similarly U max which is optimal i n M ax P C . Remark 6.6. 1. Using the tec hniqu e of [5], w e see that th ere exists optimal fu nctionals for M ax P C of the form max F w here F : { 0 , 1 } ∗ × N × P ( N ) → N is total r e cursive . This is false for M in P C . 2. T he inclusions P C ⊆ M ax P C ∩ M in P C imply that K min ( || ) ≤ ct K ( || ) and K max ( || ) ≤ ct K ( || ). Also, as is well -known, K ( || ) ≤ ct log. W e can c ho ose K min , K max so that th e constant is 0, i.e. for all x and A , K min ( x || A ) ≤ K ( x || A ) ≤ log , K max ( x || A ) ≤ K ( x || A ) ≤ log 3. In fact, th e Min/Max hierarch y Theorem 6.3 extends to the f u nctional setting. In § 8.6 we shall pr o v e a muc h stronger result, cf. Thm.8.14. 7 F unctional v ersus oracular F unctional Kolmogoro v complexities allo w f or a u niform choi ce of oracular Kolmogoro v complexities. The b eneﬁt of such a u niform choi ce was dev el- op ed in § 1.2 and is illustrated in th e hierarc hy theorem in § 8.6. Theorem 7.1. Denote K A , K A min , K A max : X → N the Kolmo gor ov c om- plexities asso ciate d to the families P R A of p artia l A -r e cursive functions and the families M in P R A , M ax P R A obtaine d by applic ation of the min and max op e r ators to P R A, X × N → N . F or al l A ⊆ N , we have K A = ct K ( || A ) , K A min = ct K min ( || A ) , K A max = ct K max ( || A ) i.e. ∀ A ∈ P ( N ) ∃ c A ∀ x    | K A ( x ) − K ( x || A ) | ≤ c A | K A min ( x ) − K min ( x || A ) | ) ≤ c A | K A max ( x ) − K max ( x || A ) | ) ≤ c A Pr o of. 1. W e let ( F k ) k ∈ N and U b e as in Po int 2 of T hm.5.4 and let F A k ( x ) = F k ( x , A ) and U A ( x ) = U ( x , A ). Th e s equ ence ( F A k ) k ∈ N is an en umeration of the family P R A, { 0 , 1 } ∗ → N of partial A -recurs ive f unctions { 0 , 1 } ∗ → X , whic h 25 is partial A -recursive as a fun ction N × { 0 , 1 } ∗ → X . Since U A (0 k 1 p ) = F A k ( p ), the classical in v ariance theorem, in its relativized version, ins ures that U A is optimal in P R A, { 0 , 1 } ∗ → N , whence K A = ct K U A . No w, U is optimal in P C P ( N ) ×{ 0 , 1 } ∗ → N , wh ence K ( || ) = ct K U ( || ). Prop.5.3 insu res K U A ( x ) = K U ( x || A ), w hence K A = ct K ( || A ). 2. The M in and M ax cases are similar. 8 Reﬁning the oracu lar Min/Max hierarc h y with the ≪ , ≪ ↑ orderings 8.1 Barzdins’ theorem in a uniform setting The next lemma is essen tially Barzdins’ result cited in § 3.1 . In ord er (p oin t 2) to get a relativized resu lt with θ recursiv e r ather than merely A -recursive, w e shall lo ok at the oracle A as a parameter and use uniform Kolmogoro v complexit y , cf. § 5.2, § 7. Lemma 8.1. 1. If ϕ : N → N is total r e cursive and tends to + ∞ then { x : K ( x ) < ϕ ( x ) } is an r.e. set which is c onstructively Σ 0 1 -dense. Mor e over, this r esult is uniform in ϕ . In fact, let ( ϕ i ) i ∈ N and ( W i ) i ∈ N b e ac c epta ble enumer ations of p artial r e cursive functions N → N and r.e. subsets of N , ther e ar e total r e cursive functions ξ : N → N and θ : N 2 → N such that i. ∀ i { x ∈ domain ( ϕ i ) : K ( x ) < ϕ i ( x ) } = W ξ ( i ) ii. ∀ i, j ( ϕ i is unb ounde d on domain ( ϕ i ) ∩ W j ⇒ ( W θ ( i,j ) is inﬁnite ∧ W θ ( i,j ) ⊆ W j ∩ { x : K ( x ) < ϕ i ( x ) } )) 2. Consider se c o nd or der Kolmo gor ov c o mplexity K ( x || A ) and an ac c eptable enumer atio n (Φ i ) i ∈ N of p a rtial c o mputable functionals N × P ( N ) → N . Using Thm.7.1 and Pr op.2.7, we shal l c onsider K ( x || A ) as uniform Kolmo gor ov r elat ivization K A ( x ) and Φ i ( x , A ) as a uniform or acle A p artia l r e cursive function ϕ A i ( x ) . We also denote W A i = domain ( ϕ A i ) . Point 1 r elat ivize s unif ormly, i.e., the ab ove total r e cursive functions ξ : N → N and θ : N 2 → N c an b e taken so as to satisfy all p ossible r elat ivi ze d c ond itions, i.e. i. ∀ i ∀ A { x ∈ domain ( ϕ A i ) : K A ( x ) < ϕ A i ( x ) } = W A ξ ( i ) ii. ∀ i, j ∀ A ( ϕ A i is unb ounde d on domain ( ϕ A i ) ∩ W A j ⇒ ( W A θ ( i,j ) is inﬁnite ∧ W A θ ( i,j ) ⊆ W A j ∩ { x : K A ( x ) < ϕ A i ( x ) } )) 26 Note 8.2. Lemma 8.1 is optimal in the sense that there is no p ossible (Π 0 1 , E )-densit y resu lt for { x : K ( x ) < ϕ ( x ) } since this set has Π 0 1 comple- men t. Pr o of. Point 1i. Let K = K U where U ∈ P R { 0 , 1 } ∗ → N . Then K ( x ) < ϕ i ( x ) ⇔ ∃ p ( | p | < ϕ i ( x ) ∧ U ( p ) = x ) whic h is a Σ 0 1 condition. Therefore { ( i, x ) : K ( x ) < ϕ i ( x ) } is r.e. and the parametrization theorem yields the desired total recursive function ξ . Point 1ii. In order to pr o v e constructiv e Σ 0 1 -densit y u n iformly in ϕ , w e ﬁr st deﬁne a partial recurs iv e function α : N 2 × { 0 , 1 } ∗ → N suc h that if there exists some x ∈ W j suc h th at ϕ i ( x ) ≥ 2 | p | th en α ( i, j, p ) is su c h an x , else α ( i, j, p ) is un deﬁned. Then we shall u se the facts that K ≤ ct K p 7→ α ( i,j, p ) , K p 7→ α ( i,j, p ) ( α ( i, j, p )) ≤ ct | p | to get an inequalit y K ( α ( i, j, p )) ≤ ct | p | from whic h K ( α ( i, j, p )) < ϕ i ( p ) can b e deduced. a. The formal deﬁnition of α as a partial recursive function is as follo ws. Denote W j,t the ﬁnite sub s et of W j obtained after t steps of its standard en umeration. L et Z t : N 2 × { 0 , 1 } ∗ → P ( N ) b e su c h that Z t ( i, j, p ) = { x ∈ W j,t : ϕ i ( x ) halts in ≤ t steps and is > 2 | p |} Clearly , { ( t, i, j, p ) : Z t ( i, j, p ) 6 = ∅} is a recursive sub set of N 3 × { 0 , 1 } ∗ . Th u s , w e can deﬁn e the p artial recursive function α as follo ws: domain ( α ) = { ( i, j, p ) : ∃ t Z t ( i, j, p ) 6 = ∅} α ( i, j, p ) = the ﬁrst elemen t in Z t ( i, j, p ) where t is least suc h that Z t ( i, j, p ) 6 = ∅ Let ( ψ i ) i ∈ N b e an acceptable enumeratio n of P R { 0 , 1 } ∗ → N . Since α is partial recursiv e, there exists a total recursive function η : N 2 → N suc h that α ( i, j, p ) = ψ η ( i,j ) ( p ) f or all i, j, p . Finally , we let θ b e a total recursiv e function su c h that W θ ( i,j ) = ψ η ( i,j ) ( { p : | p | > η ( i, j ) ∧ ( i, j, p ) ∈ domain ( ψ η ( i,j ) ) } ) Since α and ψ η ( i,j ) tak e v alues in W j , we ha ve W θ ( i,j ) ⊆ W j for all i, j . b. Let U : { 0 , 1 } ∗ → N b e such that U (0 k 1 p ) = ψ k ( p ) and U (0 k ) = ψ k ( λ ) (where λ is the empty wo rd ). The usual in v ariance theorem insures that U 27 is optimal. Th us, we can (and shall) su pp ose that K = K U . Since α ( i, j, p ) = ψ η ( i,j ) ( p ), w e ha v e α ( i, j, p ) = U (0 η ( i,j ) 1 p ). Thus, for an y ( i, j, p ) ∈ domain ( α ), K ( α ( i, j, p )) = K U ( U (0 η ( i,j ) 1 p )) ≤ | 0 η ( i,j ) 1 p | = | p | + η ( i, j ) + 1 c. S upp ose no w that ϕ i is unboun d ed on domain ( ϕ i ) ∩ W j . Then, for all p , the set Z t ( i, j, p ) is non empty for t big en ou gh , so that α ( i, j, p ) = ψ η ( i,j ) ( p ) is deﬁned for all p . Also, due to the deﬁn ition of Z t , we see that α ( i, j, p ) tends to + ∞ with the length of p . I n particular, W θ ( i,j ) is inﬁn ite. F rom the deﬁn ition of α , w e see that ϕ i ( α ( i, j, p )) > 2 | p | . Usin g b , w e see that for all | p | > η ( i, j ), we ha v e K ( α ( i, j, p )) ≤ 2 p < ϕ ( α ( i, j, p )). This prov es th at W θ ( i,j ) is included in { x : K ( x ) < ϕ i ( x ) } . Th u s , W θ ( i,j ) is an in ﬁnite r.e. set included in W j ∩ { x : K ( x ) < ϕ i ( x ) } . Point 2i. Let K ( || ) = K U where U ∈ P C P ( N ) ×{ 0 , 1 } ∗ → N . Then K A ( x ) < ϕ A i ( x ) ⇔ ∃ p ( | p | < ϕ A i x ∧ U ( p , A ) = x ) whic h is a Σ 0 1 condition. T herefore { ( i, x, A ) : K A ( x ) < ϕ A i ( x ) } is Σ 0 1 and the parametrization prop erty (cf. Def.2. 5 ) yields the desir ed total recursive function ξ . Point 2ii. The pro of is similar to that of Poin t 1ii. Just add everywhere a second order argumen t A v arying in P ( N ) and use the parametrization prop erty of Def.2.5. Thus, α is now a partial computable fun ctional α : N 2 × { 0 , 1 } ∗ × P ( N ) → N The enumeration ( ψ i ) i ∈ N no w b ecomes an en umeration (Ψ i ) i ∈ N of the partial computable f unctionals { 0 , 1 } ∗ × P ( N ) → N . T he total recursive functions η , θ are no w suc h that α ( i, j, p , A ) = Ψ η ( i,j ) ( p , A ) and W A θ ( i,j ) = { Ψ η ( i,j ) ( p , A ) : p suc h that ( i, j, p , A ) ∈ domain ( α ) ∧ | p | > η ( i, j ) } The argumen ts in b, c ab o v e go thr ough w ith the sup erscrip t A ev erywh ere and with U (cf. pro of of Poin t 2i ab ov e) in place of U . Remark 8.3. 1. Lemma 8.1 still h olds for φ ∈ M ax P R in place of ϕ ∈ P R . Ho w ev er, this do es n ot really add : an easy argument shows that if φ ∈ M ax P R and W j ⊆ domain ( φ ) is inﬁn ite then there exists an inﬁn ite W k ⊆ W j and ϕ i ∈ P R suc h that W k ⊆ domain ( ϕ i ) and ϕ i ( x ) ≤ φ ( x ) for all x ∈ domain ( ϕ i ). Moreo v er, k and i can b e giv en by tota l recurs ive functions dep ending on j and a co d e for φ in M ax P R . This also holds u niformly: r eplace ϕ by a functional Φ ∈ P C . 2. Of course, Lemma 8.1 cannot hold for φ ∈ M in P R since K is itself in M in P R . 28 8.2 Comparing K and K max ` a la Barzdins In this sub s ection and th e next on e, we now come to cen tral results of the pap er, namely , - K c an b e c omp a r e d to K max , K min via the ≪ and ≪ ↑ or d erings, - K max , K min c an b e c omp a r e d via the OftLess ↑ r elat ion. Notation 8.4. W e sh all write X is ( C 1 ∪ C 2 , D )-dense to mean X is ( C 1 , D )- dense and ( C 2 , D )-dense. Remark 8.5. Let C 1 ∨ C 2 b e the family of sets R 1 ∪ R 2 where R 1 ∈ C 1 and R 2 ∈ C 2 . If C 1 , C 2 b oth con tain the empty set (w h ic h is usually the case), then C 1 ∪ C 2 ⊆ C 1 ∨ C 2 , and therefore ( C 1 ∨ C 2 , D )-densit y (resp. constru ctiv e dens it y) alwa ys implies ( C 1 ∪ C 2 , D )-densit y (resp. constru ctiv e densit y). Con ve rs ely , ev ery inﬁnite set in C 1 ∨ C 2 con tains an inﬁnite subs et in C 1 or in C 2 , so that ( C 1 ∪ C 2 , D )-densit y implies — hence is equiv alen t to — ( C 1 ∨ C 2 , D )-densit y . Ho w ev er, this is n o more true as concerns c onst ructive density : if R 1 ∪ R 2 is inﬁ n ite one cannot decide (from co d es) which one of R 1 and R 2 is in ﬁnite. Lemma 8.6. 1. Supp ose φ : N → N is a total function in M in P R which is monotone and tends to + ∞ . Then the set { x : K max ( x ) < φ ( K ( x )) } is c onstructively (Σ 0 1 ∪ Π 0 1 , ∃ ≤ φ (Σ 0 1 ∧ Π 0 1 )) -dense (cf. Def.3.11 Point 3). Mor e over, this r esult is uniform in φ . In fact, let ( φ i ) i ∈ N and ( W i ) i ∈ N b e ac c eptable enumer atio ns of M in P R and r.e. sub sets of N . Ther e ar e total r e cursive functions θ 0 , θ 1 : N 2 → N such that, for al l i, j, k , with the notations of Def.2.12, i f φ i ∈ M in P R is total, monoto ne and tends to + ∞ then W j is inﬁnite ⇒ W ∃ ≤ φ i (Σ 0 1 ∧ Π 0 1 ) θ 0 ( i,j ) is an i nﬁnite subset of W j ∩ { x : K max ( x ) < φ ( K ( x )) } N \ W k is inﬁnite ⇒ W ∃ ≤ φ i (Σ 0 1 ∧ Π 0 1 ) θ 1 ( i,k ) is an i nﬁnite subset of ( N \ W k ) ∩ { x : K max ( x ) < φ ( K ( x )) } 2. Consider Kolmo gor ov r ela tivizations K A , K A max obtaine d fr om se c ond or d er Kolmo gor ov c omplexities K , K max (cf. Thm .7.1 ) and enumer ations ( φ A i ) i ∈ N and ( W A i ) i ∈ N of M in A P R and A -r.e. sets which c ome fr om ac c epta ble enumer atio ns of functionals in M in N × P ( N ) → N P C and of Σ 0 1 subsets of N × P ( N ) (cf. Pr op.2.7). We shal l also use notations fr om D e f.2.12. Point 1 r elativizes uniformly, i.e., the ab o ve total r e cu rsive functions θ 0 , θ 1 : N 2 → N c an b e taken so as to satisfy all p o ssible r elativize d c onditions. I.e., 29 if φ A i ∈ M in A P R is total, monoto ne and tends to + ∞ then W A j is inﬁnite ⇒ W ∃ ≤ φ A i (Σ 0 ,A 1 ∧ Π 0 ,A 1 ) θ 0 ( i,j ) ) is an inﬁnite subset of W A j ∩ { x : K A max ( x ) < φ A ( K A ( x )) } N \ W A k is inﬁnite ⇒ W ∃ ≤ φ A i (Σ 0 ,A 1 ∧ Π 0 ,A 1 ) θ 1 ( i,k ) is an i nﬁnite subset of ( N \ W A k ) ∩ { x : K A max ( x ) < φ A ( K A ( x )) } Pr o of. 1. The str ate gy. W e essen tially k eep the strateg y of th e pro of of Lemma 8.1. Th e idea is, for giv en i, j , to construct a M ax P R function α : N 2 × { 0 , 1 } ∗ suc h th at α ( i, j, p ) is in W j (or in N \ W j ) and ϕ i ( K ( α ( i, j, p ))) > 2 | p | . T hen to u se inequalities K max ≤ ct K p 7→ α ( i,j, p ) , K p 7→ α ( i,j, p ) ( α ( i, j, p )) ≤ ct | p | to get an inequalit y K max ( α ( i, j, p )) ≤ ct | p | from which K max ( α ( i, j, p )) < ϕ i ( K ( α ( i, j, p ))) can b e deduced. As w e hav e to deal with Σ 0 1 sets and with Π 0 1 sets, i.e. sets of the form W j or N \ W k , we shall deﬁ n e t wo such functions α , namely α 0 , α 1 . In ord er to get these functions in M ax P R , w e deﬁne partial recursiv e func- tions a 0 , a 1 : N 2 × { 0 , 1 } ∗ × N → N and set max a 0 = α 0 and max a 1 = α 1 . 2. A ppr oximation of φ i fr om ab ove. Let φ i ( x ) = min t ϕ i ( x, t ) wh er e ( ϕ i ) i ∈ N is an acceptable enumeration of P R N × N → N . Using the parametrizatio n theorem, let ξ : N → N b e a total r ecursiv e func- tion suc h that ϕ ξ ( i ) has domain { x : ∃ u ϕ i ( x, t ) do es halt } × N and satisﬁes ϕ ξ ( i ) ( x, 0) = ϕ i ( x, u ) where u is least su c h that ϕ i ( x, u ) do es halt ϕ ξ ( i ) ( x, t + 1) = m in( { ϕ ξ ( i ) ( x, 0) } ∪ { ϕ i ( x, v ) : v ≤ t and ϕ i ( x, v ) halts in ≤ t steps } ) Observe th at ϕ ξ ( i ) ( x, t ) is decreasing in t and φ i ( x ) = min t ϕ ξ ( i ) ( x, t ), so that ϕ ξ ( i ) ( x, t ) is a partial recursive appro ximation of φ i ( x ) fr om ab o v e. Also, for an y giv en i, x , either φ i ( x ) is un deﬁned and ϕ ξ ( i ) ( x, t ) is deﬁ n ed for no t or φ i ( x ) is deﬁned an d ϕ ξ ( i ) ( x, t ) is deﬁned for all t . 3. F unctions a ǫ and α ǫ . Denote W j,t the ﬁnite sub s et of W j obtained after t steps of its standard en umeration. Denote K t ( x ) some total, recurs ive approximati on of K ( x ) from ab o v e w hic h is decreasing in t (cf. Prop.5.6). 30 W e deﬁn e a 0 , a 1 as follo ws: a 0 ( i, j, p , 0) = the elemen t whic h app ears ﬁr s t in th e standard en umeration of W j (hence und eﬁned if W j is empty) a 0 ( i, j, p , t + 1) =                  a 0 ( i, j, p , t ) if ϕ ξ ( i ) ( K t ( a 0 ( i, j, p , t )) , t ) > 2 | p | x if ϕ ξ ( i ) ( K t ( a 0 ( i, j, p , t )) , t ) ≤ 2 | p | and x is the next element which app ears in the standard enumeratio n of W i and satisﬁes x > a 0 ( i, j, p , t ) undef ined if ϕ ξ ( i ) ( K t ( a 0 ( i, j, p , t )) , t ) is u ndeﬁned and a 1 ( i, k, p , 0) = 0 a 1 ( i, k, p , t + 1) =                  a 1 ( i, k, p , t ) if ϕ ξ ( i ) ( K t ( a 1 ( i, j, p , t )) , t ) > 2 | p | and a 1 ( i, k, p , t ) / ∈ W k ,t a 1 ( i, k, p , t ) + 1 if ϕ ξ ( i ) ( K t ( a 1 ( i, j, p , t )) , t ) ≤ 2 | p | or ( ϕ ξ ( i ) ( K t ( a 1 ( i, j, p , t )) , t ) > 2 | p | and a 1 ( i, k, p , t ) ∈ W k ,t ) undef ined if ϕ ξ ( i ) ( K t ( a 1 ( i, k, p , t )) , t ) is un deﬁned Claim. Supp ose φ i ∈ M in P R is total monotone incr e asing and tends to + ∞ . a. If W j is inﬁnite then ( p , t ) 7→ a 0 ( i, j, p , t ) and p 7→ α 0 ( i, j, p ) ar e total functions and ∀ p ( α 0 ( i, j, p ) ∈ W j ∧ φ i ( K ( α 0 ( i, j, p ))) > 2 | p | ) b. F unction a 1 is always total. If N \ W k is inﬁnite then p 7→ α 1 ( i, k, p ) is a total function and ∀ p ( α 1 ( i, k, p ) / ∈ W k ∧ φ i ( K ( α 1 ( i, k, p ))) > 2 | p | ) Pr o of of Claim. As seen in 2 ab o ve , if φ i is total so is ϕ ξ ( i ) . This in sures the total characte r of a 0 (resp. a 1 ). Fix some p . Since ϕ ξ ( i ) ( x, t ) ≥ φ i ( x ), φ i is monotone increasing and K, φ i are total and tend to + ∞ , for all large enough x and all t , w e h a v e ϕ ξ ( i ) ( K t ( x ) , t ) ≥ φ i ( K t ( x )) ≥ φ i ( K ( x )) > 2 | p | Supp ose W j is inﬁn ite. T hen th ere are elemen ts in W j whic h satisfy φ i ( K ( x )) > 2 | p | . Let x 0 ( i, j, p ) b e su c h an elemen t wh ic h app ears ﬁrst in the stan- dard enumeration of W j . It is easy to see that, for all t large en ough, w e 31 ha v e a 0 ( i, j, p , t ) = x 0 ( i, j, p ). Th us, α 0 ( i, j, p ) = x 0 ( i, j, p ) is deﬁn ed and α 0 ( i, j, p ) ∈ W j ∩ { φ i ( K ( x )) > 2 | p |} . Which pro ves P oint a of the Claim. Supp ose N \ W k is inﬁnite. Th en ther e are elemen ts in N \ W k whic h satisfy φ i ( K ( x )) > 2 | p | . Let x 1 ( i, k, p ) b e the least suc h elemen t. It is easy to see that, for all t large enough (namely , for t su c h that W k ∩ [0 , x 1 ( i, k, p )[ ⊆ W k ,t ), w e hav e a 1 ( i, k, p , t ) = x 1 ( i, k, p ). Th us, α 1 ( i, k, p ) = x 1 ( i, k, p ) is deﬁned and α 1 ( i, k, p ) ∈ ( N \ W k ) ∩ { φ i ( K ( x )) > 2 | p |} . Whic h pro ves P oint b of th e Claim. ✷ (Claim) 4. F unctions η ǫ , θ ǫ . Let ( ψ n ) n ∈ N b e an acceptable en um eration of partial r ecursiv e fu nctions { 0 , 1 } ∗ × N → N . Since a 0 , a 1 : N 2 × { 0 , 1 } ∗ × N → N are partial r ecur- siv e, the parametrization prop erty insures that there exists total r ecursiv e functions η 0 , η 1 : N 2 → N su c h th at, for all i, j, k , p , t and ǫ = 0 , 1, a ǫ ( i, j, p , t ) = ψ η ǫ ( i,j ) ( p , t ) T aking the max ov er t , and letting α ǫ = max a ǫ , we get, for all i, j, p , α ǫ ( i, j, p , t ) = (max ψ η ǫ ( i,j ) )( p ) F or all i, j, k , set Y ǫ ( i, j ) = { α ǫ ( i, j, p ) : ( i, j, p ) ∈ domain ( α ǫ ) ∧ | p | > η ǫ ( i, j ) } Using the Claim an d inequalit y K ( y ) ≤ y (wh ic h w e alw a ys can su pp ose), observ e that y ∈ Y ǫ ( i, j ) ⇔ ∃ p (2 | p | < φ i ( K ( y )) ∧ | p | > η ǫ ( i, j ) ∧ y = α ǫ ( i, j, p )) ⇔ ∃ p ( | p | < φ i ( y ) ∧ | p | > η ǫ ( i, j ) ∧ y = α ǫ ( i, j, p )) Using Prop.2.15, w e see that this is ∃ ≤ φ i (Σ 0 1 ∧ Π 0 1 ) in i, j, y (cf. Def.2.9). W e let θ 0 , θ 1 : N 2 → N b e total recursive functions suc h that W ∃ ≤ φ i (Σ 0 1 ∧ Π 0 1 )[ N ] θ ǫ ( i,j ) = Y ǫ ( i, j ) 5. P oint 1 of the L emma. Let U : { 0 , 1 } ∗ × N → N b e s u c h that U (0 n 1 p , t ) = ψ n ( p , t ) and U (0 n , t ) = ψ n ( λ, t ) (wh ere λ is the empty word). T aking th e max o v er t , w e get (max U )(0 n 1 p ) = (max ψ n )( p ) an d (max U )(0 n ) = (max ψ n )( λ ). Since the max ψ n ’s en um erate M ax { 0 , 1 } ∗ → N P R , the inv ariance th eorem 6.1 insur es that max U is optimal in M ax { 0 , 1 } ∗ → N P R . Thus, we can (and shall) supp ose th at K max = K (max U ) . Since α ǫ ( i, j, p ) = (max ψ η ǫ ( i,j ) )( p ) = (max U )(0 η ǫ ( i,j ) 1 p ), we get K max ( α ǫ ( i, j, p )) = K (max U ) ((max U )(0 η ǫ ( i,j ) 1 p ) ≤ η ǫ ( i, j ) + 1 + | p | ≤ 2 | p | in case | p | > η ǫ ( i, j ) 32 Supp ose φ i is total, monotone and tend s to + ∞ and W j (resp. N \ W k ) is inﬁnite. Usin g the last inequalit y and that from the ab o ve C laim r elativ e to ǫ = 0 (resp. ǫ = 1), we see that, for | p | > η ǫ ( i, j ), w e ha v e K (max U ) ( α ǫ ( i, j, p )) ≤ 2 | p | < φ i ( K ( α ǫ ( i, j, p )) Whic h p ro v es that W ∃ ≤ φ i (Σ 0 1 ∧ Π 0 1 )[ N ] θ ǫ ( i,j,k ) is included in { x : K max ( x ) < φ i ( K ( x )) } . Using the C laim agai n , this set is also included in W j (resp. ( N \ W k ). T his ﬁnishes the pro of of P oint 1 of the Lemma. 6. P oint 2 of the L emma. The pr o of is similar to that of P oint 1. Ju st add everywhere a second or- der argument A v a ryin g in P ( N ) and use the parametrization prop erty of Def.2.5. Thus, a 0 , a 1 are now partial computable functionals N 2 × { 0 , 1 } ∗ × P ( N ) × N → N The enumeratio n ( ψ n ) n ∈ N no w b ecomes an enumeration (Ψ n ) n ∈ N of the par- tial computable functionals { 0 , 1 } ∗ × P ( N ) → N . The total recur siv e fu nc- tions η ǫ , θ ǫ are no w such that a ǫ ( i, j, p , A, t ) = Ψ η ǫ ( i,j ) ( p , A, t ) α ǫ ( i, j, p , A ) = (max Ψ η ǫ ( i,j ) )( p , A ) W A θ ǫ ( i,j,j ) = { α ǫ ( i, j, p , A ) : ( i, j, p , A ) ∈ domain ( α ǫ ) ∧ | p | > η ǫ ( i, j ) } and U h as to b e c hanged to U ∈ P C P ( N ) ×{ 0 , 1 } ∗ → N suc h that K ( || ) = K U . The arguments for the pro of of Poin t 1 ab ov e go thr ough with the su p erscript A ev erywh ere. 8.3 Comparing K and K min ` a la Barzdins W e s hall need the f ollo wing notion to get an analog of Lemma 8.6 with K min . Deﬁnition 8.7. T he gro wth fun ction of an inﬁnite set X ⊆ N is deﬁ n ed as g r ow th X ( n ) = ( n + 1)-th p oin t of X The inﬁn ite set X h as recursive ly b ounded gro wth if g r ow th X ≤ ψ for some total recurs ive function ψ : N → N . Lemma 8.8. 1. L et’s denote g Π 0 ,A 1 the family of inﬁnite Π 0 ,A n subsets of N with A -r e cursively b ounde d gr owth . Supp o se φ : N → N is a total function in M in P R which is monotone and tends to + ∞ . Then the set { x : K min ( x ) < φ ( K ( x )) } is c onstructively (Σ 0 1 ∪ f Π 0 1 , ∃ ≤ φ (Σ 0 1 ∧ Π 0 1 )) -dense (cf. Def.3.11 Point 3). Mor e over, this r esult i s uni f orm in φ and in a r e cursive ψ b ound for the Π 0 1 33 set. In fact, let ( φ i ) i ∈ N , ( ψ m ) m ∈ N and ( W i ) i ∈ N b e ac c ep table enumer ations of M in P R , P R and of r.e. subsets of N . Ther e ar e total r e c u rsive functions θ 0 : N 2 → N and θ 1 : N 3 → N suc h that, for al l i, j, m, k , with the notations of D ef.2.12, if φ i ∈ M in P R and ψ m ∈ P R ar e total, monotone and tend to + ∞ then W j is inﬁnite ⇒ W ∃ ≤ φ i (Σ 0 1 ∧ Π 0 1 ) θ 0 ( i,j ) is an i nﬁnite subset of W j ∩ { x : K min ( x ) < φ i ( K ( x )) } N \ W k is inﬁnite and ψ m ≥ g r owt h N \ W k ⇒ W ∃ ≤ φ i (Σ 0 1 ∧ Π 0 1 ) θ 1 ( i,k ,m ) is an inﬁnite subset of ( N \ W k ) ∩ { x : K min ( x ) < φ i ( K ( x )) } 2. Consider Kolmo go r ov r elativizations K A , K A min and enu mer a tions ( φ A i ) i ∈ N and ( ψ A m ) m ∈ N and ( W A i ) i ∈ N of M in A P R , P R A and A -r.e. sets as in Point 2 of L emma 8.6. Point 1 r elativizes uniformly, i.e., the ab o ve total r e cu rsive functions θ 0 , θ 1 : N 2 → N c an b e taken so as to satisfy all p o ssible r elativize d c onditions. I.e., if φ A i , ψ A m ar e total, monotone and tend to + ∞ then W A j is inﬁnite ⇒ W ∃ ≤ φ A i (Σ 0 ,A 1 ∧ Π 0 ,A 1 ) θ 0 ( i,j ) is an inﬁnite subset of W A j ∩ { x : K A min ( x ) < φ A i ( K A ( x )) } N \ W A k is inﬁnite and ψ A m ≥ g r owt h N \ W A k ⇒ W ∃ ≤ φ A i (Σ 0 ,A 1 ∧ Π 0 ,A 1 ) θ 1 ( i,k ,m ) is an i nﬁnite subset of ( N \ W A k ) ∩ { x : K A min ( x ) < φ A ( K A ( x )) } Pr o of. 1. The str ate gy. The pro of follo ws that of Lemma 8.6 except that no w α ǫ is equal to min a ǫ and th at a 1 and α 1 also dep end on the index m of the recursiv e ma joran t ψ m of the gro wth function of the Π 0 1 set. Since α 0 (resp. α 1 ) has to b e in M in P R , i.e. is to b e recursiv ely app ro x- imated fr o m ab ove , we h a v e to force that, for giv en i, j, k, m, p , the ﬁrst deﬁned a 0 ( i, j, p , t ) (resp. a 1 ( i, k, m, p , t )) ma jorizes an elemen t x of W j (resp. N \ W k ) wh ic h is suc h that K min ( x ) < φ i ( K ( x )). T o insure this, w e c ho ose a 0 ( i, j, p , 0) (resp . a 1 ( i, k, m, p , 0)) so that the in terv al [0 , a ǫ ( i, j, p , 0)[ (resp. [0 , a ǫ ( i, k, m, p , 0)[) contai ns at least 2 2 | p | +1 p oints in { x : K min ( x ) < φ i ( K ( x )) } . 2. W e shall use the partial recurs ive app ro ximation from ab ov e ϕ ξ ( i ) ( x, t ) of φ i ( x ) deﬁ n ed in p oin t 2 of the pro of of Lemma 8.6. 3. F unctions a ǫ and α ǫ . 34 Let Z 0 ( i, j, p ) b e the set of 2 2 | p | +1 distinct elemen ts wh ic h app ear ﬁrst in the standard enumeratio n of W i . W e d eﬁne a 0 as follo ws: a 0 ( i, j, p , 0) = the largest elemen t of Z 0 ( i, j, p ) a 0 ( i, j, p , t + 1) =                a 0 ( i, j, p , t ) if ϕ ξ ( i ) ( K t ( a 0 ( i, j, p , t )) , t ) > 2 | p | x if ϕ ξ ( i ) ( K t ( a 0 ( i, j, p , t )) , t ) ≤ 2 | p | and x is the largest element of Z 0 ( i, j, p ) ∩ [0 , a 0 ( i, j, p , t )[ undef ined if ϕ ξ ( i ) ( K t ( a 0 ( i, j, p , t )) , t ) is u ndeﬁned W e now deﬁne a 1 , u sing the r ecursiv e ma joran t ψ m . a 1 ( i, k, m, p , 0) = ψ m (2 2 | p | +1 ) Let u = ϕ ξ ( i ) ( K t ( a 1 ( i, k, m, p , t )) , t ) a 1 ( i, k, m, p , t + 1) =                a 1 ( i, k, m, p , t ) if u > 2 | p | and a 1 ( i, k, m, p , t ) / ∈ W k ,t a 1 ( i, k, m, p , t ) − 1 if u ≤ 2 | p | or ( u > 2 | p | and a 1 ( i, k, m, p , t ) ∈ W k ,t ) undef ined if u is undeﬁn ed Clearly , a 0 and a 1 are partial recursive . Claim. Supp ose φ i ∈ M in P R is total monotone incr e asing and tends to + ∞ . a. If W j is inﬁnite then ( p , t ) 7→ a 0 ( i, j, p , t ) and p 7→ α 0 ( i, j, p ) ar e total functions and ∀ p ( α 0 ( i, j, p ) ∈ W j ∧ φ i ( K ( α 0 ( i, j, p ))) > 2 | p | ) b. F u nction a 1 is total. If N \ W k is inﬁnite and ψ m is a total r e cursive function such that ψ m ≥ g r ow th N \ W k then p 7→ α 1 ( i, k, m, p ) is a total function and ∀ p ( α 1 ( i, k, m, p ) / ∈ W k ∧ φ i ( K ( α 1 ( i, k, m, p ))) > 2 | p | ) Pr o of of Claim. As seen in the pr o of of Lemma 8.6, if φ i is tot al th en so are ϕ ξ ( i ) and a 0 , a 1 . Also, for an y ﬁxed p , for all large enough x and all t , w e ha v e ϕ ξ ( i ) ( K t ( x ) , t ) ≥ φ i ( K t ( x )) ≥ φ i ( K ( x )) > 2 | p | Supp ose W j is inﬁnite. Then Z 0 ( i, j, p ) con tains exactly 2 2 | p | +1 elemen ts. Let K min = K U where U ∈ M in { 0 , 1 } ∗ → N P R . S in ce there are 2 2 | p | +1 − 1 w ords 35 with length ≤ 2 | p | , there is n ecessarily some elemen t of x ∈ Z 0 ( i, j, p ) w hic h is not in U ( { q : | q | ≤ 2 | p |} ), hen ce is su ch that K min ( x ) = K U ( x ) > 2 | p | . Let x 0 ( i, j, p ) b e the largest such elemen t. It is easy to see that, for all t large enou gh , we ha v e a 0 ( i, j, p , t ) = x 0 ( i, j, p ). Thus, α 0 ( i, j, p ) = x 0 ( i, j, p ) is deﬁn ed and α 0 ( i, j, p ) ∈ W j ∩ { φ i ( K ( x )) > 2 | p |} . Whic h prov es Poi nt a of the Claim. Supp ose N \ W k is inﬁnite and ψ m is a total recursiv e fu nction suc h that ψ m ≥ g r ow th N \ W k . Th en there are 2 2 | p | +1 elemen ts of N \ W k whic h are ≤ ψ m (2 2 | p | +1 ). As ab o v e, there is n ecessarily some su c h elemen t x w h ic h is not in U ( { q : | q | ≤ 2 | p |} ), hence is suc h that K min ( x ) = K U ( x ) > 2 | p | . Let x 1 ( i, k, m, p ) b e the largest s uc h elemen t. It is easy to see that, for all t large enough (namely , for t such that W k ∩ [0 , x 1 ( i, k, m, p )[ ⊆ W k ,t ), we hav e a 1 ( i, j, m, p , t ) = x 1 ( i, j, m, p ). Thus, α 1 ( i, k, m, p ) = x 1 ( i, k, m, p ) is deﬁn ed and α 1 ( i, k, m, p ) ∈ ( N \ W k ) ∩ { φ i ( K ( x )) > 2 | p |} . Which pro v es P oin t b of the Claim. ✷ (Claim) W e conclude the p ro of of the Lemma as that of Lemma 8.6 with an alogous p oints 4,5, 6 : the sole mo diﬁ cation is to replace ev erywhere K max b y K min and the max op erator b y the min one. 8.4 Comparing K min and K max ` a la Barzdins W e shall need the follo wing result f rom [5] (Thm 7.15). Prop osition 8.9. K ≤ ct 2 K min + K max . Using Prop.8.9, Lemmas 8.6, 8.8 yield the f ollo wing corollary . Lemma 8.10. 1. L et’s denote f Π 0 1 the family of inﬁnite Π 0 1 subsets of N with r e c ursively b ounde d gr owth . Supp o se φ : N → N is a total function in M in P R which is monotone and tends to + ∞ . Then i. { x : K max ( x ) < φ ( K min ( x )) } is c onstructively (Σ 0 1 ∪ Π 0 1 , ∃ ≤ φ (Σ 0 1 ∧ Π 0 1 )) - dense. ii. { x : K min ( x ) < φ ( K max ( x )) } is c onstructively (Σ 0 1 ∪ f Π 0 1 , ∃ ≤ φ (Σ 0 1 ∧ Π 0 1 )) - dense Mor e over, this r esult is uniform in φ and, for ii, in a r e cursive b ound for the Π 0 1 set, in the sense detaile d in L emm as 8.6,8.8. 2. Consider Kolmo gor ov r elativizations K A , K A min , K A max and enumer ations ( φ A i ) i ∈ N and ( ψ A m ) m ∈ N and ( W A i ) i ∈ N of M in A P R , P R A and A -r.e. sets as in Point 2 of L emmas 8.6, 8.8. Then Point 1 r elativizes unif ormly in the sense detaile d in L emmas 8.6,8.8. 36 Pr o of. Let φ ∈ M in P R b e total, monotone increasing and unboun ded. Set θ ( x ) = min { x 4 , φ (max(0 , ⌊ x − c 2 ⌋ )) } Then θ is also a total, monotone in cr easing and unboun ded fun ction in M in P R . Also, one can recursiv ely go from a co de for φ to one for θ . Usin g Lemmas 8.6, 8.8, it suﬃces to pro ve that, for all x , K min ( x ) < θ ( K ( x )) ⇒ K min ( x ) < φ ( K max ( x )) K max ( x ) < θ ( K ( x )) ⇒ K max ( x ) < φ ( K min ( x )) W e prov e th e ﬁrst implication, the second one b eing similar. Applying Pr op.8.9, let c b e suc h that, for all x , K ( x ) < 2 K min ( x ) + K max ( x ) + c Supp ose K min ( x ) < θ ( K ( x )). Th en K min ( x ) < 1 4 K ( x ), so that K ( x ) < 2 K min ( x ) + K max ( x ) + c ≤ K ( x ) 2 + K max ( x ) + c and K ( x ) < 2 ( K max ( x ) + c ). Therefore, K min ( x ) < θ ( K ( x )) ≤ θ (2 ( K max ( x ) + c )) ≤ φ ( K max ( x )). 8.5 Syn tactical complexit y Whereas { x : K ( x ) < φ ( x ) } is r .e. wh enev er φ is partial recurs ive (cf. Lemma 8.1), the complexit y of the sets considered in Lemmas 8.6, 8.8 , 8.10 to compare K , K max , K min is muc h higher and do es in vol ve b ounded quan- tiﬁcations o v er b o olean com binations of Σ 0 1 sets as is the case in the d ensit y results obtained in th ese lemmas. Prop osition 8.11. L et φ b e a total function in M in P R . The sets { x : K max ( x ) < φ ( K ( x )) } { x : K max ( x ) < φ ( K min ( x )) } { x : K min ( x ) < φ ( K ( x )) } { x : K min ( x ) < φ ( K max ( x )) } ar e al l deﬁnable by formulas of the form ∃ ≤ log ∀ ≤ log ( A ∧ B ∧ C ) wher e A, B , C ar e Σ 0 1 ∨ Π 0 1 . In p articular, theses sets ar e ∆ 0 2 (cf. Pr op.2.11). Pr o of. Without loss of generalit y , w e can su pp ose that K max ( x ) and K min ( x ) are b oth ≤ log ( x ) for all x . Let U : N → N and V , W, ϕ : N 2 → N b e partial recursiv e fu nctions such that K = K U and K min = K α and K max = K β and φ ( x ) = min t ϕ ( x, t ) w here α ( x ) = min t V ( x, t ) and β ( x ) = min t W ( x, t ). F ollo wing a u sual conv en tion, w e shall w rite ∃ p | p |≤ x ... and ∀ p | p |≤ x ... in place 37 of ∃ p ( | p | ≤ x ∧ ... ) and ∀ p ( | p | ≤ x ⇒ ... ). Then K max ( x ) < φ ( K ( x )) if and only if ∃ p 1 | p 1 |≤ log( x ) ∃ p 2 | p 2 |≤ log( x ) ∀ q 1 | q 1 | < | p 1 | ∀ q 2 | q 2 | < | p 2 | [ U ( p 1 ) = x ∧ U ( q 1 ) 6 = x ∧ ∃ t V ( p 2 , t ) = x ∧ ∀ t ( V ( p 2 , t ) is undeﬁned or ≤ x ) ∧ ( ∀ t ( V ( q 2 , t ) is undeﬁned or 6 = x ) ∨ ∃ t V ( q 2 , t ) > x ) ∧ ∀ t ( ϕ ( | p 1 | , t ) is und eﬁ ned or | p 2 | < ϕ ( | p 1 | , t ))] Whic h is a formula of the f orm stated in the Prop osition. All th ree other cases are similar. Bounded qu an tiﬁcations o v er b o olean com b in ations of Σ 0 1 sets are also in vo lved for the s et of in tegers with K, K max , K min incompressible binary represent ations. Prop osition 8.12. The set I = { x : m in( K ( x ) , K max ( x ) , K min ( x )) ≥ ⌊ log ( x ) ⌋ − 1 } is inﬁnite and is deﬁnable b y a formula of the form ∀ ≤ log ( A ∧ B ) wher e A, B ar e Σ 0 1 ∨ Π 0 1 . In p a rticular, this set is ∆ 0 2 . Pr o of. Without loss of generalit y w e shall supp ose that K ≤ K max and K ≤ K min . The usual argumen t to get in compr essible integ ers w orks: there are P i x ) ∧ ( ∀ t ( W ( p , t ) is und eﬁned or 6 = x ) ∨ ∃ t W ( p , t ) < x )] Whic h is a formula of the f orm stated in the Prop osition. All th ree other cases are similar. Remark 8.13. I n case φ is small enough (sa y φ ( z ) ≤ z − 1), the set I is ob viously disjoin t from all fours sets considered in Prop.8.11. 38 8.6 The hierarc h y theorem W e can now pro ve the cen tral application of the OftLess ↑ relation and the ≪ and ≪ ↑ orderings. Namely , a strong h ierarc hy theorem for K, K max , K min and th eir oracular v ersions us ing the successiv e ju mps ora- cles. Whereas Thm.6.3 inv olv es the sole < ct ordering, the reﬁnm ent obtained in Thm.8.14 b elo w in volv es a c hain of more and more complex orderings which all reﬁn e < ct and are r elev a nt of Thm.4.6 and Cor.4.7. Theorem 8.14 (The hierarc hy theorem) . L et B n b e the sub class of ∆ 0 n subsets of N c onsisting of sets deﬁnable by formulas of the f orm ∃ ≤ µ (Σ 0 n ∧ Π 0 n ) wher e µ : N → N is a total function which is r e cursive in ∅ ( n − 1) . L et f Π 0 n b e the set of Π 0 n sets with ∅ ( n ) -r e c ursively b ounde d gr owt h (cf. Def.8.7). Then 1. log ≫ Σ 0 1 , Σ 0 1 PR K ≫ Σ 0 1 ∪ Π 0 1 , B 1 Min PR ↑ K max ≫ Σ 0 2 , Σ 0 2 PR ∅ ′ K ∅ ′ ... ... ≫ Σ 0 n , Σ 0 n PR ∅ (n − 1) K ∅ (n − 1) ≫ Σ 0 n ∪ Π 0 n , B n Min PR ∅ (n − 1) ↑ K ∅ (n − 1) max ≫ Σ 0 n+1 , Σ 0 n+1 PR ∅ (n) K ∅ (n) ... 2. log ≫ Σ 0 1 , Σ 0 1 PR K ≫ Σ 0 1 ∪ f Π 0 1 , B 1 Min PR ↑ K min ≫ Σ 0 2 , Σ 0 2 PR ∅ ′ K ∅ ′ ... ... ≫ Σ 0 n , Σ 0 n PR ∅ (n − 1) K ∅ (n − 1) ≫ Σ 0 n ∪ f Π 0 n , B n Min PR ∅ (n − 1) ↑ K ∅ (n − 1) min ≫ Σ 0 n+1 , Σ 0 n+1 PR ∅ (n) K ∅ (n) ... 3. Ther e is a c onstant c such that al l > ct ine qualities in 1 and 2 (which ar e inher ent to the ≫ and ≫ ↑ or d erings) ar e > ine qualities up to c . 4. Though K max and K min ar e ≤ ct inc omp ar able, we have K ∅ ( n − 1) max OftLess Σ 0 n ∪ Π 0 n ,B n M i n P R ∅ ( n − 1) ↑ K ∅ ( n − 1) min K ∅ ( n − 1) min OftLess Σ 0 n ∪ Π 0 , ≤ rec ∅ ( n − 1) n ,B n M i n P R ∅ ( n − 1) ↑ K ∅ ( n − 1) max Pr o of. 1. ≤ ct ine qualities. In equalit y log ≥ ct K is w ell-kno wn. T he inclu- sions (cf. Pr op.2.16) P R ∅ ( n ) ⊆ M in ∅ ( n ) P R ⊆ P R ∅ ( n +1) , P R ∅ ( n ) ⊆ M ax ∅ ( n ) P R ⊆ P R ∅ ( n +1) yield in equ alities K ∅ ( n ) ≥ ct K ∅ ( n ) min ≥ ct K ∅ ( n +1) , K ∅ ( n ) ≥ ct K ∅ ( n ) max ≥ ct K ∅ ( n +1) . 2. Ine qualities ... ≫ K ∅ (i) . Lemma 8.1 with A = ∅ and ϕ ◦ log in place of ϕ yields inequalit y log ≫ Σ 0 1 , Σ 0 1 PR K. Since K A max is recur siv e in A ′ (cf. Thm.6.4), Lemma 8.1 with A = ∅ ( n − 1) and ϕ ◦ K ∅ ( n − 1) max in place of ϕ yields in equ alit y K ∅ ( n − 1) max ≫ Σ 0 n+1 , Σ 0 n+1 PR ∅ (n) K ∅ (n) . Idem with K min . 39 3. Ine qualities K ∅ ( i ) ≫ ... . Direct app lication of Lemmas 8.6, 8.8. 4. P oint 3 of the the or em. This is the b eneﬁt of the uniform oracular prop- ert y obtained in Lemmas 8.1, 8.6, 8.8, 8.10. 5. Fin ally , th e OftLess relations (P oin t 4 of th e th eorem) are d irect app li- cation of Lemma 8.10. Remark 8.15. The scattered c haracter of comparisons with resp ect to the ≪ orderings is unav oidable since all complexities K, K max , K min , ..., K ∅ ( n ) are equal up to a constan t on th e inﬁnite set of in tegers w ith K ∅ ( n ) incom- pressible b inary representa tions. References [1] V. Bec her, G. Ch aitin, and S. Daicz . A h ighly rand om num b er. In C.S. Calude, M.J. Dineen, and S. Sbur lan, editors, Pr o c e e dings of the Thir d Discr ete Mathematics and The or etic al Computer Scienc e Confer enc e (DMTCS’01) , p ages 55–68. S pringer-V erlag, 2001. [2] V. Bec her , S. Figueira, A. Nies, and S. Picc hi. Program-size complexit y for p ossibly inﬁnite computations, H ∞ . 2003. T o app ear in Notre Dame J. of F ormal Logic. [3] G. Chaitin. Algorithmic ent ropy of sets. Computers and Math. with Applic. , 2:233–245 , 1976. Av ailable on Chaitin’s h ome p age. [4] M. F erbus-Zand a and S. Grigorieﬀ. M ax/ M in recursion theory . So on submitted. [5] M. F erbus-Zand a and S. Grigorieﬀ. Kolmogoro v complexities K max , K min . 2004. The or et. Comput. Sci. , 352:159–1 80, 2006. [6] R.M. F riedb erg. Three theorems on recursiv e enumerations. Journal of Symb ol ic L o gic , 23:30 9–316, 1958. [7] P . Hinman. R e c u rsion-the o r etic hier ar chies . Spr inger, 1978. [8] A.N. Kolmogoro v. Three approac hes to th e quantita tive deﬁ nition of information. Pr oblems Inform. T r ansmission , 1(1):1–7, 1965. [9] M. Li and P . Vitan yi. An intr o duction to Kolmo gor ov c omplexity and its applic atio ns . Spr inger, 1997 (2d edition). [10] Xiang Li. Eﬀectiv ely imm une sets, pr ograms index sets and eﬀectiv ely simple sets. In Chong and al., editors, Pr o c. South-e astern Asian c on- fer enc e on lo gic , p ages 97–105. North Holland, 1983. 40 [11] P . Odifreddi. Classic al R e cursion The ory , vo lume 125. North-Holland, 1989. [12] E.L. Post. Recur s iv ely en umerable sets of intege rs and their decision problems. Bul letin Americ an Math. So c. , 50:28 4–316, 1944. Also in Da vis, 1965, p.305–337. [13] E.L. P ost. Degrees of recursive unsolv abilit y . Bul letin Americ an Math. So c. , 54:641 –642, 1948. [14] H. Rogers. The ory of r e cursive func tions and e ﬀ e ctive c omp utability . McGra w-Hill, 1967. [15] J.R. Sho enﬁeld. R e cursion the ory , v olume 1 of L e ctur e Notes in L o gic . 1993. Reprinte d 2001, A.K. P eters. [16] R.M. Solo v a y . On random R.E. sets. In A.I. Arruda and al., editors, Non-classic al Lo gics, Mo del the o ry and Comp utability , pages 283–307. North-Holland, 1977. [17] A. Zv onkin and L. Levin. Th e complexity of ﬁnite ob jects and the dev elopment of the concepts of information and r an d omness by means of the theory of algorithms. Russian Math. Surveys , 6:83–124, 1970. 41

Refinment of the "up to a constant" ordering using contructive co-immunity and alike. Application to the Min/Max hierarchy of Kolmogorov complexities

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