Set theoretical Representations of Integers, I

Kolmogorov Complexit y and Set theoretical Represen tations of In tegers, I Marie Ferbus-Zanda LIAF A, Universit ´ e P aris 7 2, pl. Jussieu 7525 1 Paris C edex 05 F rance ferbus@log ique.juss ieu.fr Serg e Grigorieff LIAF A, Universit ´ e P aris 7 2, pl. Jussieu 7525 1 Paris C edex 05 F rance seg@liafa. jussieu.f r No v ember 1, 2018 Con ten ts 1 In troduction 3 1.1 Kolmogoro v complexit y and represen tations of N , Z . . . . . . . . . . . . . 3 1.2 Kolmogoro v complexities and famili es of functions . . . . . . . . . . . . . . 5 1.3 Road map of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 An abstract setting for Kol m ogoro v complexity: se lf-enumerated repre- sentat ion systems 6 2.1 Classica l Kolmogoro v complexity . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Self-enumerated representa tion systems . . . . . . . . . . . . . . . . . . . . 7 2.3 Goo d universal functions alwa y s exist . . . . . . . . . . . . . . . . . . . . . 9 2.4 Relativization of self-enumerated rep resen tation sy stems . . . . . . . . . . . 9 2.5 The Inv ariance Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Some ope rations on self-enumerated systems 11 3.1 The composition lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Produ ct of self-enumerated representation systems . . . . . . . . . . . . . . 12 4 F rom domain N to dom ai n Z 13 4.1 The ∆ op eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Z systems and N systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 Self-enumerated representation systems for r.e. se ts 14 5.1 Acceptable enumerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2 Self-enumerated representa tion systems for r.e. sets . . . . . . . . . . . . . . 15 1 6 Inﬁnite computations 18 6.1 Self-enumerated systems of max of partial recursiv e fun ctions . . . . . . . . 18 6.2 Kolmogoro v complexities K max , K ∅ ′ max , ... . . . . . . . . . . . . . . . . . . . 20 6.3 M ax 2 ∗ → N Rec and M a x 2 ∗ → N P R and inﬁ nite comput ations . . . . . . . . . . . . . . 20 6.4 M ax 2 ∗ → N P R and th e jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.5 The ∆ op eration on M ax 2 ∗ → N P R and th e jump . . . . . . . . . . . . . . . . . 22 7 Abstract representations and eﬀectivi zations 25 7.1 Some arithmetical represen tations of N . . . . . . . . . . . . . . . . . . . . . 25 7.2 Abstract rep resentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7.3 Eﬀectivizing rep resentations: why? . . . . . . . . . . . . . . . . . . . . . . . 27 7.4 Eﬀectivizations of representations and asso ciated K olmogorov complexities 27 7.5 P artial recursiv e represen tations . . . . . . . . . . . . . . . . . . . . . . . . 28 8 Cardinal representations of N 29 8.1 Basic cardin al representation and its eﬀectivizations . . . . . . . . . . . . . 29 8.2 Syntactical complexit y of cardinal rep resen tations . . . . . . . . . . . . . . . 29 8.3 Characterization of th e car d self-enumerated systems . . . . . . . . . . . . . 30 8.4 Characterization of th e ∆ c ar d represen tation system . . . . . . . . . . . . . 32 9 Index representat ions of N 32 9.1 Basic in dex rep resentation an d its eﬀectivizations . . . . . . . . . . . . . . . 32 9.2 Syntactical complexit y of ind ex representations . . . . . . . . . . . . . . . . 33 9.3 Characterization of th e index sel f-enumerated sy stems . . . . . . . . . . . . 36 9.4 Characterization of th e ∆ index sel f-enumerated sy stems . . . . . . . . . . . 42 10 F unctional represent ations of N 43 10.1 Basic Ch urch represen tation of N . . . . . . . . . . . . . . . . . . . . . . . . 43 10.2 Computable and eﬀectively con tinuous functionals . . . . . . . . . . . . . . 44 10.3 Eﬀectiveness of th e Apply functional . . . . . . . . . . . . . . . . . . . . . . 45 10.4 F unctionals o ver P R X → Y and compu tabilit y . . . . . . . . . . . . . . . . . . 46 10.5 Eﬀectivizations of Churc h rep resentation of N . . . . . . . . . . . . . . . . . 46 10.6 Some examples of eﬀectively continuous functionals . . . . . . . . . . . . . . 50 10.7 Syntactical complexit y of Church represen tation . . . . . . . . . . . . . . . 51 10.8 Characterization of t h e Chur ch representa tion system . . . . . . . . . . . . 52 10.9 Characterization of t h e ∆ Chur ch self-en umerated systems . . . . . . . . . . 53 10.10 F unctional representations of Z . . . . . . . . . . . . . . . . . . . . . . . . . 53 11 Conclusion 53 Abstract W e reco nsider some classica l natural semantics of in tegers (namely iterators o f functions, ca r dinals of se ts , index o f equiv a lence relations) in the p ersp ective o f Kolmog o rov complexity . T o each such semantics one can a ttach a simple representation of integers that we suitably ef- fectivize in or der to develop an asso c ia ted K olmogor ov theor y . Such eﬀectivizations a re particular instance s o f a gener al notio n of “self- enum erated system” that w e in tro duce in this pap er. Our main result asserts tha t, with such eﬀectivizations, Kolmo g orov theory allows to quantitativ ely dis ting uish the underlying sema ntics. W e characteriz e the families obtained by such eﬀectiviza tions and prov e that the asso - ciated Kolmogo r ov complexities co nstitute a hierarch y which coincides with that of Ko lmogorov complexities deﬁned via jump oracle s a nd/ or 2 inﬁnite c omputations (cf. [5 ]). This contrasts with the well-known fact that usual Ko lmogorov complexity do e s not depe nd (up to a c onstant) on the chosen a rithmetic r epresentation o f integers, let it b e in any base n ≥ 2 or in una ry . Also , in a conceptual point o f view, o ur result can b e se e n as a mean to measure the degree of abstra ction o f these diverse semantics. 1 In tro duction Notation 1.1. Equalit y , inequalit y and strict inequalit y up to a constant b et w een total functions D → N , where D is an y set, are denoted as follo ws: f ≤ ct g ⇔ ∃ c ∈ N ∀ x ∈ D f ( x ) ≤ g ( x ) + c f = ct g ⇔ f ≤ ct g ∧ g ≤ ct f ⇔ ∃ c ∈ N ∀ x ∈ D | f ( x ) − g ( x ) | ≤ c f < ct g ⇔ f ≤ ct g ∧ ¬ ( g ≤ ct f ) ⇔ f ≤ ct g ∧ ∀ c ∈ N ∃ x ∈ D g ( x ) > f ( x ) + c As we shall consider N -v alued partial fun ctions with d omain N , Z , 2 ∗ , N 2 ,..., the follo w ing deﬁnition is conv en ient. Deﬁnition 1.2. A basic set X is an y n on emp ty ﬁnite p ro duct of sets among N , Z or the set 2 ∗ of ﬁnite binary wo rds or the set Σ ∗ of ﬁnite wo rds in some ﬁnite or coun table alphab et Σ . Let’s also introd uce some n otations for partial recursive functions. Notation 1.3. Let X , Y b e b asic sets. W e denote P R X → Y (resp. P R A, X → Y ) the f amily of partial recursive (resp.partial A -recursive) fun ctions X → Y . In case X = Y = N ,we simply write P R and P R A . 1.1 Kolmogoro v complexit y and represen tations of N , Z Kolmogoro v complexit y K : N → N maps an intege r n on to the length of an y shortest binary program p ∈ 2 ∗ whic h outputs n . Th e inv ariance theorem asserts that, u p to an additive constan t, K do es not dep end on the p r ogram seman tics p 7→ n , pr ovided it is a universal partial recursive function. As a str aigh tforw ard corollary of the in v ariance theorem, K do es not d e- p end (again u p to a constan t) on th e r epresen tation of intege rs, i.e. w hether the program output n is really in N or is a w ord in some alphab et { 1 } or { 0 , ..., k − 1 } , for some k ≥ 2, wh ic h giv es the unary or base k representa tion of n . A result w hic h is easily extended to all p artial recurs ive r epresen tations of in tegers, cf. T h m.7.8. In this p a p er, we show that this is no mor e the c ase when (sui tably eﬀe c- tivize d) c lassic al set the or etic al r epr esentations ar e c onsider e d. W e particu- larly consider represent ations of in tegers via 3 • C hurc h iterators (Churc h [3], 1933), • cardin al equiv a lence classes (Russell [16] § IX, 1908, cf. [22] p.178), • in d ex equiv alence classes. F ollo w in g the usual wa y to deﬁn e Z f r om N , we also consid er r epresen ta- tions of a relativ e intege r z ∈ Z as p airs of represent ations of n on negativ e in tegers x, y satisfying z = x − y . In th e particular case of Churc h iterators, restricting to injective fu nctions an d considering negativ e iterations, leads to another direct wa y of repr esen ting relativ e inte gers. Programs are at the core of Kolmogoro v theory . They do n ot w ork on ab- stract entitie s bu t require f orm al r epresen tations of ob jects. Th us, we ha v e to deﬁn e eﬀect ivizations of the ab o v e abstract set theoretical notions in or- der to allo w their elemen ts to b e computed b y programs. T o do so, we use computable fu nctions and fun ctionals and recur siv ely en umerable sets. Eﬀectivized repr esentati ons of intege rs constitute particular in stances of self- enumer ate d r epr esentat ion systems (cf. Def.2.1). This is a notion of family F of partial functions f rom 2 ∗ to some ﬁxed set D for whic h an in v ari- ance theorem can b e p ro v ed using straight forw ard adaptation of original Kolmogoro v’s pro of. Wh ich leads to a notion of Kolmogoro v complexity K D F : D → N , cf. Def.2.16. The ones considered in this p ap er are K N Chur ch , K Z Chur ch , K Z ∆ Chur ch , K N c ar d , K Z ∆ c ar d , K N index , K Z ∆ index asso ciated to the sys tems obtained by eﬀecti vization of the Churc h, cardinal and index represent ations of N and the p assage to Z represen tat ions as outlined ab ov e. The m ain resu lt of this pap er states that the ab o v e Kolmogoro v complexities coincide (up to an additive constant) with those obtained via oracles and inﬁnite computations as introd u ced in [1], 2001, and our pap er [5], 2004. Theorem 1.4 (Main result) . K N Chur ch = ct K Z Chur ch ↾ N = ct K Z ∆ Chur ch ↾ N = ct K K N c ar d = ct K max K Z ∆ c ar d ↾ N = ct K ∅ ′ K N index = ct K ∅ ′ max K Z ∆ index ↾ N = ct K ∅ ′′ Thm.1.4 gathers the con ten ts of Th ms.8.5, 8.6, 9.5, 9.7, 10.24, 10.25 and § 10.10. A preliminary “ligh t” v ersion of th is result w as presen ted in [4], 2002. The strict ordering resu lt K > ct K max > ct K ∅ ′ (cf. Notatio ns 1.1) prov ed in [1, 5] and its obvious relativizatio n (cf. Prop.6.11) yield the follo wing hierarc h y theorem. 4 Theorem 1.5. log > ct K N Chur ch = ct K Z Chur ch ↾ N = ct K Z ∆ Chur ch ↾ N > ct K N c ar d > ct K Z ∆ c ar d ↾ N > ct K N index > ct K Z ∆ index ↾ N This hierarc h y result for set theoretical represent ations somewhat reﬂects their de gr e es of abstr action . Though Churc h repr esen tatio n v ia iteration functionals can b e considered as somewhat complex, we see th at, surprisin gly , the asso ciated Kolmogoro v complexities collapse to the simplest p ossible one. Also, it turns out that, for cardinal and ind ex r epresen tations, the passage from N to Z , i.e. from K N c ar d to K Z ∆ c ar d and fr om K N index to K Z ∆ index do es add complexit y . Ho w ev er, for C h urc h iterators, th e passage to Z do es not mo dify Kolmogoro v complexit y , let it b e via the ∆ op eration (for K Z ∆ Chur ch ) or restricting iterators to in jectiv e fun ctions (for K Z Chur ch ). The results ab out the ∆ car d and ∆ index classes are corollaries of those ab out the car d and index classes and of the follo wing result (Thm .6.12) whic h giv es a simple normal form to f u nctions computable relativ e to a jump oracle, and is interesting on its o wn. Theorem 1.6. L et A ⊆ N . A function G : 2 ∗ → Z is p artial A ′ -r e cursive if and only if ther e exist total A -r e cu rsive functions f , g : 2 ∗ × N → N such that, for al l p , G ( p ) = max { f ( p , t ) : t ∈ N } − max { g ( p , t ) : t ∈ N } (in p articular, G ( p ) is deﬁne d if and only if b oth max ’s ar e ﬁnite). 1.2 Kolmogoro v complexities and families of functions The equalitie s in T hm.1.4 are, in fact, corolla ries of equalities b et w een fam- ilies of f unctions 2 ∗ → N (namely , the asso ciated self-enumerated rep resen- tation sy s tems, cf. § 2.2) whic h are interesting on th eir own. F or instance (cf. T hms.8.5, 8.6, 9.5, 9.7, 10.24, 10.25 and § 10.10), Theorem 1.7. Denote X → Y the class of partial functions fr om X to Y . 1. A function f : 2 ∗ → N is the r estriction to a Π 0 2 set of a p artial r e cursive function if and only if it is of the form f = Chur ch ◦ Φ wher e - Φ : 2 ∗ → ( N → N ) ( N → N ) is a c omputable functional, - Chur ch : ( N → N ) ( N → N ) → N is the fu nctional such that Chur ch (Ψ) =  n if Ψ is the iter ator f 7→ f ( n ) undeﬁne d otherwise 5 2. A function f : 2 ∗ → N is the max of a total r e cursive (r esp. total ∅ ′ -r e cursive) se quenc e of functions (cf. Def . 6.1) if and only if it is of the form p 7→ c ar d ( W N ϕ ( p ) ) (r esp. p 7→ index ( W N 2 ϕ ( p ) ) , up to 1 ) for some total r e cursive ϕ : 2 ∗ → 2 ∗ , wher e - W N q (r esp. W N 2 q ) is the r.e. subset of N (r esp. N 2 ) with c o de q , - c ar d : P ( N ) → N is the c ar dinal fu nc tion (deﬁne d on the sole ﬁnite sets), - index : P ( N 2 ) → N is deﬁne d on e quivalenc e r elations with ﬁnitely many classes and gi ves the index (i . e. the numb er of e q uivalenc e classes). 3. A fu nc tion f : 2 ∗ → N is p artial ∅ ′ -r e cursive (r esp. ∅ ′′ -r e cursive) if and only if i t is of the form p 7→ c ar d ( W N ϕ 1 ( p ) ) − c ar d ( W N ϕ 2 ( p ) ) (r esp. p 7→ index ( W N 2 ϕ 1 ( p ) ) − index ( W N 2 ϕ 2 ( p ) ) ) for some total r e cursive ϕ 1 , ϕ 2 : 2 ∗ → 2 ∗ . 1.3 Road map of t he pap er § 2 in tro duces the notion of self-en umerated represent ation system with its asso ciated Kolmogoro v complexit y . § 3 introduce s im p le op er ations on self-enumerated systems. § 4 sets u p some connections b et w een self-en u merated representa tion systems for N and Z . § 5 considers a self-en umerated r epresen tation system for the set of recur- siv ely en umerable s ubsets of N . § 6 r ecalls material from Bec her & C haitin & Daicz, 2001 [1] and our pap er [5], 2004, ab out some extensions of Kolmogoro v complexit y inv olving inﬁ- nite computations. Th is is to make the p ap er self-con tained. § 7 introduces abstract r ep resen tations and their eﬀectiviza tions. § 8, 9, 10 develo p the set-theoretical r ep resen tatio ns mentioned in § 1.1 and pro v e all the mentioned theorems and s ome more resu lts related to the as- so ciated s elf-en umerated systems, in particular the synta ctical complexit y of universal functions for suc h systems. 2 An abstract setting for Kolmogoro v complexit y: self-en umerated represen tat ion systems 2.1 Classical K olmogorov complexit y Classical Kolmogoro v complexit y of elemen ts of a basic set X is deﬁn ed as follo w s (cf. Kolmogoro v, 1965 [7]): 1. T o ev ery ϕ : 2 ∗ → X is asso ciated K X ϕ : X → N suc h that K X ϕ ( x ) = min {| p | : ϕ ( p ) = x } 6 i.e. K X ϕ ( x ) is the shortest length of a “program” p ∈ 2 ∗ whic h is mapp ed onto x by ϕ . 2. Kolmogoro v Inv ariance Th eorem asserts that, letting ϕ v ary in P R 2 ∗ → X (cf. Notation 1.3), there is a least K X ϕ , up to an additive constan t: ∃ ϕ ∈ P R 2 ∗ → X ∀ ψ ∈ P R 2 ∗ → X K X ϕ ≤ ct K X ψ Kolmogoro v complexit y K X : N → N is such a least K X ϕ , so that it is deﬁned up to an ad d itiv e constan t. Let A ⊆ N . Th e ab o v e constru ction r elativize s to oracle A : replace P R 2 ∗ → X b y P R A, 2 ∗ → X to get the oracular K olmogoro v complexit y K A X . 2.2 Self-en umerated represen tation systems W e in trodu ce an abstract setting f or the deﬁn ition of Kolmogoro v complex- it y: self-enumer ate d r e pr esentat ion systems . As a v ariet y of Kolmogoro v complexities is considered, this allo ws to u nify the m ultiple v ariations of the in v ariance theorem, the pro ofs of whic h rep eat, m utatis m utandis, the s ame classical pro of due to Kolmogoro v (cf. Li & Vitanyi’s textb o ok [9 ] p.97). This abstract setting also leads to a study of op erations on self-en umerated systems, some of which are pr esen ted in § 4,5 and some more are deve lop ed in the con tin uatio n of this pap er. Some int uition for the n ext deﬁn ition is giv en in Note 2.2 and Rk.2.4. Deﬁnition 2.1 (Self-en umerated rep r esen tatio n systems) . 1. A self-en umerated rep resen tation system (in sh ort “self-en umerated s y s - tem”) is a pair ( D , F ) where D is a set — the domain of the system — and F is a family of p artial functions 2 ∗ → D s atisfying the f ollo win g cond itions: i. D = [ F ∈F Rang e ( F ), i.e. ev ery element of D app ears in th e range of some fu n ction F ∈ F . ii. If ϕ : 2 ∗ → 2 ∗ is a recursiv e total function an d F ∈ F then F ◦ ϕ ∈ F . iii. Th ere exists U ∈ F (called a u niv ersal fun ction f or F ) and a total recursiv e fu n ction comp U : 2 ∗ × 2 ∗ → 2 ∗ suc h that ∀ F ∈ F ∃ e ∈ 2 ∗ ∀ p ∈ 2 ∗ F ( p ) = U ( comp U ( e , p )) In other w ords, letting U e ( p ) = U ( comp U ( e , p )), the sequence of func- tions ( U e ) e ∈ N is an en umeration of F . 2. (F ull systems) In case condition ii holds for all p art ial r e cursive func- tions ϕ , the s y s tem ( D , F ) is called a self-en umerated representa tion f ul l system . 7 3. (Go o d univ ersal functions) A u niv ersal function U for F is go o d if its asso ciated comp fu nction satisﬁes th e condition ∀ e ∃ c e ∀ p | comp U ( e , p ) | ) ≤ | p | + c e i.e. for all e , w e hav e ( p 7→ | comp U ( e , p ) | ) ≤ ct | p | (cf. Notation 1.1). Note 2.2 (Intuition) . 1. The set 2 ∗ is seen as a family of pr ograms to get elemen ts of D . The c hoice of binary p rograms is a f airn ess condition in view of the d eﬁnition of K olmogoro v complexit y (cf. Def.2.1 6) based on the length of pr ograms: larger the alphab et, shorter the p r ograms. 2. Eac h F ∈ F is seen as a programming language with programs in 2 ∗ . Sp ecial r estrictions: n o input, outpu ts are elements of D . 3. Denomination comp stands for “compiler” since it m ap s a p rogram p from “language” F (with co d e p ) to its U -compiled f orm comp U ( e , p ) in the “language” U . 4. “Compilation” with a go o d unive rsal fun ction do es not increase the length of programs but for some additiv e constan t which dep ends only on the language, namely on the sole co de e . Example 2.3. If X is a basic set th en ( X , P R 2 ∗ → X ) is ob viously a s elf- en umerated repr esen tatio n system. Remark 2.4. In view of the en umerabilit y condition iii and since there is n o recursiv e enumeration of total recursiv e fun ctions, one w ould a priori rather require condition ii to b e true for all p artial r ecursiv e functions ϕ : 2 ∗ → 2 ∗ , i.e. consider the sole full systems. Ho w ever, th ere are interesting s elf-enumerated represen tation systems whic h are n ot full systems. Th e sim p lest one is M ax Rec , cf. Prop.6.2. O ther ex- amples we shall deal with in v o lv e higher ord er domains consisting of inﬁ nite ob jects, for in s tance the domain RE ( N ) of all recursively enumerable sub- sets of N , cf. § 5.2 . The p artial char acter of c omputability is alr e a dy inher ent to the obje cts in the domain or to the p articular notion of c omputability and an enumer ation the or em do es hold for a family F of total functions. F rom cond itions i and iii of Def.2.1, we immediately see that Prop osition 2.5. L et ( D , F ) b e a self-enumer ate d system. Then D and F ar e c ountable and any universal function for F is su rje ctive. Another consequence of condition iii of Def.2.1 is as follo ws. Prop osition 2.6. L et ( N , F ) b e a self-enumer ate d system. Then al l univer- sal functions for F ar e many-one e quivalent. 8 2.3 Go o d univ ersal functions alw a ys exist Let’s recall a classical w a y to co de pairs of words. Deﬁnition 2.7 (Co ding p airs of wo rds) . Let µ : 2 ∗ → 2 ∗ b e the morphism (relativ e to the monoid structur e of concatenati on pr o duct on words) suc h that µ (0) = 00 and µ (1) = 01. The function c : 2 ∗ × 2 ∗ → 2 ∗ suc h that c ( e , p ) = µ ( e )1 p is a recur siv e injection wh ich satisﬁes equation | c ( e , p ) | = | p | + 2 | e | + 1 (1) Denoting λ the empt y word, we d eﬁne π 1 , π 2 : 2 ∗ → 2 ∗ as follo ws: π 1 ( c ( e , p )) = e , π 2 ( c ( e , p )) = p , π 1 ( w ) = π 2 ( w ) = λ if w / ∈ R ang e ( c ) Remark 2.8. If we redeﬁne c as c ( e , p ) = µ ( B in ( | e | ))1 ep where B in ( k ) is the binary representati on of the intege r k ∈ N th en equation (1) can b e sharp en ed to | c ( e , p ) | = | p | + | e | + 2 ⌊ log( | e | ) ⌋ + 3 F or an optimal sh arp ening with a co d ing of pairs inv olving the function log( x ) + log log( x ) + log log log( x ) + ... see Li & Vitan yi’s b o ok [9], Example 1.11.13, p .79. Prop osition 2.9 (Existence of go o d u niv ersal fun ctions) . Every self-enumer ate d system c ont ains a go o d univ e rsal fu nction with c as asso ciate d comp function. Pr o of. The usu al pro of wo rks. Let U and comp U b e as in Def.2.1 and set U opt = U ◦ comp U ◦ ( π 1 , π 2 ) Then comp U ◦ ( π 1 , π 2 ) : 2 ∗ → 2 ∗ is total recursive and condition ii of Def.2.1 insures that U opt ∈ F . No w, we ha v e U opt ( c ( e , p )) = U ( comp U (( π 1 , π 2 )( c ( e , p )))) = U ( comp U ( e , p )) so that U opt is u niv ersal with c as asso ciated comp fun ction. 2.4 Relativization of self-en umerated representation systems Def.2.1 can b e ob viously relativized to any oracle A . Ho w ev er, con trary to what can b e a priori exp ected, this is no generalization but particularization. The main reason is Prop.2.9: there alw a ys exists a universal fu nction with c as asso ciated comp fu nction. Deﬁnition 2.10. Let A ⊆ N . A self-en umerated representati on A -system is a pair ( D , F ) w here F is a family of partial functions 2 ∗ → D satisfying condition i of Def.2.1 and the follo w ing v arian ts of conditions ii and iii : 9 ii A . I f ϕ : 2 ∗ → 2 ∗ is an A -recursive total fun ction and F ∈ F then F ◦ ϕ ∈ F . iii A . T here exists U ∈ F and a total A -recursive function comp U : 2 ∗ × 2 ∗ → 2 ∗ suc h that ∀ F ∈ F ∃ e ∈ 2 ∗ ∀ p ∈ 2 ∗ F ( p ) = U ( comp U ( e , p )) Example 2.11. If X is a basic set then ( X , P R A, 2 ∗ → X ) is obvio usly a self- en umerated repr esen tatio n A -system. Prop osition 2.12. Every self- enumer ate d r epr esentation A -system c on - tains a universal function with c as asso c iate d comp function. In p articular, ev ery such system is also a self-enumer ate d r epr esentation system. Thus, ( X , P R A, 2 ∗ → X ) is a self-enumer ate d r epr esentat ion system. Pr o of. W e rep eat the same easy argument u sed for Pr op.2.9. Let U an d comp U b e as in condition iii A of Def.2.10 and set U opt = U ◦ comp U ◦ ( π 1 , π 2 ). Then comp U ◦ ( π 1 , π 2 ) : 2 ∗ → 2 ∗ is total A -recursive and cond ition ii A insures that U opt ∈ F and we ha v e U opt ( c ( e , p )) = U ( comp U (( π 1 , π 2 )( c ( e , p )))) = U ( comp U ( e , p )) so that U opt is u niv ersal with c as asso ciated comp fun ction. 2.5 The Inv ariance Theorem Deﬁnition 2.13. Let F : 2 ∗ → D b e an y partial function. The Kolmogoro v complexit y K D F : D → N ∪ { + ∞} associated to F is the fun ction deﬁn ed as follo w s: K D F ( x ) = min {| p | : F ( p ) = x } (Con v en tion: min ∅ = + ∞ ) Remark 2.14. 1. K D F ( x ) is ﬁn ite if and only if x ∈ R ang e ( F ). Hence K D F has v alues in N (rather than N ∪ { + ∞} ) if and only if F is surjectiv e. 2. If F : 2 ∗ → D is a restriction of G : 2 ∗ → D then K D G ≤ K D F . Thanks to Pr op . 2.9 , the usual Inv ariance Theorem can b e extended to an y self-en umerated representa tion system, which allo ws to deﬁn e Kol- mogoro v complexit y for s uc h a system. Theorem 2.15 (In v ariance Th eorem, Kolmogoro v, 1965 [7]) . L et ( D , F ) b e a self-enumer a te d r epr e sentation system. 1. When F varies in the family F , ther e is a le ast K D F , up to an additive c onstant (cf . Notation 1.1): ∃ F ∈ F ∀ G ∈ F K D F ≤ ct K D G 10 Such F ’s ar e said to optimal in F . 2. Every go o d u niversal function for F is optimal. Pr o of. It suﬃces to pro v e 2. The usu al pro of w orks. Consider a goo d unive rsal en umeration U of F . Let F ∈ F and let e b e su c h that U ( comp U ( e , p )) = F ( p ) for all p ∈ 2 ∗ First, since U is s u rjectiv e (Prop.2.5), all v a lues of K D U are ﬁn ite. Thus, K D U ( x ) < K D F ( x ) for x / ∈ Ran g e ( F ) (since then K D F ( x ) = + ∞ ). F or every x ∈ R ang e ( F ), let p x b e a s mallest program su c h th at F ( p x ) = x , i.e. K D F ( x ) = | p x | . Then, x = F ( p x ) = U ( comp U ( e , p x )) and since U is goo d , K D U ( x ) ≤ | comp U ( e, p x ) | ≤ | p x | + c e = K D F ( x ) + c e and therefore K D U ≤ ct K D F . As usual, Theorem 2.15 allo ws for an intrinsic d eﬁnition of the Kol- mogoro v complexit y asso ciated to the self-enumerated system ( D , F ). Deﬁnition 2.16 (Kolmogoro v complexit y of a self-enumerate d representa- tion system) . Let ( D , F ) b e a self-en umerated r epresen tation system. The Kolmogoro v complexit y K D F : D → N is the function K D U where U is some ﬁxed go o d u niversal enumer ation in F . Up to an additiv e constant , th is deﬁn ition is indep end en t of the particular c hoice of U . The follo wing straigh tforw ard result, based on Examples 2.3 and 2.11, insures that Def.2.16 is compatible with the usu al Kolmogoro v complexit y and its r elativizations. Prop osition 2.17. L et A ⊆ N b e an or acle and let D = X b e a b asic set (cf. Def.1.2). The Kolmo g or ov c omplexities K X P R 2 ∗ → X and K X P R A, 2 ∗ → X deﬁne d ab ove ar e exactly the usual Kolmo g or ov c omp lexity K X : X → N and its r e lativization K A X (cf. § 2.1). 3 Some op erations on self-en umerated systems 3.1 The comp osition lemma The f ollo wing easy fact is a con v enient to ol to eﬀec tivize representat ions (cf. § 7.3, 7.4). W e shall also use it in § 4 to go from s y s tems with domain N to ones with d omain Z . 11 Lemma 3.1 (Th e comp osition lemma) . L et ( D , F ) b e a self-enumer at e d r epr esenta tion system and ϕ : D → E b e a surje ctive p artial function. Set ϕ ◦ F = { ϕ ◦ F : F ∈ F } . 1. ( E , ϕ ◦ F ) i s also a self- enumer ate d r epr esentation system. Mor e over, if U is universal or go o d universal for F then so is ϕ ◦ U for ϕ ◦ F . 2. F or every x ∈ E , K E ϕ ◦F ( x ) = ct min { K D F ( y ) : ϕ ( y ) = x } In p a rticular, K E ϕ ◦F ◦ ϕ ≤ ct K D F and if ϕ : D → E is a total bij e ction fr om D to E then K E ϕ ◦F ◦ ϕ = ct K D F . Pr o of. P oin t 1 is s traigh tforw a rd. As for p oint 2, let U : 2 ∗ → D b e some unive rsal function for F and observ e that, for x ∈ E , K E ϕ ◦F ( x ) = min {| p | : p su c h that ϕ ( U ( p )) = x } = min { min {| p | : p s.t. U ( p ) = y } : y s.t. ϕ ( y ) = x } = min { K D F ( y ) : y s.t. ϕ ( y ) = x } In particular, taking x = ϕ ( z ), w e get K E ϕ ◦F ( ϕ ( z )) ≤ ct K D F ( z ). Finally , observe that if ϕ is bijectiv e then z is the uniqu e y such that ϕ ( y ) = x , so th at the ab o v e min redu ces to K D F ( z ). 3.2 Pro duct of self-en umerated representation systems W e shall need a notion of pro du ct of self-en umerated representat ion systems. Theorem 3.2. L e t ( D 1 , F 1 ) and ( D 2 , F 2 ) b e self-enumer ate d r epr esentation systems We identify a p air ( F 1 , F 2 ) ∈ F 1 × F 2 with the function 2 ∗ → D 1 × D 2 which maps p to ( F 1 ( p ) , F 2 ( p )) . Then ( D 1 × D 2 , F 1 × F 2 ) is also a self-enumer ate d r epr esentat ion system. If ( D 1 , F 1 ) and ( D 2 , F 2 ) ar e ful l systems then so is ( D 1 × D 2 , F 1 × F 2 ) . If U 1 , U 2 ar e universal for F 1 , F 2 then U 1 , 2 = ( U 1 ◦ π 1 , U 2 ◦ π 2 ) is universal for F 1 × F 2 . Pr o of. Condition ii in Def.2.1 is ob vious. Condition i. Let ( d 1 , d 2 ) ∈ D 1 × D 2 . Applying condition i to ( D 1 , F 1 ) and to ( D 2 , F 2 ), we get F 1 ∈ F 1 , F 2 ∈ F 2 and p 1 , p 2 ∈ 2 ∗ suc h that d 1 = F 1 ( p 1 ) and d 2 = F 2 ( p 2 ). Therefore ( d 1 , d 2 ) = ( F 1 ◦ π 1 , F 2 ◦ π 2 )( c ( p 1 , p 2 )). Ob serv e ﬁnally that ( F 1 ◦ π 1 , F 2 ◦ π 2 ) ∈ F 1 × F 2 (condition ii for ( D 1 , F 1 ) , ( D 2 , F 2 )). 12 Condition iii. Let comp 1 , comp 2 : 2 ∗ → 2 ∗ b e the comp f u nctions asso ciated to the unive rsal functions U 1 , U 2 and set comp 1 , 2 ( e , p ) = c ( comp 1 ( π 1 ( e ) , p ) , comp 2 ( π 2 ( e ) , p )) F or ev ery ( F 1 , F 2 ) ∈ F 1 × F 2 there exist a , b ∈ 2 ∗ suc h that F 1 ( p ) = U 1 ( comp 1 ( a , p )) and F 2 ( p ) = U 2 ( comp 2 ( b , p )). Therefore ( F 1 , F 2 )( p ) = ( U 1 ( comp 1 ( a , p )) , U 2 ( comp 2 ( b , p ))) = ( U 1 ◦ π 1 , U 2 ◦ π 2 )( c ( comp 1 ( a , p ) , comp 2 ( b , p ))) = U 1 , 2 ( comp 1 , 2 ( c ( a , b ) , p )) whic h prov es that U 1 , 2 is universal for the pr o duct system F 1 × F 2 . Remark 3.3. Ob serv e that, even if U 1 , U 2 are go o d, the ab o v e u niv ersal function U 1 , 2 is not goo d since | comp 1 , 2 ( e , p ) | = 2 | comp 1 ( π 1 ( e ) , p ) | + | comp 2 ( π 2 ( e ) , p ) | + 1 whic h is ≥ 3 | p | in general. T o get a go o d function g U 1 , 2 , argue as in the pro of of P rop.2.9: g U 1 , 2 ( p ) = U 1 , 2 ◦ comp 1 , 2 ◦ ( π 1 , π 2 )( p ) = U 1 , 2 ( comp 1 , 2 ( π 1 ( p ) , π 2 ( p ))) = U 1 , 2 ( c ( comp 1 ( π 1 π 1 ( p ) , π 2 ( p )) , comp 2 ( π 2 π 1 ( p ) , π 2 ( p )))) = ( U 1 ◦ π 1 , U 2 ◦ π 2 ) ( c ( comp 1 ( π 1 π 1 ( p ) , π 2 ( p )) , comp 2 ( π 2 π 1 ( p ) , π 2 ( p )))) = ( U 1 ( comp 1 ( π 1 π 1 ( p ) , π 2 ( p ))) , U 2 ( comp 2 ( π 2 π 1 ( p ) , π 2 ( p )))) 4 F rom domain N to domain Z 4.1 The ∆ op eration Relativ e int egers are classically introdu ced as equ iv alence classes of pairs of natural inte gers of whic h they are the d iﬀeren ces. Th is give a simple w a y to go f r om a s elf-enumerated r epresen tation system with domain N to some with domain Z . Deﬁnition 4.1 (The ∆ op eration) . Let diﬀ : N 2 → Z b e the function ( m, n ) 7→ m − n . If ( N , F ) is a self-en umerated repr esen tatio n system with domain N , using notations from L emma 3.1 and Thm.3.2, we let ( Z , ∆ F ) b e the s y s tem ( Z , diﬀ ◦ ( F × F )) As a d irect corollary of Lemma 3.1 and Th m.3.2, w e hav e Prop osition 4.2. If ( N , F ) is a self-enumer ate d r epr esentation system (r esp. ful l system) with domain N then so is ( Z , ∆ F ) . 13 4.2 Z systems and N systems The f ollo win g prop ositions coll ect some easy facts ab out self-enumerated systems with d omain Z and their asso ciated Kolmogoro v complexities. Prop osition 4.3. L et ( Z , G ) b e a self-enumer ate d system. 1. L et F = { G ↾ G − 1 ( N ) : G ∈ G } . Then ( N , F ) is also a self-e nu mer ate d system and K N F = K Z G ↾ N . 2. Denote opp : Z → Z the function n 7→ − n . If G ◦ opp = G then K Z G = ct K Z G ◦ opp . Pr o of. 1. Cond itions i-ii of Def.2.1 are ob vious. As for iii, observe th at if U ∈ G is un iv ersal for G then U ↾ U − 1 ( N ) is in F and is universal for F with the same asso ciated comp function. No w, K U ↾ U − 1 ( N ) = K U ↾ N . Whence K N F = K Z G ↾ N . 2. Observe that if ϕ, F ∈ G and K ϕ ≤ ct K F then K ϕ ◦ opp ≤ ct K F ◦ opp . Since G ◦ opp = G , we see that if ϕ is optimal then so is ϕ ◦ opp . Whence K ϕ = ct K ϕ ◦ opp , and therefore K Z G = ct K Z G ◦ opp . Prop osition 4.4. L et A ⊆ N . 1. P R A, 2 ∗ → N = P R A, 2 ∗ → Z ∩ ( N → N ) = { G ↾ G − 1 ( N ) : G ∈ P R A, 2 ∗ → Z } . In p articular, K A, Z ↾ N = ct K A, N . 2. P R A, 2 ∗ → Z = P R A, 2 ∗ → Z ◦ opp = ∆ P R A, 2 ∗ → N . In p articular, K A, Z = ct K A, Z ◦ opp . 5 Self-en u merated represen ta tion systems for r.e. sets W e no w come to examples of self-en umerated systems of a somewhat diﬀer- en t kind, wh ic h will b e used in the eﬀectivizatio n of set theoretical r epre- sen tations of intege rs. 5.1 Acceptable enum erations Let’s r ecall the notion of acceptable en umeration of partial recursive func- tions (cf. Rogers [15] Ex. 2.10 p .41, or Odifrredd i [12], p.215) Deﬁnition 5.1. Let X , Y b e some b asic sets and A ⊆ N . 1. An en umeration ( φ A e ) e ∈ 2 ∗ of partial A -recursiv e functions X → Y is ac c e ptable if i. it is partial A -recursiv e as a function 2 ∗ × X → Y 14 ii. and it s atisﬁes the parametrization (also called s -m-n ) prop ert y: f or ev ery basic set Z , th ere exists a total A -recursiv e function s Z X : 2 ∗ × Z → 2 ∗ suc h that, for all e ∈ 2 ∗ , z ∈ Z , x ∈ X , φ A e ( h z , x i ) = φ A s Z X ( e , z ) ( x ) where h z , x i is the im age of the pair ( z , x ) by some ﬁxed total recursive bijection Z × X → X . 3. An en umeration ( W A e ) e ∈ 2 ∗ of A -recursivel y enumerable subsets of X is ac c epta ble if, for all e ∈ 2 ∗ , W A e = domain ( φ A e ) for some acceptable en umeration ( φ A e ) e ∈ 2 ∗ of partial A -recursiv e fun ctions. W e shall need Rogers’ theorem (cf. O difreddi [12] p .219). Theorem 5.2 (Rogers’ theorem) . If ( φ A e ) e ∈ 2 ∗ and ( ψ A e ) e ∈ 2 ∗ ar e two ac c ept- able enumer ations of p artial A -r e cursive functions X → Y , then ther e exists some A -r e cursive bije ction θ : 2 ∗ → 2 ∗ such that ψ A e = φ A θ ( e ) for al l e ∈ 2 ∗ . Corollary 5.3. L e t ( W ′ A e ) e ∈ 2 ∗ and ( W ′′ A e ) e ∈ 2 ∗ b e two ac c epta ble enu mer- ations of A -r.e. subsets of X . Then ther e exists an A - r e c u rsive bije ction θ : 2 ∗ → 2 ∗ such that W ′′ A e = W ′ A θ ( e ) for al l e ∈ 2 ∗ . Pr o of. Apply Roger’s theorem to acceptable enumerations ( φ A e ) e ∈ 2 ∗ , ( ψ A e ) e ∈ 2 ∗ of p artial A -recursive functions suc h that W ′ A e = domain ( φ A e ) and W ′′ A e = domain ( ψ A e ). 5.2 Self-en umerated represen tation systems for r .e. sets Cor.5.3 allo ws to get a natural in trinsic notion of “partial A -computable” map 2 ∗ → R E A ( X ). Prop osition 5.4. L et R E A ( X ) b e the family of A -r e cursively enumer able subsets of X and let ( W ′ A e ) e ∈ 2 ∗ and ( W ′′ A e ) e ∈ 2 ∗ b e two ac c eptable enumer a- tions of A -r.e. subsets of X . L et G : 2 ∗ → R E A ( X ) . 1. The f ol lowing c ond itions ar e e quivalent: i. Ther e e xi sts a total A -r e cursive fu nction f : 2 ∗ → 2 ∗ such that G ( p ) = W ′ A f ( p ) for al l p ∈ 2 ∗ ii. Ther e exists a total A -r e cursive func tion f : 2 ∗ → 2 ∗ such that G ( p ) = W ′′ A f ( p ) for al l p ∈ 2 ∗ 2. The f ol lowing c ond itions ar e e quivalent: i. Ther e exists a p artial A -r e cursive function f : 2 ∗ → 2 ∗ such that, for al l p ∈ 2 ∗ , G ( p ) =  W ′ A f ( p ) if f ( p ) is deﬁne d undeﬁne d otherwise 15 ii. Ther e exi sts a p artial A -r e cursive function f : 2 ∗ → 2 ∗ such that, for al l p ∈ 2 ∗ , G ( p ) =  W ′′ A f ( p ) if f ( p ) is deﬁne d undeﬁne d otherwise Pr o of. Applying Cor.5.3, we get W ′′ A f ( p ) = W ′ A θ ( f ( p )) and W ′ A f ( p ) = W ′ A θ − 1 ( f ( p )) . T o conclude, ob s erv e that θ ◦ f and θ − 1 ◦ f are b oth total (p oin t 1) or partial (p oin t 2) A -recursiv e as is f . W e can no w come to the notion of self-enumerate d systems for r.e. sets. Deﬁnition 5.5 (Self-en umerated systems for r.e. sets) . Let R E A ( X ) b e the class of A -r.e. subs ets of the b asic set X . Let ( W A e ) e ∈ 2 ∗ b e some ﬁxed acceptable en umeration of A -r.e. su bsets of X . C or.5.3 insur es that the families deﬁned hereafter do not dep end on the c hosen acceptable enumeration. 1. W e let F RE A ( X ) b e the f amily of all total fu nctions 2 ∗ → R E A ( X ) of the form p 7→ W A f ( p ) where f : 2 ∗ → 2 ∗ v aries o v er total A -recursiv e fun ctions. 2. W e let P F RE A ( X ) b e the family of all p art ial functions 2 ∗ → RE A ( X ) of the form p 7→  W A f ( p ) if f ( p ) is deﬁned undeﬁn ed o therwise where f : 2 ∗ → 2 ∗ v aries o v er p artial A -recurs ive functions. The follo w ing prop osition shows that, in th e d eﬁnition of F RE A ( X ) , one can either r elax the total “ A -recursiv e” condition on f to “partial A -recursive” with a sp ecial con v en tio n (diﬀerent f rom that considered in the deﬁn ition of P F RE A ( X ) ) or restrict it to some particular A -recursiv e sequence of total functions. Prop osition 5.6. F or any ac c eptable enumer ation ( W A e ) e ∈ 2 ∗ of A -r.e. sub- sets of X ther e exists a total A -r e cursive function σ : 2 ∗ × 2 ∗ → 2 ∗ such that, for any total function G : 2 ∗ → R E A ( X ) , the fol lowing c onditions ar e e quiv- alent: a. G is of the form p 7→ W A σ ( e , p ) for some e ∈ 2 ∗ b. G ∈ F RE A ( X ) c. F or al l p , G ( p ) =  W A g ( p ) if g ( p ) is deﬁne d ∅ otherwise . Pr o of. Since a ⇒ b ⇒ c is trivial whatev er b e th e total r ecursiv e fun ction σ , it remains to deﬁne σ such that c ⇒ a holds. Let ( φ A e ) e ∈ 2 ∗ b e an acceptable en umeration of partial A -recursiv e fun ctions X → N su c h that W A e = domain ( φ A e ). 16 Let ( ψ A e ) e ∈ 2 ∗ b e an enumeration of partial A -recursiv e fu nctions 2 ∗ → 2 ∗ and let a b e such that φ A ψ A e ( p ) ( x ) = φ A a ( h ( e , p ) , x i ) for all e , p ∈ 2 ∗ , x ∈ X . The parameter theorem insures that there exists a total A -recursive fun ction s : 2 ∗ × ( 2 ∗ × 2 ∗ ) → 2 ∗ suc h that φ A ψ A e ( p ) ( x ) = φ A a ( h ( e , p ) , x i ) = φ A s ( a , e , p ) ( x ) = φ A σ ( e , p ) ( x ) where σ ( e , p ) = s ( a , e , p ). Whence the equalit y W A ψ A e ( p ) = W A σ ( e , p ) whic h is also v al id when ψ A e ( p ) is un d eﬁned, in the sense th at b oth sets are empt y . Let G, g b e as in c. Sin ce g : 2 ∗ → 2 ∗ is A -recursiv e, there exists e suc h that g ( p ) = ψ A e ( p ) for an y p ∈ 2 ∗ . Thus, W A g ( p ) = W A ψ A e ( p ) = W A σ ( e , p ) an equalit y v alid also if g ( p ) is un d eﬁned, in th e s en se th at all s ets are emp ty . This prov es c ⇒ a . Theorem 5.7. ( RE A ( X ) , F RE A ( X ) ) and ( RE A ( X ) , P F RE A ( X ) ) ar e self-enumer ate d r epr esentation systems. Pr o of. Conditions i, ii A of Def.2.1, 2.10 are obvio us for b oth systems. If U satisﬁes iii A for P R A, 2 ∗ → X then p 7→  W A U ( p ) if U ( p ) is deﬁn ed undeﬁn ed o therwise satisﬁes iii A for P F RE A ( X ) with the same asso ciated comp function. Prop.5.6 pro v es that the fun ction p 7→ W A e satisﬁes condition iii A with σ as comp fu nction. Thus, ( RE A ( X ) , F RE A ( X ) ) and ( RE A ( X ) , P F RE A ( X ) ) are self-en umerated A -systems. W e conclude using Prop.2.12. Remark 5.8. I t is p ossible to impr o v e P r op.5.6 so as to get σ total recursive (rather than A -recursiv e) in condition a . This w ill hold for particular acce pt- able enumeratio ns of A -r.e. sets, with the same total recursive σ w hatev er b e A . W e sk etc h h ow this can b e obtained (for more details ab out this t ype of argument , cf. our pap er [6] § 2.3, 2.4.). Using p artial computable functionals X × P ( N ) → N , we can view p artial A -recursiv e f unctions as fu nctions ob tained by freezing the second order ar- gumen t in such functionals. W e can also also consider A -r.e. subsets of X as obtained f r om domains of su ch fu nctionals by freezing the second order argumen t. 17 When freezing the second order argumen t to A ⊆ N , acceptable enumer- ations of partial computable fun ctionals giv e acceptable en umerations of partial A -recursive functions. In this wa y , consider an acceptable enumeration (Φ e ) e ∈ 2 ∗ of partial com- putable functionals X × P ( N ) → N and let W A e = { x : ( x , A ) ∈ domain (Φ e ) } . Arguing as in the pr o of of Prop.5.6 (with an acceptable en umeration (Ψ e ) e ∈ 2 ∗ of partial computable fu nctionals 2 ∗ × P ( N ) → 2 ∗ ) we get Φ Ψ e ( p ,A ) ( x , A ) = Φ a ( h ( e , p ) , x i , A ) = Φ s ( a , e , p ) ( x , A ) = Φ σ ( e , p ) ( x , A ) where s is the total r ecur siv e fun ction in v olv ed in the parameter prop ert y for the acceptable enumeratio n (Φ e ) e ∈ 2 ∗ and σ ( e , p ) = s ( a , e , p ). No w, let G ∈ F RE A ( X ) and let g : 2 ∗ → 2 ∗ b e total A -recursiv e such that G ( p ) = W A g ( p ) . Let e ∈ 2 ∗ b e su c h that g = Ψ( e , A ). Th en Φ g ( p ) ( x , A ) = Φ Ψ e ( p ,A ) ( x , A ) = Φ σ ( e , p ) ( x , A ) and G ( p ) = W A g ( p ) = W A σ ( e , p ) 6 Inﬁnite computations Chaitin, 1976 [2], and Solo v a y , 1977 [20], considered inﬁ nite computations pro du cing inﬁ nite ob jects (namely r ecursiv ely en umerable sets) so as to de- ﬁne Kolmogoro v complexity of suc h inﬁnite ob jects. F ollo w in g the idea of p ossibly inﬁnite compu tations leading to ﬁnite output (i.e. remov e the sole halting condition), Bec her & Ch aitin & Daicz, 2001 [1] in tro duced a v arian t K ∞ of Kolmogoro v complexit y . In our pap er [5], 20 04, we intro d uced t w o v arian ts K max , K min of Kol- mogoro v complexit y and pro v ed that K ∞ = K max . These v arian ts are based on t w o self-en umerated representat ion systems, namely the classes of max and m in of p artial recursive sequences of partial recursive functions. 6.1 Self-en umerated systems of max of partial recursiv e func- tions Notation 6.1. Let A ⊆ N . 1. Let X b e a basic set. Extend ing Notation 1.3, we denote Rec A, 2 ∗ → X the family of total functions 2 ∗ → X whic h are r ecursiv e in A . 2. Let X b e N or Z . If f : 2 ∗ × N → X , we denote max f the fun ction (max f )( p ) = max { f ( p , t ) : t ∈ N } (with the con v en tion that max X is undeﬁn ed if X is empty or inﬁnite). W e deﬁn e the f amilies of f u nctions M ax 2 ∗ → X P R A = { max f : f ∈ P R A, 2 ∗ × N → X } M ax 2 ∗ → X Rec A = { max f : f ∈ Rec A, 2 ∗ × N → X } In case A is ∅ , we simply write M ax 2 ∗ → X P R and M ax 2 ∗ → X Rec . 18 Prop osition 6.2. L et A ⊆ N . Then ( N , M ax 2 ∗ → N P R A ) , ( Z , M ax 2 ∗ → Z P R A ) , ( N , M ax 2 ∗ → N Rec A ) ar e self- enumer ate d r e pr esentation systems. Pr o of. First consider th e n o oracle case (i.e. A = ∅ ). Conditions i-ii in Def.2.1 are trivial. Th e classical en umeration theorem easily extends to M ax 2 ∗ → X P R (cf. [5 ], Thm .4.1), proving condition iii for ( X , M ax 2 ∗ → X P R ) wh ere X is N or Z . It remains to sh o w condition iii for M ax 2 ∗ → N Rec . W e use the f ollo wing straigh t- forw ard fact (cf. [5], Thm .3.6): F act 6.3. If f ∈ P R 2 ∗ × N → N and g ( p , t ) = max ( { 0 } ∪ { f ( p , i ) : i ≤ t ∧ f ( p , i ) c onver ges in at most t steps } ) then g ∈ R ec 2 ∗ × N → N and max g i s an extension of m ax f with value 0 on domain (max g ) \ domain (max f ) (which is the set of n ’s such that f ( n, t ) is deﬁne d for no t ). Let U ∈ M ax 2 ∗ → N P R b e go o d u n iv ersal for M ax 2 ∗ → N P R and let V b e an ex- tension of U in M ax 2 ∗ → N Rec giv en by the ab o v e f act. If F ∈ R ec 2 ∗ → N then it is in P R 2 ∗ → N and there exists e such that F ( p ) = U ( comp U ( e , p )) f or all p ∈ 2 ∗ . Since V extends U and F is total, w e also ha v e F ( p ) = V ( comp U ( e , p )). Th us, V is go o d universal for M ax 2 ∗ → N Rec with th e same asso ciated comp function. Relativiza tion to oracle A pro v es conditions ii A , iii A , (cf. Def.2.10) for ( X , M ax 2 ∗ → X P R A ) and ( N , M ax 2 ∗ → N Rec ). W e conclud e using Prop .2.12 . Remark 6.4. 1. F act 6.3 imp lies that M ax 2 ∗ → X P R and M ax 2 ∗ → N Rec con tain the same total functions. Ho w ev er, considering partial functions, the inclusion M ax 2 ∗ → X Rec ⊂ M ax 2 ∗ → N P R is strict (cf. [5] Thm .3.6, p oint 1). 2. Let X b e N or Z and let M in 2 ∗ → X P R A , M in 2 ∗ → X Rec A b e deﬁned with min ins tead of max as in Poin t 2 of the ab ov e d eﬁnition (with the same conv entio n that min ∅ is und eﬁ ned). T hen ( X , M in 2 ∗ → X P R A ) is also a self-enumerated repr esen- tation system. W e shall not use an y min based sy s tem in this pap er b ecause they hav e no simple set theoretical counterparts. 3. None of the systems ( Z , M ax 2 ∗ → Z Rec A ), ( N , M in 2 ∗ → N Rec ) and ( Z , M in 2 ∗ → Z Rec ) is self-en umerated (cf. [5], Thm.4.3). 19 6.2 Kolmogoro v complexities K max , K ∅ ′ max , ... W e apply Def.2.16 to the self-en umerated represent ation systems considered in § 6.1. Deﬁnition 6.5 (Kolmogoro v complexities) . Let X b e N or Z . W e denote K A, X max : X → N the Kolmogoro v complexit y of the self-en umerated repre- sen tation system ( X , M ax 2 ∗ → X P R A ). In case X = N , we omit the sup ers cript N . In case X = N and A is ∅ w e simp ly write K max . Using Remark 2.14, p oin t 2, and F act 6.3, it is not h ard to p r o v e the follo w ing result (cf. [5], Prop.6.3). Prop osition 6.6. L et A ⊆ N . Then K A max is also the Kolmo gor ov c omplex- ity of the self-enumer at e d system ( N , M ax 2 ∗ → N Rec A ) . I.e. K N M ax 2 ∗ → N Rec A = K N M ax 2 ∗ → N P R A Remark 6.7. T he ab ov e prop osition has no analog with Z since M ax 2 ∗ → Z Rec A is not self-en umerated (cf. Remark 6.4, p oin t 3). 6.3 M ax 2 ∗ → N Rec and M ax 2 ∗ → N P R and inﬁnite c omputations The follo wing simple result giv es a mac hine charac terization of fu nctions in M ax 2 ∗ → N Rec A (resp. M ax 2 ∗ → N P R A ) whic h will b e u sed in the pro of of Thm.9.5. Deﬁnition 6.8. Let M b e an oracle T uring m ac hine suc h that 1. the alphab et of the in put tap e is { 0 , 1 } , plus an end-mark er to delim- itate the input, 2. the output tap e is w rite-only and has unary alphab et { 1 } , 3. there is no h alting s tate (r esp . but there are some distinguished states). The p artial fun ction F A : 2 ∗ → N computed by M with oracle A through inﬁnite computation (resp. with d istinguished states) is deﬁn ed as follo ws: F A ( p ) is deﬁned with v alue n if and only if the in ﬁnite computation (i.e. whic h lasts f orever) of M on input p outputs exactly n letters 1 (resp. and at some step the cu rrent state is a distinguished one). Prop osition 6.9. L et A ⊆ N b e an or acle. A function F : 2 ∗ → N i s in M ax 2 ∗ → N Rec A (r esp. M ax 2 ∗ → N P R A ) if and only if ther e e xists an or acle T u ring machine M which, with or acle A , c omputes F thr ough inﬁnite c omputation (r esp. with distinguishe d states) in the sense of Def.6.8. 20 Pr o of. ⇐ . The function asso ciated to an oracle T uring mac hine thr ough inﬁnite computation (resp. with distinguish ed states) is clearly max f where f ( p , t ) is the current output at step t (resp. and is undeﬁned while the mac hine has n ot b een in some distingu ish ed state). ⇒ . Su pp ose f : 2 ∗ × N → N is total (resp . partial) A -recur siv e and set X ( p , t ) = { f ( p , t ′ ) : t ′ < t ∧ f ( p , t ′ ) con v erge s in ≤ t steps } ) Observe that X ( p , 0) = ∅ , so that the follo wing is ind eed an A -recursive deﬁnition: e f ( p , t ) =    0 (resp. und eﬁned) if X ( p , t ) = ∅ e f ( p , t − 1) + 1 if X ( p , t ) 6 = ∅ ∧ e f ( p , t − 1) < max X ( p , t ) e f ( p , t − 1) otherwise Then max e f = max f . Also, the un ary r epresen tation of e f ( p , t ) can b e simp ly in terpreted as the current output at step t of the inﬁnite compu tation (resp. with distinguished states) of an oracle T uring machine with inp ut p . So that max e f is the function asso ciated to that mac hine. 6.4 M ax 2 ∗ → N P R and t he jump The follo wing prop osition is easy . Prop osition 6.10. L et A ⊆ N and let X b e N or Z . Then M ax 2 ∗ → X P R A ⊂ P R A ′ , 2 ∗ → X Pr o of. 1. Let f : 2 ∗ → X b e p artial A -recursive. A partial A ′ -recursiv e deﬁnition of (max f )( p ) is as follo ws: i. First, c hec k wh ether there exists t s u c h that f ( p , t ) is deﬁned. If the c hec k is negativ e then (max f )( p ) is undeﬁn ed. ii. If chec k i is p ositive then start successiv e steps of the follo wing pro cess. - At step t , chec k w h ether f ( p , t ) is d eﬁned, - if deﬁned, compute its v al ue, - and chec k wh ether there exists u > t suc h that f ( p , u ) is greater than the maximum v alue computed up to that step. iii. If at some step the last c hec k in ii is n egativ e then halt and output th e maxim um v alue computed u p to n o w. Clearly , oracle A ′ allo w s for the chec ks in i and ii. Also, the ab o v e pr o cess halts if and only if f ( p , t ) is deﬁned for some t and { f ( p , t ) : t ∈ N } is b ound ed, i.e. if and only if (max f )( p ) is deﬁn ed . In that case it outputs exactly (max f )( p ). 21 2. T o see that the inclusion is strict, observ e that the graph of an y function in M ax 2 ∗ → X P R A is Σ 0 ,A 1 ∧ Π 0 ,A 1 since y = (max f )( p ) ⇔ (( ∃ t f ( p , t ) = y ) ∧ ¬ ( ∃ u ∃ z > y f ( p , u ) = z )) Whereas th e graph of functions in P R A ′ , 2 ∗ → X can b e Σ 0 ,A ′ 1 and n ot ∆ 0 ,A ′ 1 , i.e. Σ 0 ,A 2 and not ∆ 0 ,A 2 . In the v ein of Prop.6.10, let’s mention the follo wing result, cf. [1] (where the pro of is for K ∞ , cf. start of § 6 ab o v e) and [5] Prop.7.2-3 & Cor.7.7. Prop osition 6.11. L et A ⊆ N . 1. K A and K A max ar e r e c ursive in A ′ . 2. K A > ct K A max > ct K A ′ . 6.5 The ∆ op eration on M ax 2 ∗ → N P R and the jump The follo wing v ariant of Prop.6.10 is a n ormal form for partial A ′ -recursiv e Z -v alued f unctions. W e sh all use it in § 8-9. Theorem 6.12. L et A ⊆ N . Then P R A ′ , 2 ∗ → Z = ∆( M ax 2 ∗ → N P R A ) = ∆( M ax 2 ∗ → N Rec A ) Thus, eve ry p artial A ′ -r e cursive function is the diﬀer enc e of two functions in M ax Rec A (cf. N otation 6.1). Before en tering the pro of of Thm .6.12, let’s recall t w o w ell-kno w n facts ab out oracular computation and appro ximation of the ju mp. Lemma 6.13. L et ( B t ) t ∈ N b e a se quenc e of subsets of N which c onver g es p ointwise to B ⊆ N , i.e. ∀ n ∃ t n ∀ t ≥ t n B t ∩ { 0 , 1 , ..., n } = B ∩ { 0 , 1 , ..., n } L et X , Y b e b asic sets and let ψ : X → Y b e a p artial B - r e cursive function c ompute d by some or acle T u ring machine M with or acle B . L et x ∈ X . Then, ψ ( x ) is deﬁne d if and only if ther e exists t x such that i. the c o mputation of M on input x with or acle B t x halts in at most t x steps, ii. for al l t ≥ t x the c omputa tion of M on input x with or acle B t is step by step exactly the same as that with or acle B t x (in p articular, it asks the same questions to the or acle, gets the same answers and halts at the same c omputation step ≤ t x ). 22 Lemma 6.14. L et A ⊆ N and let A ′ ⊆ N b e the jump of A . Ther e ex- ists a total A -r e cursive se q u enc e ( Appr ox ( A ′ , t )) t ∈ N of subsets of N which i s monotone incr e as ing with r esp e ct to set inc lu si on and which has u nion A ′ . In p articular, this se quenc e c onver ges p o intwise to A ′ . W e can no w pr o v e Thm.6.12. Pr o of of Thm.6.12. Using Prop.6.10 and Prop.4.4, we get ∆( M ax 2 ∗ → N Rec A ) ⊆ ∆( M ax 2 ∗ → N P R A ) ⊆ ∆( P R A ′ , 2 ∗ → N ) = P R A ′ , 2 ∗ → Z Since M ax 2 ∗ → N Rec A is closed by su m s, we hav e ∆(∆( M ax 2 ∗ → N Rec A ) = ∆( M ax 2 ∗ → N Rec A ). Th us, to get the wan ted equalit y , it suﬃces to prov e inclusion P R A ′ , 2 ∗ → N ⊆ ∆( M ax 2 ∗ → N Rec A ) Let M b e an oracle T uring mac hine with inp uts in 2 ∗ , w h ic h, with oracle A ′ , computes the partial A ′ -recursiv e fun ction ϕ A ′ : 2 ∗ → N . T o prov e that ϕ A ′ is in ∆( M ax 2 ∗ → N Rec A ), we d eﬁne total A -recursiv e fun ctions f , g : 2 ∗ × N → N w hic h are (n on strictly) m on otone increasing and suc h that ϕ A ′ = max f − max g . The idea to get f , g is as follo ws. W e consider A -recurs ive approximati ons of oracle A ′ (as giv en by Lemma 6.14 ) and use them as fak e oracles. F unction f is obtained b y letting M run with the f ak e oracles and restart its compu- tation eac h time some b etter approxi mation of A ′ sho ws the previous fak e oracle has giv en an incorrect answe r. F unction g collects all the outputs of the computations wh ich ha v e b een recognized as incorrect in the computing pro cess for f . W e now formally d eﬁne f , g . First, since we do not care ab out computation time and space, we can sup- p ose without loss of generalit y , that, at an y step t , M asks to the oracle ab out the intege r t and writes d o wn the oracle answ er on the t -th cell of some dedicated tap e. Consider t + 1 steps of the compu tation o f M on inpu t p w ith oracle Appr ox ( A ′ , t ) (cf. Lemma 6.14). W e d enote C p ,t +1 this limited computa- tion. W e say that C p ,t +1 halts if M (with that fak e oracle) halts in at most t + 1 steps. W e denote output ( C p ,t ) the current v alue (whic h is in Z ) of the output tap e after step t . The A -recursive deﬁnition of f , g is as follo ws. i. f ( p , 0) = g ( p , t ) = 0 ii. S upp ose Appr ox ( A ′ , t + 1) ∩ { 0 , ..., t } = Appr ox ( A ′ , t ) ∩ { 0 , ..., t } . T h en, up to the halting step of C p ,t or up to step t in case C p ,t do es not halt, b oth compu tations C p ,t , C p ,t +1 are step wise identica l. 23 (a) If C p ,t halts then so do es C p ,t +1 at the same step. And b oth computations hav e the s ame output. In that case, we set f ( p , t + 1) = f ( p , t ) , g ( p , t + 1) = g ( p , t ). (b) If C p ,t do es not h alt th en let δ t +1 = outp ut ( C p ,t +1 ) − output ( C p ,t ), and set f ( p , t + 1) = f ( p , t ) + 1 + max(0 , δ t +1 ) g ( p , t + 1) = g ( p , t ) + 1 + max(0 , − δ t +1 ) i.e. w e add | δ t +1 | to f or g according to th e sign of δ t +1 . iii. Su pp ose Appr ox ( A ′ , t + 1) ∩ { 0 , ..., t } 6 = Appr ox ( A ′ , t ) ∩ { 0 , ..., t } . Since these approximati ons are monotone in creasing, we necessarily h a v e Appr ox ( A ′ , t ) ∩ { 0 , ..., t } 6 = A ′ ∩ { 0 , ..., t + 1 } . Th us, th e fak e oracle in C p ,t has giv en answ ers w hic h are not compatible with A ′ . In that case, w e set f ( p , t + 1) = f ( p , t ) + g ( p , t ) + 1 + max(0 , output ( C p ,t +1 )) g ( p , t + 1) = f ( p , t ) + g ( p , t ) + 1 + m ax(0 , − output ( C p ,t +1 )) i.e. we up rise f , g to a common v alue (namely f ( p , t ) + g ( p , t )) and then add | output ( C p ,t +1 ) | to f or g according to the sign of outp ut ( C p ,t +1 ). F rom th e ab o v e indu ctiv e deﬁnition, we see that, f or eac h t > 0, f ( p , t ) − g ( p , t ) = output ( C p ,t ) Supp ose ϕ A ′ ( p ) is deﬁne d. Applying Lemmas 6.13, 6.14, w e see that there exist s p ≤ t p suc h that - M , on input p , with oracle A ′ , halts in s p steps, - Appr ox ( A ′ , t p ) ∩ { 0 , ..., t p } = A ′ ∩ { 0 , ..., t p } . Th us, for all t ≥ t p , f p ,t = f p ,t p and g p ,t = g p ,t p and f p ,t − g p ,t = ϕ A ′ ( p ). Supp ose ϕ A ′ ( p ) is not deﬁne d. Observe that, eac h time the “fak e” computation C p ,t with oracle Appr ox ( A ′ , t ) do es not halt or app ears not to b e the “right” one with oracle A ′ (b ecause Appr ox ( A ′ , t + 1) ∩ { 0 , ..., t } d iﬀers from Appr ox ( A ′ , t ) ∩ { 0 , ..., t } ), w e strictly increase b oth f , g (this is why w e pu t +1 in the equations of iib and iii). Applying Lemmas 6.13, 6.14, we see that, if ϕ A ′ ( p ) is not deﬁned then C p ,t do es n ot halt for inﬁ nitely many t ’s, so that f ( p , t ) and g ( p , t ) incr ease in- ﬁnitely often. T herefore, (max f )( p ) and (max g )( p ) are b oth und eﬁned, and so is th eir diﬀerence. This p ro v es that ϕ A ′ = max f − max g . Sin ce the sequence ( Appr ox ( A ′ , t )) t ∈ N is A -recursive , so are f , g . Th us, max f , max g are in M ax 2 ∗ → N Rec A and their diﬀerence ϕ A ′ is in ∆( M ax 2 ∗ → N Rec A ). ✷ 24 7 Abstract represen tations and eﬀectivizat ions 7.1 Some arithmetical represen tations of N As p oin ted in § 1.1, abstract en tities such as n um b ers can b e repr esented in man y diﬀerent w a ys. In fact, eac h r ep resen tatio n illuminates some p artic- ular role and/or prop ert y , i.e. some p ossible semantics c hosen in order to eﬃcien tly access sp ecial op erations or stress sp ecial p rop erties of integ ers. Usual arithmetical representati ons of N using w ords on a digit alphab et can b e lo ok ed at as a (total) sur jectiv e (non n ecessarily injectiv e) fu nction R : C → N where C is some simple fr ee algebra or a quotient of some f r ee algebra. Suc h representat ions are the “degree zero” of abstraction f or representa tions and, as exp ected, their asso ciated Kolmogoro v complexities all coincide (cf. Thm.7.8 b elo w). Example 7.1 (Base k r epresen tations) . 1. Intege rs in unary representa tion corresp ond to elemen ts of th e free alge- bra built u p from one generator and one unary f u nction, namely 0 and the successor function x 7→ x + 1. The asso ciated function R : 1 ∗ → N is simply the length f unction. 2. The v a rious base k (with k ≥ 2) repr esen tatio ns of intege rs also inv olv e term algebras, not n ecessarily free. They diﬀer by the set A ⊂ N of d igits they u se but all are based on the usu al in terpretation R : A ∗ → N suc h that R ( a n . . . a 1 a 0 ) = P i =0 ,...,n a i k i . Which, written ` a la H¨ orner, k ( k ( . . . k ( k a n + a n − 1 ) + a n − 2 ) . . . ) + a 1 ) + a 0 is a comp osition of applications S a 0 ◦ S a 1 ◦ . . . ◦ S a n (0) wh ere S a : x 7→ k x + a . If a representat ion uses d igits a ∈ A then it corresp ond s to the alge bra generated by 0 and the S a ’s where a ∈ A . i. T he k -adic represent ation uses d igits 1 , 2 , . . . , k and corresp onds to a free algebra built up from one generator and k un ary functions. ii. Th e usual k -ary represen tatio n u ses d igits 0 , 1 , . . . , k − 1 and corre- sp onds to the quotien t of a free algebra built up from one generator and k un ary functions, namely 0 and the S a ’s wh ere a = 0 , 2 , . . . , k − 1, b y the r elation S 0 (0) = 0. iii. Avizienis base k rep r esen tatio n uses digits − k + 1 , . . . , − 1 , 0 , 1 , . . . , k − 1 (it is a m uc h redund an t representat ion used in compu ters to p erform additions without carry propagation) and corresp onds to the quotient of th e free algebra b u ilt up f rom one generator and 2 k − 1 unary functions, namely 0 and the S a ’s where a = − k + 1 , . . . , − 1 , 0 , 1 , . . . , k − 25 1, by the relations ∀ x ( S − k + i ◦ S j +1 ( x ) = S i ◦ S j ( x )) wh ere − k < j < k − 1 and 0 < i < k . Somewhat exotic repr esentati ons of int egers can also b e asso ciated to deep results in n um b er theory . Example 7.2. 1. R : N 4 → N su ch that R ( x, y , z , t ) = x 2 + y 2 + z 2 + t 2 is a representati on based on Lagrange’s four squ ares theorem. 2. R : ( P r ime ∪ { 0 } ) 7 → N such that R ( x 1 , . . . , x i ) = x 1 + . . . + x i is a represent ation based on Schnirelman’s theorem (1931) in its last improv ed v ersion obtained b y R amar´ e, 1995 [13], whic h insur es that ev ery eve n num ber is the sum of at most 6 prim e num b ers (hence ev ery num b er is the sum of at most 7 primes). Suc h representa tions app ear in the stud y of the expressional p o w er of some w eak arithmetics. F or instance, the represen tation as sums of 7 primes allo ws for a v ery simple pro of of the deﬁnability of multiplic ation with addition and the divisibility p redicate (a result v alid in fact with successor and divisibility , (Julia Robins on, 1948 [14])). 7.2 Abstract representations F oundational questions, going back to Russell, [16] 1908, an d Ch urc h, [3] 1933, lead to qu ite d iﬀeren t r epresen tations of N : set theoretical r ep resen- tations in v olving abstract sets and fun ctionals m uc h more complex than the in tegers they r epresen t. W e shall consider the follo wing simp le and general notion. Deﬁnition 7.3 (Abstract repr esen tatio ns) . A representa tion of an inﬁnite set E is a pair ( C , R ) where C is some (nec- essarily inﬁ nite) set and R : C → E is a surje ctive p artial function. Remark 7.4. 1. Though R really op erates on the sole subset domain ( R ), the un derlying set C is qu ite signiﬁcant in the eﬀectiviza tion p r o cess which is n ecessary to get a self-enumerate d systen and then an asso ciated Kolmogo ro v complexit y . 2. W e shall consider repr esen tati ons with arbitrarily complex domains in the P ost hierarc h y (cf. Prop.8.4, 9.3, 10.23, and coming pap ers). In fact, the sole cases in this pap er where R is a total f unction are the usu al recursive represent ations. 3. Represent ations can also inv olve a prop er class C (cf. Rk. 8.3). Ho w ev er, w e shall stic k to the case C is a set. 26 7.3 Eﬀectivizing r epresen tations: wh y? T urn in g to a computer s cience (or recursion theoretic) p oin t of view, there are some ob jections to the consideration of abstract sets, functions and functionals as w e did in § 1.1 and 7.2: • W e cannot appr ehend abstract sets, functions and functionals bu t solely programs to compute them (if they are computable in some sense). • Moreo v er, programs dealing with sets, f unctions and functionals h a v e to go th rough some in tensional r ep resen tatio n of these ob jects in order to b e able to compu te with suc h ob jects. T o get eﬀectiv eness, w e turn f rom set theory to compu tabilit y theory . W e shall d o that in a somewh at abstract wa y u sing s elf-en umerated r epresen ta- tion systems (cf. Def.2.1). W e shall consider higher ord er r ep resen tatio ns and shall “eﬀectivize” ab- stract sets, functions and fun ctionals via recursiv ely enumerable s ets, partial recursiv e f unctions or max of total or partial recurs iv e functions, and partial computable fu nctionals. 7.4 Eﬀectivizations of representations and asso ciated Kol- mogoro v complexities A formal represen tatio n of an in tege r n is a ﬁnite ob j ect (in general a word) whic h describ es some characte ristic prop ert y of n or of some abstract ob ject whic h characte rizes n . T o eﬀectivize a represent ation R : C → E , we shall pro cess as f ollo ws: 1. Restrict the s et C to a sub family D of elemen ts wh ic h, in some sense, are computable or p artial computable. O f course, we w an t the restric- tion of R to D to b e still surjectiv e. 2. Cons ider a self-enumerate d representat ion system for D . This leads to the f ollo win g deﬁnition. Deﬁnition 7.5. 1. A set D is adapted to th e repr esen tatio n R : C → E if D ⊆ C and the partial fun ction R ↾ D : D → E is still su rjectiv e. 2. [Eﬀectivization] An eﬀectivization of the r epresen tation R : C → E of the set E is an y self-en umerated r epresen tation sys tem ( D , F ) for a d omain D adapted to the r epresen tation R : C → E . Using the Comp osition Lemma 3.1, we immediately get 27 Prop osition 7.6. L et R : C → E b e a r epr esentation of E and ( D , F ) b e some eﬀe ctivization of R . Then ( E , ( R ↾ D ) ◦ F ) is a self-enumer ate d r epr esentation system and the asso ciate d Kolmo gor ov c omplexity K E ( R ↾ D ) ◦F (cf. Def.2.16) satisﬁes K E ( R ↾ D ) ◦F ( x ) = min { K D F ( y ) : R ( y ) = x } for al l x ∈ E Remark 7.7. Whereas abstract repr esentati ons are quite natural and con- ceptually simple, the fun ctions ( R ↾ D ) ◦ F , for F ∈ F , in the self-en umerated represent ation families of their eﬀectivized v ersions ma y b e qu ite complex. In the examples w e shall consid er, their domains in v olv e lev els 2 or 3 of the arithmetical hierarc h y . In particular, su ch repr esentati ons are not T urin g reducible one to the other. 7.5 P artial recursive represen tations W e already men tioned in § 7.1 that all usual arithmetic represen tatio ns lead to the same Kolmogoro v complexity (up to an additive constan t). Th e follo w ing result extend s this assertion to all partial r ecursiv e representa tions. Theorem 7.8. W e ke ep the notations of Notations 1.3 and Def.2.16. L et A ⊆ N b e an or acle. If C, E ar e b a sic sets and R : C → E is p artial r e c ursive (r esp. p artial A -r e cursive) then R ◦ P R 2 ∗ → C = P R 2 ∗ → E (r esp. R ◦ P R A, 2 ∗ → C = P R A, 2 ∗ → E ) K E R ◦ P R 2 ∗ → C = K E (r esp. K E R ◦ P R A, 2 ∗ → C = K A E ) Thus, al l Kolmo gor ov c omp lexities asso ciate d to p artial r e c ursive (r esp. p ar - tial A -r e cursive) r e pr esentat ions of E c oincide with the u sual (r esp. A - or acular) Kolmo gor ov c omp lexity on E . Pr o of. It suﬃces to pro v e that R ◦ P R A, 2 ∗ → C = P R A, 2 ∗ → E Inclusion R ◦ P R A, 2 ∗ → C ⊆ P R A, 2 ∗ → E is trivial. F or the other inclusion, we use the fact that R : C → E is surjectiv e p artial A -recursive . First, deﬁne a partial A -recursiv e S : E → C su c h that, for x ∈ E , S ( x ) is the elemen t y ∈ C satisfying R ( y ) = x which app ears ﬁrst in an A - recursiv e en umeration of the graph of R . Clearly , S is a righ t inv erse of R , i.e. R ◦ S = I d E where I d E is the iden tit y on E . Using the trivial inclusion S ◦ P R A, 2 ∗ → E ⊆ P R A, 2 ∗ → C w e get P R A, 2 ∗ → E = R ◦ S ◦ P R A, 2 ∗ → E ⊆ R ◦ P R A, 2 ∗ → C 28 8 Cardinal represen tati ons of N 8.1 Basic cardinal represen tation and its eﬀectivizations Among the conceptual representa tions of int egers, the most basic one go es bac k to Russell, [16] 1908 (cf. [22] p.178), and considers non negativ e in tege rs as equiv ale nce classes of sets relativ e to cardinal comparison. Deﬁnition 8.1 (Cardinal rep resen tatio n of N ) . Let c ar d ( Y ) d en ote the cardinal of Y , i.e. the num b er of its elements. The cardinal representati on of N relativ e to an inﬁnite set X is the partial function c ar d X : P ( X ) → N with domain P <ω ( X ), su ch that c ar d X ( Y ) =  c ar d ( Y ) if Y is ﬁnite undeﬁn ed o therwise Deﬁnition 8.2 (Eﬀectivizations of the cardinal r epresen tation of N ) . W e ef- fectivize the cardinal r epresen tation b y r eplacing P ( X ) by RE ( X ) or RE A ( X ) where X is some b asic set an d A ⊆ N is some oracle. Tw o kind s of self-en umerated represen tation systems can b e naturally asso- ciated to these domains (cf. § 5.2 and the Comp osition Lemma 3.1): ( RE ( X ) , c ar d ◦ F RE ( X ) ) or ( RE A ( X ) , c ar d ◦ F RE A ( X ) ) ( RE ( X ) , c ar d ◦ P F RE ( X ) ) or ( RE A ( X ) , c ar d ◦ P F RE A ( X ) ) Remark 8.3. 1. Historically , the cardinal repr esen tatio n of N considered the whole class of sets rather than some P ( X ). Ho w ev er, the ab o v e eﬀectivizatio n m ak es suc h an extension unsigniﬁcant for our study . 2. One can also consider the total repr esen tatio n obtained by r estriction to the set P <ω ( X ) of all ﬁ nite subsets of X . Bu t this amoun ts to a partial recursiv e repr esen tatio n an d is relev a n t to § 7.5. 8.2 Syn tactical complexit y of cardinal represen tations The follo wing prop osition giv es the synta ctical complexit y of the ab o v e ef- fectivizat ions of th e cardinal representa tions. Prop osition 8.4 (Syntac tical complexity) . The family { domain ( ϕ ) : ϕ ∈ c ar d ◦ F RE A ( X ) } is exactly the family of Σ 0 ,A 2 subsets of 2 ∗ . Idem with c ar d ◦ P F RE A ( X ) . In p articular, any universal function for c ar d ◦F RE A ( X ) or for c ar d ◦P F RE A ( X ) is Σ 0 ,A 2 -c omplete. 29 Pr o of. Let ( W A e ) e ∈ 2 ∗ b e an acceptable enumeration of RE A ( X ). 1. I f g : 2 ∗ → 2 ∗ is p artial A -recursiv e then domain ( p 7→ c ar d ( W A g ( p ) ) = { p : W A g ( p ) is ﬁnite } is clearly Σ 0 ,A 2 . 2. Let X ⊆ 2 ∗ b e a Σ 0 ,A 2 set of the f orm X = { p : ∃ u ∀ v R ( p , u, v ) } where R ⊆ 2 ∗ × N 2 is A -recursive . Set σ p =  { u ′ : u ′ < u } if u is least such that ∀ v R ( p , u, v ) N if there is no u suc h that ∀ v R ( p , u, v ) It is easy to c hec k that σ p ⊆ N is an A -r.e. set whic h can b e deﬁned b y the follo w ing en umeration pr o cess describ ed in P ascal-li k e instru ctions: { Initializa tion } u := 0 ; v := 0 ; { Loop } DO FOREVER BEGIN WHILE R ( p , u, v ) DO v := v + 1 ; output u in σ p ; u := u + 1 ; v := 0 ; END; Clearly , car d ( σ p ) is ﬁnite if and only if p ∈ X . No w, th e set { ( p , n ) : n ∈ σ p } is also A -r.e., hence of the form W 2 ∗ × N a for some a . The parameter prop erty yields a total A -recursive function g : 2 ∗ → 2 ∗ suc h that σ p = W g ( a , p ) . Finally , th e fun ction p 7→ c ar d ( W g ( a , p ) ) is in c ar d ◦ F RE A ( X ) and has domain X . 8.3 Characterization of the c ar d self-en umerated systems Theorem 8.5. F or any b asic set X and any or a cle A ⊆ N , 1i. c ar d ◦ F RE A ( X ) = M ax 2 ∗ → N Rec A ii . c ar d ◦ P F RE A ( X ) = M ax 2 ∗ → N P R A 2. K N c ar d ◦F RE A ( X ) = ct K N c ar d ◦P F RE A ( X ) = ct K A max We shal l simply write K N ,A c ar d in plac e of K N c ar d ◦F RE A ( N ) . When A = ∅ we simply write K N c ar d . Pr o of. P oin t 2 is a direct corollary of Poin t 1 and Pr op.6.6. Let’s prov e p oint 1. 1i. Inclusion ⊆ . Let g : 2 ∗ → 2 ∗ b e total A -recursive . W e deﬁ n e a total A -recursive fun ction u : 2 ∗ × N → N such that ( ∗ ) { u ( p , t ) : t ∈ N } = ( { 0 , ..., n } if W A g ( p ) con tains exactly n p oin ts N if W A g ( p ) is inﬁ nite 30 The deﬁnition is as f ollo ws. First, set u ( p , 0) = 0 for all p . Consid er an A - recursiv e en umeration of W A g ( p ) . If at step t , some new p oin t is enumerated then set u ( p , t + 1) = u ( p , t ) + 1, else set u ( p , t + 1) = u ( p , t ). F rom ( ∗ ) we get c ar d ( W p ) = (max f )( p ), so that p 7→ c ar d ( W A g ( p ) ) is in M ax 2 ∗ → N Rec A . 1ii. Inclusion ⊆ . No w g : 2 ∗ → 2 ∗ is partial A -recurs ive and w e d eﬁne u : 2 ∗ × N → N as a partial A -recursive function such that { u ( p , t ) : t ∈ N } =      ∅ if g ( p ) is u ndeﬁned { 0 , ..., n } if W A g ( p ) con tains exactly n p oints N if W A g ( p ) is inﬁn ite The deﬁ n ition of u is as ab o v e except that, f or any t , w e requ ir e that u ( p , t ) is deﬁned if and only if g ( p ) is. 1i. Inclusion ⊇ . An y function in M ax 2 ∗ → N Rec A is of the form max f : 2 ∗ → N where f : 2 ∗ × N → N is total A -recursiv e. The idea to pro v e that max f is in c ar d ◦ F RE A ( X ) is quite simp le. F or ev ery p , we deﬁne an A -r.e. s u bset of X whic h collects some new elemen ts eac h time f ( p , t ) gets greater than max { f ( p , t ′ ) : t ′ < t } . F ormally , let ψ : 2 ∗ × N → N b e the partial A -recur siv e fu nction suc h that ψ ( p , t ) =  0 if ∃ u f ( p , u ) > t undeﬁn ed o therwise Clearly , domain ( ψ p ) =  { t : 0 ≤ t < (max f )( p ) } if (max f )( p ) is deﬁned N otherwise W e d eﬁne ϕ : 2 ∗ × X → N su c h that ϕ ( p , x ) = ψ ( p , θ ( x )) where θ : X → N is some ﬁ x ed total recurs iv e bijection. Let’s d enote ψ p and ϕ p the fu nc- tions t 7→ ψ ( p , t ) and x 7→ ϕ ( p , x ). Let e b e su c h that W A e = {h p , x i : ( p , x ) ∈ domain ( ϕ ) } (where h , i is a bijection 2 ∗ × X → X ). T h e p aram- eter prop ert y yields an A -recurs iv e function s : 2 ∗ × 2 ∗ → 2 ∗ suc h that W A s ( e , p ) = domain ( ϕ p ) for all p . Thus, letting g : 2 ∗ → 2 ∗ b e the A -recursiv e function such that g ( p ) = s ( e , p ), we ha v e c ar d ( W A g ( p ) ) = c ar d ( domain ( ϕ p )) = c ar d ( domain ( ψ p )) = (max f )( p ) Whic h prov es that m ax f is in c a r d ◦ F RE A ( X ) . 1ii. Inclusion ⊇ . 31 W e argue as in the ab o v e p ro of of i. ⊇ . Ho wev er, f : 2 ∗ × N → N is no w partial A -recursive and there are tw o reasons for which (max f )( p ) may b e undeﬁn ed: ﬁrst, if t 7→ f ( p , t ) is u n b ounded , second if it has empt y d omain. Keeping ψ and ϕ as deﬁn ed as ab ov e, we no w ha v e, domain ( ψ p ) =    { v : 0 ≤ v < (max f )( p ) } if (max f )( p ) is deﬁned N if r ang e ( t 7→ f ( p , t )) is inﬁnite ∅ if f ( p , t ) is deﬁned for no t W e let e , s, g b e as ab o v e and d eﬁne h : 2 ∗ → 2 ∗ suc h that h ( p ) =  g ( p ) if f ( p , t ) is deﬁned for some t undeﬁn ed o therwise Observe that - if t 7→ f ( p , t ) has empt y domain then h ( p ) is undeﬁn ed, - if t 7→ f ( p , t ) is un b ound ed then car d ( W A h ( p ) ) = car d ( W A g ( p ) ) is inﬁnite, - otherwise car d ( W A h ( p ) ) = car d ( W A g ( p ) ) = (max f )( p ). Whic h prov es that m ax f is in car d ◦ P F RE A ( X ) . 8.4 Characterization of the ∆ c ar d represen tation system W e no w lo ok at the self-delimited system with domain Z obta ined from car d ◦ F RE A ( X ) b y the op eration ∆introd uced in § 4.1. Theorem 8.6. L et A ⊆ N and let A ′ b e the jump of A . L et X b e a b asic set. Then ∆( car d ◦ F RE A ( X ) ) = ∆( car d ◦ P F RE A ( X ) ) = P R A ′ , 2 ∗ → Z Henc e K Z ∆( card ◦F RE A ( X ) ) = ct K A ′ , Z . We shal l simply write K N ,A ∆ car d in plac e of K Z ∆( card ◦F RE A ( N ) ) ↾ N . When A = ∅ we simply write K Z ∆ car d . Pr o of. The equalities ab out the s elf-enumerated systems is a direct corolla ry of T h m.8.5 and Thm.6.12. T he equalities ab out Kolmogoro v complexities are trivial corollaries of those ab out self-en umerated systems. 9 Index represen tat ions of N 9.1 Basic index represen tation and its eﬀectivizations A v arian t of the cardinal representat ion considers indexes of equiv alence relations. More precisely , it views an integ er as an equiv alence class of equiv alence relations relativ e to ind ex comparison. 32 Deﬁnition 9.1 (Index representat ion) . The index repr esen tati on of N relativ e to an inﬁnite set X is the partial function index N P ( X 2 ) : P ( X 2 ) → N with domain the family of equ iv alence relations on subsets of X wh ic h hav e ﬁnite index, s uc h that index N P ( X 2 ) ( R ) =    index ( R ) if R is an equiv alence relation with ﬁn ite index undeﬁn ed o therwise (where index ( R ) denotes th e num ber of equiv alence classes of R ). 9.2 Syn tactical complexit y of index representations Deﬁnition 9.2 (Eﬀectivization of the index representat ion of N ) . W e eﬀec- tivize the index representa tion b y replacing P ( X 2 ) b y RE ( X 2 ) or R E A ( X 2 ) where X is some b asic set an d A ⊆ N is some oracle. Tw o kind s of self-en umerated represen tation systems can b e naturally asso- ciated (cf. § 5.2 and the C omp osition Lemma 3.1): ( RE ( X 2 ) , index ◦ F RE ( X 2 ) ) or ( RE A ( X 2 ) , index ◦ F RE A ( X 2 ) ) ( RE ( X 2 ) , index ◦ P F RE ( X 2 ) ) or ( RE A ( X 2 ) , index ◦ P F RE A ( X 2 ) ) The follo wing prop osition give s the syntact ical complexit y of the ab o v e eﬀectiviza tions of the ind ex representa tions. Prop osition 9.3 (Syntac tical complexity) . The family { domain ( ϕ ) : ϕ ∈ index ◦ F RE A ( X ) } is exactly the family of Σ 0 ,A 3 subsets of 2 ∗ . Idem with i ndex ◦ P F RE A ( X ) . In p articular, any universal function for index ◦ F RE A ( X ) or for index ◦ P F RE A ( X ) is Σ 0 ,A 3 -c omplete. Pr o of. W e trivially r ed uce to the case X = N and only consider the case A = ∅ , relativization b eing straigh tforw ard. 1. Let ( W N 2 e ) e ∈ 2 ∗ b e an acceptable enumeratio n of R E ( N 2 ) and g : 2 ∗ → 2 ∗ b e a partial recursive fun ction and ψ : 2 ∗ → N b e suc h th at ψ ( p ) = index ( W N 2 g ( p ) ). T o see that domain ( ψ ) is Σ 0 3 , observe that p ∈ domain ( ψ ) if and only if i. g ( p ) is deﬁned. Whic h is a Σ 0 1 condition. 33 ii. W N 2 g ( p ) is an equiv ale nce relation on its domain, i.e. ∀ x ∀ y (( x , y ) ∈ W N 2 g ( p ) ⇒ (( x , x ) ∈ W N 2 g ( p ) ∧ ( y , x ) ∈ W N 2 g ( p ) )) ∧ ∀ x ∀ y ∀ z ((( x , y ) ∈ W N 2 g ( p ) ∧ ( y , z ) ∈ W N 2 g ( p ) ) ⇒ ( x , z ) ∈ W N 2 g ( p ) ) Whic h is a Π 0 2 form ula (since ( u , v ) ∈ W N 2 g ( p ) is Σ 0 1 ). iii. W N 2 g ( p ) has ﬁ nitely many classes, i.e. ∃ n ∀ k ∃ m ≤ n ( k , m ) ∈ W N 2 g ( p ) . Whic h is a Σ 0 3 form ula. 2. Let X ⊆ 2 ∗ b e Σ 0 3 . W e construct a total r ecursiv e function g : 2 ∗ → 2 ∗ suc h that X = { p : index ( W N 2 g ( p ) ) is ﬁ nite } . A. Supp ose X = { p : ∃ u ∀ v ∃ w R ( p , u, v , w ) } wh ere R ⊆ 2 ∗ × N 3 is recursive . Let θ : 2 ∗ × N 2 → N b e the total recursive function such that θ ( p , u, t ) = largest v ≤ t such that ∀ v ′ ≤ v ∃ w ≤ t R ( p , u, v ′ , w ) } Observe that θ is monotone increasing with resp ect to t . Also, ( ∗ ) if p / ∈ X then, for all u , m ax t ∈ N θ ( p , u, t ) is ﬁnite, ( ∗∗ ) if p ∈ X and u is least s u c h that ∀ v ∃ w R ( p , u, v , w ) then  max t ∈ N θ ( p , u, t ) = + ∞ max t ∈ N θ ( p , u ′ , t ) is ﬁnite for all u ′ < u F ollo w in g this observ ation, giv en p ∈ 2 ∗ , we d eﬁne a monotone in creasing sequence of equiv alence relations ρ t p on ﬁnite initial in terv als of N such that ρ t p has t + 1 equiv alence classes I t p , 0 , I t p , 1 , ... , I t p ,t whic h are s uccessiv e ﬁnite interv als [0 , n t p , 0 ] , [ n t p , 0 + 1 , n t p , 1 ] , [ n t p , 1 + 1 , n t p , 2 ] , . . . , [ n t p ,t − 1 + 1 , n t p ,t ] where n t p , 1 < n t p , 2 < . . . < n t p ,t − 1 < n t p ,t . The intuitio n is as follo ws: i. th e class I t p ,u is related to θ ( p , u, t ), i.e. to the b est we can say at step t ab out th e truth v alue of ∀ v ∃ w R ( p , u, v , w ). ii. if and when θ ( p , u, t ) increases, i.e. θ ( p , u, t + 1) > θ ( p , u, t ) for some u , then w e increase the class I t p ,u for the least such u . Of course, an equiv al ence class which gro w s and remains an interv al either is the righ tmost one or has to agg regate some of its neighbor class(es). Whence the follo wing ind uctiv e deﬁnition of the ρ t p ’s and n t p ,u ’s, u ≤ t : 34 i. (Base c ase). ρ 0 p is the equiv ale nce relation with one class { 0 } , i.e. n 0 p , 0 = 0. ii. (Inductive c ase. Su b c ase 1). S upp ose θ ( p , u, t + 1) = θ ( p , u, t ) for all u ≤ t . Then ρ t +1 p is obtained from ρ t p b y add in g a new singleton class on the righ t: (a) F or all u ≤ t we let n t +1 p ,u = n t p ,u , hen ce I t +1 p ,u = I t p ,u . (b) n t +1 p ,t +1 = n t p ,t + 1, hence I t +1 p ,t +1 = { n t p ,t + 1 } . ii. (Inductive c ase. Su b c ase 2). Su pp ose θ ( p , u, t + 1) > θ ( p , u, t ) for some u ≤ t . Let u b e least suc h. Then, (a) for u ′ < u , classes I t p ,u ′ are left unc hanged: n t +1 p ,u ′ = n t p ,u ′ and I t +1 p ,u ′ = I t p ,u ′ , (b) class I t +1 p ,u aggrega tes all classes I t p ,u ′′ for u ≤ u ′′ ≤ t , (c) t + 1 − u singleton classes are add ed: I t +1 p ,u + i = { n t p ,t + i } w here i = 1 , ..., t + 1 − u . I.e. n t +1 p ,u ′ = n t p ,u for all u ′ ≤ u n t +1 p ,u + i = n t p ,t + i for all s ∈ { i, ..., t + 1 − u } B. Let ρ p = S t ∈ N ρ p ,t . Case p ∈ X . Let u b e least such that ∀ v ∃ w R ( p , u, v , w ). F or u ′ < u , let V u ′ = max { v : ∀ v ′ ≤ v ∃ w R ( p , u ′ , v ′ , w ) } t = min { t ′ : ∀ u ′ < u ( V u ′ ≤ t ′ ∧ ∀ v ′ ≤ V u ′ ∃ w ≤ t ′ R ( p , u ′ , v ′ , w ) } Then • ∀ u ′ < u ∀ v ( ∀ v ′ ≤ v ∃ w R ( p , u ′ , v ′ , w ) ⇒ ( v ≤ t ∧ ∀ v ′ ≤ v ∃ w ′ ≤ t R ( p , u ′ , v ′ , w ′ ))) • n t ′ p ,u ′ , = n t p ,u ′ and I t ′ p ,u ′ = I t p ,u ′ for all u ′ < u and t ′ ≥ t . • n t ′ p ,u tends to + ∞ with t ′ and I t ′ p ,u = [ n t ′ p ,u − 1 + 1 , n t ′ p ,u ] tends to the coﬁnite inte rv al [ n t p ,u − 1 + 1 , + ∞ [. • for u ′′ > u , classes I t ′ p ,u ′′ are in terv als the left end p oin ts of whic h tend to + ∞ w ith t ′ , hen ce they v anish at inﬁn it y . Th us, ρ p , w h ic h is the limit of the ρ t p ’s, h as u + 1 classes, h ence has ﬁ nite index. Case p / ∈ X . F or eve ry u ∈ N , the class I t p ,u stabilizes as t tend s to + ∞ . Th us, ρ p has inﬁn ite index. 35 C. Clearly , the sequence ( ρ t p ) p ∈ 2 ∗ ,t ∈ N is recursive. Thus, ρ = { ( p , m, n ) : ∃ t ( m, n ) ∈ ρ t p } is r.e. Let a ∈ 2 ∗ b e suc h th at ρ = W 2 ∗ × N 2 a . Applyin g the parametrization prop erty , let s : 2 ∗ × 2 ∗ → 2 ∗ b e a total recurs ive function suc h that ρ p = { ( m, n ) ∈ N 2 : ( p , m, n ) ∈ W 2 ∗ × N 2 a } = W N 2 s ( a , p ) Let g : 2 ∗ → 2 ∗ b e total recur siv e su c h th at g ( p ) = s ( a , p ). Using p oint B, w e see th at p ∈ X if and on ly if index ( W N 2 g ( p ) ) is ﬁ nite. 9.3 Characterization of the i ndex self-en umerated systems W e n o w come to the charact erization of the index self-enumerated fami- lies. It turns out that these families are almost equal to M ax 2 ∗ → N Rec A ′ , almost meaning h er e “up to 1”. Notation 9.4. If G is a family of fun ctions 2 ∗ → N , w e let G + 1 = { f + 1 : f ∈ G } Theorem 9.5. 1. F or any b as ic set X and any or acle A ⊆ N , the fol lowing strict inclusions hold: M ax 2 ∗ → N Rec A ′ + 1 ⊂ index ◦ F RE A ( X 2 ) ⊂ index ◦ P F RE A ( X 2 ) ⊂ M ax 2 ∗ → N Rec A ′ 2. K N index ◦F RE A ( X 2 ) = ct K N index ◦P F RE A ( X 2 ) = ct K A ′ max . We shal l simply write K N ,A index in plac e of K N index ◦F RE A ( N ) . When A = ∅ we simply write K N index . Pr o of. Observe th at if F is a self-en umerated system with domain D and with U as a goo d universal function, then F + 1 is also a self-enumerated system with U + 1 as a go o d universal fun ction. In particular K D F = K D F +1 . P oin t 2 is a dir ect corollary of Po in t 1 and Prop.6.6 and th e pr evious obser- v ation. Let’s pr o v e p oint 1. The cent ral inclusion index ◦ F RE A ( X 2 ) ⊂ index ◦ P F RE A ( X 2 ) is trivial. A. Non strict inclusion index ◦ P F RE A ( X 2 ) ⊆ M ax 2 ∗ → N Rec A ′ . Let G ∈ index ◦ P F RE A ( X 2 ) and let g : 2 ∗ → 2 ∗ b e partial A -recursiv e such that G ( p ) =      index ( W A, X 2 g ( p ) ) if g ( p ) is deﬁn ed and W A, X 2 g ( p ) is an equiv alence relation with ﬁnite ind ex undeﬁn ed otherwise 36 W e deﬁn e a total A ′ -recursiv e f unction u : 2 ∗ × N → N such that ( ∗ ) { u ( p , t ) : t ∈ N } =  { 0 , ..., n } if G ( p ) is d eﬁned and G ( p ) = n N if G ( p ) is undeﬁn ed The deﬁnition is as follo ws. Since g is p artial A -recursive and w e lo ok for an A ′ -recursiv e d eﬁnition of u ( p , t ), we can u s e oracle A ′ to c hec k if g ( p ) is deﬁned. If g ( p ) is u ndeﬁned then we let u ( p , t ) = t for all t . Whic h ins u res ( ∗ ). Supp ose n ow that g ( p ) is deﬁned. First, set u ( p , 0) = 0. Consider an A -recursive enumeration of W A, X 2 g ( p ) . Let R t b e the set of pairs en umerated at steps < t and D t b e the set of x ∈ X wh ic h app ear in pairs in R t (so that R 0 and D 0 are emp t y). Sin ce at most one new pair is enumerated at eac h step, the set R t con tains at most t pairs and D t con tains at most 2 t p oint s. A t step t + 1, use oracle A ′ to c hec k the follo wing p rop erties: α t . F or ev ery x ∈ D t +1 the pair ( x , x ) is in W A, X 2 g ( p ) . β t . F or ev ery p air ( x , y ) ∈ R t +1 the p air ( y , x ) is in W A, X 2 g ( p ) . γ t . F or ev ery p airs ( x , y ) , ( y , z ) ∈ R t +1 the p air ( x , z ) is in W A, X 2 g ( p ) . δ t . F or every x ∈ D t +1 there exists y ∈ D t suc h that the pair ( x , y ) is in W A, X 2 g ( p ) . Since R t +1 , D t +1 are ﬁn ite, all th ese pr op erties α t - δ t are ﬁnite b o olean com- binations of Σ 0 ,A 1 statemen ts. Hence oracle A ′ can decide th em all. Observe that if W A, X 2 g ( p ) is an equiv ale nce relation then answ ers to α t - γ t are p ositiv e for all t . And if W A, X 2 g ( p ) is not an equiv alence relatio n th en, for s ome π ∈ { α, β , γ } , answe rs to π t are negativ e for all t large en ough . Also, if W A, X 2 g ( p ) is an equiv alence relation then a new equiv alence class is rev ealed eac h time δ t is false. And every equiv alence class is so revea led. Th us, in case g ( p ) is deﬁ ned, w e insur e ( ∗ ) by letting u ( p , t + 1) =  u ( p , t ) if all answers to α t - δ t are p ositiv e u ( p , t ) + 1 otherwise F rom ( ∗ ), w e get G = max u . Since u is total A ′ -recursiv e, this pro v es that G is in M ax 2 ∗ → X Rec A ′ . B. Non strict inclusion M ax 2 ∗ → N Rec A ′ + 1 ⊆ index ◦ F RE A ( X 2 ) . W e redu ce to the case X = N . Let F ∈ M ax 2 ∗ → N P R A ′ . Using Prop.6.9, let M b e an oracle T urin g mac hine 37 whic h on inpu t p and oracle A ′ computes F ( p ) through an inﬁ nite compu- tation. The idea to pro v e that F is in in dex ◦ F RE A ( N 2 ) is as follo ws. W e consider A -recursiv e appro ximations of oracle A ′ and use them as f ak e oracles. F or eac h p w e build an A -r.e. equiv alence relation ρ p ⊆ N 2 with domain N which consists of one big class contai ning 0 and some sin gleton classes. Eac h time the compu tation with the fak e oracle outputs a n ew digit 1, we put some new singleton class in ρ p . When, with a b etter approximati on of A ′ , we see that the fak e oracle has giv en an incorrect answer, all singleton classes which w ere p ut in ρ p b ecause of th e oracle incorrect answer are annihilated: they are aggregated to th e class of 0. S ince we are going to consider index ( ρ p ), this pro cess will lead to the correct v a lue F ( p ) + 1. F ormally , w e consid er an A -recursive monotone increasing sequence ( Appr ox ( A ′ , t )) t ∈ N suc h that A ′ = S t ∈ N Appr ox ( A ′ , t ) (cf. Lemma 6.14). Though all oracles Appr ox ( A ′ , t ) are false appr o ximatio ns of oracle A ′ , they are n ev ertheless “less and less false” as t in cr eases. Without loss of generalit y , we can supp ose th at at eac h compu tation step of M there is a qu estion to th e oracle (p ossibly the same one many times). Let C p ,t b e the computation of M on in put p w ith oracle Appr ox ( A ′ , t ), reduced to the sole t ﬁrs t steps. Increasing parts of oracle Appr ox ( A ′ , t ) are qu estioned d uring C p ,t . Let Ω p ,t : { 1 , ..., t } → P f in ( N ) (where P f in ( N ) is the set of ﬁnite subsets of N ) b e such that Ω p ,t ( t ′ ) is the set of k su c h that the oracle has b een ques- tioned ab out k dur in g the t ′ ﬁrst steps, 1 ≤ t ′ ≤ t . Clearly , Ω p ,t is (non strictly) m on otone increasing with resp ect to set inclusion. Let 1 n p ,t b e the outpu t of C p ,t (recall that M outputs a ﬁn ite or inﬁ nite sequence of d igits 1’s). The successiv e digits of this outpu t are written d own at increasing times (all ≤ t ). Let O T p ,t : { 0 , ..., n p ,t } → { 0 , ..., t } b e su c h that O T p ,t ( n ) is the least step at w hic h the current outpu t is 1 n ( O T stands for output time). Clearly , O T p ,t (0) = 0. W e construct A -recursiv e sequences ( ρ p ,t ) p ∈ 2 ∗ ,t ∈ N and ( w p ,t ) p ∈ 2 ∗ ,t ∈ N (where w stands for witness) s u c h that i t . ρ p ,t is an equ iv alence relation on { 0 , ..., 2 t − 1 } with index equal to 1 + n p ,t (there is nothing essent ial with 2 t , it is merely a large enough b ound con v enien t f or the construction), ii t . all equiv ale nce classes of ρ p ,t are s ingleton sets except p ossib ly the equiv alence class of 0. iii t . if t > 0 then ρ p ,t con tains ρ p ,t − 1 . 38 iv t . w p ,t is a b ijection b et w een { 1 , ..., n p ,t } and the set of p oint s ∈ { 1 , ..., 2 t − 1 } su c h that { s } is a singleton class of ρ p ,t (in case n p ,t = 0 then w p ,t is the empt y map). First, w p , 0 is the empt y map and ρ p , 0 = { (0 , 0) } , i.e. the trivial equiv ale nce relation on { 0 } . The ind uctiv e construction of the ρ p ,t ’s uses the ab o v e conditions i t - iv t as an indu ction h yp othesis. Case Appr ox ( A ′ , t + 1) ∩ Ω p ,t ( t ) = Appr ox ( A ′ , t ) ∩ Ω p ,t ( t ) . Then the computation C p ,t is totally compatible with C p ,t +1 . No w, that last computation m a y p ossibly output one more digit 1, i.e. n p ,t +1 = n p ,t or n p ,t +1 = n p ,t + 1. Hence the t w o follo wing sub cases. Sub c ase n p ,t +1 = n p ,t . Th en ρ p ,t +1 is obtained from ρ p ,t b y putting 2 t , 2 t + 1 , ..., 2 t +1 − 1 as new p oints in the class of 0. In particular, ρ p ,t +1 and ρ p ,t ha v e the same index. W e also set w p ,t +1 = w p ,t . Sub c ase n p ,t +1 = n p ,t + 1 . T hen ρ p ,t +1 is obtained from ρ p ,t as follo ws: • Ad d a new singleton class { 2 t } . • Pu t 2 t + 1 , ..., 2 t +1 − 1 as new p oint s in th e class of 0. W e also set w p ,t +1 = w p ,t ∪ { ( n p ,t +1 , 2 t ) } . In b oth sub cases, conditions i t +1 - iv t +1 are clearly s atisﬁed. Case Appr ox ( A ′ , t + 1) ∩ Ω p ,t ( t ) 6 = Appr ox ( A ′ , t ) ∩ Ω p ,t ( t ) . Let τ ≤ t b e least suc h that App r ox ( A ′ , t + 1) ∩ Ω p ,t ( τ ) 6 = Appr ox ( A ′ , t ) ∩ Ω p ,t ( τ ). Though the computation C p ,t is not en tirely compatible with C p ,t +1 , it is compatible up to step τ − 1. Let n ≤ n p ,t b e greatest suc h that O T p ,t ( n ) < τ . Th en the n ﬁ rst digits output b y C p ,t are also output b y C p ,t +1 at th e same compu tation steps. I n particular, n p ,t +1 ≥ n . Then ρ p ,t +1 , w p ,t +1 are obtained from ρ p ,t , w p ,t as follo ws: • Pu t all w p ,t ( m ), w here n < m ≤ n p ,t , as n ew p oin ts in the class of 0. This ann ihilates the singleton classes of ρ p ,t corresp ondin g (via w p ,t ( m )) to the part of the outp ut which was created by answ ers of oracle Appr ox ( A ′ , t ) wh ich are known to b e f alse at step t + 1. • Ad d a n ew singleton class { 2 t − 1 + i } for eac h i > 0 suc h th at n + i ≤ n p ,t +1 . T ogether with the singleton classes of ρ p ,t whic h ha v e not b een aggragated by the ab o v e p oint, this allo ws to get exactly n p ,t +1 singleton classes in ρ p ,t +1 Accordingly , s et w p ,t +1 = ( w p ,t ↾ { 1 , ..., n } ) ∪ { ( n + i, 2 t − 1 + i ) : 0 < i ≤ n p ,t +1 − n } 39 • Pu t the 2 t − 1 + j ’s, where j ≥ max(1 , n p ,t +1 − n ), as new p oin ts in the class of 0. Again, conditions i t +1 - iv t +1 are clearly satisﬁed. Let ρ p = S t ∈ N ρ p ,t . Condition iii t insures th at ρ p is also an equiv alence relation. Condition ii t go es thr ou gh the limit wh en t → + ∞ , so that all classes of ρ p are singleton sets except the class of 0. The computation we are r eally interesting in is that wh ic h giv es F ( p ), i.e. the inﬁnite computation of M on in p ut p with oracle A ′ . Let denote it C p . When t in creases, the common part of C p with computation C p ,t gets larger and larger (though not monotonously). W e now pro v e the equalit y ( † ) index ( ρ p ) =  1 + F ( p ) if F ( p ) is deﬁned + ∞ otherwise Case F ( p ) is deﬁne d and F ( p ) = z . Let τ b e the computation time at wh ic h C p has output z . Let Ω p b e the set of k such that oracle A ′ has b een questioned ab out dur ing the ﬁrst τ steps of C p . F or t large enough, sa y t ≥ t z , w e ha v e Appr ox ( A ′ , t ) ∩ Ω p = A ′ ∩ Ω p . In p articular, the τ ﬁrst steps of C p ,t and C p will b e exactly the same and b oth compu tations output z . The same with the τ ﬁ rst steps of C p ,t and C p ,t +1 . Th us, w p ,t +1 ↾ { 1 , ..., z } = w p ,t ↾ { 1 , ..., z } . Let w p = w p ,t +1 ↾ { 1 , ..., z } . Then all sin gleton sets { w p ( i ) } , where 1 ≤ i ≤ z , are equiv ale nce classes for the ρ p ,t ’s, hen ce for ρ p . No w, if n p ,t > z then oracle Appr ox ( A ′ , t ) has b een questioned on Ω p ,t ( n p ,t ) and diﬀers fr om A ′ on that s et. Let u > t b e ﬁrst su c h th at Appr ox ( A ′ , u ) agrees with A ′ on Ω p ,t ( z + 1). Then the singleton class { w p ,t ( z + 1) } of ρ p ,t is aggregate d at step u to the class of 0 in ρ p ,t +1 , hence also in ρ p . Th us, th e { w p ( i ) } ’s, w here 1 ≤ i ≤ z , are the sole singleton equ iv alence classes of ρ p . And the class of 0 con tains all other p oin ts in N . In particular, in dex ( ρ p ) = 1 + F ( p ). Case F ( p ) is undeﬁne d b e c ause the output of M on input p with or acle A ′ is inﬁnite. As in the ab ov e case, we see that there are more and more sin gleton set classes of ρ p ,t whic h are never annihilated. Th us, the index of ρ p is inﬁn ite. This prov es ( † ). Observing that all the constru ction of the ρ p ,t ’s is A -recursiv e, w e see that ρ = [ p ∈ 2 ∗ ρ p 40 is A -r.e. Thus, ρ = W A, 2 ∗ × N 2 a for some a . Th e parameter prop erty giv es a total A -recursive function s : 2 ∗ × 2 ∗ → 2 ∗ suc h that ρ p = W A, N 2 s ( a , p ) Th us, p 7→ index ( ρ p ) is indeed in index ◦ F RE A ( X 2 ) . Thanks to ( † ), th e same is true of 1 + F . C. Inclusion M ax 2 ∗ → N Rec A ′ + 1 ⊆ index ◦ F RE A ( X 2 ) is strict. The constant 0 function is an obvious coun terexample to equalit y . D. Inclusion index ◦ P F RE A ( X 2 ) ⊆ M ax 2 ∗ → N Rec A ′ is strict. W e exhibit a fu nction κ X in P R A ′ \ index ◦ P F RE A ( X 2 ) 6 = ∅ . Let X ⊂ 2 ∗ b e A ′ -recursiv e, i.e. ∆ 0 ,A 2 , but not a b o olean com bination of Σ 0 ,A 1 sets. Let κ X : 2 ∗ → N b e the { 0 , 1 } -v alued c haracteristic function of X . Then κ X is A ′ -recursiv e (hence in M ax 2 ∗ → N Rec A ′ ) and κ − 1 X (0) = X is a ∆ 0 ,A 2 set wh ich is not a b o olean combinatio n of Σ 0 ,A 1 sets. No w, supp ose G is in index ◦ P F RE A ( X 2 ) and G = index ( W A, X 2 g ( p ) ) where g : 2 ∗ → 2 ∗ is in P R A . Th en G ( p ) = 0 ⇔ ( g ( p ) is d eﬁned ∧ W A, X 2 g ( p ) = ∅ ) ⇔ ( g ( p ) is deﬁned ∧ ∀ t ∀ e ( g ( p ) con v erges to e in t steps ⇒ W A, X 2 e = ∅ ) so that G − 1 (0) is Σ 0 ,A 1 ∧ Π 0 ,A 1 . This sho ws that no G ∈ index ◦ P F RE A ( X 2 ) can b e equal to the ab o v e κ X . Therefore, the considered inclusion cannot b e an equalit y . Let’s ﬁn ally observ e a s imple fact con trasting inclusions in Thm .9.5. Prop osition 9.6. 1 + P R A ′ , 2 ∗ → 2 ∗ (a fortior i 1 + M ax 2 ∗ → N P R A ′ ) is not include d in index ◦ P F RE A ( X 2 ) . Pr o of. The pro of is analog to that of p oin t D in the p ro of of Th m.9.5. 1. W e sh o w that G − 1 (1) is Π 0 ,A 2 for ev ery G ∈ index ◦ P F RE A ( X 2 ) . Supp ose G = index ( W A, X 2 g ( p ) ) where g : 2 ∗ → 2 ∗ is partial A -recurs iv e. Let’s denote W A, X 2 e ,t the ﬁn ite part of W A, X 2 e obtained after t steps of its en umeration. Let’s also d en ote C V g ( p , e , t ) the A -recursive relation stating 41 that g ( p ) con v erges to e in ≤ t steps. Th en G ( p ) = 1 ⇔ ( g ( p ) is d eﬁned ∧ W A, X 2 g ( p ) 6 = ∅ ∧ W A, X 2 g ( p ) is an equiv ale nce relation w ith index 1) ⇔ ( g ( p ) is deﬁned ∧ W A, X 2 g ( p ) 6 = ∅ ∧ ∀ t ∀ e ( C V g ( p , e , t ) ⇒ W A, X 2 e is an equiv alence relation with in dex 1) The ﬁrst tw o conjuncts are clearly Σ 0 ,A 1 . As for the last one, observe that W A, X 2 e is an equiv alence r elation if and only if ∀ x , y ∈ X (( x , y ) ∈ W A, X 2 e ⇒ ( x , x ) ∈ W A, X 2 e ∧ ( y , x ) ∈ W A, X 2 e ) ∧ ∀ x , y , z ∈ X (( x , y ) ∈ W A, X 2 e ∧ ( y , z ) ∈ W A, X 2 e ) ⇒ ( x , z ) ∈ W A, X 2 e ) Whic h is Π 0 ,A 2 since W A, X 2 e is Σ 0 ,A 1 . Also, if W A, X 2 e is a non emp ty equiv ale nce relation then it has in d ex 1 if and only if ∀ x , y , x ′ , y ′ ∈ X (( x , x ′ ) ∈ W A, X 2 e ∧ ( y , y ′ ) ∈ W A, X 2 e , ) ⇒ ( x , y ) ∈ W A, X 2 e ) Whic h is again Π 0 ,A 2 . This prov es that G − 1 (1) is indeed Π 0 ,A 2 . 2. No w, let X ⊂ X b e Σ 0 ,A ′ 1 and not A ′ -recursiv e. Thus, X is Σ 0 ,A 2 and not Π 0 ,A 2 . Let π X : 2 ∗ → N b e suc h that π X ( p ) =  1 if p ∈ X undeﬁn ed o therwise Then π X ∈ 1 + P R A ′ , 2 ∗ → N . Since π − 1 X (1) = X is not Π 0 ,A 2 , π X cannot b e in index ◦ P F RE A ( X 2 ) . 9.4 Characterization of the ∆ in dex self-en umerated systems Theorem 9.7. L et A ⊆ N and let A ′′ b e the se c ond jump of A . L et X b e a b asic set. 1. ∆( index ◦ F RE A ( X ) )) = ∆( index ◦ P F RE A ( X ) )) = P R A ′′ , 2 ∗ → Z 2. K Z ∆( index ◦F RE A ( X ) ) = ct K A ′′ , Z . We shal l simply write K N ,A ∆ index in plac e of K Z ∆( index ◦F RE A ( N ) ) ↾ N . When A = ∅ we simply write K Z ∆ index . 42 Pr o of. P oin t 2 is a d irect corollary of Po in t 1. Let’s prov e p oin t 1. Using Thm.9.5, and applying the ∆ op erator, we get ∆( M ax 2 ∗ → N Rec A ′ + 1) ⊆ ∆( index ◦ F RE A ( X 2 ) ) ⊆ ∆( index ◦ P F RE A ( X 2 ) ) ⊆ ∆( M ax 2 ∗ → N Rec A ′ ) But, f or any family G of functions 2 ∗ → N , w e trivially ha v e ∆( G +1) = ∆( G ). This pro v es th at the ab o v e in clus ions are, in fact, equalities. W e conclude with Thm .6.12. 10 F unctional represen tati ons of N Notation 10.1 (F un ctions sets) . W e denote - Y X the set of total fu nctions from X in to Y . - X → Y the set of partial fun ctions from X int o Y . - X 1 − 1 → X the set of inj ectiv e p artial functions f r om X in to X . - I d X the identit y fun ction o v er X . 10.1 Basic Churc h represen tation of N First, let’s in tro d uce some s im p le notations related to function iteration. Deﬁnition 10.2 (Iteration) . 1) If f : X → X is a p artial f unction, we ind u ctiv ely deﬁ ne for n ∈ N the n -th iterate f ( n ) : X → X of f as th e partial function such that: f (0) = I d X , f ( n +1) = f ( n ) ◦ f 2) I t ( n ) X : ( X → X ) → ( X → X ) is the total functional f 7→ f ( n ) . I t N X : N → ( X → X ) ( X → X ) is the total fu nctional n 7→ I t n X . The follo wing Prop osition is easy . Prop osition 10.3. The total functional I t N X : N → ( X → X ) ( X → X ) is inje ctive (henc e admits a left inverse) if and only i f X is an inﬁnite set. W e can now come to the functional representa tion of integ ers in tro duced b y Churc h, 1933 [3]. Deﬁnition 10.4 (Ch urc h representati on of N ) . If X is an inﬁ nite set, the Churc h representat ion of N relativ e to X is the function Chur ch N X : ( X → X ) ( X → X ) → N whic h is the uniqu e left inv erse of I t N X with domain Rang e ( I t N X ) = { I t n X : n ∈ N } , i.e. Chur ch N X ◦ I t N X = I d N Chur ch N X ( F ) =  n if F = I t n X undeﬁn ed i f ∀ n ∈ N F 6 = I t n X 43 F or future use in Def.10.17 , let’s introdu ce the follo wing v ariant of Chur ch N X . Deﬁnition 10.5. W e denote church N ,A X : ( P R A, X → X ) P R A, X → N the functional whic h is the u nique left inv ers e of the r estriction of I t N X to ( P R A, X → X ) P R A, X → X , i.e. chur ch N ,A X ( F ) = ( n if F = I t n X ↾ ( P R A, X → X ) P R A, X → X undeﬁn ed i f ∀ n ∈ N F 6 = I t n X ↾ ( P R A, X → X ) P R A, X → X 10.2 Computable and eﬀectiv ely c ontin uous functionals W e recall th e tw o classical n otions of partial computability for fun ctionals, cf. Odifredd i’s b o ok [12] p.178, 188, 197. Deﬁnition 10.6 (Kleene partial computable fun ctionals) . 1. Let X , Y , S , T b e s ome b asic s p ace and ﬁx some suitable represent ations of their element s by w ords. An ( X → Y )-oracle T urin g mac hine with inputs and outputs resp ectiv ely in S , T is a T uring mac hine M which has a sp ecial oracle tap e and is allo wed at certain states to ask an oracle f ∈ ( X → X ) what are the successiv e digits of the v alue of f ( q ) where q is the elemen t of X curr ently written on the oracle tap e. The fu nctional Φ M : (( X → Y ) × S ) → T asso ciated to M maps the pair ( f , s ) on the outpu t (when deﬁned) computed by M when f is giv en as th e partial fun ction oracle and s as the input. If on inpu t x and oracle f the compu tation asks the oracle its v alue on an elemen t on wh ic h f is und eﬁ ned then M gets stuc k, so th at Φ M ( f , x ) is undeﬁn ed. 2. A fu nctional Φ : (( X → Y ) × S ) → T is partial compu table (also called partial recursive) if Φ = Φ M for some M . A functional obtained via curr y ﬁ cations from such a f unctional is also called partial computable. W e denote PC τ the family of partial computable fun ctionals with t yp e τ . If A ⊆ N , w e denote A - PC τ the analog family with th e extra oracle A . Deﬁnition 10.7 (Usp enskii (eﬀectiv ely) con tin uous fu nctionals) . Denote F in ( X → Y ) the class of partial functions X → Y with ﬁ nite domains. Observe that, for α, β ∈ F in ( X → Y ) are compatible if and only if α ∪ β ∈ F in ( X → Y ). 1. Let’s say that the relation R ⊆ F in ( X → Y ) × S × T is fun ctional if α ∪ β ∈ F in ( X → Y ) ∧ ( α, s , t ) ∈ R ∧ ( β , s , t ′ ) ∈ R ⇒ t = t ′ T o su c h a fu nctional relation R can b e asso ciated a functional Φ R : (( X → Y ) × S ) → T 44 suc h that, for ev ery f , s , t , ( † ) Φ( f , s ) = t ⇔ ∃ u ⊆ f R ( u, s , t ) 2. (Usp enskii [21], Nero de [11]) A functional Φ : (( X → Y ) × S ) → T is con tin uous if it is of the form Φ R for some functional relation R . Φ is eﬀectiv ely contin uous (resp. ( A -eﬀectiv ely contin uous) if R is r.e. (resp . A -r.e.). Eﬀectiv ely con tin u ous fu nctionals are also called recursiv e op er ators (cf. Rogers [15], Od ifreddi [12]). A functional obtained via curr y ﬁ cations from such a f unctional is also called eﬀectiv ely con tin uous. W e denote E ﬀCont τ the family of eﬀectiv ely conti n uous fun ctionals with t yp e τ . If A ⊆ N , we denote A - EﬀCont τ the analog family with the extra oracle A . Eﬀectiv e con tin uit y is m ore general than p artial computabilit y (cf. [12] p.188). Theorem 10.8. L et A ⊆ N . 1. (Usp enskii [21], Nero de [11]) P artial A -c omp utable fu nc tionals ar e A -eﬀe c tively c ontinuous. 2. (Sasso [17 , 18]) Ther e ar e A -eﬀe c tively c ontinuous functionals which ar e not p artial A -c omp utable. Ho w ever, restricted to total fun ctions, b oth notions coincide. Prop osition 10.9. A functional Φ : ( Y X ) × S → T is the r estriction of a p artial A -c omputable functional (( X → Y ) × S ) → T if and only if it is the r estriction of an A - eﬀe ctively c ontinuous functional. 10.3 Eﬀectiv eness of the Apply functional The follo wing result will b e used in § 10.7-10.5. Prop osition 10.10 . L et φ : 2 ∗ → P R A, X → X b e p artial A -r e cursive (as a function 2 ∗ × X → X ) and Φ : 2 ∗ → A -EﬀCont 2 ∗ → (( X → X ) → ( X → X )) b e eﬀe ctive ly c ontinuous. Ther e exists a p artial A -r e cursive function g : 2 ∗ × 2 ∗ × X such that, for al l e , p ∈ 2 ∗ and x ∈ X , ( ∗ ) g ( p , e , x ) = (Φ( e )( φ ( p )))( x ) Pr o of. Let R ⊆ 2 ∗ × F in ( X → X ) × X × X b e an A -r.e. set suc h that, for all e , R ( e ) = { ( α, x , y ) : ( e , α, x , y ) ∈ R } is fun ctional and Φ( e ) = Φ R ( e ) . W e deﬁne g ( p , e , x ) as f ollo w s : i. A -eﬀectiv ely en umerate R ( e ) and the graph of φ ( p ) u p to the momen t w e get ( α, x , y ) ∈ R ( e ) and a ﬁnite part γ of φ ( p ) such that α ⊆ γ . ii. If and wh en i halts then outpu t y . It is clear that g is partial A -recurs iv e and satisﬁes ( ∗ ). 45 10.4 F unctionals o v er P R X → Y and computabilit y Using indexes, one can also consider computabilit y for f u nctionals op erating on the sole partial r ecursiv e or A -recursive fun ctions. Deﬁnition 10.11. Let A ⊆ N and let ( ϕ X → Y ,A e ) e ∈ 2 ∗ denote s ome acceptable en umeration of P R A, X → Y (cf. Def.5.1 ). 1. A functional Φ : P R A, X → Y × S → T is an A -eﬀectiv e functional on partial A -recursiv e functions if there exists some partial A -recur siv e function f : 2 ∗ → 2 ∗ suc h that, for all s ∈ S , e ∈ 2 ∗ , Φ( ϕ X → Y ,A e ) = f ( e ) W e denote A - Eﬀ P R A, X → Y × S → T the family of such functionals. 2. W e denote A - E ﬀ P R A, X → Y × S 1 → P R A, S 2 → T the family of functionals ob tained b y curryﬁ cation of the ab o v e class with S = S 1 × S 2 . An easy application of the parameter p rop ert y shows that these functionals are exactly those for wh ic h there exists some partial A -recursive fun ction g : 2 ∗ × S 1 → 2 ∗ suc h that, for all s 1 ∈ S 1 , e ∈ 2 ∗ , Φ( ϕ X → Y ,A e , s 1 ) = ϕ S 2 → T ,A g ( e , s 1 ) Note 10.12. 1. Thanks to Rogers’ theorem (cf. Th m.5.2), the ab o v e d eﬁnition do es n ot dep end on the c hosen acceptable enumeratio ns. 2. The ab o v e fu nctions f , g should hav e the f ollo w in g prop erties: ϕ X → Y ,A e = ϕ X → Y ,A e ′ ⇒ f ( e , s ) = f ( e ′ , s ) ϕ X → Y ,A e = ϕ X → Y ,A e ′ ⇒ ϕ S 2 → T ,A g ( e , s 1 ) = ϕ S 2 → T ,A g ( e ′ , s 1 ) As shown b y the follo w ing remark a ble resu lt, such functionals essentially reduce to th ose of Def.10.7 (cf. Od ifreddi’s b o ok [12] p.206–20 8). Theorem 10.13 (Usp enskii [21], Myhill & Shepherd son [10]) . L et A ⊆ N . The A -eﬀe ctive functionals P R A, X → Y → P R A, S → T ar e exactly the r estrictions to P R A, X → Y of A -eﬀ e ctively c ontinuous functionals ( X → Y ) → ( S → T ) . 10.5 Eﬀectivizations of Churc h r epresen tation of N Observe the follo w ing trivial fact (whic h uses notatio ns from Def.10.6,10.7). Prop osition 10.14. L et A ⊆ N and τ b e any 2d or der typ e. F unctionals in A -PC 2 ∗ → τ (r esp. A -EﬀCont 2 ∗ → τ ) ar e total maps 2 ∗ → A -PC τ (r esp. 2 ∗ → A -E ﬀ Cont τ ). 46 Theorem 10.15. L et τ b e any 2d or der typ e. The systems ( A -PC τ , A -PC 2 ∗ → τ ) , ( A -EﬀCont τ , A -EﬀCont 2 ∗ → τ ) ar e self- enumer ate d r e pr esentation A -systems. Pr o of. P oin ts i-ii of Def.2.1 are trivial. As for p oin t iii, we use the classical en umeration theorem for partial computable (resp. eﬀectiv ely con tin uous) functionals: consider a function V ∈ A - PC 2 ∗ → ( 2 ∗ → τ ) whic h enumerates A - PC 2 ∗ → τ and set U ( c ( e , p )) = V ( e )( p ). Idem with A - EﬀCont . As an easy corollary of T hms.10.15 and 10.13, w e get the follo wing result. Theorem 10.16. L et A ⊆ N . L et A -Eﬀ 2 ∗ → ( P R A, X → Y × S → T ) b e obtaine d by curryﬁc ation fr om A -Eﬀ ( P R A, X → Y × S × 2 ∗ ) → T . The systems ( A -Eﬀ P R A, X → Y × S → T , A -Eﬀ 2 ∗ → ( P R A, X → Y × S → T ) ) ( A -Eﬀ P R A, X → Y → P R A, X → Y , A -Eﬀ 2 ∗ → ( P R A, X → Y → P R A, S → T ) ) ar e self- enumer ate d r e pr esentation A -systems. Deﬁnition 10.17 (Eﬀectivizations of Churc h representat ion of N ) . W e ef- fectivize the Churc h rep resen tatio n by replacing ( X → X ) → ( X → X ) by one of th e follo wing classes: A - PC ( X → X ) → ( X → X ) , A - Eﬀ Cont ( X → X ) → ( X → X ) , A - Eﬀ P R A, X → Y → P R A, X → Y where X is some basic set. an d A ⊆ N is some oracle. Usin g Def.10.5, this leads to three self-en umerated systems with domain N : F 1 = ( N , Chur ch N X ◦ A - P C 2 ∗ → (( X → X ) → ( X → X )) ) F 2 = ( N , Chur ch N X ◦ A - E ﬀCont 2 ∗ → (( X → X ) → ( X → X )) ) F 3 = ( N , chur ch N ,A X ◦ A - Eﬀ 2 ∗ → ( P R A, X → Y → P R A, X → Y ) ) The follo wing result greatly simpliﬁes the land scap e. Theorem 10.18. The thr e e systems F 1 , F 2 , F 3 of Def.10.17 c oincide. Before proving th e theorem (cf. the end of this subsection), we state some conv enient to ols in the next three pr op ositions, the ﬁrst of whic h will also b e used in § 10.7. Prop osition 10.19. Supp ose R ⊂ F in ( X → X ) × X × X is functional (cf. Def.10.7). The fol lowing c ond itions ar e e quiv alent i. Φ R = I t ( n ) X ii. Φ R ↾ F in ( X → X ) = I t ( n ) X ↾ F in ( X → X ) 47 iii. ∀ α ∈ F in ( X → X ) ∀ x ( α ( n ) ( x ) is deﬁne d ⇒ ( α ↾ { α ( i ) ( x ) : 0 ≤ i < n } , x , α ( n ) ( x )) ∈ R ) and ∀ α ∈ F in ( X → X ) ∀ x ∀ y (( α, x , y ) ∈ R ⇒ ( α ( n ) ( x ) is deﬁne d ∧ y = α ( n ) ( x ))) Pr o of. iii ⇒ i and i ⇒ ii are trivial. ii ⇒ iii. Assu me ii . Supp ose ( α, x , y ) ∈ R then Φ R ( α )( x ) = y . Since α ∈ F in ( X → X ), ii ins u res that α ( n ) ( x ) is d eﬁned and α ( n ) ( x ) = y . This pro v es the second part of iii . Supp ose α ( n ) ( x ) is d eﬁned and let α ( n ) ( x ) = y . Then Φ R ( α ↾ { α ( i ) ( x ) : 0 ≤ i < n } )( x ) = I t ( n ) X ( α ↾ { α ( i ) ( x ) : 0 ≤ i < n } )( x ) = I t ( n ) X ( α )( x ) = y So that there exists a restriction β of α ↾ { α ( i ) ( x ) : 0 ≤ i < n } suc h that ( β , x , y ) ∈ R . T h us, Φ R ( β )( x ) = y . Applying ii , this yields that β ( n ) ( x ) is deﬁned and β ( n ) ( x ) = y . Since β is a restriction of α ↾ { α ( i ) ( x ) : 0 ≤ i < n } , this insures that β = α ↾ { α ( i ) ( x ) : 0 ≤ i < n } . Th is p ro v es th e ﬁrst part of iii . Prop osition 10.20. L et n ∈ N . If Φ R ( f ) i s a r estriction of f ( n ) for eve ry f : X → X then e ither Φ R = I t ( n ) X or Φ R is not an iter ator. Pr o of. W e redu ce to the case X = N . Let S ucc : N → N b e the su ccessor function. S in ce Φ R ( S ucc ) is a restriction of S ucc ( n ) , either Φ R ( S ucc )(0) is undeﬁn ed or Φ R ( S ucc )(0) = n . In b oth cases it is diﬀerent from S ucc ( p ) (0) for any p 6 = n . Whic h pr o v es that Φ R 6 = I t ( p ) N for ev ery p 6 = n . Hence the prop osition. Prop osition 10.21. 1. L et ( W e ) e ∈ 2 ∗ b e an ac c epta ble enumer ation of r.e. subsets of F in ( X → X ) × X × X . Ther e exists a total r e cursive function ξ : 2 ∗ → 2 ∗ such that, for al l e , a. W ξ ( e ) ⊆ W e and W ξ ( e ) is functional (cf . Def.10.7, p oint 1), b. W ξ ( e ) = W e whenever W e is functional. 2. Ther e exists a p artial r e cursive fu nc tion λ : 2 ∗ → N such that if R e is functional and Φ R e is an iter ator then λ ( e ) is deﬁne d and Φ R e = I t ( λ ( e )) X . (However, λ ( e ) may b e deﬁne d even if R e is not functional or Φ R e is not an iter ator). 3. Ther e exists a total r e cursive function θ : 2 ∗ → 2 ∗ such that, for al l e ∈ 2 ∗ , 48 a. i f Φ R e is an iter ator then the ( X → X ) - or acle T uring machine M θ ( e ) with c o de θ ( e ) (cf. Def.10.6) c omp utes the functional Φ R e , b. if Φ R e is not an iter ator then ne i ther i s the functional c ompute d by the ( X → X ) -or acle T uring machine M θ ( e ) with c o de θ ( e ) . In other wor ds, C hur ch (Φ R e ) = C hur ch (Φ M θ ( e ) ) 4. The ab ove p oints r elativize to any or acle A ⊆ N . Pr o of. 1. This is the classical f act underlying the en umeration th eorem f or eﬀectiv ely con tin uous fun ctionals. T o get W ξ ( e ) , enumerate W e and retain a trip le if and only if, together w ith the already r etained ones, it do es not con tradict fun ctionalit y (cf. Od ifreddi’s b o ok [12] p.197). 2. W e red uce to the case X = N . Let α n : N → N b e suc h that domain ( α n ) = { 0 , ..., n } , α n ( i ) = i + 1 for i = 0 , ..., n Supp ose R is functional and Φ R = I t ( n ) N . Prop .10.19 insu r es ( α n , 0 , n ) ∈ R . Also, f or m 6 = n , since α m and α n are compatible and R is fun ctional, R cannot cont ain ( α m , 0 , m ). Th us, if Φ R = I t ( n ) N then n is the uniqu e integ er suc h that R con tains ( α n , 0 , n ). This leads to the follo wing deﬁnition of the w an ted partial recursive fun ction λ : 2 ∗ → N : - enumerate R e , - if and when some triple ( α n , 0 , n ) app ears, halt and output λ ( e ) = n . 3. Giv en a co de e of a functional relatio n R e , we let θ b e th e total recursiv e function whic h giv es a cod e for the oracle T uring mac hine M whic h acts as follo w s: i. First, it compu tes λ ( e ). ii. If λ ( e ) is d eﬁned then, on inp ut x and oracle f , M tries to compu te I t ( λ ( e )) X ( f )( x ) in th e obvious wa y: ask the oracle the v alues of f ( i ) ( x ) for i ≤ λ ( e ). iii. Finally , in case i and ii halt, M en umerates R e and halts and accepts (with th e outpu t computed at ph ase ii) if and only if ( f ↾ { f ( i ) ( x ) : i ≤ λ ( e ) } , x , f ( λ ( e )) ( x )) app ears in R e . I.e. if and only if f ( λ ( e )) ( x ) = Φ R ( f )( x ) Clearly , the fun ctional Φ M computed by M is suc h that Φ M ( f ) is equal to or is a restriction of I t ( λ ( e )) X ( f ). If Φ R e is an iterator then p oint 2 insu res that Φ R e = I t ( λ ( e )) X and Pr op .10.19 insures that phase iii is no p roblem, so that M compu tes exactly Φ R e . 49 Supp ose Φ R e is n ot an iterator. If λ ( e ) is un deﬁned then M computes the constant functional with v alue the no where deﬁned fun ction. Thus, M do es not compute an iterator. If λ ( e ) is deﬁ ned then, on inpu t x , M computes f ( λ ( e )) ( x ) and halt and accepts if and only f ( λ ( e )) ( x ) = Φ R ( f )( x ). Since Φ R is n ot an iterator, there exists f and x such that f ( λ ( e )) ( x ) is deﬁ ned and Φ R ( f )( x ) 6 = f ( λ ( e )) ( x ). Hence Φ M ( f ) is a strict restriction of I t ( λ ( e )) X ( f ), s o th at Φ M 6 = I t ( λ ( e )) X . Finally , P r op.10.20 insur es that Φ R e cannot b e an iterator. Pr o of of The or em 10.18. 1. Since F in ( X → X ) ⊂ P R A, X → Y , condition ii of Prop.10.19 and Thm.10.13 pro v e equalit y F 2 = F 3 . 2. Inclusion F 1 ⊆ F 2 is a corollary of T hm.10.8, p oin t 1. Let’s pro v e the con v erse in clusion. Su pp ose Φ : ( 2 ∗ × ( X → X )) → ( X → X ) is ef- fectiv el y con ti n uous an d let R ⊆ 2 ∗ × F in ( X → X ) × X × X b e a fu n c- tional r.e. set s uc h that Φ = Φ R . Using the parameter prop er ty , let h : 2 ∗ → 2 ∗ b e a total recurs ive function such th at h ( e ) is an r.e. co de for R ( e ) = { ( α, x , y ) : ( e , α, x , y ) ∈ R } . Prop.10.21, p oint 3, giv es a total recursiv e θ : 2 ∗ → 2 ∗ suc h that C hur ch (Φ R ( e ) ) = C hurch (Φ M θ ( e ) ). Thus, e 7→ C hur ch (Φ R ( e ) ) is partial compu table with a ( X → X )-oracle T urin g mac hine havi ng in puts in 2 ∗ × X . ✷ 10.6 Some examples of eﬀectively con tin uous functionals F or futu re use in sectio ns § 10.7-10.8 , let’s get the follo wing examples of eﬀectiv ely con tin uous fun ctionals. Prop osition 10.22. If ϕ : 2 ∗ → N is p artial A - r e c u rsive and S ⊆ 2 ∗ is Π 0 ,A 2 then ther e exists an A - eﬀe ctive ly c ontinuous functional Φ : 2 ∗ → ( X → X ) X → X such that, for al l p , ( ∗ ) p ∈ S ∩ domain ( ϕ ) ⇒ Φ( p ) = I t ( ϕ ( p )) X ( ∗∗ ) p / ∈ S ∩ domain ( ϕ ) ⇒ Φ( p ) is not an i ter ato r Pr o of. W e consid er the sole case A = ∅ , relativizat ion b eing straigh tforw a rd. Let S = { e : ∀ u ∃ v ( e , u, v ) ∈ σ } where σ is a recursiv e su bset of 2 ∗ × N × N . W e construct a total recursive fu nction g : 2 ∗ → 2 ∗ suc h that, for all p , W g ( p ) is fu nctional and p ∈ S ∩ domain ( ϕ ) ⇒ Φ W g ( p ) = I t ( ϕ ( p )) X p / ∈ S ∩ domain ( ϕ ) ⇒ Φ W g ( p ) is not an iterator 50 Let S n = { ( α, x , y ) : α ∈ F in ( X → X ) ∧ α ( n ) ( x ) is d eﬁned ∧ y = α ( n ) ∧ domain ( α ) = { α ( i ) : i ≤ n }} Let γ : N 2 → S n ∈ N S n b e a total r ecursiv e fu nction su c h that, for all n , u 7→ γ ( n, u ) is a b ijection N → S n . Set ρ e = { γ ( ϕ ( e ) , u ) : ϕ ( e ) is deﬁned ∧ ∃ v ( e , u, v ) ∈ σ } Clearly , ρ e is functional. Also, the construction of the ρ e ’s is eﬀectiv e and the parametrization prop ert y yields a total r ecursiv e function g : 2 ∗ → 2 ∗ suc h that ρ e = W g ( e ) . If ϕ ( e ) is not d eﬁ ned then ρ e = ∅ s o that Φ ρ e is the constant functional whic h maps any fu nction to the n o where deﬁn ed f unction. In p articular, Φ ρ e is not an iterator. Supp ose ϕ ( e ) is d eﬁned. Condition iii of Prop.10.19 and th e deﬁnition of ρ e sho w that Φ ρ e is an iterator ⇔ Φ ρ e = I t ( ϕ ( n )) X ⇔ ρ e ⊇ r ang e ( u 7→ γ ( ϕ ( n ) , u )) ⇔ ∀ u ∃ v ( e , u, v ) ∈ σ ⇔ e ∈ S Since ρ e = W g ( e ) , the functional Φ : e 7→ Φ ρ e is eﬀect iv ely contin u ous. Clearly , it satisﬁes ( ∗ ) and ( ∗∗ ). 10.7 Syn tactical complexit y of Ch urc h r epresen tation Prop osition 10.23 (Synt actical complexit y) . The family { domain ( ϕ ) : ϕ ∈ Chur ch N X ◦ A -EﬀCont 2 ∗ → (( X → X ) → ( X → X )) } is exactly the family of Π 0 ,A 2 subsets of 2 ∗ . Thus, any unive rsal function for Chur ch N X ◦ A -EﬀCont 2 ∗ → (( X → X ) → ( X → X )) has Π 0 ,A 2 -c omplete domain. Pr o of. T o simplify notations, w e only consider the case A = ∅ . Relativiza- tion b eing straigh tforw ard. 1. Pr op.10.22 insur es that ev ery Π 0 2 set is the domain of C hur ch N X ◦ Φ for some eﬀectiv ely con tin u ous f unctional Φ. 2. Conv ersely , we prov e that eve ry fun ction in Chur ch N X ◦ EﬀCont 2 ∗ → (( X → X ) → ( X → X )) has Π 0 2 domain. Supp ose Φ : ( 2 ∗ × ( X → X )) → ( X → X ) is eﬀectiv ely con tin uous and let 51 R ⊆ 2 ∗ × F in ( X → X ) × X × X b e a fu n ctional r.e. set su c h that Φ = Φ R . F or e ∈ 2 ∗ , let R e = { ( α, x , y ) : ( e , α, x , y ) ∈ R } . Th en domain ( Chur ch N X ◦ Φ) = { e : Φ R e is an iterator } No w, an r.e. co de for the fun ctional relation R e is given b y a tota l recursiv e function h : 2 ∗ → 2 ∗ . Applying Prop.10.20, p oin t 2, the partial recursive function λ ◦ h is suc h that if Φ R e is an iterator then Φ R e = I t ( λ ( h ( e )) ) X . Th us, Φ R e is an iterator if and only if a. λ ( h ( e )) is conv ergent , b. condition iii of Prop.10.19 with n = λ ( h ( e )) holds. Condition a is Σ 0 1 and condition b is Π 0 2 . Thus, domain ( Chur c h N X ◦ Φ ) is Π 0 2 . 10.8 Characterization of the C hur ch represen tation system Theorem 10.24. L et’s denote P R A, 2 ∗ → N ↾ Π 0 ,A 2 the family of r estrictions to Π 0 ,A 2 subsets of p artial A -r e cursive f u nctions 2 ∗ → N . L et X b e some b asic set and A ⊆ N b e some or a cle. 1. Chur ch ◦ A - E ﬀCont 2 ∗ → (( X → X ) → ( X → X )) = P R A, 2 ∗ → N ↾ Π 0 ,A 2 2. K N Chur ch ◦ A - EﬀCont 2 ∗ → (( X → X ) → ( X → X )) = ct K A We shal l simply write K N ,A Chur ch , or K N Chur ch when A = ∅ . Pr o of. 1A. First, we prov e that, for an y A -eﬀectiv ely con tin uous fu nctional Φ : 2 ∗ → ( X → X ) X → X , the fun ction Chur ch ◦ Φ : 2 ∗ → N h as a partial A -recursiv e extension. W e reduce to the case X = N . Let S ucc : N → N b e the su ccessor function. Obser ve that, for all n ∈ N , ( I t ( n ) N ( S ucc ))(0) = n Th us, if C hur ch (Φ( e ) is d eﬁned then C hur ch (Φ( e )) = (Φ( e )( S ucc ))(0). Ap- plying Prop.10.10, we see th at e 7→ (Φ( e )( S ucc ))(0) is a partial A -recursive extension of Chur ch ◦ Φ : 2 ∗ → N . 1B. Prop.10.23 insures that Chur ch ◦ Φ : 2 ∗ → N has Π 0 ,A 2 domain. T ogether with p oin t 1A, this insu res that Chur ch ◦ Φ : 2 ∗ → N is the restriction of a partial A -recursive function to a Π 0 ,A 2 set. T his pr ov es the inclusion Chur ch ◦ A - E ﬀCont 2 ∗ → (( X → X ) → ( X → X )) ⊆ P R A, 2 ∗ → N ↾ Π 0 ,A 2 1C. Th e con verse inclus ion is P rop.10.22. 2. Inclusion P R A, 2 ∗ → N ⊆ Chur ch ◦ A - E ﬀCont 2 ∗ → (( X → X ) → ( X → X )) yields the 52 inequalit y K N Chur ch ◦ A - EﬀCont 2 ∗ → (( X → X ) → ( X → X )) ≤ ct K A . Consider a function φ ∈ Chur ch ◦ A - EﬀCont 2 ∗ → (( X → X ) → ( X → X )) . Let b φ b e a partial A -recurs ive extension of φ . T h en K φ ≥ K b φ . This prov es inequalit y K N Chur ch ◦ A - EﬀCont 2 ∗ → (( X → X ) → ( X → X )) ≥ ct K A . 10.9 Characterization of the ∆ Ch ur ch self-en umerated sys- tems Theorem 10.25. L et X b e some b asic set and A ⊆ N b e some or acle. 1. ∆( Chur ch ◦ A - E ﬀCont 2 ∗ → (( X → X ) → ( X → X )) ) = P R A, 2 ∗ → Z ↾ Π 0 ,A 2 2. K Z ∆( Chur ch ◦ A - EﬀCont 2 ∗ → (( X → X ) → ( X → X )) ) = ct K A Z We shal l simply write K Z ,A ∆ Chur ch , or K Z ∆ Chur ch when A = ∅ . Pr o of. 1. Observe that ∆( P R A, 2 ∗ → N ↾ Π 0 ,A 2 ) = P R A, 2 ∗ → Z ↾ Π 0 ,A 2 and app ly Thm.10.24. 2. Ar gue as in p oint 2 of the p ro of of Thm.10.24. 10.10 F unctional represen tations of Z Sp eciﬁc to Churc h repr esentati on, there is another approac h f or an exten- sion to Z : p ositive and ne gative iter ations of inj ectiv e fu nctions o v er s ome inﬁnite set X . F ormally , I.e., letting X 1 − 1 → X d enote the family of injectiv e functions, consider the Z -iterator fun ctional I t Z X : Z → ( X 1 − 1 → X ) X 1 − 1 → X suc h that, for n ∈ N , I t Z X ( n )( f ) = f ( n ) and I t Z X ( − n )( f ) = I t Z X ( n )( f − 1 ). Eﬀectivizat ion can b e don e as in § 10.5. Th m.10.18, Prop.10.23 and Thm.10.24 go through the Z conte xt. 11 Conclusion W e hav e c haracte rized Kolmogoro v complexities asso ciated to some set the- oretical representati ons of N in terms of the K olmogoro v complexities asso- ciated to oracular and/or inﬁn ite computations (Thm.1.4). As a corollary , w e got a hierarc h y result (Th m.1.5). These results can b e impro v ed in tw o directions. First, one can consider higher ord er (higher than t yp e 2) eﬀectivizatio ns of set theoretical representa tions of N . T his is the con ten ts of a forthcoming con tin uati on of th is pap er. Second, using the r esults of our pap er [6 ], the hierarch y result Thm.1.5 can 53 b e improv ed with ﬁ n er orderings than < ct . Th ese ord erings ≪ C , D F are su c h that f ≪ C , D F g if and only if 1. f ≤ ct g 2. F or ev ery in ﬁnite set X ∈ C and ev ery total m onotone increasing f u nc- tion φ ∈ F there exists an in ﬁnite set Y ∈ D su c h that Y ⊆ { z ∈ X : f ( z ) < φ ( g ( x )) } 3. Th e ab o v e prop ert y is eﬀectiv e: relativ e to standard en umerations of C , D , F , a co d e for Y can b e recursively computed from co des for X and φ . Thm.1.5 can b e restated in the follo w ing impro v ed form. Theorem 11.1. Denote M in P R (r esp. 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