math.LO 2008-01-07 0

Kolmogorov complexities Kmax, Kmin on computable partially ordered sets

We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes Max^{X\to D}_{PR} and Max^{X\to D}_{Rec} of functions X \to…

Authors: Marie Ferbus-Z, a (LIAFA), Serge Grigorieff (LIAFA)

Kolmogoro v complexitie s K m ax , K m in on computabl e partially ord e red sets Marie F erbus-Zanda LIAF A, Universit´ e Paris 7 & CNRS, F r anc e ferbus@l ogique.ju ssieu.fr Serge Grigo r ieff LIAF A, Universit´ e Paris 7 & CNRS, F r anc e seg@liaf a.jussieu .fr Abstract W e in t ro duce a mac hine free mathematical framework to get a n atural form aliza- tion of s ome general notions of infinite computation in the context of Kolmogoro v complexit y . Namely , the c lasses M ax X →D P R and M ax X →D R ec of functions X → D which are p oint wise maxim um o f partial or total compu table sequences of functions where D = ( D , < ) is some computable partially ordered set. The en um er ation theorem and the in v ariance theorem alw a ys hold for M ax X →D P R , leading to a v arian t K D m ax of Kolmogoro v complexit y . W e c haracterize the orders D suc h that the en umera- tion theorem (resp. the in v ariance theorem) also holds for M ax X →D R ec . It turns out that M ax X →D R ec ma y satisfy the in v ariance theorem but n ot th e enumeration theo- rem. Also, when M ax X →D R ec satisfies the in v ariance theorem then the Kolmogoro v complexities asso ciated to M ax X →D R ec and M ax X →D P R are equal (up to a constan t). Letting K D m in = K D r ev m ax , where D r ev is the rev erse order, w e pro ve th at either K D m in = ct K D m ax = ct K D (= ct is equalit y up to a constant) or K D m in , K D m ax are ≤ ct incomparable and < ct K D and > ct K ∅ ′ ,D . W e c haracterize the orders leading to eac h case. W e also sho w that K D m in , K D m ax cannot b e b oth m uch smaller than K D at an y p oin t. These results are pro v ed in a m ore general setting with t wo orders on D , one extending the other. Con t en ts 1 In tro duction 3 1.1 Non halting programs for whic h the curr en t output is ev en tually th e w anted ob ject, but one do es not kno w wh en... 3 1.2 F unctions appro ximable from below (resp. from ab ov e) as decompressors for v ariant s of Kolmogoro v complexit y 4 1.3 Main theorems 5 Preprint submitted to Elsevier 30 Ma y 2018 1.4 The M ax R ec and M in R ec classes 6 1.5 Notatio ns 6 2 The M ax and M in classes of functions 7 2.1 Infinite computations and monotone machines 7 2.2 Mathematica l mo delization: the M ax and M in classes 8 2.3 Domains of fun ctions in the M ax/ M in classes 9 2.4 Examples of functions in the M ax and M in classes 9 2.5 Normalized representati ons 13 3 Kolmogoro v complexities K D m ax , K D m in 14 3.1 Kolmogoro v complexit y K D 14 3.2 En umeration theorem for M ax X →D P R 15 3.3 Kolmogoro v complexit y K D m ax and K D m in 16 4 Main theorems: comparing K , K D m ax , K D m in , K ∅ ′ 17 4.1 The < ct hierarc hy theorem 17 4.2 ( ∗ ) is an effectiv e ve rsion of the n egation of ( ∗∗ ) 19 4.3 K D m ax , K D m in are not sim u ltaneously m uc h smaller than K D 20 4.4 K D m ax , K D m in and the jump 21 4.5 Pro of of Theorem 4.1 (1st h ierarc hy theorem) 22 4.6 Inequalities K D st m ax ≤ ct K D wk m ax and K D st m in ≤ ct K D wk m in 22 4.7 If ( ∗ ) holds: pro of of P oin t 1 of T heorem 4.3 (3rd hierarc hy theorem) 22 4.8 Pro of of Po in t 1 of Th eorem 4.2 (2d hierarc hy theorem) 24 4.9 If ( ∗∗ ) holds: proof of Po in t 2 of T h eorem 4.3 (3d hierarc hy theorem) 24 4.10 Pro of of P oin t 2 of Th eorem 4.2 (2d hierarc hy theorem) 25 5 Complemen tary results ab out the M ax and M in classes 26 5.1 T otal functions in M ax X →D R ec and M ax X →D P R 26 5.2 Comparing M ax X →D P R , M ax X →D R ec and P R X →D , Rec X →D 26 5.3 P ost hierarc hy and the M ax/ M in classes 27 5.4 M ax ∩ M in classes 29 6 M ax 2 ∗ →D R ec and M in 2 ∗ →D R ec and Kolmogoro v complexit y 30 6.1 M ax X →D R ec and the enumeratio n theorem 30 6.2 M ax 2 ∗ →D R ec and the in v ariance theorem 31 References 34 2 1 In t r o duction 1.1 Non halting pr o g r ams for whi c h the curr ent output is eve ntual ly the wante d obje ct, but one do es no t know w hen... In this pap er, w e conside r a particular kind o f description metho ds in order to define v ar ia n ts of Kolmogorov complexit y . Let’s start with tw o paradigmatic examples. Given n ∈ N and u ∈ Σ ∗ (where Σ b e some finite alphab et), ho w do w e get - the v alue B B ( n ) of the busy b eav er function B B : N → N , - the v alue K Σ ∗ ( u ) of Ko lmogorov complexit y K Σ ∗ : Σ ∗ → N ? The definitions of B B ( n ) and K Σ ∗ ( u ) lead to the following mec hanisms. - run a ll T uring mac hines with ≤ n states and all programs with length ≤ n , - for each t , consider those mac hines and programs halting in ≤ t steps, - lo ok a t the maxim um n um b er of cells visited by these mac hines, - lo ok a t the minim um length of these programs. In this wa y , one gets tw o computable functions bb : N × N → N and k : Σ ∗ × N → N , with o ne more integer argumen t (f or time steps) suc h that, f or ev ery fixed n ∈ N and u ∈ Σ ∗ , the ma ps t 7→ bb ( n, t ) and t 7→ k ( u, t ) are resp ectiv ely monotone increasing a nd decreasing and are b o th eve n tua lly con- stan t with resp ectiv e v a lues B B ( n ) and K Σ ∗ ( u ). Since neither B B nor K Σ ∗ is computable, there is no computable functions of n or u whic h b ound the momen t t hese maps b ecome constan t. These examples lead us to introduce the follo wing notion of description meth- o ds for ob jects of a partially ordered set D with a computable structure (cf. Definition 2.3, 2.4). A c o m putable a p pr oximation fr om b elow (r esp. fr om ab ove) of obje cts of D is a pr o gr am for a c om putable function f : X × N → D (wher e X is some r e ason- able set such as N or 2 ∗ , cf. § 1.5 Notations) such that, for every fixe d x ∈ X , the map t 7→ f ( x, t ) is monotone inc r e as ing (r esp. de cr e asing) and eventual ly c onstant. Nothing is assume d ab out the m o ment t 7→ f ( x, t ) b e c omes c onstant: ther e may b e no c omputable function of x majorizing it. The asso ciate d de c ompr essor — or descrip tion metho d — is the function F : X → D such that F ( x ) is the limit value of f ( x, t ) when t → + ∞ , i.e. the max- imum (r esp. mini m um) value of the finite set { f ( x, t ) : t ∈ N } . W e shall call suc h functions F computably approximable f rom b elo w (resp. from ab ov e). Consider Σ ∗ with the prefix or dering. The contex t of non halting (hence in- finite) computations, cf. Chaitin, 1975 [4], and Solo v ay , 1977 [19], leads to functions F : 2 ∗ → Σ ∗ whic h are computably appro ximable fr o m b elo w. In fact, if the output alphab et is Σ, the current output f ( x, t ) at time t is a function f : 2 ∗ × N → Σ ∗ whic h is a computable appro ximation from b elow for w ords in Σ ∗ suc h that F ( x ) is the max of the f ( x, t )’s . 3 Observ e that, in case the ordered s et D is no etherian (resp. w ell-founded), the notion of appro ximation fr om b elo w (resp. fr o m abov e) of ob jects of D redu ces to that of computable function f : X × N → D whic h is monotone increas- ing (r esp. decreasing) with respect to its second a rgumen t. This is indeed t he case with the a ppro ximation from ab o v e of the v alues of K Σ ∗ since ( N , < ) is w ell-founded. Cf. also § 2.4.6. Other examples a r e dev elopp ed in § 2.4. In particular, there is one in v olving quotien ts of regular languages b y a fixed computably enume rable la nguage. 1.2 F unctions appr oximabl e f r om b elow (r esp. fr om ab ov e) as de c ompr essors for v ariants of Kolmo gor ov c omplexity The abov e men tioned contex t of non halting computations has rece n tly led to in teresting v arian ts K ∞ Σ ∗ : Σ ∗ → N , K ∞ N : N → N of Kolmogorov complexit y in tro duced (in their prefix-complexit y v ersion H ∞ ) in Bec her & Chaitin [1]. This last Kolmogorov complexit y K ∞ N has also pro ved to b e equal to the Kolmogorov complexit y K c ard in tro duced in F erbus & Grigorieff, 2002 [9 ,8] where w e compare some natural set theoretical seman tics of in tegers, namely Ch urc h iterators of functions, cardinals of computably en umerable sets, in- dexes of computably en umerable equiv alence r elat io ns. Comparison of these seman tics is do ne via asso ciated Ko lmogorov complexities whic h somehow constitute measures of their “abstractio n degree” a nd are defined in terms of infinite or/and oracular computations. The cornerstone of Kolmogorov complexit y , namely the inv ariance theorem, really deals with partial computable f unctions, not T uring mac hines. In fact, T uring machines do not constitute suc h an abstract structured mathematical framew ork as partial computable functions do. Going to this last framew ork op ens new natural considerations whic h w ould not b e simply view ed w ith T ur- ing mac hines. In this pap er w e abstract from non halting computatio ns on T uring ma chines and dev elop a general mac hine-free mathematical framew ork using a partially ordered set D . Namely , letting X b e a basic space (cf. § 1.5 Notations), we in tro duce the classes of functions F : X → D M ax X →D P R , M in X →D P R whic h are partial computably approx imable from b elow (resp. from ab o v e). Whic h means that the f : 2 ∗ × N → D suc h that F ( x ) is the max or min of the f ( x, t )’s is partial computable rather than computable. Of c ourse, the M in c lasses are the M ax clas ses associated to the rev erse order. 4 W e also introduce the sub classes of functions M ax X →D R ec , M in X →D R e c whic h are computably appro ximable f r o m b elo w (resp. from ab ov e). It happ ens that the M ax X →D R ec class is closely related with the class based on non ha lt ing T uring machines computations with o utputs in D (modulo adequate co ding of D ). As for the a b o ve examples, the bus y beav er function B B : N → N is in M ax D R ec and Kolmogor ov complexit y K Σ ∗ : Σ ∗ → N a nd its prefix-free v ariant H Σ ∗ : Σ ∗ → N are in M in D R ec with D = ( N , < ). These classes lead to new v arian ts of Kolmogoro v complex it y whic h w o uld just b e ignored when considering T uring mac hines. 1.3 Main the or ems The dev elopmen t of Kolmogorov complexities K D m ax , K D m in asso ciated to the classes M ax 2 ∗ →D P R and M in 2 ∗ →D P R is straigh tforw a rd (cf. § 3). The main results of the pap er deal with the comparison of K D m ax , K D m in with the classical Kol- mogorov complexit y K D and its relativized v ersion K ∅ ′ ,D to oracle ∅ ′ . In § 4, w e prov e three theorems whic h giv e the main comparison relations (relativ e to the “up to a constan t” order ≤ ct , cf. § 1.5 Notations) b et w een these com- plexities. The first theorem (Thm.4.1) is v alid whatev er b e the partia l order on D . It states that K ∅ ′ ,D < ct inf ( K D m ax , K D m in ) and that K D m ax , K D m in , though obv iously ≤ ct K D , cannot b e simultaneous ly m uc h smaller than K D since K D ≤ ct ( K D m ax + log( K D m ax )) + ( K D m in + log( K D m in )) The second theorem (Thm.4.2 ) prov es that either K D m ax = ct K D m in = ct K D or K D m ax , K D m in are ≤ ct incomparable and b oth are < ct to K D . This dic hotomy is also c haracterized by a simple prop erty on the order. The third theorem (Thm.4.3) considers t w o partial or ders < wk and < st on D , the second extending the first. W e giv e conditions ( ∗ ) and ( ∗∗ ) on the orders suc h that - ( ∗ ) insures t ha t K D st m ax = ct K D wk m ax and K D st m in = ct K D wk m in , - ( ∗∗ ) insures that K D st m ax < ct K D wk m ax and K D st m ax < ct K D wk m ax and neither K D st m ax nor K D st m in is ≤ ct min( K D wk m ax , K D wk m in ). These c onditions are a lmo st complemen tary: ( ∗∗ ) is an effectiv e v ersion of the negation of ( ∗ ). An in teresting case of this theorem is o btained when Σ = { 1 , ..., k } with the 5 ob vious order and < wk , < st are the prefix and the lexicographic orders on Σ ∗ (the last one b eing isomorphic to the order on k -a dic ratio na l reals in [0 , 1]) . 1.4 The M ax R e c and M in R e c classes In § 5.2 and 5.3 w e come bac k to the f o ur class es M ax 2 ∗ →D P R , M in 2 ∗ →D P R and M ax 2 ∗ →D R ec , M in 2 ∗ →D R ec . W e compare t hem to that of partial computable func- tions X → D and look at the syn tactical complex it y of their domains and graphs. In § 5.4 w e compute M ax X →D P R ∩ M in X →D P R under simple conditions ab out t he partial order on D . In § 6, w e c onsider the p ossible dev elopmen t of Kolmogoro v complexities based on the classes M ax 2 ∗ →D R ec and M in 2 ∗ →D R ec . This leads to lo ok at the t w o follo wing problems: - the existence of an en umeration, - the inv ariance theorem. F or each problem, w e characterize the orders D for whic h there is a p ositive answ er (cf. § 6.1, 6.2). It turns out (cf. Thm.6.2) that when t he in v a r iance t heorem ho lds for M ax 2 ∗ →D R ec then ev ery function in M ax 2 ∗ →D P R has an extension (not necessarily to t a l) in M ax 2 ∗ →D R e c . This insures that the Kolmogorov complexities associated to M ax 2 ∗ →D R ec and M ax 2 ∗ →D P R coincide. In particular, K ∞ Σ ∗ is the complexit y asso- ciated to M ax 2 ∗ →D R ec and M ax 2 ∗ →D P R when D is Σ ∗ with the prefix o r der. Surprisingly , there ar e orders suc h that the in v ar iance theorem ho lds for M ax 2 ∗ →D R ec whereas the enumeration theorem fa ils (compare Thm.6.1 and Thm.6.2 ). 1.5 Notations 1. Equalit y , inequalit y and strict inequ alit y up to a constan t b et w een tota l functions S → N are denoted as follo ws: f ≤ ct g ⇔ ∃ c ∀ s f ( s ) ≤ g ( s ) + c f = ct g ⇔ f ≤ ct g ∧ g ≤ ct f ⇔ ∃ c ∀ s | f ( s ) − g ( s ) | ≤ c f < ct g ⇔ f ≤ ct g ∧ ¬ ( g ≤ ct f ) ⇔ f ≤ ct g ∧ ∀ c ∃ s g ( s ) > f ( s ) + c 2. [Basic sp ac es] 2 ∗ denotes the set of binary w ords. W e call ba sic spaces the pro ducts of non e mpt y finite families of s paces o f the form N or Z or A ∗ where A is some finite alphab et. Basic spaces are denoted b y S , X , Y , ... 3. [Partial r e cursive (or c omputable) func tion s ] P R X → Y (resp. Rec [ X → Y ]) 6 denotes the family o f pa r tial ( resp. total) computable functions from X to Y . 2 The M ax and M in classes of functions 2.1 Infinite c om p utations and m onotone machines Recall that a T uring mac hine is monotone if its curren t output ma y only increase with r espect to the prefix order o n w or ds: no o v erwriting is allo w ed. This is indeed T uring’s origina l assumption [20], insuring that, in the limit of time, the output o f a non halting computat io n alw ay s con v erg es, either to a finite or to a n infinite sequenc e. This concept was also considere d b y Levin [14] and Sc hnorr [16,17], see [15] p.276. Suc h infinite computations with p ossibly infin ite o utputs can be us ed to obtain highly random reals, cf. Bec her & Chaitin [1] and Bec her & Grigorieff [3]. In this p ap er, when c onside ring infinite c omputations, we r e tain the sole li m it outputs that ar e finite. The following easy prop osition links infinite computations, a s considered for the definition of K ∞ and its prefix v ersion H ∞ in tro duced in [2], with t he general approac h whic h is the sub ject of this pap er. Prop osition 2.1 L et F : 2 ∗ → Σ ∗ wher e Σ is so me non empty fini te alphab et. The fol lo wing c onditions ar e e quivalent: i. F c an b e c omp ute d via p ossibly infinite c omputations on some monotone T uring machine with output alphab et Σ , ac c or ding to the fol lowing c on- vention: F ( s ) is defin e d if and only if the output r emain s c onstant af ter some step. ii. Ther e exists a total c o mputable function f : 2 ∗ × N → Σ ∗ such that - f ( s , t ) is monotone incr e asing in t w ith r esp e ct to the pr efix or der on Σ ∗ , - s ∈ dom ( F ) if and o nly if { f ( s , t ) : t ∈ N } is fini te and n on empty, - F ( s ) is the maximum value of { f ( s , t ) : t ∈ N } . iii. L et λ de n ote the empty wor d. Idem as ii, with f such that f ( s , 0) = λ , f ( s , t + 1) ∈ { f ( s , t ) } ∪ { f ( s , t ) σ : σ ∈ Σ } PR OOF. ii i ⇒ ii is trivial; i ⇔ iii : let f ( s , t ) b e the curren t output at time t when the input is s . As for ii ⇒ iii , let e f ( s , 0) = λ and e f ( s , t + 1 ) b e the prefix of f ( s , t + 1) with length min( | e f ( s , t ) | + 1 , | f ( s , t + 1 ) | ). Then { e f ( s , t ) : t ∈ N } and { f ( s , t ) : t ∈ N } are simultaneously finite or infinite and, when finite, their maximum elemen t s are equal. ✷ 7 2.2 Mathematic al m o delization: the M ax and M in classes Prop osition 2.1 and the a rgumen tation in § 1.1 – 1.2 in vite to a mathem ati- cal, machine-free mo delization of the notion of function defined b y infinite computations. Namely that o f f unction obta ined as p oint wise maximum of a computable sequenc e of tota l computable f unctions. A c on struction which makes sense for map s fr om a b asic set X into any c omputable p artial ly or der e d set D = ( D , < ), and leads to the class M ax X →D R ec . It is a lso quite natural – in fact, it is even much mo r e natur al fr om a mathe- matic al p oint of view – to consider the v ersion of the abov e mo delization using p artial c o m putable functions instead of total computable ones. This leads to the class M ax X →D P R . Natural and in teresting imp ortan t examples (cf. § 2.4) are obtained when D is among the fo llo wing (obviously computable) par tially ordered sets: ( N , < ) , ( Z , < ) , (Σ ∗ , < p re f ix ) , (Σ ∗ , < l exico ) and the rev erse orders obtained b y replacing < b y > , where < l exico on Σ ∗ dep ends on a t o tal or partia l o r der on the alphab et Σ. Definition 2.2 (The max D and min D op erators) L et X b e some b asi c set and D = ( D , < ) b e some p artial ly or der e d set. L et f : X × N → D b e monotone incr e asing in its se c ond ar gument on its domain. We defi ne max D f : X → D (r esp. min D f : X → D ) as the function i. define d on the x ’s in X for which the map t 7→ f ( x , t ) has finite non empty r ange, ii. and s uch that ( ma x D f )( x ) (r esp. (min D f )( x ) ) is the maximum (r esp. mi n - imum) eleme nt of { f ( x , t ) : t ∈ N } . Definition 2.3 1. A c omputable p artial ly or der e d set D is a triple ( D , <, ρ ) such that ρ : N → D is a bije ctive total map (in p articular, D is infinite c ountable) and < is a p artial or der on D such that { ( m, n ) : ρ ( m ) < ρ ( n ) } is c o m putable. 2. L et X b e a b asic sp ac e. A function F : X → D is p artial (r esp. total) c omputable if so is ρ − 1 ◦ F : X → N . A set Z ⊆ X × D k is c omp utable if so is ( I d X , ρ, ..., ρ ) − 1 ( Z ) as a subset of X × N k , w h er e I d X is the identity function on X . Of course, w e shall omit an y reference to ρ w hen D is N or Z with the natural order, o r Σ ∗ with the prefix o r the lexicographic order (with r espect to some partial or total order of the elemen ts o f Σ). Definition 2.4 ( M ax and M in classes) L et X b e a b asic sp ac e and D = ( D , <, ρ ) b e a c o m putable p artial ly or d er e d set. We let 8 M ax X →D R e c = { max D f : f : X × N → D is total c o m putable } M ax X →D P R = { max D f : f : X × N → D is p artial c omputable } We r e s p e ctively denote by M in X →D P R and M in X →D R ec the analo g classes define d with the min D op er ator, i.e. the classes M ax X →D r ev P R and M ax X →D r ev P R wher e D r ev = ( D, > ) . Prop osition 2.1 can b e rephrased in terms of the prefix ordering on Σ ∗ . Prop osition 2.5 If Σ is a fi n ite alp h a b et then M ax 2 ∗ → (Σ ∗ ,< pr efix ) R e c is the c l a s s of functions c ompute d via p oss i b ly infinite c om p utations on monotone T uring machines (cf. Pr op osition 2.1 i) w i th Σ a s output alphab et. 2.3 Domains of functions in the M ax/ M in classes W e denote by Σ 0 1 ∧ Π 0 1 the family of conjunctions of Σ 0 1 and Π 0 1 form ula s. Let X b e a basic set and D b e a computable ordered set. The arithmetical hierarc h y on N induces a hierarc h y on D and X × D : a relation R ⊆ X × D is Σ 0 n or Π 0 n or Σ 0 n ∧ Π 0 n if so is ( I d X , ρ ) − 1 ( R ) ⊆ X × N . Prop osition 2.6 L et X b e a b asic se t and D b e a c omputable or der e d se t. Every p artial function in M ax X →D P R or in M in X →D P R has Σ 0 1 ∧ Π 0 1 gr aph a nd Σ 0 2 domain. PR OOF. Let f : X × N → D b e partial computable, monotone increasing in its second argumen t on its domain. Then (max D f )( x ) = z ⇔ ∃ t ( f ( x , t ) is defined ∧ f ( x , t ) = z ) ∧ ∀ t ( f ( x , t ) is defined ⇒ f ( x , t ) ≤ z ) x ∈ dom (max D f ) ⇔ ∃ z F ( x ) = z Idem with M in X →D P R . ✷ 2.4 Examples o f functions in the M ax a nd M in classes The classes M ax X →D R e c , M in X →D R ec con tain man y fundamen tal non computable functions. T o see t ha t some functions are not in such classes, w e shall use Theorem 5.5 below (the pro o f o f which do es not dep end on an y result of this § ). 9 2.4.1 Kolmo gor ov and Chaitin-L evin pr o gr am-size c om plexities Prop osition 2.7 L et D b e ( N , < ) . Kolm o gor ov and Chaitin-L evin pr o gr am- size c omplexities K N , H N : N → N (r esp. K Σ ∗ , H Σ ∗ : Σ ∗ → N ) ar e in M in N →D R ec \ M ax N →D P R (r esp. in M in Σ ∗ →D R e c \ M ax Σ ∗ →D P R ). PR OOF. T hat K , H b elong to M in N →D R ec is a mere reformulation of the w ell- kno wn fact that they are computably appro ximable from ab o ve, i.e. they are limits of decre asing computable sequences of total computable functions. That these total functions ar e not in M ax N →D P R is an ob vious application of The orem 5.5 b elo w. ✷ 2.4.2 Busy b e ave r Prop osition 2.8 L et D b e ( N , < ) . L et B B : N → N b e the busy b e aver func- tion, i.e. B B ( n ) is the maximum numb er of c el ls visite d by the input he ad o f a T uring machine w i th n + 1 states which halts with no input. Then B B ∈ M ax N →D R e c \ M in N →D P R . PR OOF. Obs erv e that B B = max bb where bb is the total computable func- tion suc h that bb ( n, t ) is t he maxim um among 0 and the num b ers of cells visited b y T uring mac hines with n + 1 states whic h halt in at most t steps. An ob vious application of Theorem 5.5 b elo w sho ws that B B is not in M in N →D P R . ✷ Remark 2.9 V ariants of the busy b e av e r function c an b e very natur al ly de- fine d with r anges over various typ es of data s tructur es. F or instanc e, finite gr aphs r elative to the inclusion or emb e dding or derin g. 2.4.3 Car dinality of finite c omputably enumer able sets The follo wing example is completely inv estigated in [9,8]. Prop osition 2.10 L et D b e ( N , < ) . L et c ar dRE : N → N b e such that c ar dRE ( n ) =      car d ( W n ) if W n is finite undefine d otherwise wher e car d ( W n ) is the numb er of elements o f the c om putably enumer able set W n with c o de n . Then c ar dRE ∈ M ax N →D R ec \ M in N →D P R . 10 PR OOF. Obs erv e that c ar dRE = max h where h ( n, t ) is t o tal computable and coun ts the nu m b er of elemen ts of W n obtained after t computation steps. The do main o f the partia l f unction c ar dRE is kno wn to b e Σ 0 2 complete, hence not Σ 0 1 ∧ Π 0 1 . Applying Theorem 5.5 b elo w, w e see that c ar dRE cannot b e in M in N →D P R . ✷ 2.4.4 Inter acting fini te sets with a fixe d c omputably enumer able set If X, Y ⊆ N , let’s denote X − Y and X \ Y the sets X − Y = { x − y : x ∈ X ∧ y ∈ Y ∧ x ≥ y } , X \ Y = { z : z ∈ X ∧ z / ∈ Y } Prop osition 2.11 L et D b e b e the family P <ω ( N ) of finite subsets of N , o r- der e d by set in clusion. If A ⊆ N is a fixe d c om p utably enumer able set which is non c omputable then i. the ma p s X 7→ X ∩ A and X 7→ X − A ar e in M ax D →D R ec \ M in D →D P R . ii. the map X 7→ X \ A is in M in D →D R ec \ M ax D →D P R . PR OOF. Let A = ϕ ( N ) where ϕ : N → N is total computable. Define total computable maps f , g , h : D × N → D suc h that f ( X , t ) = X ∩ ϕ ( { 0 , ..., t } ) g ( X, t ) = X − ϕ ( { 0 , ..., t } ) h ( X , t ) = X \ ϕ ( { 0 , ..., t } ) It is easy to see that X ∩ A = (max D f )( X ) and X − A = (max D g ) ( X ) and X \ A = (min D h )( X ) . ✷ 2.4.5 Quotients of r e gular languages by a fixe d c omp utably enumer able lan- guage W e now come to a v ery differen t example. The family R e g of regular languages o ver alphab et Σ can b e defined by regular expressions whic h are w o r ds in the alphab et e Σ obtained b y enric hing Σ with sym b ols + , ∗ , · , ( , ) . Let ζ : e Σ ∗ → R e g b e the surjectiv e map suc h that, if u is a regular expression then ζ ( u ) is the asso ciated regula r language, else ζ ( u ) = ∅ . Since equalit y of regular languages is dec idable, there exists a computable map η : N → e Σ ∗ suc h that ρ = ζ ◦ η : N → R e g is bijectiv e. Using decidabilit y of inclusion of regular languages, w e see that ( R e g , ⊆ , ρ ) is a computable partially ordered set in the sense of Definition 2.3. 11 It is k no wn that, if L is a regular language and M ⊆ Σ ∗ is an y language (ev en non computable) then M − 1 L = { u ∈ Σ ∗ : ∃ v ∈ M v u ∈ L } is alw ays regular and M − 1 L = M ′− 1 L for some finite subset M ′ ⊆ M . Recall the core of the easy pro of: if L is the set of w ords leading from state q 0 to a final state of automaton A and if the w ords in M lead from state q 0 to the states in X , then M − 1 L is the set of w ords leading from a state in X to a final state. Prop osition 2.12 L et M ⊆ Σ ∗ b e a fixe d c omputably enumer ab l e language which is non c omputable. L et F M : R e g → R e g b e such that F M ( L ) = M − 1 L . Then F M is i n M ax R e g → Re g R e c \ M in R e g → Re g P R . PR OOF. Let M = ϕ ( N ) where ϕ : N → Σ ∗ is a total computable function. Observ e t hat F M = max R e g f M where f M : R e g × N → R e g is suc h that f M ( L, t ) = ( ϕ ( { 0 , ..., t } ) − 1 L Observ e tha t M is computable with o racle F since u ∈ M if and only if M − 1 { u } = { λ } . Since M is not computable, F cannot b e computable. Using Theorem 5.5 p oin t 1 (and the fact that F is total), w e see that F is not in M in R e g → Re g P R . ✷ Using the ab ov e surjection ζ : e Σ ∗ → R e g , one can reformulate the ab o v e result in terms of a pa rtial computable preordering on w ords quite differen t of the usual ones . This nece ssitates a straigh tforward extension to preorderings o f the material ab out the M ax and M in classes . Let µ : R e g → ∆ ∗ b e the map whic h associates to a regular langua g e L t he regular expression (obtained via some fixed algorithm) describing its minimal automaton. Observ e that ζ is a retraction of t he injectiv e map µ , i.e. ζ ◦ µ is the iden tit y map on R e g . Prop osition 2.13 L et D b e e Σ ∗ with the fol lowing c om putable pr e or dering: u  v ⇔ ζ ( u ) ⊆ ζ ( v ) L et M ⊆ e Σ ∗ b e a fix e d c omputably enumer able la nguage which is non r e cuc o m- putablersive. Then the map u 7→ µ ( M − 1 ζ ( u ) ) (which maps a r e gular expr ession for L to one for M − 1 L ) is in M ax D →D R e c \ M in D →D P R . 12 PR OOF. Let F , f be as in the pro o f of Prop o sition 2.12. Sinc e ζ ◦ µ = I d R e g , w e see that e F = µ ◦ F ◦ ζ make s the follo wing diagram comm ute: R e g F − − − → R e g ζ x    x    ζ e Σ ∗ e F − − − → e Σ ∗ whic h allows to transfer the results of Prop osition 2.12. ✷ 2.4.6 No etherian or wel l-founde d or derin g s Supp ose D is No etherian (resp. w ell- f ounded) and let f : X × N → D . If t 7→ f ( x, t ) is monotone increasing (resp. decreasing) then it is nece ssarily ev en t ua lly constant. In that case, the considered notio n of approximation from b elo w (resp. from ab o ve) coincides with monotone approxim ation. Fix n ≥ 1 . An imp ortant case is the no etherian se t ( D , ⊆ ) of ideals in the ring of n -v ariables polynomials with real algebraic coefficien ts (this last h yp othesis insures that D is coun table with a computable ordering). 2.5 Normalize d r epr esentations It sometimes prov es useful to normalize the f in max D f . Prop osition 2.14 L et X b e a b asic set and D = ( D , <, ρ ) b e a c omputable or der e d set. 1. Every F ∈ M ax X →D P R is of the form F = max D f f o r som e p artial c om- putable f : X × N → D , monotone in cr e a sing i n its se c ond ar gument, such that dom ( f ) = Z × N wher e Z is some Σ 0 1 subset of D . 2. If F ∈ M ax X →D P R has Σ 0 1 domain then one c an supp ose Z = dom ( F ) . PR OOF. 1. Let g : X × N → D b e partial computable, monotone increasing in its second argumen t, suc h that F = max D g . Let Z = { x : ∃ t ( x , t ) ∈ dom ( g ) } b e the first pro j ection of dom ( g ). Let θ : X → D b e the pa rtial computable function with domain Z suc h that θ ( x ) is the v alue first o bt a ined in { g ( s , t ) : t ∈ N } b y dov etailing o v er computat io ns of g ( s , 0) , g ( s , 1) , ... . Let also ∆ x ,t = { g ( x , u ) : u ≤ t ∧ g ( x , u ) halt s in ≤ t steps } and define f with doma in Z × N suc h that f ( x , t ) is t he greatest eleme n t of { θ ( x ) } ∪ ∆ x ,t 13 2. O bserv e tha t Z necessarily contains d o m ( F ). If dom ( F ) is Σ 0 1 then b f = f ↾ ( dom ( F ) × N ) is also partial computable and max D b f = max D f . ✷ 3 Kolmogoro v complexities K D m ax , K D m in Kolmogorov complexity theory go es through with the M ax X →D P R and M in X →D P R classes with no difficult y . First, w e recall Kolmogo ro v complexit y o v er elemen ts of D . 3.1 Kolmo gor ov c ompl e x ity K D Classical Ko lmogorov complexit y for elemen ts in D is defined as f o llo ws (cf. Kolmogorov, 1 9 65 [11], or Li & Vita n yi [15], Downey & Hirsc hfeldt [6], G ` acs [10] or Shen [18 ]). Definition 3.1 L et ϕ : 2 ∗ → D . We denote K ϕ : D → N the p artial function with d o main r ang e ( ϕ ) such that K ϕ ( d ) = min {| p | : ϕ ( p ) = d } I.e., c onsid e ring w o r ds in 2 ∗ as pr o gr ams, K ϕ ( d ) is the shortest length o f a pr o gr am p m a pp e d onto d by ϕ . Theorem 3.2 (In v ariance theorem, Kolmogoro v, 1965 [11]) L et X b e a b asic sp ac e and D = ( D , <, ρ : N → D ) b e a c omputable p artial ly or der e d set. When ϕ varies in the fami l y P R 2 ∗ → D of p artial c o m putable functions 2 ∗ → D , ther e is a le a st K ϕ , up to an additive c on s tant: ∃ ϕ ∈ P R 2 ∗ → D ∀ ψ ∈ P R 2 ∗ → D K ϕ ≤ ct K ψ Such ϕ ’s ar e said to b e optimal in P R 2 ∗ → D . Definition 3.3 Kolmo gor ov c omple xity K D : D → N is K ϕ wher e ϕ i s some fixe d optimal function in P R 2 ∗ → D . Th us, K D is define d up to an additive c onstant. Of course, K D and K N are related. Prop osition 3.4 K D ◦ ρ = ct K N . PR OOF. Sinc e P R 2 ∗ → D = { ρ ◦ ψ : ψ ∈ P R 2 ∗ → N } a nd K ψ ( n ) = K ρ ◦ ψ ( ρ ( n )) for all ψ ∈ P R 2 ∗ → N , w e see that if ϕ is o ptimal in P R 2 ∗ → N then ρ ◦ ϕ is optimal 14 in P R 2 ∗ → D . ✷ W e also observ e the f ollo wing simple fact: Prop osition 3.5 s up { K D ( d ) : d ∈ X } = + ∞ for every infin i te X ⊆ D . PR OOF. T he result is w ell-kno wn for K N and it transfers to K D using Prop o - sition 3.4. ✷ 3.2 Enumer ation the or em for M ax X →D P R The classical en umeration theorem for partial computable functions go es through the max o p erator, leading to an en umeration of M ax X →D P R . F irst, w e recall a folklore result on enumeration of m o notone partial computable functions. Prop osition 3.6 L et X b e a b asic set and D = ( D, <, ρ ) b e a c omputable or der e d set. L et P R X × N → D , ↑ b e the family of p artial c omputable functions X × N → D which ar e mono tone incr e asing in their last ar gument. Ther e exists a p artial c omp utable function ψ : N × X × N → D such that { ψ n : n ∈ N } = P R X × N →D , ↑ wher e ψ n : X × N → D deno tes the function ( x , t ) 7→ ψ ( n, x , t ) . PR OOF. Let φ : N × X × N → D b e a partial computable function which en umerates the family P R X × N → D of partia l computable functions X × N → D , i.e, { φ n : n ∈ N } = P R X × N → D W e mo dify φ to ψ so as to get an en umeratio n of P R X × N → D , ↑ . Consider an injectiv e computable enum eration ( n i , x i , t i , d i ) i ∈ N of the gr a ph of φ . Let Z = { ( n i , x i , t i , d i ) : ∀ j < i ( n j = n i ∧ x j = x i ∧ t j < t i ⇒ d j ≤ d i ) } Let ψ : N × X × N → D be the partial computable function with graph Z . It is clear that ψ is monotone increasing in its last arg ument, so that so are all ψ n ’s. Also, if φ n is monotone increasing in its last argumen t t hen { n } × g r aph ( φ n ) is included in Z , so that ψ n = φ n . Thus , the ψ n ’s en umerate P R X × N → D , ↑ . ✷ Theorem 3.7 (En umeration theorem for M ax X →D P R ) L et X b e a b asic set and D = ( D , <, ρ ) b e a c omputable or der e d se t. Ther e e x ists a function E : 15 N × X → D in M ax N × X → D P R such that { E n : n ∈ N } = M ax X →D P R wher e E n : X → D de notes the function s a tisfying E n ( x ) = E ( n, x ) . PR OOF. Let ψ : N × X × N → D b e a partia l computable function whic h en umerates P R X × N → D , ↑ . Let E : N × X → D b e suc h tha t E n = max D ψ n for all n . F or any F : X → D in M ax X →D P R there exists n suc h that F = max D ψ n . W e then hav e x ∈ dom ( F ) ⇔ { ψ n ( x , t ) : t s.t. ψ n ( x , t ) is defined } is finite non empt y ⇔ { ψ ( n, x , t ) : t s.t. ψ ( n, x , t ) is defined } is finite non empt y ⇔ ( n, x ) ∈ dom ( E ) ⇔ x ∈ dom ( E n ) F ( x ) = greatest elemen t of { ψ n ( x , t ) : t s.t. ψ n ( x , t ) is defined } = greates t elemen t of { ψ ( n, x , t ) : t s.t. ψ ( n, x , t ) is defined } = E ( n, x ) = E n ( x ) Whic h prov es that E en umerates M ax X →D P R . ✷ 3.3 Kolmo gor ov c ompl e x ity K D m ax and K D m in The in v ariance theorem extends easily to M ax 2 ∗ →D P R , leading to Kolmog oro v complexit y K D m ax : D → N . Theorem 3.8 (In v ariance theorem for M ax 2 ∗ →D P R ) L et X b e a b asic sp ac e and D = ( D , <, ρ : N → D ) b e a c omp utable p artial ly or der e d set. When F varies in the famil y M ax 2 ∗ →D P R ther e is a le ast K F , up to an additive c ons tant: ∃ U ∈ M ax 2 ∗ →D P R ∀ F ∈ M ax 2 ∗ →D P R K U ≤ ct K F Such U ’s a r e said to b e optimal in M ax 2 ∗ →D P R . PR OOF. T he usual pro of w orks. Let E : N × 2 ∗ → D in M ax N × 2 ∗ →D P R b e an en umeration of M ax 2 ∗ →D P R . Define U : 2 ∗ → D suc h that U (0 n 1 p ) = E ( n, p ) and U ( q ) is undefined if q is not of the f o rm 0 n 1 p for some n ∈ N and p ∈ 2 ∗ . If F ∈ M ax 2 ∗ →D P R and F = E n then 16 K F ( d ) = min {| p | : F ( p ) = d } = min {| p | : E ( n, p ) = d } = min {| p | : U (0 n 1 p ) = d } = min {| 0 n 1 p | : U ( 0 n 1 p ) = d } − n − 1 ≥ min {| q | : U ( q ) = d } − n − 1 = K U ( d ) − n − 1 ✷ Definition 3.9 Kolmo gor ov c omplexity K D m ax is define d up to an additive c on- stant as any K U wher e U is optimal in M ax 2 ∗ →D P R . Kolmo gor ov c omplexity K D m in is K D ′ m ax wher e D ′ is the r everse o r der of D . 4 Main t heorems: comparing K , K D m ax , K D m in , K ∅ ′ 4.1 The < ct hier ar chy the or em The main motiv ation of this section is to compare the Kolmo g oro v complexi- ties K D , K D m ax , K D m in , K ∅ ′ : D → N Comparisons of K D m ax , K D m in and K D turn o ut to b e a particular application of more general results dealing with b oth K D m ax and K D m in complexities relativ e to t w o computable orders D st = ( D , < st , ρ ) and D wk = ( D, < wk , ρ ) on the same set D , the strong one < st b eing a n extension of the w eak one < wk . A question with naturally arises when considering f or instance the prefix and lexicographic orders on Σ ∗ . In the case of N with the natural order or of Σ ∗ with the prefix order, the inequalities K ∅ ′ ,D < ct K D m ax < ct K D w ere obtained (mo dulo Prop osition 2.5) for the prefix vers ion H ∞ Bec her & Figueira & Nies & Picci, 200 5 [2], W e state our results as three theorems, the pro ofs of whic h are giv en in § 4.5 to 4.10. Theorem 4.1 (1st hierarc hy theorem) L et D = ( D, < , ρ ) b e a c omputable or der e d set. 1. K ∅ ′ ,D < ct inf ( K D m ax , K D m in ) 2. K D m ax , K D m in ar e ≤ ct smal ler than K D but not s imultane ously much smal ler: K D ≤ ct ( K D m ax + log( K D m ax )) + ( K D m in + log( K D m in )) Theorem 4.2 (2d hierarc h y theorem) 1. I f ( D , < ) c ontains arbitr arily lar ge finite chains then K D m ax and K D m in ar e 17 ≤ ct inc omp ar able and b oth ar e < ct smal ler than K D . In fact, a much str onger pr o p erty holds: i. K D is n ot majorize d by a c omputable function of K D m in , ii. K D is n ot majorize d by a c omputable function of K D m ax , iii. K D m ax is not majorize d by a c omputable function of K D m in , iv. K D m in is n ot maj o rize d by a c omputable function of K D m ax , I.e., for any total c omputable function α : N → N , the fol low i n g sets ar e infinite { d ∈ D : K D m ax ( d ) ≥ α ( K D m in ( d )) } , { d ∈ D : K D ( d ) ≥ α ( K D m in ( d )) } { d ∈ D : K D m in ( d ) ≥ α ( K D m ax ( d )) } , { d ∈ D : K D ( d ) ≥ α ( K D m ax ( d )) } 2. If ( D , < ) do es n o t c ontain arbitr arily lar ge finite chains then K D m in = ct K D m ax = ct K D Theorem 4.3 (3d hierarc h y theorem) L et D st = ( D , < st , ρ ) and D wk = ( D , < wk , ρ ) b e two c o m putable or ders on the sam e set D (“wk” and “st” stand for “ we ak” and “str on g”) such that < st is an extension o f < wk . 1. L et ( ∗ ) b e the fol lowing c ondition ( ∗ ) F or al l k ther e exists a str ong cha in w ith k elements which is a we ak antichain. If ( ∗ ) holds then K D st m ax < ct K D wk m ax and K D st m in < ct K D wk m in . In fact, a much str onge r pr o p erty holds: inf ( K D wk m in , K D wk m ax ) is not majorize d by a c omputable function of K D st m ax or K D st m in . I .e., for any total c omputable f unction α : N → N , the fol lowing sets ar e infin ite { d ∈ D : inf ( K D wk m in ( d ) , K D wk m ax ( d )) ≥ α ( K D st m ax ( d )) } { d ∈ D : inf ( K D wk m in ( d ) , K D wk m ax ( d )) ≥ α ( K D st m in ( d )) } 2. L e t ( ∗∗ ) b e the fol low ing c ondition (which is an e ffe ctive v e rs i on, tailor e d for i n finite c omputations, of the ne gation of ( ∗ ) , cf . § 4.2). ( ∗∗ ) Ther e exists k such that for every p artial c om p utable f : 2 ∗ × N → D which is monotone incr e asing in its se c ond a r gument r elative t o the str ong or der < st ther e exist p a rtial c omputable function s f 1 , ..., f k : 2 ∗ × N → D which ar e monotone incr e asing in their se c ond ar gument r elative to the we ak or der < wk such that { f ( p, t ) : t ∈ N } = [ i =1 ,...,k { f i ( p, t ) : t ∈ N } If ( ∗∗ ) holds then K D st m in = ct K D wk m in and K D st m ax = ct K D wk m ax . 18 Corollary 4.4 L et Σ b e a finite or infinite c ountable alphab et, let < 1 , < b e c omputable or ders on Σ such that < 1 is p artial but non trivial and < is a total extension of < 1 . Conside r on Σ ∗ the fol lowing or ders: the pr efix or der < pr efix , the lex ic o gr a p hic or ders < lexic o 1 and < lexic o asso ciate d to < 1 and < . Then K D pr efix m ax < ct K D < lexic o 1 m ax < ct K D < lexic o m ax , K D pr efix m in < ct K D < lexic o 1 m in < ct K D < lexic o m in PR OOF. Let a, b, c, d ∈ Σ b e suc h t ha t a < 1 b and c < d but c 6 < 1 d . Since < extends < 1 , c and d are < 1 incomparable. Observ e that { a n b : n ∈ N } is an infinite inc reasing c hain f or < lexic o 1 and a n antic ha in for the prefix order. Also, { c n d : n ∈ N } is an infinite increasing c hain for < lexic o and an antic hain for the < lexic o 1 order. This giv es condition ( ∗ ) relativ e to the pairs ( < pr efix , < lexic o 1 ) and ( < lexic o 1 , < lexic o ) o f orders on Σ ∗ . ✷ 4.2 ( ∗ ) is an effe ctive version of the ne gation of ( ∗∗ ) Recall Dilw orth’s theorem. Theorem 4.5 (Dilw or th, 1950 [5]) L et D = ( D , < ) b e an or der e d set and k ∈ N . If every antichain in D has at most k elements then D i s the union of k chains. Dilw or t h’s theorem leads to an equiv alen t form ( † ) of ( ∗ ) a nd condition ( ∗∗ ) app ears as an effectiv e ve rsion of ¬ ( † ), tailored for infinite computations. Prop osition 4.6 L et D st = ( D , < st ) and D wk = ( D , < wk ) b e two or ders on the sa m e set D such that < st is a n extension of < wk . Then ( ∗ ) is e quival e nt to the fo l lowing c ondition ( † ) : ( † ) F or al l k ther e exists a fin ite str ong chain X which is not the union of k we ak chains PR OOF. ( ∗ ) ⇒ ( † ). Apply ( ∗ ) with k + 1 and observ e that a w eak antic hain with k + 1 elemen ts cannot b e the union of k w eak chains . ¬ ( ∗ ) ⇒ ¬ ( † ). Let k b e an in t eger whic h con tradicts ( ∗ ). Then, in a ny strong c hain, an y w eak antic hain has < k elemen t s. Apply Dilworth’s the orem to get ¬ ( † ). ✷ Remark 4.7 1. Cle arly ( ∗∗ ) ⇒ ¬ ( † ) . We do not know whe ther the c onverse implic ation holds or not. The pr oblem is that the pr o of o f Dilworth’s the or em is not in cr emental as we now detail. L et X ∪ { d } b e a str ong chain with d > st x for a l l x ∈ X and such that every we ak antichain in clude d in X has at most k 19 elements. If X is c over e d by k we ak chains C 1 , ..., C k then d may b e inc omp a- r able to the top elemen ts of al l these k chains. Thus, though X ∪ { d } is also the union of k we ak chains, such chains may b e q uite differ ent fr om the C i ’s. Condition ( ∗ ∗ ) (a s c ontr a ste d to ¬ ( † ) ), do e s insur e such an incr emental char- acter. 2. I n c ase < wk has a smal lest element d , c on d ition ( ∗∗ ) is e quivalent to the analo g c o n dition in which functions f , f 1 , ..., f k ar e r eplac e d by total c omputable g , g 1 , ..., g k . This c an b e se en by defining g , g 1 , ..., g k fr om f , f 1 , ..., f k as fol low s g ( p, 0) = d , g ( p, t + 1) =      f ( p, t ) i f f ( p, t ) c onver ges in ≤ t steps g ( p, t ) otherwise and the same wi th g 1 , ..., g k fr om f 1 , ..., f k . 4.3 K D m ax , K D m in ar e not simultane ously much smal ler than K D Lemma 4.8 L et D = ( D , <, ρ ) b e a c om p utable or d er e d s e t. L et c : 2 ∗ × 2 ∗ → 2 ∗ b e a total c omputable inje ctive map and let J : N × N → N and M ∈ N b e such that | c ( p, q ) | ≤ J ( | p | , | q | ) + M fo r al l p, q ∈ 2 ∗ . T h en K D ≤ ct J ( K D m in , K D m ax ) In p articular (with the sp e cial c onvention log(0) = 0 ), K D ≤ ct ( K D m ax + log( K D m ax )) + ( K D m in + log( K D m in )) PR OOF. Let U, V : 2 ∗ → D b e optimal in M ax 2 ∗ →D P R and M in 2 ∗ →D P R , i.e. K D m ax = K U and K D m in = K V . Let f , g : 2 ∗ × N → D be partial computable, resp ectiv ely monot o ne increasing and decreasing with respect to their 2d ar- gumen t such tha t U = max D f and V = min D g . Define a partia l computable f unction ϕ : 2 ∗ → D as follo ws: • If r is not in r ang e ( c ) then ϕ ( r ) is undefined. Else, from input r , get p and q suc h that c ( p, q ) = r . • Do v etail computations of t he f ( p, t )’s and g ( q , t )’s for t = 0 , 1 , 2 , ... . • If and when there are t ′ , t ′′ suc h that f ( p, t ′ ) and g ( q , t ′′ ) are b oth defined and hav e the same v alue t hen output their common v alue and halt. By the inv ariance theorem, there is a constan t N suc h that K D ≤ K ϕ + N . Let d ∈ D and let p, q b e shortest programs suc h that U ( p ) = V ( q ) = d , i.e. K D m ax ( d ) = | p | and K D m in ( d ) = | q | . Observ e that, whenev er f ( p, t ′ ) and g ( q , t ′′ ) are both defined, w e hav e f ( p, t ′ ) ≤ 20 d ≤ g ( p, t ′′ ). Also, since U ( p ) = max D { f ( p, t ) : t ∈ N } and V ( q ) = min D { g ( q , t ) : t ∈ N } , there are t ′ , t ′′ suc h that f ( p , t ′ ) = d = g ( q , t ′′ ). Therefore, ϕ ( c ( p, q )) halts and outputs d . Therefore K D ( d ) ≤ K ϕ ( d ) + N ≤ | c ( p, q ) | + N ≤ J ( K D m in ( d ) , K D m ax ( d )) + N The last assertion o f the Lemma is obtained with the injectiv e map c ( p, q ) =      0 | B in ( | p | ) | 1 B in ( | p | ) pq if | p | ≤ | q | 1 | B in ( | q | ) | 0 B in ( | q | ) pq if | p | > | q | (where B in ( x ) denotes t he binary represen tatio n of x ) since | c ( p, q ) | = | p | + | q | + 2 ⌊ log(min( | p | , | q | ) ) ⌋ + 3 ≤ ( | p | + log( | p | )) + ( | q | + lo g( | q | )) + 3 ✷ 4.4 K D m ax , K D m in and the jump Prop osition 4.9 1. L et X b e a b asic sp ac e. A l l functions in M ax X →D P R and M in X →D P R ar e p artial c omputable in ∅ ′ . In p articular, K D is r e cur c omputablesi v e in ∅ ′ . 2. K D m in and K D m ax ar e c omputable in ∅ ′ . PR OOF. 1. Prop o sition 2.6 insures t hat an y F : X → D in M ax X →D P R or M in X →D P R has Σ 0 1 ∧ Π 0 1 graph. Therefore t wo calls to oracle ∅ ′ suffice to decide F ( x ) = d . 2. Let p 0 , p 1 , . . . b e a length incr e a s ing en umeration of 2 ∗ and let U : 2 ∗ → D b e optimal in M ax 2 ∗ →D P R , i.e. K U = K D m ax . One can compute K D m ax ( d ) with oracle ∅ ′ as follo ws: i. Using oracle ∅ ′ , test success iv e equalities U ( p ) = d (cf. P oint 1) for pro- grams p = p 0 , p 1 , . . . . ii. When suc h an equalit y holds (whic h necessarily do es happ en) then output | p | and halt. Idem with K D m in . ✷ 21 4.5 Pr o o f of T he or em 4.1 (1st hier ar chy the o r em) 1. L ar ge ine quality K ∅ ′ ,D ≤ ct inf ( K D m in , K D m ax ) . P oin t 1 of Prop osition 4.9 in- sures that M ax D P R and M in D P R are included in P R ∅ ′ . Therefore K ∅ ′ ,D ≤ ct K D m in and K ∅ ′ ,D ≤ ct K D m ax , i.e. K ∅ ′ ,D ≤ ct inf ( K D m in , K D m ax ). Strict ine quality K ∅ ′ ,D < ct inf ( K D m in , K D m ax ) . P oin t 2 of Prop osition 4.9 insures that inf ( K D m in , K D m ax ) is computable in ∅ ′ . No w, t he w ell-kno wn fact that if ψ = ct K D then ψ is no t computable relativizes : if ψ = ct K ∅ ′ ,D then ψ is not computable in ∅ ′ . In particular, inf ( K D m in , K D m ax ) 6 = ct K ∅ ′ ,D . 2. This is the con ten ts of L emma 4.8. ✷ 4.6 Ine qualities K D st m ax ≤ ct K D wk m ax and K D st m in ≤ ct K D wk m in The follo wing result is straightforw ard. Prop osition 4.10 Wit h the notations of The or em 4.3, K D st m in ≤ ct K D wk m in , K D st m ax ≤ ct K D wk m ax PR OOF. Sinc e < st extends < wk , ev ery par t ial computable function 2 ∗ → D whic h is mono t o ne increasing in its second argument relativ e to < wk is also monotone increasing relativ e t o < st . So that M ax D wk P R ⊆ M ax D st P R . Whic h yields K D st m ax ≤ ct K D wk m ax . ✷ 4.7 If ( ∗ ) holds: pr o of of Poi nt 1 of The or em 4.3 (3r d hier ar chy the or em) W e use the notations of Theorem 4.3. Lemma 4.11 L et α : N → N b e a total c omputable function. If c ondition ( ∗ ) holds then ther e exists total functions F , G : N → D r esp e c- tively in M ax N →D st R ec and M in N →D st R ec and a c onstant c such that, for al l i ∈ N , K D wk m ax ( F ( i )) ≥ α ( i ) , K D wk m in ( F ( i )) ≥ α ( i ) , K D st m ax ( F ( i )) ≤ log( i ) + c K D wk m ax ( G ( i )) ≥ α ( i ) , K D wk m in ( G ( i )) ≥ α ( i ) , K D st m in ( G ( i )) ≤ log( i ) + c PR OOF. 1. Since ( ∗ ) holds, for all i ∈ N , there exists a finite strong chain with 2 α ( i )+1 elemen ts whic h is a w eak antic hain. Dov etailing o v er subsets of D with 2 α ( i )+1 elemen ts, one can effectiv ely find suc h a strong c hain Z i . Th us, 22 there exists a total computable function σ : N × N → D suc h that, for all i ∈ N , • σ ( i, 0) < st σ ( i, 1) < st ... < st σ ( i, 2 α ( i )+1 − 1) • Z i = { σ ( i, j ) : j = 0 , ..., 2 α ( i )+1 − 1 } is a w eak an tic hain. 2. L et f and g are partial computable functions 2 ∗ × N → D suc h that U = max D wk f and V = min D wk g are optimal in M ax 2 ∗ →D wk P R and M in 2 ∗ →D wk P R , i.e. K U = K D wk m ax and K V = K D wk m in . W e observ e that inequalities K D wk m ax ( F ( i )) ≥ α ( i ) and K D wk m in ( F ( i )) ≥ α ( i ) are equiv alen t to dis equalities U ( p ) 6 = F ( i ) and V ( p ) 6 = F ( i ) for ev ery p suc h that | p | < α ( i ). W e define F , G : N → D as F = max D st ℓ and F = min D st ℓ for some to t a l computable ℓ : N × N → D . Let X p = { f ( p, t ) : t s.t. f ( p, t ) con v erges } Y p = { g ( p, t ) : t s.t. g ( p, t ) con ve rges } X t p = { f ( p, t ′ ) ∈ Z i : t ′ ≤ t and f ( p, t ′ ) con ve rges in ≤ t steps } Y t p = { g ( p, t ′ ) ∈ Z i : t ′ ≤ t and g ( p , t ′ ) con ve rges in ≤ t steps } Since Z i is a w eak a ntic hain and X p , Y p are w eak c hains, each one of the sets Z i ∩ X p and Z i ∩ Y p has at most one elemen t. Th us, S | p | <α ( i ) ( X p ∪ Y p ) has at most 2(2 α ( i ) − 1) = 2 α ( i )+1 − 2 elemen ts in Z i . Since Z i has 2 α ( i )+1 elemen ts and the σ ( i, j )’s are in Z i , the follo wing definition mak es sense: ℓ ( i, t ) = σ ( i, j ) where j is least suc h that σ ( i, j ) / ∈ [ | p | <α ( i ) ( X t p ∪ Y t p ) No w, F ( i ) = (max D st ℓ )( i ) and G ( i ) = (min D st ℓ )( i ) are o f the form ℓ ( i, t ′ i ) a nd ℓ ( i, t ′′ i ) for some t ′ i , t ′′ i , hence they are not in S | p | <α ( i ) ( X p ∪ Y p ). In particular, since U ( p ) = max D wk X p is in X p and V ( p ) = min D wk Y p is in Y p , w e see that F ( i ) and G ( i ) are not in { U ( p ) , V ( p ) } for any | p | < α ( i ). Which prov es that K D wk m ax ( F ( i )), K D wk m in ( F ( i )), K D wk m ax ( G ( i )) and K D wk m in ( G ( i )) are a ll ≥ α ( i ). 3. Since F ∈ M ax N →D st R e c , the in v ariance theorem insures that K D st m ax ≤ ct K F . No w, K F ( F ( i )) ≤ ct log( i ), hence the inequalit y K D st m ax ( F ( i )) ≤ log ( i ) + c for some constant c . Idem with K D st m in ( G ( i )). ✷ Pro of of Poin t 1 of Theorem 4.3 . Apply Lemma 4.11 with α ′ suc h that α ′ is monotone increasing and α ′ ( i ) ≥ max( α ( i ) , i ) f or all i . Since α ′ ( i ) tends to + ∞ with i , so do es F ( i ). Let i 0 b e such tha t lo g( i ) + c ≤ i for all i ≥ i 0 . Since α ′ is increasing and α ′ ≥ α , for all i ≥ i 0 w e ha v e K D wk m ax ( F ( i )) ≥ α ′ ( i ) ≥ α ′ ( ⌊ log( i ) + c ⌋ ) ≥ α ′ ( K D st m ax ( F ( i ))) ≥ α ( K D st m ax ( F ( i ))) 23 Similarly , w e hav e K D wk m in ( F ( i )) ≥ α ( K D st m ax ( F ( i ))) and K D wk m ax ( G ( i )) ≥ α ( K D st m in ( G ( i ))) and K D wk m in ( G ( i )) ≥ α ( K D st m in ( G ( i ))). Finally , observ e that { F ( i ) : i ≥ i 0 } and { G ( i ) : i ≥ i 0 } a re infinite. Whic h concludes the pro of o f Poin t 1 of Theorem 4.3. ✷ 4.8 Pr o o f of Po int 1 of Th e or e m 4.2 (2d hie r ar chy the or em) Comp aring K D to K D m ax and K D m in . Let < st b e < a nd < wk b e the empt y order. Then K D st m ax = K D m ax , K D st m in = K D m in , K D wk m ax = K D wk m in = K D The conditio n (in P oint 1 of Theorem 4.2) that D con tains arbitrarily large c hains insures condition ( ∗ ) ab out < st and < wk . Thus , w e can apply (the just pro ved) Poin t 1 of Theorem 4.3. This giv es prop erties i and ii o f P oint 1 of Theorem 4.2. Comp aring K D m ax and K D m in . W e shall pro ve prop erties iii and iv of P oin t 1 of Theorem 4.2 us ing prop erties i and ii and also Lemma 4.8. Applying Lemma 4.8, let c b e suc h tha t, ( † ) K D ≤ 2 ( K D m ax + K D m in ) + c Prop ert y iii applied to α ′ ( i ) = 2 ( α ( i ) + i ) + c insures that the set X = { d : K D ( d ) ≥ 2 ( α ( K D m in ( d )) + K D m in ( d )) + c } is infinite. No w, using ( † ), w e see that, for d ∈ X , 2 ( α ( K D m in ( d )) + K D m in ( d )) + c ≤ K D ( d ) ≤ 2 ( K D m ax ( d ) + K D m in ( d )) + c hence K D m ax ( d ) ≥ α ( K D m in ( d )). Whic h prov es iii. The pro of of iv is similar. ✷ 4.9 If ( ∗∗ ) holds: pr o of of Point 2 o f Th e or em 4.3 (3 d hier ar chy the or em) Lemma 4.12 With the notations of The or em 4.3, if c ondition ( ∗∗ ) holds then K D st m in ≥ ct K D wk m in , K D st m ax ≥ ct K D wk m ax PR OOF. 1. Let k b e as in ( ∗∗ ). Let U st b e optimal in M ax D st P R and f : 2 ∗ × N → D b e partial computable suc h that max D st f = U st . Due t o Pro p osition 2.14 , w e can supp o se that f has domain of the form Z × N 24 and is monotone increasing in its se cond argumen t, with respect t o the strong order. Applying ( ∗∗ ) to f , w e get k partial computable functions f 1 , ..., f k , monotone increasing in their second argumen t, with resp ect to the w eak order, such that ( ♯ ) { f ( p, t ) : t ∈ N } = [ i =1 ,...,k { f i ( p, t ) : t ∈ N } Define g : 2 ∗ × N → D suc h that g ( q , t ) =      f i ( p, t ) if q = 0 i 1 k − i p for some p and 1 ≤ i ≤ k undefined otherwise Clearly , g is partial computable and monotone increasing in its second argu- men t relat ive to the w eak order < wk . If p ∈ dom ( U st ), then { f ( p, t ) : t ∈ N } is finite and non empt y . Let f ( p, t p ) b e its < st greatest elemen t. Condition ( ♯ ) insures that there exists i suc h tha t { g (0 i 1 k − i p, t ) : t ∈ N } is finite a nd con tains f ( p, t p ). Since g is < wk increasing in t , the set { g (0 i 1 k − i p, t ) : t ∈ N } is a w eak c hain. Since < st extends < wk , f ( p, t p ) is neces sarily its < wk greatest elemen t. Th us, U st ( p ) = f ( p, t p ) = (max wk g ) (0 i 1 k − i p ) This pro ves that, for all d ∈ D , K D st m ax ( d ) = least | p | suc h that U st ( p ) = d = least | p | suc h that (max wk g ) (0 i 1 k − i p ) = d for some i ≥ least | q | − k suc h t ha t (max wk g ) ( q ) = d = K max D wk g ( d ) − k Since, b y the in v a riance theorem , K max D wk g ≥ ct K D wk m ax , we get the desired inequalit y K D st m ax ≥ ct K D wk m ax . 2. Considering t he rev erse orders, w e get the inequalit y K D st m in ≥ ct K D wk m in . ✷ Pro of of Poin t 2 of Theorem 4.3 . Straigh tforw ard from the a b o ve L emma 4.12 and Prop osition 4.10. ✷ 4.10 Pr o of of Point 2 of Th e or em 4.2 (2d hier ar chy the or em) As in § 4.8 , let < st b e < a nd < wk b e ∅ , so that K D st m ax = K D m ax , K D st m in = K D m in , K D wk m ax = K D wk m in = K D 25 Supp ose all c ha ins in ( D , < ) ha v e length ≤ k . W e shall prov e condition ( ∗∗ ) for the a b o ve orders < st and < wk . Let f : 2 ∗ × N → D b e partial computable, monotone increasing in its 2 d argumen t for the strong or der, i.e. for the < order. Compute f ( p, t ) for t = 0 , 1 , ... to get the ≤ k distinct elemen ts of t he c hain { f ( p, t ) : t ∈ N } (not necessarily in increasing order) a nd let f i ( p ) b e the i -th elemen t so obtained (if there is some). Then f 0 , ..., f k : 2 ∗ → D are partial computable and { f ( p, t ) : t ∈ N } = { f i ( p ) : i s.t. f i ( p ) is defined } whic h insures condition ( ∗ ∗ ) . Applying P oin t 2 of Theorem 4.3 (prov ed ab ov e), we get = ct equalities whic h are exactly those o f P oin t 2 o f Theorem 4.2. ✷ 5 Complemen tary results ab out the M ax and M in classes In this section w e further in ves tigate the differen t M ax and M in classes. The results do not inv olv e as man y tec hnicalities as t ho se o f § 4. 5.1 T otal functions in M ax X →D R e c and M ax X →D P R As a straightforw ard corollary of P oint 2 of Prop osition 2.14, w e get the fol- lo wing result. Theorem 5.1 The classes M ax X →D R ec and M ax X →D P R c ontain the sam e total functions: M ax X →D P R ∩ D X = M ax X →D R e c ∩ D X 5.2 Comp aring M ax X →D P R , M ax X →D R ec and P R X →D , Rec X →D Prop osition 5.2 L et X b e a b asic set and D = ( D, <, ρ ) b e a c omputable or der e d set. 1. If < is empty then P R X →D = M ax X →D P R and Rec X →D = M ax X →D R e c . 2. If < is not empty then M ax X →D R e c c ontains non c omp utable total functions. In p a rticular, P R X →D ⊂ M ax X →D P R and Rec X →D ⊂ M ax X →D R e c (wher e ⊂ den otes strict inclusio n ). 3. Whatever b e < , P R X →D is n ot incl ude d in M in X →D R e c ∪ M ax X →D R e c . 26 PR OOF. 1. Straightforw ard. 2. Inclusions P R X →D ⊆ M ax X →D P R and R ec X →D ⊆ M ax X →D R ec are o bvious. Supp ose there exists compar a ble distinct elemen ts a < b in D . Let Z b e some computably en umerable non computable subset of X and let θ : N → X b e a total computable map with range Z . Define f : X × N → D total computable, monotone increasing in t , suc h that f ( x , t ) =      a if x / ∈ { θ ( n ) : n ≤ t } b otherwise Then max f is total and (max f ) − 1 ( b ) = Z and (max f ) − 1 ( a ) = X \ Z . Since Z is not computable, max f is not computable. Whic h pro v es Rec X →D ⊂ M ax X →D R ec 3. First, w e consider the case where ( D , < ) has a minimal elemen t d . Let π Z d : X → D b e the p artial computable function w ith do main Z (as in P oin t 2 of this pro of ) whic h is constan t on Z with v alue d . W e show that π Z d is not in M ax D R ec . Supp ose f : X × N → D is total computable, monotone in its sec ond argumen t, suc h that max D f = π Z d . Since d is minimal in D , (ma x D f )( x ) = d if and o nly if ∀ t f ( x , t ) = d . Th us, the computably enumerable set Z w ould b e Π 0 1 , hence computable, con tradiction. W e no w consider the case where ( D , < ) has no minimal elemen t. Let γ : D → D be t he t otal computable function whic h associates to each d ∈ D the elemen t ρ ( k d ) where n d is the least k suc h t ha t ρ ( k ) < d . L et ( φ ) e ∈ X b e an en umeration of P R X × N → D whic h is partial computable as a function Φ : X × X × N → D . W e consider an en umeration ( e n , x n , t n , d n ) n ∈ N of the gr a ph of Φ and define a partial computable function ϕ : X → D as follow s: ϕ ( x ) =      γ ( d n ) if n is least suc h that e n = x n = x undefined if there is no suc h n It is clear that, for ev ery e , if φ e ( e , t ) is defined for some t then ϕ ( e ) is defined and ϕ ( e ) < φ e ( e , t ). In particular, if φ e is t otal then ϕ ( e ) < (max D φ e )( e ), hence ϕ 6 = max D φ e . Whic h prov es that ϕ is not in M ax X →D R ec . Arguing with D r ev w e get some function in P R X →D whic h is not in M in X →D R ec . Considering ϕ 0 , ϕ 1 ∈ P R X →D suc h that ϕ 0 / ∈ M ax X →D R ec and ϕ 1 / ∈ M in X →D R ec and a computable bijection σ : X × { 0 , 1 } → X w e g et a partia l computable function ϕ : X → D whic h is not in M ax X →D R ec ∪ M in X →D R ec b y setting ϕ ( σ ( x , 0)) = ϕ 0 ( x ) and ϕ ( σ ( x , 1 )) = ϕ 1 ( x ). ✷ 5.3 Post hie r ar chy and the M ax/ M in classes W e ke ep notations of § 2.3. 27 Theorem 5.3 L et X b e a b asic set and D b e a c o m putable o r der e d set. 1. L et D ′ b e an initial se gment of D (i.e. d ′ ∈ D ′ ∧ e < d ′ ⇒ e ∈ D ′ ). Supp ose D ′ is Π 0 1 and do es not c on tain any strictly i n cr e asing infinite se quenc e d ′ 0 < d ′ 1 < ... . Then i. Ev e ry D ′ -value d function in M ax X →D P R has Σ 0 1 ∧ Π 0 1 domain. ii. Every D ′ -value d function in M ax X →D R ec has Π 0 1 domain. 2. L et D ′ b e a final s e gm ent of D (i.e. d ′ ∈ D ′ ∧ e > d ′ ⇒ e ∈ D ′ ). Supp ose D ′ is Σ 0 1 and do es not c ontain any strictly inc r e asing infinite se quenc e. The n i. Ev e ry D ′ -value d function in M ax X →D P R has Σ 0 1 domain. ii. Every D ′ -value d function in M ax X →D R ec is total. PR OOF. 1. Supp ose that max D f is D ′ -v alued. Since D ′ is an initial seg- men t a nd f c an b e supp osed monotone increasing in its second a r g umen t, if (max D f )( x ) is defined then, for all t , f ( x , t ) is e ither undefined or in D ′ . No w, since D ′ has no infinite increasing sequence , the set { f ( x , t ) : t ∈ N s.t. f ( x , t ) ∈ D ′ } cannot be infinite. Th us, x ∈ dom (max D f ) if and o nly if ∃ t f ( x , t ) is defined ∧ ∀ t ( f ( x , t ) is defined ⇒ f ( x , t ) ∈ D ′ ) In case f is tota l computable, t hen the ab ov e equiv alence is simply x ∈ dom (max D f ) ⇔ ∀ t f ( x , t ) ∈ D ′ 2. Since D ′ is a final segmen t and f can b e supposed monoto ne incre asing in its second a rgumen t, if (max D f )( x ) is defined then, for all t large enough, f ( x , t ) is either undefined or in D ′ . Now, since D ′ has no infinite increasing sequence , the set { f ( x , t ) : t ∈ N s.t. f ( x , t ) ∈ D ′ } cannot b e infinite. Thus , x ∈ dom (max D f ) ⇔ ∃ t ( f ( x , t ) is defined ∧ f ( x , t ) ∈ D ′ ) ✷ The next coro llary is an application of the a b o v e theorem with the rev erse of the follo wing D ’s: • D is the natural o rder on Z and D ′ = N , • D is the natural o rder on N or of the prefix order on Σ ∗ and D ′ = D , Corollary 5.4 1. Every N -value d function in M in X → Z P R (r esp. M in X → Z R e c ) has Σ 0 1 ∧ Π 0 1 (r esp. Π 0 1 ) d o main. 2. L et D b e N with the natur al or der or Σ ∗ with the pr efix p artial or der. Then every func tion in M in X →D P R (r esp. M in X →D R ec ) h a s Σ 0 1 domain ( r esp. is total). 28 5.4 M ax ∩ M in classes Theorem 5.5 L et X b e a b asic s et and D = ( D , <, ρ ) b e a c omputable or der e d set. 1. Every function F : X → D in M ax X →D P R ∩ M in X →D P R is the r estriction of a p artial c omp utable function X → D to some Σ 0 1 ∧ Π 0 1 subset of X . In p articular, every total function in M ax X →D P R ∩ M in X →D P R is c omputable. 2. Supp ose D has no maximal (r esp . minimal) e l e m ent. T hen the r estriction of any p artial c om p utable function X → D to any Σ 0 1 ∧ Π 0 1 subset of X is in M ax X →D P R (r esp. M in X →D P R ). 3. Supp ose D has no ma x i m al or m inimal element. Then M ax X →D P R ∩ M in X →D P R c oincides with the fa mily of r estrictions of p artial c omputable functions X → D to Σ 0 1 ∧ Π 0 1 subsets of X . PR OOF. 1. Let F = max D f = min D g where f , g : X × N → D are partial computable a nd f (resp. g ) is mo no tone increasing (resp. decreasing) in its second argumen t. Let’s che c k tha t F ( x ) is defined if and only if ( ∗ ) ( ∃ t ′ , t ′′ f ( x , t ′ ) = g ( x , t ′′ )) ∧ ( ∀ u , v f ( x , u ) ≤ g ( x , v )) In fact, if F ( x ) is defined then F ( x ) = f ( x , t ′ ) = g ( x , t ′′ ) for some t ′ , t ′′ , g ( x , u ) ≤ F ( x ) ≤ f ( x , v ) for all u, v suc h that g ( x , u ) , f ( x , v ) a r e defined. Con ve rsely , fr om ( ∗ ) w e see that, for u ≥ t ′ and v ≥ t ′′ , f ( x , u ) = f ( x , t ′ ) = g ( x , t ′′ ) = g ( x , v ). Hence the finiteness of { f ( x , u ) : u } and { g ( x , v ) : v } . This pro ves that the domain of F is Σ 0 1 ∧ Π 0 1 . Let G : X → D b e the partial computable function defined as follo ws: Do v etail computations of f ( x , 0) , f ( x , 1) , . . . , g ( x , 0) , g ( x , 1 ) , . . . un til w e get t ′ , t ′′ suc h that f ( x , t ′ ) , g ( x , t ′′ ) are b oth define d a nd equal. O utput this c om- mon v alue. Applying ( ∗ ), if F ( x ) is defined, then so is G ( x ) and F ( x ) = G ( x ). Th us, F is the restriction of a partial computable function to some Σ 0 1 ∧ Π 0 1 set. 2. Supp ose there is no maximal elemen t. Since the order < is computable, by do vetailing, one can define a total computable function γ : D → D suc h that γ ( d ) > d for all d ∈ D . Let F : X → D b e partial computable and let Z ⊆ X b e Σ 0 1 ∧ Π 0 1 definable: x ∈ Z ⇔ ( ∃ t R ( x , t )) ∧ ( ∀ t S ( x , t )) 29 where R , S ⊆ X × N are computable. Letting γ ( t ) denote the t -th iterate of γ , w e define f : X × N → D as follows : f ( x , t ) =                            F ( x ) if F ( x ) conv erges in ≤ t steps and ( ∃ t ′ ≤ t R ( x , t ′ )) ∧ ( ∀ t ′ ≤ t S ( x , t ′ )) γ ( t ) ( F ( x )) if F ( x ) conv erges in ≤ t steps and ∃ t ′ ≤ t ¬ S ( x , t ′ ) undefined otherwise It is easy to c hec k that max D f is the restriction o f F to Z . The assertion with M in X →D P R is obtained with the o rder reve rse to D . 3. Str aigh t f orw ard fro m P oints 1 and 2. ✷ Remark 5.6 The or em 5.3 shows that Points 2, 3 of the ab ove the or em do not hold for gener al or der e d sets D . 6 M ax 2 ∗ →D R e c and M in 2 ∗ →D R e c and Kolmogorov complexit y Since there is no computable en umeration of tota l computable functions, it seems a priori desp erate to get an in v a riance theorem for the class M ax X →D R ec . Nev ertheless, there are imp orta nt cases where suc h a result do es hold. F or instance, when D is N with its usual ordering. The purp ose of this section is to characterize the orders D such tha t an in- v a r iance theorem holds f o r the class M ax 2 ∗ →D R e c (resp. M in 2 ∗ →D R ec ). First, w e deal with the en umeration theorem. 6.1 M ax X →D R ec and the enumer ation the or em Theorem 6.1 (En umeration theorem for M ax X →D R ec ) L et X b e a b asic set and D = ( D , <, ρ ) b e a c omputable or der e d set. The fol l o wing c onditions a r e e quivalent: i. T her e exists a smal le s t element in D . ii. Ther e exists a function e E : N × X → D in M ax N × X → D R ec such that { e E n : n ∈ N } = M ax X →D R ec wher e e E n : X → D de notes the function x 7→ e E ( n, x ) . 30 PR OOF. i ⇒ ii . Let α ∈ D b e the smallest eleme n t of D . As in § 3.2, let ψ : N × X × N → D b e pa r t ia l computable monoto ne increasing in its last argumen t suc h that E = max D ψ is an en umeration of M ax X →D P R . Consider an injectiv e computable en umeration ( n i , x i , t i , d i ) i ∈ N of the graph of ψ . Since α is the smallest elemen t, we can define a total computable function e ψ : N × X × N → D as follows: X ( n, x , t ) = { d i : i ≤ t ∧ n i = n ∧ x i = x ∧ t i ≤ t } e ψ ( n, x , t ) = g reatest elemen t of { α } ∪ X ( n, x , t ) Supp ose ψ n is total, w e show that max D e ψ n = m ax D ψ n . Fix some x . Observ e that { e ψ n ( x , t ) : t ∈ N } is { ψ n ( x , t ) : t ∈ N } or { α } ∪ { ψ n ( x , t ) : t ∈ N } . Th us, { e ψ n ( x , t ) : t ∈ N } and { ψ n ( x , t ) : t ∈ N } are sim ultaneously finite or infinite, and when finite t hey ha v e the same greatest elemen t. Since ψ n is total, this pro ves that (max D e ψ n )( x ) = (ma x D ψ n )( x ). Th us, ev ery function in M ax X → D R ec is of the form max D e ψ n for some n . Set e E = max D e ψ . Then e E is in M ax N × X → D R ec and t he e E n ’s enum erate M ax X → D R ec . ii ⇒ i . W e pro v e ¬ i ⇒ ¬ ii . Supp ose D has no minim um elemen t. By do v e- tailing one can define a total computable map γ : D → D suc h that d 6≤ γ ( d ) for all d . Let E = max D g : N × X → D where g : N × X × N → D is to tal computable monotone increasing in its last argumen t. W e define a total computable map f : X → D such that f 6 = E n for all n . Let θ : N → X b e some computable bijection. Set f ( θ ( n )) = γ ( g ( n, θ ( n ) , 0)). Then g ( n, θ ( n ) , 0) 6≤ f ( θ ( n )) and g ( n, θ ( n ) , 0 ) ≤ (max D g ) ( θ ( n )) = E n ( θ ( n )) Th us, f ( θ ( n )) 6 = E n ( θ ( n )). Hence f 6 = E n for all n . ✷ 6.2 M ax 2 ∗ →D R e c and the invaria n c e the or em If D con tains a smallest elemen t then the en umeration theorem o f § 6.1 allows to get a n inv ariance result for the class M ax 2 ∗ →D R e c . Surprisingly , it turns out that an in v ariance result can b e pro v ed for pa r t ia lly ordered sets with no smallest elemen t, hence whic h fail t he en umeration the- orem. Also, in case the class M ax 2 ∗ →D R e c has optimal functions then they pro ve to b e also optimal f or the bigg er class M ax 2 ∗ →D P R . Theorem 6.2 L et X b e a b asic sp ac e and D = ( D , <, ρ : N → D ) b e a c omputable p artial ly or der e d set. L et ( ∗ ) b e the fol lo wing c ondition on D : 31 ( ∗ ) The set of minimal elements of D is fin ite and every el e ment of D do m- inates a minimal element 1. If D satisfie s ( ∗ ) then i. Every function in M ax 2 ∗ →D P R has an extension (not ne c essarily total) in M ax 2 ∗ →D R ec . ii. The invarianc e the or em holds for M ax 2 ∗ →D R ec . iii. Every U in M ax 2 ∗ →D R ec which is optimal for M ax 2 ∗ →D R ec is a lso optimal for the cla ss M ax 2 ∗ →D P R . In p articular, the Kolmo gor ov c omplexity asso ciate d to M ax 2 ∗ →D R ec c oin- cides (up to a c onstant) w ith that asso ciate d to M ax 2 ∗ →D P R . 2. If D d o es not satisfy ( ∗ ) then the invarian c e the or em fails for M ax 2 ∗ →D R e c . Mor e o v e r, c ounter examples c an b e taken in the class Rec 2 ∗ →D of total c om- putable func tion s 2 ∗ → D : ∀ G ∈ M ax 2 ∗ →D R e c ∃ F ∈ Rec 2 ∗ →D K G 6≤ ct K F PR OOF. 1. Suppo se ( ∗ ) holds and let M = { m 0 , ..., m k } b e the set of min- imal elemen ts. F or i ≤ k , let D i = { d ∈ D : d ≥ m i } . Some of the D i ’s ma y be finite, thoug h not all of them (else D w ould be finite). Let ℓ ≤ k b e suc h that D i is infinite for i ≤ ℓ and finite for ℓ < i ≤ k . Since the D i are computable, for i ≤ ℓ , there exists a computable map ρ i : N → D i suc h that D i = ( D i , < ∩ ( D i × D i ) , ρ i ) is a computable partially ordered set. A. Since D i has a smallest elemen t, namely m i , M ax D i R e c satisfies the enume r- ation theorem (cf. Theorem 6.1) . The pro of o f Theorem 3.8 applies, insuring that M ax D i R e c satisfies the inv ariance theorem. Let g i : 2 ∗ × N → D i b e total computable such that max D i g i = U i : 2 ∗ → D i is optimal in M ax D i R ec . Let’s c hec k that U i is also optimal in M ax D i P R . Let F i ∈ M ax D i P R and F i = max D i f i where f i : 2 ∗ × N → D i is partia l computable monotone inc reasing in its second argumen t a nd has domain Z i × N w here Z i ⊆ 2 ∗ is computably en u- merable ( cf. Proposition 2.14). Define a total computable map e f i : 2 ∗ × N → D i suc h that e f i ( p, t ) =      f i ( p, t ) if p is seen to b e in Z i in ≤ t steps m i otherwise Set f F i = max D i e f i . If p ∈ Z then e f i ( p, t ) = f i ( p, t ) fo r t larg e enough, so that F i ( p ) = (max D i f i )( p ) = (max D i e f i )( p ) = f F i ( p ). Th us, f F i extends F i . Whic h trivially yields K e F i ≤ K F i . Since f F i ∈ M ax D i R e c , w e ha ve K U i ≤ ct K e F i . Hence K U i ≤ ct K F i . B. W e group the functions g i and U i of Poin t A to get a to tal computable 32 g : 2 ∗ × N → D and the asso ciated U = max D g in M ax D R e c . D efine g a s follo ws: g ( q , t ) =      g i ( p, t ) if q is of the form 0 i 1 p with i ≤ ℓ , p ∈ 2 ∗ m 0 otherwise F or i ≤ ℓ and d ∈ D i , we hav e K U ( d ) ≤ K U i ( d ) + i + 1 for all i ≤ ℓ and d ∈ D i Supp ose F is in M ax D P R is of the form F = max D f where f : 2 ∗ × N → D is partial computable. F or i ≤ ℓ , let F i = max D i f i where f i : 2 ∗ → D i is suc h that f i ( p, t ) =      f ( p, t ) if f ( p, t ) is defined a nd is in D i undefined otherwise Clearly , F i is the restriction of F to F − 1 ( D i ). Thus , K F ( d ) = K F i ( d ) for all d ∈ D i . Since F i ∈ M ax D i P R and U i is optimal in M ax D i P R , there exists c i suc h that K U i ≤ K F i + c i . Thus , for d ∈ D i , w e ha v e K U ( d ) ≤ K U i ( d ) + i + 1 ≤ K F i ( d ) + c i + i + 1 ≤ K F ( d ) + c i + i + 1 Let a be the maximum v alue o f K F on the finite set S ℓ

Original Paper

Loading high-quality paper...

Comments & Academic Discussion

Loading comments...

Leave a Comment