Church, Cardinal and Ordinal Representations of Integers and Kolmogorov complexity

Chur ch, Cardinal and Ordinal Repres ent a tions of Inte gers and K olmo goro v comple xity Denis Richard’s 60th Bi ir thda y Conference Marie Ferbus-Zanda Universit ´ e P aris 7 2, pl. Jussieu 75251 Paris Cedex 05 F rance ferbus@log ique.jussi eu.fr Serg e Grigorieff LIAF A, Universi t´ e P aris 7 2, pl. Jussieu 75251 Paris Cedex 05 F rance seg@liafa. jussieu.fr Mai 2002 Con te n ts 1 Kolmogoro v complexities 2 1.1 Kolmogoro v complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Infinite computations and oracles . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Prefix Kolmogoro v comp lex ities . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Representations of in tegers 4 2.1 Abstract representations of integers . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 F ormal representations of integers . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Effectivization of sets of typ e ≤ 2 ob jects . . . . . . . . . . . . . . . . . . . 7 2.4 Recursion-theoretic representations of integers . . . . . . . . . . . . . . . . . 8 2.5 Kolmogoro v complexity and representations of integers . . . . . . . . . . . . 10 3 Main Theorem 10 3.1 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Representations ρ such that K ρ = K . . . . . . . . . . . . . . . . . . . . . . 11 3.3 K N car d and K ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 K Z car d ↾ N and K ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5 K or d and K ′ ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Abstract W e co nsider classica l repres e n tations of in tegers: Chu rch’s func- tion iterator s, cardina l equiv alence classe s of sets, ordina l equiv alence classes of totally order ed sets. Since pr ograms do not work on ab- stract entities and r e quire formal represe ntations of o b jects, we effec- tivize these abs tract notio ns in order to a llow them to be c o mputed by progra ms. T o any such effectivized representation is then as so ciated 1 a notion of Kolmo g orov complexity . W e prove that these Kolmo gorov complexities fo r m a strict hierar ch y which coincides with that obtained by r elativization to jump o racles a nd/or allow ance of infinite co mpu- tations. 1 Kolmogoro v complexiti es W e shall use the follo wing notations. Notation 1.1 . 1) In equalit y , s trict inequ alit y and equalit y up to a constant b et w een fu n c- tions N → N are denoted as follo ws: f ≤ ct g ⇔ ∃ c ∀ n ( f ( n ) ≤ g ( n ) + c ) f < ct g ⇔ f ≤ ct g ∧ ∀ c ∃ n ( f ( n ) < g ( n ) − c ) f = ct g ⇔ ∃ c ∀ n ( | f ( n ) − g ( n ) | ≤ c ) ⇔ f ≤ ct g ∧ g ≤ ct f 2) Y X (resp. X → Y ) denotes th e set of total (resp. partial) functions from X in to Y . 3) W e den ote ϕ e the partial recursive f unction N → N with co de e . 4) W e denote car d ( X ) the n um b er of elemen ts of X in case X is a finite s et. 1.1 Kolmogoro v c omplexit y Definition 1.1 (Kolmogo ro v, 1965 [5]) . Kolmogoro v co mplexit y K : N → N is defined as follo ws: K ( n ) is the shortest length of a pr o gr am which halts an d outputs n T o mak e Def. 1.1 m eanin gfu l, some p oin ts h av e to b e p r ecised: (Q1) Wher e ar e pr o gr ams taken fr om? In which alp hab et? Bigger the alphab et, shorter the programs. W e shall therefore fix the alphab et of all programming languages to b e b in ary . No w, Kol- mogoro v’s in v ariance theorem insures that there exist optimal uni- v ersal programming languages U suc h that, for an y programming lan- guage V , the asso ciated complexit y functions K U , K V satisfy K U ≤ ct K V (cf. No tations 1.1). In particular, if U 1 , U 2 are t w o optimal u niv ersal programming languages then K U 1 = ct K U 2 . (Q2) What do e s it me an that a pr o gr am outputs an inte ger n ? A pr ogram can only outpu t a formal ob j ect suc h as a word in some alphab et wh ic h r epresen ts n . Ho w ev er, th er e is again an in v ariance prop erty relativ e to th e u sual rep r esen tations of the outpu t in teger n : up to a constant, the same complexit y functions are obtained when considering unary represent ation or base k representat ion for k ≥ 2 (cf. [7] or [4 ] or [10]). 2 The question aroused b y ( Q 2) is th e core of th is pap er. W e sh all reconsider it in § 2 an d § 3. 1.2 Infinite computations and oracles Chaitin, 1976 [3], and S olov a y , 1977 [11], considered Kolmogoro v complexit y of in finite ob jects (namely recursiv ely en umerable sets) pr o duced b y infinite computations. Allo wing programs leading to p ossibly infin ite compu tations bu t finite out- put (i.e. remo v e the sole halting condition) , we get a v arian t of K olmogoro v complexit y for wh ic h th e results m entionned in ( Q 1) ab o v e also app ly . Definition 1.2. Allo wing programs with p ossib ly infin ite computations, Kolmogoro v com- plexit y K ∞ : N → N is defined as f ollo ws: K ∞ ( n ) is the shortest length of a (p ossibly non halting) pr o gr am which outputs n i n unary r epr esentation R emark 1.1 . The definition of K ∞ ( n ) is d ep endent on the u nary represen tation of out- puts: ( Q 2) do es n ot apply . Kolmogoro v complexit y can also b e considered for computabilit y relativ e to some oracle. Definition 1.3. Considering p artial recursiveness r elativ e to some oracle A , Def. 1.1, 1.2 lead to relativ e Kolmogoro v complexities K A : N → N and K A, ∞ : N → N . In case A = ∅ i is the i -th ju mp (i.e a Σ 0 i -complete set of intege rs), we simply write K i , K i, ∞ . F or explicit v alues i = 1 , 2 , . . . w e also wr ite K ′ , K ′′ . . . , K ′ ∞ , K ′′ ∞ . . . . Jump oracle ∅ ′ (resp. ∅ ′′ ,. . . ) a llo ws the computatio n to decide for free (in a single compu tation step) an y Σ 0 1 or Π 0 1 (resp. Σ 0 2 or Π 0 2 ,. . . ) statemen t ab out intege rs. As exp ected and is well kno wn, suc h an oracle leads to an extended notion of programs. Whic h allo ws to • compute non recursive sets and functions, • get m uc h shorter pr ograms to compute finite ob jects or recursive sets and fun ctions, n amely K > ct K ′ > ct K ′′ > ct . . . . Infinite computations also lead to shorter programs but, as pro v ed by Bec her, 2001 [1], th ey d o not help as muc h as the j ump oracle. Prop osition 1.1 ([1]) . K > ct K ∞ > ct K ′ > ct K ′ ∞ > ct K ′′ > ct . . . 3 1.3 Prefix K olmogorov complexities In tro duced by Levin [6] and Ch aitin [2] (cf. [7]), prefi x complexit y H : N → N is the analog of Kolmogoro v complexit y obtained by restricting program- ming languages to b e p refix-free: tw o d istin ct pr ograms h av e to b e incompa- rable with resp ect to the prefix ordering. V ariants H ′ , H ′′ . . . , H ∞ , H ′ ∞ , . . . in v olving in finite computations and/or r elativiza tion are d efined in th e ob- vious w a y . As concerns all questions considered in this pap er, eve rything go es through with straigh tforw ard c hanges. S o that we shall deal exclusive ly with th e K complexit y and its v a rian ts K ∞ , K ′ , K ′ ∞ , K ′′ , . . . . 2 Represen tations of in tegers The pu rp ose of the p ap er is to consider particular rep resen tations of in tegers and to stud y their in fluence on the defi n ition of Kolmogoro v complexit y as p oint ed in question ( Q 2) r elativ e to Def. 1.1 ab o v e. This will lead to a strict hierarc h y of K olmogoro v complexitie s wh ic h happ ens to coincide with that obtained in P r op. 1.1. 2.1 Abstract represen tations of in tegers A rep resen tation of intege rs in v olv e s some abs tract ob ject C (in practice m uc h more complex than N itself ) suc h that s ome of its elemen ts c har- acterize th e diverse integ ers through some prop ert y . Eac h rep r esen tation illuminates some r ole and/or prop erties of in tegers. Definition 2.1. A repr esen tatio n of in tegers is a pair ( C , R ) where C is some (necessarily infinite) set and R is a surje ctive p artial fun ction R : C → N . R emark 2.1 . In practice, D omain ( R ) will b e a strict su bset of C , in fact a very sm all part of C . Example 2.1. 1) The unary represen tation of in tegers corresp ond s to the f ree algebra built up from one generator and one unary f unction, namely 0 and the su ccessor function x 7→ x + 1. 2) Th e v ario us base k (with k ≥ 2) represen tations of in teg ers also inv olve term algebras, n ot necessarily free. They differ b y the sets of digits they use bu t all are b ased on the u s ual inte rpretation d n . . . d 1 d 0 7→ P i =0 ,...,n d i k i whic h, seen in Horner style : k ( k ( . . . k ( k d n + d n − 1 ) + d n − 2 ) . . . ) + d 1 ) + d 0 is a comp osition of app lications S d 0 ◦ S d 1 ◦ . . . ◦ S d n (0) wher e S d : x 7→ k x + d . 4 If a representat ion uses digits d ∈ D th en it corresp onds to the algebra generated by 0 and the S d ’s where d ∈ D . i. The k -adic represen tation uses d igits 1 , 2 , . . . , k and corresp onds to a free algebra b uilt up fr om one generator and k unary functions. ii. The us u al k -ary repr esen tatio n uses digits 0 , 1 , . . . , k − 1 and corre- sp onds to the quotien t of a free algebra bu ilt up from one generator and k u nary functions, namely 0 and the S d ’s where d = 0 , 2 , . . . , k − 1, b y the relation S 0 (0) = 0. iii. Avizienis base k represent ation u ses digits − k + 1 , . . . , − 1 , 0 , 1 , . . . , k − 1, (it is a muc h redun dan t representa tion u sed to p erform add itions without carry p ropagation) and corresp ond s to the quotien t of the free algebra b uilt u p f rom one generator a and 2 k − 1 unary functions, namely 0 and th e S d ’s where d = − k + 1 , . . . , − 1 , 0 , 1 , . . . , k − 1, b y the relations ∀ x ( S − k + i ◦ S j +1 ( x ) = S i ◦ S j ( x )) where − k < j < k − 1 and 0 < i < k . 3) R : N 4 → N such th at R ( x, y , z , t ) = x 2 + y 2 + z 2 + t 2 is a representa tion based on Lagrange’s four squares theorem 4) R : P rime ≤ 7 → N suc h that R ( x 1 , . . . , x i ) = x 1 + . . . + x i (with i ≤ 7) is a representat ion based on Schnirelman’s theorem (as improv ed b y Ra- mar, 1995) wh ic h insur es that ev ery num ber is the su m of at most 7 p rime n um b ers. Besides such num b er th eoretic representat ions of int egers, we shall con- sider classical set theoretic repr esen tatio ns inv o lving higher order ob jects. Example 2.2. 1) Churc h’s representation. In tegers are viewed as fu nction iterators (whic h are t yp e 2 ob jects) f 7→ f ( n ) where f (0) = I d and f ( n +1) = f ( n ) ◦ f . Thus, C is the class con t aining all functional sets ( X → X ) X → X (cf. Notations 1.1) and R is defined on the prop er sub class of C constituted of functionals which are fin ite iterators on some X → X and R ( F ) = n if and only if F ( f ) = f ( n ) for all f : X → X . 1bis) Z-Churc h’s representation. Negativ e ite rators f 7→ f ( − n ) are defin ed as follo ws: i. D omain ( f ( − n ) ) = R ang e ( f ( n ) ) ii. f ( − n ) ( x ) = y if y is the smallest suc h that f ( n ) ( y ) = x (relativ e to some fixed well -order on X ) The defin itions of C and R are analog to th at in p oin t 1. 2) Cardinal represen tation. In tegers are view ed as equiv ale nce classes of sets r elativ e to card in al compar- ison. Thus, C is the class of all sets and R is defin ed on the prop er sub class 5 of C constituted of fi nite sets and R ( X ) is the cardin al of the s et X . 3) Z-Cardinal represen tation. Relativ e int egers are viewe d as differences of natural in tegers whic h are them- selv es view ed via cardinal representati on. Thus, C is th e class of all pairs of sets and R is defined on the prop er sub class of C constituted of fin ite sets and R ( X , Y ) is the d ifference of th e card inals of X and Y . 4) Ordinal represen tation. In tegers are viewed as equiv alence classes of totally ordered sets. Th us, C is the class of totally ordered sets and R is defined on the prop er su b class of C constituted of finite totally ordered sets and R ( X ) is the ord er t yp e of X . 2.2 F ormal represen tations of in tegers A f ormal represent ation of an inte ger n is a finite ob ject (in general a word) whic h describ es some c haract eristic prop erty of n or some abstract ob ject whic h characte rizes n . In fact, eac h particular representa tion is really a c hoice made in order to access sp ecial op erations or str ess sp ecial pr op erties of in tegers. The computer science (or r ecursion theoretic) p oin t of view brings an ob- jection to the consideration of abstract sets, functions and f unctionals as we did in E x amp le 2.2: • We c annot appr ehend abstr act sets, functions and functionals but solely pr o gr ams to c omp ute them (if they ar e c omp utable i n some sense). • Mor e over, pr o g r ams de a ling with sets, functions and functionals have to go thr ough some intensional r epr esenta tion of these obje cts in or der to b e able to c ompute with such obje cts. Th us, to get effectiv eness, we turn from set theory to r ecursion theory and “effectivi ze” abstract s ets: • sets of intege rs w ill b e recursiv ely en umerable (r.e.), i.e. domains of partial recursive fu nctions, • functions on intege rs will b e p artial recur s iv e, • functionals will b e partial recur s iv e in the sen se of higher typ e r ecur- sion theory (cf. u sual textb o oks [9] or [8]). Though abstract rep r esen tations are quite natural and conceptually simple, their effectivized versions are quite complex: the sets of pr ograms computing ob jects in their domains inv olve lev els 2 or 3 of the arithmetic al h ierarc h y . In particular, such r epr esentations ar e not al l T uring r e ducible one to the other . In the sequel w e shall only consider type ≤ 2 r epresen tations. In order to get the adequate notion of r ecursion-theoretic representa tions of in tegers, we h av e to r eview some h igher order recurs ion concepts. 6 2.3 Effectivization of sets of type ≤ 2 ob jects First, we recall the notion of t yp e 2 recursion that we sh all use. Definition 2.2 (Effectiv e op erations) . Let X , Y , Z, T b e t yp e 0 spaces (i.e. N , N k , { 0 , 1 } ,. . . ). W e denote P R ( X → Y ) the set of partial recursiv e fu n ctions X → Y . An effectiv e op eration F : P R ( X → Y ) → P R ( Z → T ) or F : P R ( X → Y ) → Z is an op eration which can b e defined via p artial recursive op erations on the G¨ odel num b ers of partial recursive functions. In other wo rds, letting U P R ( X → Y ) denote a partial recursive enumeration of P R ( X → Y ), there exists f such that the follo wing diagram comm utes P R ( X → Y ) F − − − − → P R ( Z → T ) U P R ( X → Y ) x   x   U P R ( Z → T ) { 0 , 1 } ∗ f − − − − → { 0 , 1 } ∗ W e denote E f f (( X → Y ) → ( T → Z )) (resp . E f f (( X → Y ) → Z )) the family of effectiv e op erations f rom P R ( X → Y ) into P R ( Z → T ) (resp. in to Z ). Let’s recall the follo wing fact. Theorem 2.1 (Myhill & Sh ep herdson, 1955) . Effe ctive op er ations F : P R ( X 7→ Y ) → Z (or F : P R ( X 7→ Y ) → P R ( Z 7→ T ) ) ar e exactl y the r estrictions of effe ctively c ontinuous functionals F : Y X → Z (or F : Y X → T Z ) in the sense of U sp enskii, 1955 (cf. R o g ers [9], or Odifr e dd i [8]). Definition 2.3 (Effectiviz ation of h igher t yp e sets) . 1) The effectivization of the t yp e 1 s ets Y X and X → Y is the su b set P R ( X → Y ) of partial recursive f unctions. 2) The effectivizatio n of the t yp e 2 set ( T Z ) Y X (resp. Z Y X ) is the sub set of effectiv e op erations P R ( X → Y ) → P R ( Z → T ) (resp. P R ( X → Y ) → Z ). W e ca n no w define str ongly universal partial recursive functions and str ongly universal effectiv e op erations. Definition 2.4 (Univ ersal enumerations) . Let X , Y , Z, T b e t yp e 0 sets and let E b e X or P R ( X → Y ) (resp. E f f (( X → Y ) → ( Z → T ))). 1) A partial r ecursiv e f unction (resp . effectiv e op eration) U : { 0 , 1 } ∗ → E is universal if there is a recursive function comp : { 0 , 1 } ∗ × { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that if we denote U e the fu nction su c h that U e ( p ) = U ( comp ( e , p )), 7 then the family ( U e ) e ∈{ 0 , 1 } ∗ is an enumeration of the class of p artial recur- siv e functions (resp . effectiv e op erations) from { 0 , 1 } ∗ to E . Intuition : W ords in { 0 , 1 } ∗ are seen as programs. A partial r ecursiv e f unc- tion (resp. effectiv e op eration) { 0 , 1 } ∗ → E maps a p rogram to the ob ject it computes (which lies in E ). F unction p 7→ comp ( e , p ) is th erefore seen as a compiler. W e say that e is a G¨ odel num b er f or F ∈ E if F = U e . 2) U is str ongly universal if it is univ ersal and for eac h index e , there is a constan t c ( e ) s u c h that for all p ∈ { 0 , 1 } ∗ , w e hav e | comp ( e , p ) | ≤ | p | + c ( e ) The follo wing theorem is a cla ssical result of recursion theory which is crucial for the definition of Kolmogoro v complexit y in § 2.5. Theorem 2.2. Ther e exists a str ongly unive rsal p art ial r e cursive fu nc tion (r esp. eff e ctive op er ation). Mor e over, one c a n supp ose that comp is th e p airing function h , i define d by h ǫ, p i = 1 p , h e 1 e 2 · · · e n , p i = 0 e 1 0 e 2 · · · 0 e n 1 p which satisfies the e quality |h e , p i| = | p | + 2 | e | + 1 2.4 Recursion-theoretic represen tations of integers Finally , we are in a p osition to introduce the wa n ted defin ition. Definition 2.5 (Recursion-theoretic representa tions) . A recursion theoretical represen tation of N (resp. Z ) is an y surje ctive function ρ : C → N (resp. ρ : C → Z ) where C is the effectiviz ation E of some higher typ e s et. W e u se Example 2.2 to illustrate the effectivization p ro cesses describ ed in § 2.3. Example 2.3. 1) Effective Churc h and Z-Churc h representations. Iterators I t n : P R ( N → N ) → P R ( N → N ) of partial recursiv e functions are indu ctiv ely d efined as follo ws for n ∈ N : i. I t 0 ( f ) = f ii. I t n +1 ( f ) = I t n ( f ) ◦ f Negativ e iterators I t − n : P R ( N → N ) → P R ( N → N ) are d efined as follo ws: iii. I t − n ( f ) has d omain Rang e ( I t n ( f ) iv. I t − n ( f )( x ) = y if I t n ( f )( y ) = x and I t n ( f )( y ) is the first h alting computation among all compu tations I t n ( f )(0) , I t n ( f )(1) , . . . . 8 W e let C hur ch N : E f f (( N → N ) → ( N → N )) → N b e so that C hurch N ( F ) = n if F = I t n for some n ∈ N , o therwise u ndefined and C hur ch Z : E f f (( N → N ) → ( N → N )) → Z b e so that C hurch Z ( F ) = n if F = I t n for some n ∈ Z , otherwise un defined 2) Effective cardinal and Z-cardinal represen tations. W e let car d N : P R ( N → N ) → N b e so that i. car d N ( f ) is defined if and only if domain ( f ) is finite ii. car d N ( f ) = car d ( domain ( f )) W e let car d Z : P R ( N → N ) × P R ( N → N ) → Z b e so that iii. car d Z ( f , g ) is defin ed if and only if f , g ha v e fin ite d omains iv. car d Z ( f , g ) = car d ( domain ( f )) − car d ( domain ( g )) 3) Effective ordinal represen tation. W e let or d N : P R ( N 2 → N ) → N b e s o that i. or d N ( f ) is defin ed if a nd only if the quotient order asso ciated to the transitiv e closure of domain ( f ) is fin ite ii. or d N ( f ) is the order type of this qu otien t order. The f ollo wing result measures th e syntac tical complexit y of th e d omain and the graph of the f unctionals C hur ch N , C hurch Z , car d N , car d Z , or d in terms of the asso ciated index sets. Prop osition 2.1 (Syn tactical complexit y of represent ations) . 1) Churc h represen tations. The set of p airs ( e , n ) such that n ∈ N (r e sp. n ∈ Z ) and e is a G¨ odel numb er for the iter ation functional I t n is Π 0 2 -c o mplete. The set of G¨ odel numb ers of effe ctive functionals in the domain of C hur ch N (r esp. C hur ch Z ) is Σ 0 3 -c o mplete. 2) Cardinal represen tations. The set of p airs ( e , n ) such that n ∈ N (r esp. n ∈ Z ) and e i s the G¨ od el numb er of a p artial r e cursive function the doma in of which is finite with n elements is Σ 0 2 -c o mplete. The set of G¨ odel numb ers of p art ial r e c ursive func tions with finite do mains is Σ 0 2 -c o mplete. 3) Ordinal representation. The se t of p airs ( e , n ) such that n ∈ N and e is the G¨ odel numb er of a p a r- tial r e cursive function N 2 → N such that the quotient or der asso ci ate d to the tr ansitive closur e of domain ( f ) is finite with n elements is Σ 0 3 -c o mplete. The set of G¨ odel numb e rs of p artial r e cursive functions su ch that the quo- tient or der asso ciate d to the tr ansitive closur e of domain ( f ) is finite is Σ 0 3 - c omplete. 9 2.5 Kolmogoro v c omplexit y and represen tations of in tegers The usual defi nition of Kolmogoro v complexit y c an b e extended to any recursion-theoretic rep resen tation of inte gers. Definition 2.6 (Kolmogoro v complexit y relativ e to a representa tion) . Let E b e the effectiviz ation of some h igher type set and let ρ : E → N (resp. ρ : E → Z ) b e a r ecursion-theoretic repr esentati on of int egers. Considering the d iagram { 0 , 1 } ∗ U E − → E ρ − → N where U E is some strongly univ ersal en umeration of E , Kolmogoro v com- plexit y K ρ : N → N is defined as K N ρ ( n ) = min {| p | : ρ ( U E ( p )) = n } The defin ition of K ρ : Z → N is analog. Theorem 2.2 implies an inv ariance theorem for strongly u niv ersal enu- merations. Whic h insu r es that the ab o v e definition do es not dep end (up to a constant) of the particular c hoice of th e strongly universal en umeration U E of E . R emark 2.2 . The domain of ρ ◦ U E is not recursively en umerable in general (cf. Prop. 2.1). 3 Main Theorem 3.1 Main theorem Reconsidering the answer to ( Q 2) given after Def. 1.1, the main theorem of this pap er insu res the follo wing. • Kolmogoro v complexit y is muc h dep endent on the c hosen higher ord er effectiv e represen tations of inte gers, • The K olmogoro v complexitie s associated to represent ations of E x am- ple 2.3 constitute a hierarc h y wh ic h coincides with that obtained with infinite computations and relativization to the ju mps (cf. Prop. 1.1). Th us, Kolmogoro v complexity measures the complexit y of higher order ef- fectiv e representat ions and allo ws to classify them. Theorem 3.1 (Ma in Th eorem) . L et K N C hur ch , K Z C hur ch , K N car d , K Z N car d , K or d b e the Kolmo gor o v c omp lexities as- so ciate d to the effe ctive versions of the higher or der r epr esentations describ e d in Example 2.3. Then K N C hur ch = ct K Z C hur ch ↾ N = ct K 10 K N car d = ct K ∞ K Z car d ↾ N = ct K ′ K N or d = ct K ′ ∞ So that we have K N C hur ch = ct K Z C hur ch ↾ N > ct K N car d > ct K Z car d ↾ N > ct K or d 3.2 Represen tations ρ suc h that K ρ = K The follo wing theorem gives simple s ufficien t conditions on repr esen tatio ns ρ so that th e asso ciated Kolmogoro v complexit y K ρ b e equ al, up to a constan t, to usual Kolmogoro v complexity K . In particular, these conditions will apply to C h urc h’s representati ons. Theorem 3.2. L et E b e P R ( X → Y ) or E f f (( X → Y ) → ( Z → T ) and U E : { 0 , 1 } ∗ → E b e str ongly universal. L et ρ : E → N (r esp. ρ : E → Z ) b e a r epr esentation of inte gers. Consider the f ol lowing c onditions: (*) ρ ◦ U E is the r estriction to its domain of some p artial r e cursive function f : { 0 , 1 } ∗ → N ( f : { 0 , 1 } ∗ → Z ). (**) ρ is effectiv ely sur jectiv e : ther e exists a total recursiv e function g : N → { 0 , 1 } ∗ (r esp. g : Z → { 0 , 1 } ∗ ) such that ρ ( U E ( g ( n ))) = n for al l n ∈ N (r esp. n ∈ N ) 1) Condition (*) implies K ≤ ct K ρ . 2) Condition (**) implies K ρ ≤ ct K . Pr o of. 1) Let n ∈ N and let p ∈ { 0 , 1 } ∗ b e a minimal length pr ogram suc h that ρ ( U E ( p )) = n . Then K ρ ( n ) = | p | . Condition (*) imp lies f ( p ) = n . There- fore, viewing f as a programming language and using K olmogoro v’s inv ari- ance theorem th er e is a constant c indep endent of n such that K ( n ) ≤ K f ( n ) + c ≤ | p | + c = K ρ ( n ) + c Th us, K ≤ ct K ρ . 2) Let U N : { 0 , 1 } ∗ → N b e a s trongly un iv ersal en umeration of N . The strong universali t y of U E insures that there exists e suc h that ∀ p U E ( g ( U N ( p ))) = U E ( h e , p i ) Let n ∈ N and let p ∈ { 0 , 1 } ∗ b e a minimal length pr ogram su c h that U N ( p ) = n . Th en K ( n ) = | p | . 11 Condition (**) implies ρ ( U E ( g ( U N ( p )))) = n . Thus, ρ ( U E ( h e , p i )) = n , whence (using Th m 2.2) K ρ ( n ) ≤ |h e , p i| = | p | + 2 | e | + 1 = K ( n ) + 2 | e | + 1 and therefore K ρ ≤ ct K . Corollary 3.1. K N C hur ch = ct K Z C hur ch ↾ N = ct K Pr o of. W e show th at conditions (*) and (**) of Theorem 3.2 are satisfied for the ρ asso ciated to K N C hur ch . T h e argument also applies with straigh tforw ard mo difications to K Z C hur ch ↾ N . T o get condition (**), just design a program for fun ctional I t n . As for condition (*), let E = E f f (( N → N ) → ( N → N )) and S ucc : N → N b e the s u ccessor fun ction and define f { 0 , 1 } ∗ → N as follo ws: f ( p ) = U E ( p )( S ucc )(0) If ρ ( U E ( p )) = n th en U E ( p ) = I t n so that U E ( p )( S ucc ) is the fun ction x 7→ x + n and f ( p ) = U E ( p )( S ucc )(0) = n . R emark 3.1 . 1) Conditions (*) and (**) b oth h old trivially in case ρ ◦ U E is a partial recursiv e fun ction or an effectiv e fu nctional. 2) Theorem 3.2 relativizes to compu tabilit y with an oracle. 3.3 K N car d and K ∞ Theorem 3.3. K N car d = ct K ∞ Pr o of. 1) E = P R ( N → N ). L et n ∈ N and let p ∈ { 0 , 1 } ∗ b e a minimal length program which outputs n in u n ary th r ough a p ossibly in fnite computation. Then K ∞ ( n ) = | p | . Define h : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that h ( p ) is th e fol- lo wing program for a partial recurs ive f unction ϕ h ( p ) : N → N : ϕ h ( p ) ( t ) = IF at step t p r ogram p outpu ts 1 THEN 1 EL SE u ndefined Clearly , car d N ( U E ( h ( p )) = n . The strong un iv ersalit y of U E insures th at there exists e such that ∀ p U E ( h ( p )) = U E ( h e , p i ) This leads to K N car d ( n ) ≤ K ∞ ( n ) + 2 | e | + 1, wh ence K N car d ≤ ct K ∞ . 12 2) Let n ∈ N and let p ∈ { 0 , 1 } ∗ b e a minimal length pr ogram suc h that car d N ( U E ( p )) = n . Define h : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that h ( p ) b eha v es as follo ws: • h ( p ) em ulates some do v etailing of all compu tations U E ( p )( i ) for i = 0 , 1 , 2 , . . . , • eac h time some compu tation U E ( p )( i ) halts (i.e. a new p oin t i is pro v ed to b e in the domain of U E ( p )) then h ( p ) output 1. It is clear that h ( p ) outputs n in u nary . Th us, K ∞ ( n ) ≤ | h ( p ) | . But th ere is some constant c such th at | h ( p ) | = | p | + c , whence K ∞ ( n ) ≤ | p | + c and therefore K ∞ ≤ ct K N car d . 3.4 K Z car d ↾ N and K ′ Theorem 3.4. K Z car d ↾ N = ct K ′ Pr o of. No w E = ( P R ( N → N )) 2 and U E is a pair of fun ctions ( U E 1 , U E 2 ). 1) Let n ∈ N and let p ∈ { 0 , 1 } ∗ b e a minimal length pr ogram suc h that car d Z ( U E ( p )) = car d N ( U E 1 ( p )) − car d N ( U E 2 ( p )) = n . Define h : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that h ( p ) is a p r ogram whic h u ses oracle ∅ ′ and b eha v es as follo ws: • h ( p ) em ulates some do v etai ling of all computations U E 1 ( p )( i ) , U E 2 ( p ) for i = 0 , 1 , 2 , . . . , • A t ea c h computation step, h ( p ) asks the oracle w h ether there is still some computation that will halt. I f th e answ er is “NO” then h ( p ) halts and outputs the difference of the n um b er of p oints i ’s on which U E 1 ( p ) has b een c hec k ed to conv erge and that for U E 2 ( p ) It is clear that h ( p ) outputs n . Thus, K ′ ( n ) ≤ | h ( p ) | . But there is some constan t c such that | h ( p ) | = | p | + c , w hence K ′ ( n ) ≤ | p | + c and therefore K ′ ≤ ct K Z car d ↾ N . 2) Let n ∈ N and let p ∈ { 0 , 1 } ∗ b e a minimal length program using oracle ∅ ′ whic h outpu ts n in unary . Then K ′ ( n ) = | p | . T o em ulate computations using oracle ∅ ′ , we sh all u se Chaitin’s h armless o v ersho ot tec hnique [3]: • If an ∃ x . . . assertion is true then a compu tation lo op will c hec k that it is true. • Ho w ev er, if it is false, there is no w a y to c hec k it in fin ite time. • Whence the strategy to systematically ans wer “NO” to eac h ∃ x . . . question and then c hec k this answ er via a computation lo op. 13 • Ev ery false answ er “NO” will b e p ro v ed false at some time dur ing this lo op, giving a p ossib ilit y to correct it. • Ev ery compu tation u sing oracle ∅ ′ whic h halts uses finitely man y qu es- tions to th e oracle. Th e ab o v e strategy will therefore b e corrected only finitely many times so that it ev en tually leads for a f air em ulation. Define h : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h th at U E 1 ( h ( p )) and U E 2 ( h ( p )) b ehav e as follo ws: 1. U E 1 ( h ( p )) em ulates p and w h enev er p outputs 1 then U E 1 ( h ( p )) will con v erge on t where t is th e current computation step. 2. Eac h time p asks a question to oracle ∅ ′ of the form ∃ x . . . ? then U E 1 ( h ( p )) answers “NO”. 3. Cautiously , U E 1 ( h ( p )) chec ks eac h of its oracle answers starting a com- putation lo op w h ic h will halt only if th e r igh t answer was “YES” (in- stead of “NO”). 4. In case some answer was false, then U E 1 ( h ( p )) r estarts th e whole em- ulation of p (correcting its p ast answers) and U E 2 ( h ( p )) will co n v erge on a set of p oints in bijection with that on wh ich U E 1 ( h ( p )) was mad e con v erging b efore the answer wa s r ecognized to b e false. Corrections br ough t in p oint 4 make the final differen ce car d N ( U E 1 ( h ( p ))) − car d N ( U E 2 ( h ( p ))) equal to n . The strong un iv ersalit y of U E insures that there exists e suc h that ∀ p ( U E 1 ( h ( p )) = U E ( h e 1 , p i ) ∧ U E 2 ( h ( p )) = U E ( h e 2 , p i )) This leads to K Z car d ( n ) ≤ K ′ ( n ) + 2 | e | + 1, whence K Z car d ↾ N ≤ ct K ′ . 3.5 K or d and K ′ ∞ Theorem 3.5. K or d = ct K ′ ∞ Pr o of. No w E = P R ( N 2 → N ). 1) Let n ∈ N and let p ∈ { 0 , 1 } ∗ b e a minimal length pr ogram suc h that or d ( U E ( p )) = n . Define h : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that h ( p ) uses orac le ∅ ′ and b eha v es as follo ws: • h ( p ) initializes a set X of in tegers to ∅ . 14 • Step 0 . h ( p ) c hec ks if th ere is s ome pair w ith 0 as a comp onen t which is in the domain of U E ( p ). If s o it outputs a 1 and puts 0 in the set X . • Step t > 0 . h ( p ) chec ks if there is some c hain (constituted of p airs in the domain of U E ( p )) con taining t and all el emen ts of X . If so it outputs a 1 and puts t in the set X . It is clear th at, thr ough an in fi nite computation, h ( p ) outputs the un ary represent ation of the order t yp e of the transitive closure of the domain of U E ( p ) in case it is a finite ordered type. Thus, K ′ ∞ ( n ) ≤ | h ( p ) | . But there is some constan t c suc h th at | h ( p ) | = | p | + c , whence K ′ ∞ ( n ) ≤ | p | + c and therefore K ′ ∞ ≤ ct K or d . 2) Let n ∈ N , n > 0 and let p ∈ { 0 , 1 } ∗ b e a m inimal length pr ogram u sing oracle ∅ ′ whic h outputs n in unary thr ough an infinite co mputation. Then K ′ ∞ ( n ) = | p | . W e shall ag ain use Chaitin’s harmless o v ershoot tec hnique. Define h : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that U E ( h ( p )) b eha v es as f ollo w s: 1. U E ( h ( p )) initializes a set X of intege rs to { 0 } . This set X will alwa ys b e fin ite. 2. U E ( h ( p )) emulate s p an d wheneve r p outpu ts 1 then a new p oint k is added to X and U E ( h ( p )) w ill conv er ge on ev ery p air ( x, k ) where x ∈ X . 3. Eac h time p asks a question to oracle ∅ ′ of the form ∃ x . . . ? then U E ( h ( p )) answers “NO”. 4. Cautiously , U E ( h ( p )) chec ks eac h one of its oracle answ ers starting computation lo ops whic h w ill halt only if some righ t answer w as “YES” (instead of “NO”). 5. In case some answer was false, then U E ( h ( p )) r estarts th e whole em- ulation of p (correcting its p ast answers) and U E ( h ( p )) will co n v erge on all p airs ( x, y ) ∈ X 2 and the set X is r einitialize d to { 0 } . Corrections brought in p oint 5 make the fin al transitive closure of the domain of U E ( h ( p )) a preord ered set the qu otien t of which has exactly n p oin ts. The strong un iv ersalit y of U E insures that there exists e suc h that ∀ p ( U E ( h ( p )) = U E ( h e , p i ) This leads to K or d ( n ) ≤ K ′ ∞ ( n ) + 2 | e | + 1, wh ence K or d ≤ ct K ′ ∞ . 15 References [1] V. Bec her, S. Daicz, and G. C haitin. A highly random num b er. In Combinatorics, Computa bility and L o g i c: Pr o c e e dings of the Thir d Discr ete M athematics a nd The or e tic al Computer Scienc e Confer enc e (DMTCS’01) , pages 55–68. Spr in ger-V erlag, 2001. [2] G. Chaitin. A theory of program s ize formally identical to information theory . Journal of the ACM , 22:329–3 40, 19 75. Av ailable on Chaitin’s home page. [3] G. Chaitin. Information theoretic charact erizations of infinite strin gs. The or et. Comp ut. Sci. , 2:4 5–48, 197 6. Av ailable on C haitin’s home page. [4] M. F erbus -Zanda and S . Grigorieff. Is r an d omnes n ativ e to compu ter science? Bul l. E A TCS , 74:78– 118, 2001. Av ailable on the author’s home page. [5] A.N. Kolmogoro v. Three approac hes to the quant itativ e definition of information. Pr oblems Inform. T r ansmission , 1(1):1–7 , 19 65. [6] L. L evin. On the n otion of rand om sequence. Soviet Math. Dokl. , 14(5): 1413–1 416, 1973. [7] M. Li an d P . Vita n yi. An intr o duction to Kolmo gor ov c omplexity and its applic ations . Sprin ger, 1997 (2d edition). [8] P . Odifred di. Classic al R e cursion The or y , v olume 125. North-Holland, 1989. [9] H. Rogers. The ory of r e cursive functions and effe ctive c omputa bility . McGra w-Hill , 1967. [10] A. Shen. Kolmogoro v complexity and its app lications. L e c- tur e Notes, Uppsala University, Swe den , pages 1–23, 2000. h ttp://www.csd.uu.se/˜v orob y o v/ Courses/K C/2000 /all.ps. [11] R.M. Solo v a y . On random R.E. sets. In A.I. Arru da and al., editors, Non-classic al Lo gics, Mo del the or y and Computa bility , p ages 283– 307. North-Holland, 1977. 16