Limit complexities revisited

Symposium on Theoretical Aspects of Computer Science 2008 (Bordeaux), pp. 73-84 www .stacs-conf .org LIMIT COMPLEXITIES REVISITED LAURENT BIENVENU 1 , ANDREJ MUCHNIK 2 , ALEX ANDER SHEN 3 , AND N IKOLA Y VERESHCHAGIN 4 1 Lab oratoire d ’Informatique F ondamentale CNRS & Universit ´ e de Pro vence, 39 ru e Joliot Curie, F-13453 Marseille cedex 13 E-mail addr ess : Laurent.Bi envenu@lif.univ-mrs.fr 2 Andrej Muchnik (24.02.1958 – 18.03.2007 ) w orked in t h e Institute of New T echnolo gies in Education, Mosco w 3 Lab oratoire d ’Informatique F ondamentale, P oncelet Lab oratory , CNRS, I ITP RAS, Mosco w E-mail addr ess : Alexander. Shen@lif.univ- mrs.fr 4 Mosco w State Lomonosov Universit y , Russia E-mail addr ess : ver@mccme. ru Abstra ct. The main goal of th is p ap er is to p u t some known results in a common p er- sp ective and to simplify their pro ofs. W e start with a simple pro of of a result from [7] saying that lim sup n C ( x | n ) (here C ( x | n ) is conditional (plain) Kolmogoro v complexity of x when n is kn o wn) equals C 0 ′ ( x ), the plain Kolmogoro v complexit y with 0 ′ -oracle. Then we use the same argument to prove similar results for prefix complexity (and also improv e results of [4] ab out limit frequencies), a priori p robabilit y on binary tree and measure of effectively open sets. As a by-pro d uct, we get a criterion of 0 ′ Martin-L¨ of randomness (called also 2-randomness) prov ed in [3]: a sequence ω is 2-rand om if and only if there exists c such t h at any prefix x of ω is a prefix of some string y suc h t h at C ( y ) > | y | − c . (In the 1960 ies this property w as suggested in [1] as one of p ossible randomness defi n itions; its equiv alence to 2-randomness was shown in [3] while proving another 2-randomn ess criterion (see also [5]): ω is 2-random if and only if C ( x ) > | x | − c for some c and infinitely many prefixes x of ω . Finally , we sho w that th e low - basis theorem can b e used to get alternative proofs for these results an d to improv e the result ab out effectively op en sets; th is stronger version implies the 2-randomness criterion mentioned in th e p revious sentence. Key wor ds and phr ases: K olmogoro v complexity , limit complexities, limit frequencies, 2-randomness, low basis. c  L. Bienv en u, An. Muchnik, A. Shen, and N. V ereshch agin CC  Creative Commons Attribution-NoDer ivs License 74 L. BIENVENU, AN. MUCH NIK, A. SHEN, AND N. VERESHCH AGI N 1. Plain complexit y By C ( x ) w e mean the p lain complexit y of a b inary str in g x (the length of the shortest description of x when an optimal d escription m etho d is fixed, see [2 ]; no r equiremen ts ab out prefixes). By C ( x | n ) w e mean conditional complexit y of x when n is giv en [2]. S up e rs cript 0 ′ in C 0 ′ means that we consider the relativize d (with oracle 0 ′ , the universal enumerable set) v ersion of complexity . The follo wing r esult was p ro ve d in [7]. W e pro vide a simple pro of for it. Theorem 1.1. lim sup n →∞ C ( x | n ) = C 0 ′ ( x ) + O (1) . Pr o of. W e start with the easy part. Let 0 n b e the (fin ite) part of the un iv ersal enumerable set th at app eared after n steps. If C 0 ′ ( x ) 6 k , then there exists a d escrip tion (program) of size at m ost k that generates x using 0 ′ as an oracle. Only fin ite part of the oracle can b e u s ed, so 0 ′ can b e rep laced b y 0 n for all sufficient ly large n , and oracle 0 n can b e reconstructed if n is giv en as a condition. Therefore, C ( x | n ) 6 k + O (1 ) for all sufficien tly large n , and lim sup n →∞ C ( x | n ) 6 C 0 ′ ( x ) + O (1) . No w fix k and assume that lim sup C ( x | n ) < k . This means that for all sufficien tly large n th e string x b elongs to the s et U n = { u | C ( u | n ) < k } . The family U n is an enumerable family of sets (give n n and k , w e generate U n ); eac h of these sets has less than 2 k elemen ts. W e need to constru ct a 0 ′ -computable pro cess that giv en k generates at most 2 k elemen ts, and among them all elements that b elong to U n for all sufficien tly large n . (Then strings of length k ma y b e assigned as 0 ′ -computable co d es of all generated elemen ts.) T o d escrib e this pr o cess, consider the follo wing op eration: for some u and N add u to all U n suc h that n > N . (In other terms, we add a horizonta l ra y starting from ( N , u ) to the set U = { ( n, u ) | u ∈ U n } .) This op eration is ac c eptable if all U n still ha ve less than 2 k elemen ts after it (i.e., if b efore this op er ation all U n suc h that n > N either cont ain u or ha ve less than 2 k − 1 elemen ts). F or giv en u and k we can fin d out using 0 ′ -oracle wh ether th is op eration is acceptable. No w for all pairs ( N , u ) (in some computable order) w e p erform ( N , u )-op eration if it is acceptable. (The elements add ed to some U i remain there and are take n int o accoun t w hen next op erati ons are attempted.) This p ro cess is 0 ′ -computable since after an y finite n u m b er of operations the family U is enumerable (with ou t any oracle) and its en umeration algorithm can b e 0 ′ -effectiv ely foun d (un iformly in k ). Therefore th e set of all element s u that participate in acceptable op erations durin g th is pro cess is un if orm ly 0 ′ -en umerable. This set cont ains less than 2 k elemen ts (otherwise U n w ould b ecome to o b ig for large n ). Finally , this set cont ains all u s uc h th at u b elongs to the (initial) U n for all sufficien tly large n . Indeed, the op eration is alwa ys acceptable if all added elemen ts are already p resen t. LIMIT COMPLEXITIES REVISI TED 75 The pro of has the follo win g structure. W e h a v e an enumerable family of sets U n that ha ve less than 2 k elemen ts. This implies that the set U ∞ = lim in f n →∞ U n has less than 2 k elemen ts (the lim inf of a sequence of sets is the set of element s that b elong to almost all sets of the sequence). If this set were 0 ′ -en umerable, we w ould b e done. Ho w eve r , this may b e not the case: the criterion u ∈ U ∞ ⇔ ∃ N ( ∀ n > N ) [ u ∈ U n ] has ∃ ∀ pr efix b efore an enumerable (not necessarily decidable) relatio n , that is, one quan- tifier more than we wan t (to guaran tee that U ∞ is 0 ′ -en umerable). Ho wev er, in our pro of w e managed to co v er U ∞ b y a set that is 0 ′ -en umerable an d still has less than 2 k elemen ts. 2. Prefix complexit y and a priori probability No w we pro v e similar result f or prefi x complexit y (or, in other terms, for a priori probabilit y). Let us recall the d efinition. Th e function a ( x ) on bin ary strings (or in tegers) with non-negativ e real v alues is called a semime asur e if P x a ( x ) 6 1. The fu n ction a is lower semic omputable if there exists a compu table total fu nction ( x, n ) 7→ a ( x, n ) with rational v alues su ch that for ev ery x th e sequence a ( x, 0) , a ( x, 1) , . . . is a non-decreasing sequence that has limit a ( x ). There exists a m aximal (up to a constan t f actor) lo wer semicomputable semimeasure m . T he v alue m ( x ) is sometimes called the a priori pr ob ability of x . In the same w a y w e can define c onditional a priory prob ab ility m ( x | n ) and 0 ′ - r elativize d a priori probabilit y m 0 ′ ( x ). Theorem 2.1. lim inf n →∞ m ( x | n ) = m 0 ′ ( x ) up to a Θ(1) f actor. (In other terms, t wo inequ alities with O (1) factors hold.) Pr o of. If m 0 ′ ( x ) is greater that some ε , th en f or some k the increasing sequence m 0 ′ ( x, k ) that h as limit m 0 ′ ( x ) b eco mes greate r than ε . The computation of m 0 ′ ( x, k ) uses on ly finite amount of information ab out the oracle, thus for all sufficiently large n w e ha ve m 0 n ( x ) > m 0 n ( x, k ) > ε . So, similar to the previous theorem, we ha ve lim inf n →∞ m ( x | n ) > lim inf n →∞ m 0 n ( x ) > m 0 ′ ( x ) up to O (1 ) factors. In the other direction the pro o f is also similar to the p revious one. In stead of enumer- able fin ite sets U n no w w e ha v e a sequence of (uniformly) lo w er semicomputable functions x 7→ m n ( x ) = m ( x | n ). Eac h of m n is a semimeasure. W e n eed to construct a lo wer 0 ′ -semicomputable semimeasure m ′ suc h that m ′ ( x ) > lim in f n →∞ m n ( x ) 76 L. BIENVENU, AN. MUCH NIK, A. SHEN, AND N. VERESHCH AGI N Again, the lim inf itself cannot b e u sed as m ′ : though P x lim inf n m n ( x ) < 1 if P x m n ( x ) 6 1 for all n , b ut, unfortun ately , the equiv alence r < lim in f n →∞ a n ⇔ ( ∃ r ′ > r )( ∃ N ) ( ∀ n > N ) [ r ′ < a n ] has too man y quan tifier alternations (o n e more than needed; n ote that lo wer semicom- putable a n mak es [ . . . ] condition en umer ab le). The similar trick h elps . F or a triple ( r , N , u ) consider an incr e ase op er ation that increases all v alues m n ( u ) such that n > N up to a giv en r ational num b er r (not c hanging th em if they we r e greate r than or equ al to r ). This op eration is ac c eptable if all m n remain semimeasures after the in crease. The qu estion w hether op erati on is acceptable is 0 ′ -decidable; if it is, we get a new (uniformly) lo wer semicomputable (withou t an y oracle) sequ ence of semimeasures and can rep eat an attempt to p erform an increase op eration for some other triple. Doing that for all triples (in some computable ordering), w e can then defin e m ′ ( u ) as the up p er b ound of r for all successful ( r, N , u ) increase op e r ations (for all N ). This give s a 0 ′ -lo w er semicomputable function; it is a semimeasure since w e v erify the semimeasure in equalit y for ev ery su ccessful increase attempt; finally , m ′ ( u ) > lim inf m n ( u ) since if m n ( u ) > r for all n > N , then ( r , N , u )-increase do es not c hange an ything and is guaran teed to b e acceptable. The expression − log m ( x ) equals the s o-calle d pr efix complexit y K ( x ) (up to O (1) term; see [2]). The same is tru e f or relat ivized and conditional versions, an w e get the follo wing r eformulation of th e last theorem: Theorem 2.2. lim sup n →∞ K ( x | n ) = K 0 ′ ( x ) + O (1 ) . Another corollary impro ves a result of [4]. F or an y (partial) fu nction f from N to N we define the limit fr e qu e ncy of an integ er x as q f ( x ) = lim in f n →∞ # { i < n | f ( i ) = x } n In other wo rd s, we look at the fraction of x -terms in f (0) , . . . , f ( n − 1) (undefined v alues are also listed) and tak e lim inf of these frequencies. It is easy to see that for a total compu table f the function q f is a lo wer 0 ′ -semicomputable semimeasure. The argument ab o ve pr o v es the follo wing result: Theorem 2.3. F or any p artial c omputable f the function q f is upp er b ounde d by a lower 0 ′ -semic omputable semime asur e. In [4] it is sho wn that for some total computable f the function q f is a maximal lo w er 0 ′ - semicomputable semimeasure and ther efore 0 ′ -relativize d a priori probability can b e defined as maximal limit frequency for total computable functions. No w we see that th e same is true for partial computable f unctions: allo wing them to b e partial do es not increase the maximal limit frequency . The similar argumen t also is applica b le to the so-called a pr iori c omplexity defined as negativ e logarithm of a maximal lo wer semicompu table semimeasure on the b inary tree (see [8 ]). Th is complexit y is sometimes denoted as KA ( x ) and w e get th e follo wing state- men t: Theorem 2.4. lim sup n →∞ KA ( x | n ) = K A 0 ′ ( x ) + O (1 ) . LIMIT COMPLEXITIES REVISI TED 77 (T o pro v e this we defin e an increase op erat ion in s u c h a w a y that it in cr eases n ot only a ( x ) but also a ( y ) for y that are prefixes of x , if n ecessary . The increase is acceptable if a (Λ) still do es not exceed 1.) It w ould b e inte r esting to find out wh ether similar results are true for monotone com- plexit y or not (the authors do not kno w this). 3. Op en sets of small measure W e n o w try to apply the same tric k in a sligh tly different situation, f or effectiv ely op en sets. The Can tor space Ω is a set of all infi n ite sequence of zeros and on es. An interval Ω x (for a b inary string x ) is formed b y all sequences that hav e prefix x . Op en sets are unions of in terv als. An effe ctively op en subset of Ω is an enumerable union of interv als, i.e., the union of in terv als Ω x where x are tak es fr om some enumerable set of s trings. W e consider stand ard (uniform Bernou lli) measure on Ω: the in terv al Ω x has measure 2 − l where l is the length of x . A classica l theorem of measure theory says: if U 0 , U 1 , U 2 , . . . ar e op en sets of me asur e at most ε , then lim inf n U n has me asur e at most ε , and this implies that for eve ry ε ′ > ε ther e exists an op en set of me asur e at most ε ′ that c overs lim inf n U n . Indeed, lim inf n →∞ U n = [ N \ n > N U n , and the m easure of the union of an increasing sequence V N = \ n > N U n , equals the limit of measures of V N , and all these measures do not exceed ε since V N ⊂ U N . It r emains to note th at for an y measurable set X its measure is the infimum of the measures of op en sets that co v er X . W e no w can try to “effecti vize” this statemen t in the s ame wa y as w e did b e fore. First w e started with an (evident) statemen t: if U n ar e finite sets of at most 2 k elements, then lim in f n U n has at most 2 k elements and p ro ved its effect ive v ersion: for a uniformly enumer able family of op en sets U n that have at mos t 2 k elements, the set lim inf n U n is c ontaine d in a u niformly 0 ′ -enumer able set that has at most 2 k elements. Then w e d id similar thing with semimeasures (aga in, the non-effectiv e ve rs ion is trivial: it says that if P x m n ( x ) 6 1 for every n , then P x lim inf n m n ( x ) 6 1). No w th e effectiv e ve rs ion could lo ok lik e this. L et ε > 0 b e a r ational numb er and let U 0 , U 1 , . . . b e an enumer able f amily of effe ctively op en sets of me asur e at most ε e ach. Then for every r ational ε ′ > ε ther e exists a 0 ′ -effe ctively op en set of me asur e at most ε ′ that c ontains lim inf n →∞ U i = S N T n > N U n . Ho w eve r , the authors d o n ot kno w whether this is alwa ys true. Th e argument that w e ha ve us ed can nevertheless b e applied do pro ve the follo wing w eak er v ersion: Theorem 3.1. L et ε > 0 b e a r ational numb er and let U n b e an enumer able family of effe ctiv ely op en sets of me asur e at most ε e ach. Then ther e exists a uniformly 0 ′ -effe ctively op en set of me asur e at most ε that c ontains [ N In t  \ n > N U n  78 L. BIENVENU, AN. MUCH NIK, A. SHEN, AND N. VERESHCH AGI N Here In t( X ) d enotes the inte rior part of X , i.e., the un ion of all op en subsets of X . In this case w e d o n ot need ε ′ (whic h one could exp e ct since th e un ion of op e n sets is op en). Pr o of. F ollo wing the same sc heme, for ev ery string x an d intege r N we consider ( x, N )- op eration that add s Ω x to all U n suc h that n > N . T h is op eration is ac c eptable if measur es of all U n remain at most ε for eac h n . This can b e c h ec k ed using 0 ′ -oracle (if the op eration is n ot acceptable, it b e comes kno wn after a fin ite num b er of steps). W e attempt to p erform this op erati on (if acceptable) for all p airs in some compu table order. The union of all added interv als for all accepted p airs is 0 ′ -effectiv ely op en. If some sequence b elongs to the u nion of the interior parts, then it is co vered b y some inte rv al Ω u that is a s ubset of U n for all su fficien tly large n . Th en some ( u, N )- op erat ion is acceptable since it actually d o es not c hange anything and therefore Ω u is a part of an 0 ′ -op en set that w e hav e constr u cted. 4. Kolmogoro v and 2 -randomness This result has an h istorically remark able corollary . Wh en Kolmogoro v tried to define randomness in 1960ie s, he started with the follo win g app roac h. A sequence x of length n is “random” if its complexit y C ( x ) (or conditional complexit y C ( x | n ); in fact, these requirement s are almost equ iv alen t) is close to n : the r andomness deficiency d ( x ) is defined as the difference | x | − C ( x ) (here | x | stands for the length of x ). This sou n ds reasonable, bu t if w e then defin e a r andom sequence as a s equ ence whose pr efixes ha ve d eficiencies b o u n ded b y a constan t, suc h a sequence do es not exist at all: Ma r tin-L¨ of sho wed that ev ery infinite sequence has prefixes of arb itrarily large d eficiency , and suggested a differen t definition of randomness using effectiv ely null sets. Later more refined v ersions of randomness deficiency (using monotone or prefix complexit y) app eared that mak e the criterion of r andomness in terms of d eficiencies p o ssib le. But b efore that, in 1968, Kolmogoro v wrote: “Th e most natural d efinition of infi nite Bernoulli sequence is the follo wing: x is considered m -Bernoulli t yp e if m is such that all x i are initial se gments of the finite m -Bernoulli sequences. Martin- L¨ of give s another, p o ssib ly narrow er definition” ([1 ], p . 663). Here Kolmogoro v sp eaks ab out “ m -Bernoulli” finite sequence x (this means that C ( x | n, k ) is greater than log  n k  − m where n is the length of x and k is the num b er of ones in x ). F or the case of u niform Bern oulli measure (where p = q = 1 / 2) one would reformulate this definition as follo ws. L et u s define ¯ d ( x ) = inf { d ( y ) | x is a prefix of y } and require that ¯ d ( x ) is b oun d ed for all pr efixes of an in fi nite sequence ω . It is sho wn by J. Miller in [3 ] that this defi n ition is equiv alent to Martin-L¨ of r an d omness r elativized to 0 ′ (called also 2- r andomness ): Theorem 4.1. A se q u enc e ω is Martin-L¨ of 0 ′ -r andom if and only if the quantities ¯ d ( x ) for al l pr efixes x of ω ar e b ounde d by a ( c ommon ) c onstant. In turns out that this result (in one direction) easily follo ws from the previous theorem. Pr o of. Assume that ¯ d -deficiencies for pr efixes of ω are not b o un ded. According to Martin- L¨ of definition, w e ha ve to construct for a give n c an 0 ′ -effectiv ely op en set that co vers ω and has measure at m ost 2 − c . LIMIT COMPLEXITIES REVISI TED 79 Fix some c . F or eac h n consider the set D n of all sequences u of length n suc h that C ( u ) < n − c (i.e., sequences u of length n suc h that d ( u ) > c ). It has at most 2 n − c elemen ts. The r equiremen t ¯ d ( x ) > c means that ev ery string extension y of x b el ongs to D m where m is its length. This imp lies that Ω x is con tained in ev ery U m where m > | x | and U m is the set of all s equ ences that ha ve pr efixes in D m (this set has measure at most 2 − c ). Therefore, in this case the in terv al Ω x is a subset of T m > | x | U m and (b eing op en) is a sub set of its in terior. T hen we conclude (using the result prov ed ab o ve) that Ω x (=ev ery sequence with prefix x ) is co vered by an 0 ′ -effectiv ely op en set of measure at most 2 − c constructed as explained ab ov e. So if some ω has p refixes of arbitrarily large ¯ d -deficiency , then ω is not 0 ′ Martin-L¨ of random. Note that this argumen t w orks also for conditional complexit y (with length as co nd ition) and giv es a sligh tly stronger r esult. F or the s ak e of completeness w e repro duce (from [3]) th e pro of of the reverse impli- cation (essential ly u nc han ged). Assume that a sequence ω is co vered (for eac h c ) b y a 0 ′ -computable sequence of interv als I 0 , I 1 , . . . of total measure at most 2 − c . (W e omit c in our notation, but all these constructions dep e nd on c .) Using the appr o ximations 0 n instead of full 0 ′ and p erforming at most n steps of computation f or eac h n we get another (no w computable) family of interv als I n, 0 , I n, 1 , . . . suc h that I n,i = I i for ev ery i and sufficien tly large n . W e ma y assume without loss of generalit y that I n,i either has size at least 2 − n (i.e., is determined by a string of length at most n ) or equals ⊥ (a sp ecial v alue th at denotes the empty set) since only the limit b ehavio r is prescrib e d. Moreo ve r, w e m ay also assum e that I n,i = ⊥ for i > n and that the total measure of all I n, 0 , I n, 1 , . . . do es n ot exceed 2 − c for ev ery n (by deleting the excessiv e in terv als in this order; the stabilization guarantees that all limit interv als w ill b e even tually let th rough). Since I n,i is defined by inte rv als of size at least 2 − n , we get at m ost 2 n − c strings of length n co v ered b y in terv als I n,i for giv en n and all i . This set is decidable (recall that only i not exceeding n are u sed), therefore eac h string in this set can b e defined (assumin g c is known) by a string of length n − c , binary representa tion of its ordin al n umb er in this set. (Note that this strin g also determines n if c is known.) Returning to the sequence ω , w e note that it is co v ered by some I i and th erefore is co v ered by I n,i for this i and all suffi cien tly large n (after the v alue is stabilized), sa y , f or all n > N . Let u b e a prefix of ω of length N . All con tinuatio n s of u of an y length n are co v ered by I n,i and hav e complexit y less th an n − c + O (1). In fact, this is a conditional complexit y with condition c ; we get n − c + 2 log c + O (1), s o ¯ d ( u ) > c − 2 log c − O (1). Suc h a strin g u can b e found for ev ery c , therefore ω has p refixes of arbitrarily large ¯ d -deficiency . In f act a stronger statemen t th an Th eorem 4.1 is p r o v ed in [3 , 5]; our to ols are still to o w eak to get this statemen t. Ho we ver, the lo w b asis theorem h elps. 5. The low basis theorem This is a classical resu lt in recurs ion theory (see, e.g., [6]). It was used in [5] to pro ve 2-randomness criterion; analyzing this pr o of, we get theorems ab out limit complexities as b ypr o ducts. F or the sak e of completeness w e repro d uce the statemen t and the pro of of lo w-basis theorem her e; th ey are qu ite simple. 80 L. BIENVENU, AN. MUCH NIK, A. SHEN, AND N. VERESHCH AGI N Theorem 5.1. L et U ⊂ Ω b e an effe ctively op en set that do es not c oincide with Ω . Then ther e exists a se quenc e ω / ∈ U which is low, i.e., ω ′ = 0 ′ Here ω ′ is the jum p of ω ; the equ ation ω ′ = 0 ′ means th at the un iv ersal ω -e numerable set is 0 ′ -decidable. Theorem 5.1 sa ys that an y effectiv ely closed non-empt y set contai n s a lo w ele ment. F or example, if P , Q ⊂ N are enumerable inseparable sets, then the set of all separating sequences is an effectiv ely closed set that do es not conta in computable sequences. W e conclude, therefore, that there exists a non-computable lo w separating sequ en ce. Pr o of. Assume that an oracle mac h ine M and an in put x are fixed. The computation of M with oracle ω on x ma y terminate or not d ep endin g on oracle ω . Let us consider the set T ( M , x ) of all ω s u c h that M ω ( x ) terminates (for fixed mac hin e M and inp ut x ). T his set is an effectiv ely op en set (if termination happ ens, it happ ens due to finitely man y oracle v alues). This set together with U ma y co v er the ent ire Ω; this means that M ω ( x ) terminates for al l ω / ∈ U . If it is not the case, we can add T ( M , x ) to U and get a b igger effecti vely op en set U ′ that still has non-empty complement suc h that M ω ( x ) do es not terminate for al l ω ∈ U ′ . This op erati on guarantee s (in one of t w o w ays) that termination of the computatio n M ω ( x ) does not dep end on the c hoice of ω (in the remaining non-empt y effectiv ely closed set). This op eration can b e p erformed for all p airs ( M , x ) sequ entially . Note that if U ∪ T ( M , x ) co v ers the en tire Ω, this happ en s on some fi nite stag e (compactness), so 0 ′ is enough to fin d out whether it happ ens or not, and on the n ext step we ha ve again some effectiv ely op en (without any oracle) set. So 0 ′ -oracle is enou gh to sa y which of th e compu tations M ω ( x ) termin ate (as we ha ve said, this do e s not dep end of the c hoice of ω ). Therefore an y suc h ω is lo w (the univ ersal ω -en um erable set is 0 ′ -decidable). And suc h an ω exists since th e intersectio n of the d ecreasing sequence of non-empt y closed sets is non-empty (compactness). 6. Using the lo w basis theorem Let us sho w ho w Theorem 1.1 can b e pro v ed using the lo w basis theorem. As w e ha ve seen, w e ha v e an en um er ab le family of sets U n that ha v e at most 2 k elemen ts and need to constru ct effectiv ely a 0 ′ -en umerable s et that has at m ost 2 k elemen ts and con tains U ∞ = lim in f n U n . If the sets U n are (uniformly) decidable, then U ∞ is 0 ′ -en umerable an d w e d o not need an y other set. The lo w basis theorem allo w s us to reduce general case to this sp ec ial one. Let us consid er the family of all “up p er b o u nds” for U n : by an upp er b ound we mean a sequence V n of finite sets that con tain U n and s till ha ve at m ost 2 k elemen ts eac h. The sequence V 0 , V 1 , . . . can b e enco ded as an infi nite binary sequence (first we enco de V 0 , then V 1 etc.; n ote that eac h V i can b e enco d ed by a finite n umber of b its though this num b e r dep end s on V i ). F or a binary sequence the prop erty “to b e an encod ing of an up p er b ound for U n ” is effectiv ely closed (the restriction # V n < 2 k is decidable and the restriction U n ⊂ V n is co-en umerable). T herefore th e lo w basis theorem can b e app lied. W e get an up p er b ound V that is lo w. Then V ∞ = lim in f V n is (uniformly in k ) V ′ -en umerable (as w e ha v e said: with V -o r acle the family V n is un iformly decidable), but since V is lo w , V ′ -oracle can b e replaced by 0 ′ -oracle, an d w e get the desired result. LIMIT COMPLEXITIES REVISI TED 81 This pro of though b eing simple looks r ather myste rious : we get something almost out of nothing! (As far as we know, this idea in a more adv anced cont ext app eared in [5].) The same tric k can b e used to prov e Theorem 2.1: here “upp er b ounds” are distrib u- tions M n with rational v alues and fi n ite supp ort that are greater than m ( x | n ) bu t still are semimeasures. (T ec hnical correction: fir st we ha v e to assume th at m ( x | n ) = 0 if x is large, and th en w e ha v e to we ake n the restriction P M n ( x ) 6 1 replacing 1 b y , say , 2; th is is needed since the v alues m ( x | n ) ma y b e irr ational.) Theorem 2.4 can b e also pro ved in this wa y (up p er b ou n ds should b e semimeasures on tree with rational v alues and fin ite supp ort). As to Theorem 3.1 , here the application of the lo w b asis th eorem allo ws us to get a stronger result than b efore (though n ot the most strong version we mentio n ed as an op en question): Theorem 6.1. L et ε > 0 b e a r ational numb er and let U n b e an unif ormly enumer able family of effe ctively op en sets, i.e., U n = ∪{ Ω x | ( n, x ) ∈ U } for some enumer able set U ⊂ N × { 0 , 1 } ∗ . Assume that U n has me asur e at most ε for every n . A ssume also that U i has “effe ctively b ounde d gr anularity”, i.e., al l strings x such that ( n, x ) ∈ U have length at most c ( n ) wher e c is a total c omputable function. Then f or every ε ′ > ε ther e exists a 0 ′ -effe ctively op en set W of me asur e at most ε ′ that c ontains lim inf n →∞ U n = [ N \ n > N U n and this c onstruction is uniform. Pr o of. First we u s e th e low basis theorem to r educe the general case to the case w here U is decidable and for ev ery ( n, x ) ∈ U the length of x is exactly c ( n ). Indeed, define an “upp er b ou n d” as a sequence V of sets V n where V n is a set of strings of length c ( n ) suc h that U n is co v ered b y the interv als generated by elemen ts of V n . Again V can b e encod ed as an infin ite sequence of zeros and ones, and the prop ert y “to b e an upp er b ound” is effectiv ely closed. Ap p lying the lo w basis theorem, we c ho ose a lo w V and add it is an oracle. Since V ′ is equiv alen t to 0 ′ , for our purp ose we ma y assume that V is decidable. No w w e ha ve to deal with the d ecidable case. Let us rep resen t the s et U ∞ as a un ion of the disjoin t sets F 0 = \ i U i , F 1 = \ i > 1 U i \ U 0 , F 2 = \ i > 2 U i \ U 1 , . . . (for eac h elemen t x in U ∞ w e consider the last U i that do es n ot con tain x ). Each of F i is (in the d ecidable case) an effecti vely closed set (recall th an U i is op en-cl osed due to the restriction on c ( i )). Moreo ve r , the F i are pairwise disjoin t and the family F i satisfies lim inf n → + ∞ U n = [ i F i and th us X i µ ( F i ) = µ (lim in f n → + ∞ U n ) . 82 L. BIENVENU, AN. MUCH NIK, A. SHEN, AND N. VERESHCH AGI N The measure of eac h of F i is 0 ′ -computable, and usin g 0 ′ -oracle we can fi nd a finite set of in terv als that co v ers F i and has measure µ ( F i ) + ( ε ′ − ε ) / 2 i +1 Putting all these in terv als together, we get the desired set W . So th e decidable case (and therefore the general one, thanks to lo w b asis th eorem) is completed. 7. Corollary on 2-random ness Theorem 6.1 can b e used to pro ve 2-randomness criterion f rom [3, 5 ]. In fact, this giv es exactly the pro of from [5]; the on ly thin g w e did is structur in g the pr o of in t wo p arts (form ulating Th eorem 6.1 explicitly and putting it in the con text of other resu lts on limits of complexities). Theorem 7.1 ([3, 5]) . A se q uenc e ω is 0 ′ Martin-L¨ of r andom if and only if C ( ω 0 ω 1 . . . ω n − 1 ) > n − c for some c and for infinitely many n . Pr o of. Let us first un derstand the r elation b et we en this theorem and T heorem 4.1. I f C ( ω 0 ω 1 . . . ω n − 1 ) > n − c for infi nitely man y n and giv en c , then ¯ d ( x ) 6 c for eve ry prefix x of ω (indeed, one can find the requir ed con tin u ation of x among p refixes of ω ). As we kno w, th is guarant ees that ω is 0 ′ Martin-L¨ of rand om. It remains to pro ve that if for all c we ha ve C ( ω 0 ω 1 . . . ω n − 1 ) < n − c for all s ufficien tly large n , then ω is not 0 ′ -random. Using the s ame notation as in the p r o of of Theorem 4.1, w e can sa y that ω has a prefix in D n and therefore b elo n gs to U n for all sufficien tly large n . W e can apply then T heorem 6.1 sin ce U n is defined u sing strings of length n (so c ( n ) = n ) and co ve r U ∞ (and therefore ω ) b y a 0 ′ -effectiv ely op en set of small measure. Sin ce this can b e uniformly done for all c , th e sequen ce ω is not 0 ′ -random. Remark . The results ab o ve may b e considered as sp ecial cases of an effectiv e ve rsion of a classical theorem in measur e theory: F atou’s lemma. This lemma guarante es that if R f n ( x ) dµ ( x ) 6 ε for µ -measurable f u nctions f 0 , f 1 , f 2 , . . . , then Z lim inf n → + ∞ f n ( x ) dµ ( x ) 6 ε. The constructiv e version assumes that f i are low er semicomputable and satisfy some ad- ditional conditions; it says that for every ε ′ > ε there exists a lo w er 0 ′ -semicomputable function ϕ suc h that lim in f f n ( x ) 6 ϕ ( x ) for eve r y x and R ϕ ( x ) dµ ( x ) 6 ε ′ . LIMIT COMPLEXITIES REVISI TED 83 References [1] Kolmogoro v A.N., Logical Basis for I nformation Theory and Probability Theory . IEEE T r ansactions on I nformation The ory , v. I T-14, N o. 5, S ept. 1968. (Russian version was pu blished in 1969.) [2] Li M., Vit´ an yi P ., An Intr o duction to Kolmo gor ov Complexity and Its Applic ations , Second Edition, Springer, 1997. (638 pp.) [3] Mille r J., Every 2-random real is Kolmogoro v random, Journal of Symb olic L o gic , 69 (2):555–584 (2004). [4] Muc hnik A n .A., Low er limits of frequencies in computable sequences an d relativized a priori probabilit y , SIAM T he ory Pr ob ab. Appl., 1987, vol. 32, p. 513–514 . 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