Equivalent characterizations of partial randomness for a recursively enumerable real

Equiv alen t c haracterizations of partial randomness for a recursiv ely en umerable real Koh taro T adaki Researc h and Dev elopment Initiativ e, Chuo Univ ersit y 1–13– 27 Kasuga, Bunkyo- ku, T oky o 112-8 551, Japan E-mail: tadaki@k c.c huo-u.ac.jp Abstract. A real num b er α is called recursively en u merable if there exists a com- putable, increasing sequence of rational n um b ers which con verges to α . The randomness of a recursiv ely enumerable real α can b e c haracterized in v arious w a ys using eac h of the notions; program-size complexit y , Martin-L¨ of test, Chaitin’s Ω n umber, the domination and Ω-lik eness of α , the universali t y of a compu table, in cr easing sequence of rational n um b ers whic h conv er ges to α , and u n iv ersal prob ab ility . In this pap er, w e generalize these c haracterizations of ran d omness o ve r the notion of partial rand omness by pa- rameterizing eac h of the notions ab o ve b y a r eal num b er T ∈ (0 , 1]. W e th u s present sev eral equiv alent c h aracterizat ions of partial randomness for a recurs iv ely en umerable real num b er . Key wor ds : algorithmic rand omness, recursiv ely enumerable real n u m b er, partial ran- domness, Chaitin’s Ω n u m b er, program-size complexit y , unive rsal pr obabilit y 1 In tro duction A real num b er α is called r e cursively enumer able (“r.e.” for s h ort) if th er e exists a computable, increasing sequence of rational num b ers whic h con verges to α . The randomness of an r.e. real α can b e c haracterized i n v arious w ays using eac h of th e notions; pr o gr am-size c omplexity , Martin-L¨ o f test , Chaitin ’s Ω numb er , the domination and Ω -likeness of α , the u niversality of a computable, increasing s equ ence of rational n um b ers wh ic h con verges to α , and univ ersal pr ob ability . These equiv alen t c haracterizatio ns of rand omness for an r .e. real n u m b er are summ arized in Th eorem 3.4 (see Section 3), wh ere the equiv alences are established b y a series of wo rks of S c hn orr [13], Chaitin [4], S olo v a y [14], Calude, Hertling, K houssaino v and W ang [1], Ku ˇ cera and Slaman [8], and T adaki [17]. In this pap er, w e generalize these c haracterizations of r andomness o v er the notion of p artial r andomness , which was in tro duced b y T adaki [15, 16]. W e in tro duce s everal c h aracterizat ions of partial rand omn ess f or an r.e. real num b er by p arameterizing eac h of the notions ab o ve on randomness b y a real num b er T ∈ (0 , 1]. W e prov e the equiv alence of all these c h aracteriza tions of partial randomn ess in Theorem 4.6, our main r esult, in S ection 4. The pap er is organized as follo ws . W e b egin in Section 2 with some p reliminaries to algorithmic information theory and partial randomness. In Section 3, w e review the previous r esults on the equiv alen t c haracterizations of r andomness for an r.e. r eal n u m b er. Our main result on partial randomness of an r.e. real n um b er is presente d in Section 4, and its pro of is completed in Section 5. In Section 6, we inv estigate some pr op erties of the notion of T -c onver ge nc e for an in creasing sequence of real n umbers, whic h pla ys a cru cial r ole in our c haracterizatio ns of partial r an d omness. W e conclude this p ap er with a m ention of the fu ture direction of this work in Section 7. 1 2 Preliminaries 2.1 Basic notation W e start with s ome notation ab out num b er s and strings whic h will b e used in this pap er. # S is the cardinalit y of S for an y s et S . N = { 0 , 1 , 2 , 3 , . . . } is the set of natur al num b ers , and N + is the set of p ositiv e in tegers. Q is the s et of rational n um b ers, and R is the set of real num b ers . A sequence { a n } n ∈ N of n um b ers (rational num b ers or real n umb er s ) is called i ncr e asing if a n +1 > a n for all n ∈ N . { 0 , 1 } ∗ = { λ, 0 , 1 , 00 , 01 , 10 , 11 , 00 0 , 001 , 010 , . . . } is the set of finite bin ary strings where λ de- notes the empty string , and { 0 , 1 } ∗ is ordered as indicated. W e id entify any s tring in { 0 , 1 } ∗ with a natural num b er in this order, i.e., w e consid er ϕ : { 0 , 1 } ∗ → N s uc h that ϕ ( s ) = 1 s − 1 w here th e concatenati on 1 s of strings 1 and s is regarded as a dya dic intege r, and then w e ident ify s with ϕ ( s ). F or any s ∈ { 0 , 1 } ∗ , | s | is the length of s . A sub s et S of { 0 , 1 } ∗ is called a pr efix-fr e e set if no string in S is a pr efix of another string in S . F or any partial function f , the domain of definition of f is d enoted b y dom f . W e write “r.e.” instead of “recursively en umerable.” Normally , o ( n ) denotes an y fun ction f : N + → R such that lim n →∞ f ( n ) /n = 0. O n the other hand, O (1) denotes an y fu nction g : N + → R su c h that there is C ∈ R with the prop erty that | g ( n ) | ≤ C for all n ∈ N + . Let α b e an arbitrary real num b er. W e denote α − ⌊ α ⌋ by α mo d 1, where ⌊ α ⌋ is the greatest in teger less than or equal to α . Hence, α mo d 1 ∈ [0 , 1). Normally , ⌈ α ⌉ denotes th e smallest in teger greater than or equal to α . W e denote by α n ∈ { 0 , 1 } ∗ the fi rst n bits of the base-t wo exp an s ion of α mo d 1 with infin itely man y zeros. Th us, in particular, if α ∈ [0 , 1), then α n denotes the firs t n bits of the base-t wo expansion of α with in finitely many zeros. F or example, in th e case of α = 5 / 8, α 6 = 10100 0. A r eal num b er α is called r.e. if there exists a computable, increasing s equ ence of rational n um b ers which con verges to α . An r.e. real num b er is also called a left- c omputable real num b er . On the other han d , a r eal num b er α is called right-c omputable if − α is left-computable. W e s ay that a real num b er α is c omputable if there exists a computable sequence { a n } n ∈ N of rational n um b ers suc h that | α − a n | < 2 − n for all n ∈ N . It is then easy to see that, for every α ∈ R , α is computable if and only if α is b oth left-computable and right-c omputable. A sequence { a n } n ∈ N of real num b ers is called c omputable if there exists a total recursive function f : N × N → Q such th at | a n − f ( n, m ) | < 2 − m for all n, m ∈ N . See e.g. Pour-El and Ric h ards [11] and W eihrauc h [20] for the detail of the treatment of the compu tabilit y of real num b ers and sequen ces of real num b ers. 2.2 Algorithmic information theory In the follo wing w e concisely review some definitions and results of algorithmic inform ation the- ory [4, 5]. A c omputer is a partial recursiv e fu nction C : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that dom C is a prefix-free set. F or eac h computer C and eac h s ∈ { 0 , 1 } ∗ , H C ( s ) is defined by H C ( s ) = min  | p |   p ∈ { 0 , 1 } ∗ & C ( p ) = s  . A computer U is said to b e optimal if for eac h computer C there exists a constant sim( C ) with the follo wing prop ert y; if C ( p ) is d efi ned, then there is a p ′ for wh ic h U ( p ′ ) = C ( p ) and | p ′ | ≤ | p | + sim( C ). It is easy to see that there exists an optimal computer. W e choose a p articular optimal computer U as th e stand ard one for use, and d efine H ( s ) as H U ( s ), whic h is referred to as the pr o gr am-size c omplexity of s , the information c ontent of s , or the Kolmo gor ov c omplexity of s [7, 9, 4]. Thus, H ( s ) ≤ H C ( s ) + sim( C ) for eve ry computer 2 C . Let V b e a n arbitrary optimal computer. F or eac h s ∈ { 0 , 1 } ∗ , P V ( s ) is defined as P V ( p )= s 2 −| p | . Chaitin ’s halting pr ob ability Ω V of V is defined by Ω V = X p ∈ dom V 2 −| p | . Th us, Ω V = P s ∈{ 0 , 1 } ∗ P V ( s ). Definition 2.1 (wea k Chaitin r andomness, Ch aitin [4, 5]) . F or any α ∈ R , we say that α is we akly Chaitin r andom if ther e exists c ∈ N such that n − c ≤ H ( α n ) for al l n ∈ N + . Chaitin [4] sho w ed that, for every optimal computer V , Ω V is wea kly Ch aitin random. Definition 2.2 (Martin-L¨ of randomness, Martin-L¨ of [10]) . A subset C of N + × { 0 , 1 } ∗ is c al le d a Martin-L¨ of test if C is an r.e. set an d ∀ n ∈ N + X s ∈ C n 2 −| s | ≤ 2 − n , wher e C n =  s   ( n, s ) ∈ C  . F or any α ∈ R , we say that α is Martin-L¨ of r andom if f or every Martin-L¨ of test C , ther e exists n ∈ N + such that, for every k ∈ N + , α k / ∈ C n . Theorem 2.3 (Schnorr [13]) . F or every α ∈ R , α is we akly Chaitin r andom if and only if α is Martin-L¨ of r andom. It follo w s fr om Theorem 2.3 that Ω V is Martin-L¨ of random for ev ery optimal computer V . The program-size complexit y H ( s ) is originally defined using the concept of program-size, as stated ab ov e. How ev er, it is p ossible to define H ( s ) without referring to su c h a concept, i.e., as in the follo wing, w e fi rst int ro duce a universal pr ob ability m , and th en define H ( s ) as − log 2 m ( s ). A unive rsal pr obabilit y is defined as follo ws [21]. Definition 2.4 (u n iv ersal p robabilit y) . A function r : { 0 , 1 } ∗ → [0 , 1] is c al le d a lower-c omputable semi-me asur e if P s ∈{ 0 , 1 } ∗ r ( s ) ≤ 1 and the set { ( a, s ) ∈ Q × { 0 , 1 } ∗ | a < r ( s ) } is r.e. We say that a lower-c omputable semi- me asur e m i s a unive rsal pr ob ability if for every lower-c omputable semi-me asur e r , ther e exists c ∈ N + such that, for al l s ∈ { 0 , 1 } ∗ , r ( s ) ≤ cm ( s ) . The follo win g th eorem can b e then sho wn (see e.g. Theorem 3.4 of Ch aitin [4] for its pro of ). Theorem 2.5. F or every optimal c omputer V , b oth 2 − H V ( s ) and P V ( s ) ar e universal pr ob abilities. By Theorem 2.5, w e see that H ( s ) = − log 2 m ( s ) + O (1) for ev ery u niv ers al probabilit y m . Thus it is p ossible to d efine H ( s ) as − log 2 m ( s ) with a particular un iv ersal probabilit y m instead of as H U ( s ). Note that the difference u p to an additiv e constant is nonessentia l to algo rithmic in formation theory . Any univ ersal probabilit y is not compu table, as corresp ond s to the u n computabilit y of H ( s ). As a resu lt, w e see th at 0 < P s ∈{ 0 , 1 } ∗ m ( s ) < 1 for every unive rsal pr obabilit y m . 3 2.3 P artial r andomness In the w orks [15, 16], we generalized the n otion of th e rand omness of a r eal num b er so that the de gr e e of the r andomness , which is often referred to as the p artial r andomness recen tly [2, 12, 3], can b e c h aracterized by a real num b er T w ith 0 < T ≤ 1 as follo ws. Definition 2.6 (w eak Chaitin T -rand omness) . L et T ∈ R with T ≥ 0 . F or any α ∈ R , we say that α is we akly Chaitin T -r andom if ther e exists c ∈ N such that T n − c ≤ H ( α n ) for al l n ∈ N + . Definition 2.7 (Martin-L¨ of T -randomness) . L et T ∈ R with T ≥ 0 . A su b set C of N + × { 0 , 1 } ∗ is c al le d a Martin-L¨ of T - test if C is an r.e. set and ∀ n ∈ N + X s ∈ C n 2 − T | s | ≤ 2 − n . F or any α ∈ R , we say that α is M artin-L¨ of T -r andom if for e very M artin-L¨ of T -test C , ther e exists n ∈ N + such that, for e v ery k ∈ N + , α k / ∈ C n . In the case where T = 1, the weak Chaitin T -randomness and Martin-L¨ of T -randomness result in wea k Chaitin randomness and Martin-L¨ of randomness, resp ectiv ely . T adaki [16] generalized Theorem 2.3 o ver the n otion of T -randomness as follo ws. Theorem 2.8 (T adaki [16]) . L et T b e a c omputable r e al numb er with T ≥ 0 . Then, for every α ∈ R , α is we akly Chaitin T -r andom if and only i f α is M artin-L¨ of T -r andom. Definition 2.9 ( T -compressibilit y) . L et T ∈ R with T ≥ 0 . F or any α ∈ R , we say that α is T -c ompr essible if H ( α n ) ≤ T n + o ( n ) , which is e quivalent to lim n →∞ H ( α n ) n ≤ T . F or every T ∈ [0 , 1] and eve ry α ∈ R , if α is weakly Chaitin T -random and T -compressible, then lim n →∞ H ( α n ) n = T , (1) and therefore the c ompr ession r ate of α b y the p rogram-size complexit y H is equal to T . Note, ho wev er, that (1) do es not necessarily imp ly that α is w eakly Chaitin T -random. In the works [15, 16], we generalized Chaitin’s halting pr obabilit y Ω to Ω( T ) as f ollo ws. F or eac h optimal computer V and eac h real n um b er T > 0, th e gener alize d halting pr ob ability Ω V ( T ) of V is defined by Ω V ( T ) = X p ∈ dom V 2 − | p | T . Th us, Ω V (1) = Ω V . I f 0 < T ≤ 1, then Ω V ( T ) con verges and 0 < Ω V ( T ) < 1, s in ce Ω V ( T ) ≤ Ω V < 1. The follo win g th eorem holds for Ω V ( T ). Theorem 2.10 (T adaki [15, 16]) . L et V b e an op timal c omputer and let T ∈ R . (i) If 0 < T ≤ 1 and T is c omputable, then Ω V ( T ) is we akly Chaitin T -r andom and T -c ompr essible. 4 (ii) If 1 < T , then Ω V ( T ) diver ges to ∞ . Note also that the compu tabilit y of Ω V ( T ) giv es a su fficien t condition for a real n um b er T ∈ (0 , 1) to b e a fixe d p oint on p artial r andomness as follo ws . Theorem 2.11 (T adaki [18]) . L et V b e an optimal c omputer. F or every T ∈ (0 , 1) , if Ω V ( T ) is c omputable, then T is we akly Chaitin T -r andom and T - c ompr essible, and ther efor e lim n →∞ H ( T n ) n = T . 3 Previous results on the randomness of an r.e. real In this section, w e review the previous results on the randomness of an r.e. r eal num b er. First we review some notions on r .e. real num b ers. Definition 3.1 (Ω-lik eness) . F or any r.e. r e al numb ers α and β , we say that α dominates β if ther e ar e c omputable, incr e asing se quenc es { a n } and { b n } of r ational numb ers and c ∈ N + such that lim n →∞ a n = α , lim n →∞ b n = β , and c ( α − a n ) ≥ β − b n for al l n ∈ N . An r.e. r e al numb er α is c al le d Ω -like if it dominates al l r.e. r e al numb ers. Solo v a y [14] sh o wed the follo wing theorem. F or its pro of, s ee also T h eorem 4.9 of [1]. Theorem 3.2 (Solo v a y [14]) . F or ev ery r.e. r e al numb ers α and β , if α dominates β then H ( β n ) ≤ H ( α n ) + O (1) . Definition 3.3 (u n iv ersality) . A c omputable, incr e asing and c onver ging se quenc e { a n } of r ational numb ers is c al le d universal if for every c omputable, incr e asing and c onver g i ng se quenc e { b n } of r ational numb ers ther e exists c ∈ N + such tha t c ( α − a n ) ≥ β − b n for al l n ∈ N , wher e α = lim n →∞ a n and β = lim n →∞ b n . The previous results on the equiv alent c haracterizations of ran d omness for an r .e. real n umb er are summ arized in the follo win g th eorem. Theorem 3.4 ([13, 4 , 14, 1, 8, 17]) . L et α b e an r.e. r e al numb er with 0 < α < 1 . Then the fol lowing c onditions ar e e quivalent: (i) The r e al numb er α is we akly Chaitin r andom. (ii) The r e al numb er α is Martin-L¨ o f r andom. (iii) The r e al numb er α is Ω -like. (iv) H ( β n ) ≤ H ( α n ) + O (1) for every r.e. r e al numb er β . (v) Ther e exists an optimal c omputer V su ch that α = Ω V . (vi) Ther e exists a universal pr ob ability m such that α = P s ∈{ 0 , 1 } ∗ m ( s ) . 5 (vii) Every c omputable, incr e asing se que nc e of r ational numb ers which c onver ges to α i s universal. (viii) Ther e exi sts a u niversal c omputable, i ncr e asing se quenc e of r ational numb ers which c onver ges to α . The historical remark on the pro ofs of equiv alences in Theorem 3.4 is as follo ws. S c hn orr [13] sho wed that (i) and (ii) are equiv alent to eac h other. Ch aitin [4] sh ow ed that (v) implies (i). Solo v a y [14] sho wed that (v) implies (iii), (iii) implies (iv), and (iii ) implies (i). Calude, Hertling, Khoussaino v, and W ang [1] show ed that (iii) implies (v), and (v) implies (vii). Kuˇ cera and Slaman [8] s h o wed that (ii) im p lies (vii). Finally , (vi) was inserted in the course of the deriv ation from (v) to (viii) by T adaki [17]. 4 New results on the partial randomness of an r.e. real In this sectio n, w e generalize Theorem 3.4 ab o ve o v er the notion of partial rand omn ess. F or that purp ose, we firs t in tro d uce some new notio ns. L et T b e an arbitrary real n um b er w ith 0 < T ≤ 1 throughout the rest of this p ap er. These notions are parametrized b y the real n umb er T . Definition 4.1 ( T -conv ergence) . An incr e asing se quenc e { a n } o f r e al numb ers is c al le d T -c onver gent if P ∞ n =0 ( a n +1 − a n ) T < ∞ . An r.e. r e al numb er α is c al le d T -c onver gent if ther e e xi sts a T - c onver gent c omputable, incr e asing se quenc e of r ational numb ers which c onver ges to α . Note that ev ery increasing and con v erging sequence of real n u mb ers is 1-con v ergent, and th u s ev ery r.e. real n u m b er is 1-con v ergent. In general, based on th e follo wing lemma, we can fr eely switc h from “ T -con v ergen t computable, increasing sequence of real num b ers” to “ T -con v er gent computable, increasing sequence of ratio nal n umbers .” Lemma 4.2. F or e very α ∈ R , α is an r.e. T -c onver gent r e al numb er if and only if ther e exists a T -c onver gent c omputable, incr e asing se quenc e of r e al numb e rs which c onver ges to α . Pr o of. The “only if ” p art is obvious. W e show the “if ” part. Supp ose that { a n } is a T -con ve rgen t computable, increasing sequence of real num b ers whic h con verges to α . Then, w e first see that there exists a computable sequ en ce { b n } of rational num b ers such that a n < b n < a n +1 for all n ∈ N . Ob viously , { b n } is an increasing sequence of rational num b ers wh ic h con ve rges to α . O n the other hand, using the inequalit y ( x + y ) t ≤ x t + y t for real n umb ers x, y > 0 and t ∈ (0 , 1], we see that ( b n +1 − b n ) T < ( a n +2 − a n ) T ≤ ( a n +2 − a n +1 ) T + ( a n +1 − a n ) T . Thus, since P ∞ n =0 ( a n +2 − a n +1 ) T and P ∞ n =0 ( a n +1 − a n ) T b oth con verge, the increasing sequen ce { b n } of rational num b ers is T - con verge n t. The follo win g argument illustrates the wa y of usin g Lemm a 4.2 : Let V b e an optimal com- puter, and let p 0 , p 1 , p 2 , . . . b e a recur siv e enumeration of the r.e. set dom V . Then Ω V ( T ) = P ∞ i =0 2 −| p i | /T , and the increasing sequence  P n i =0 2 −| p i | /T  n ∈ N of real num b ers is T -con ve rgen t since Ω V = P ∞ i =0 2 −| p i | < 1. If T is computable, then this sequence of real num b ers is compu table. Th us, by Lemma 4.2 w e ha v e Theorem 4.3 b elo w. Theorem 4.3. L et V b e an optimal c omputer. If T is c omputable, then Ω V ( T ) is an r.e. T - c onver gent r e al numb er. 6 Definition 4.4 (Ω( T )-lik eness) . An r.e. r e al numb er α is c al le d Ω( T ) -like if it dominates al l r.e. T - c onver gent r e al numb ers. Note that an r.e. real num b er α is Ω(1)-lik e if and only if α is Ω-like . Definition 4.5 ( T -u niv ers alit y) . A c omputable, incr e asing and c onver ging se que nc e { a n } of r atio- nal numb e rs is c al le d T -universal if for every T -c onver gent c omputable, incr e asing and c onver ging se quenc e { b n } of r ational numb ers ther e exists c ∈ N + such that c ( α − a n ) ≥ β − b n for al l n ∈ N , wher e α = lim n →∞ a n and β = lim n →∞ b n . Note th at a computable, increasing and con verging sequence { a n } of rational n u m b ers is 1- unive rsal if and only if { a n } is un iv ersal. Using the n otions in tro du ced ab o v e, Theorem 3.4 is ge neralized as f ollo ws. Theorem 4.6 (main result) . L et α b e an r.e. r e al numb er with 0 < α < 1 . Supp ose that T is c omputable. Then the fol lowing c onditions ar e e quivalent: (i) The r e al numb er α is we akly Chaitin T - r andom. (ii) The r e al numb er α is Martin-L¨ o f T -r andom. (iii) The r e al numb er α is Ω( T ) -like . (iv) H ( β n ) ≤ H ( α n ) + O (1) for every r.e. T -c onver gent r e al numb er β . (v) F or every r.e. T -c onver gent r e al numb er γ > 0 , ther e e xi st an r.e. r e al numb er β ≥ 0 and a r ational numb er q > 0 such tha t α = β + q γ . (vi) Ther e exist an optimal c omputer V and an r.e. r e al numb er β ≥ 0 such that α = β + Ω V ( T ) . (vii) Ther e e xi sts a universal pr ob ability m such that α = P s ∈{ 0 , 1 } ∗ m ( s ) 1 T . (viii) Every c omputable, incr e asing se quenc e of r ational numb ers which c onver ges to α is T -u niversal. (ix) Ther e exists a T -universal c omputable, incr e asing se quenc e of r ational numb ers which c on- ver ges to α . The condition (vi) of Theorem 4.6 corresp onds to the condition (v) of Th eorem 3.4 . Note, ho wev er, that, in the condition (vi) of Theorem 4.6 , a n on -n egativ e r.e. real num b er β is n eeded. The r eason is as follo w s: In the case of β = 0, the p ossibilit y that α is w eakly Chaitin T ′ -random with a real n u m b er T ′ > T is excluded by the T -compressibility of Ω V ( T ) imp osed by Th eorem 2.10 (i). How ev er, this exclusion is inconsisten t with th e condition (i) of Theorem 4.6. Theorem 4.6 is pro ved as follo ws, partially b ased on Theorems 5.3, 5.4, 5.5, and 5.6, wh ic h will b e pr o ve d in the n ext sectio n. Pr o of of The or em 4.6. W e pro v e the equiv alences in Th eorem 4.6 by sho w in g the t w o paths [A] and [B] of implicatio ns b elo w. [A] Th e implications (i) ⇒ (ii) ⇒ (v) ⇒ (vi) ⇒ (i): First, b y T heorem 2.8, (i) implies (ii) ob viously . It follo ws from Theorem 5.3 b elo w that (ii) implies (v), and also it follo ws from Theorem 5.4 b elo w that (v) implies (vi). F or the forth implication, let V b e an optimal computer, and let β 7 b e an r.e. r eal num b er. It is then easy to show that β + Ω V ( T ) dominates Ω V ( T ) (see the condition 2 of Lemma 4.4 of [1]). It f ollo ws from Theorem 3.2 and Theorem 2.10 (i) that the condition (vi) results in the condition (i) of Theorem 4.6. [B] The implications (v) ⇒ (vii) ⇒ (viii) ⇒ (ix) ⇒ (iii) ⇒ (iv) ⇒ (i): First, it follo ws f r om Theorem 5.5 b elo w that (v) implies (vii), and also it f ollo ws from Theorem 5.6 b elo w that (vii) implies (viii). Obviously , (viii) implies (ix) and (ix) implies (iii). It follo w s from Theorem 3.2 th at (iii) implies (iv). Finally , n ote that Ω U ( T ) is an r .e. T -con v ergen t real num b er wh ic h is w eakly Chaitin T -random b y Theorem 2.10 (i) and Theorem 4.3. Thus, b y s etting β to Ω U ( T ) in the condition (iv), the condition (iv) results in the condition (i). As a consequence of Theorem 4.6, we obtain the follo w ing co rollary , for example. Corollary 4.7. Supp ose that T is c omputable. Then, for every two op timal c omputers V and W , H ((Ω V ( T )) n ) = H ((Ω W ( T )) n ) + O (1) . Pr o of. Corollary 4.7 follo w s immediately from Theorem 4.3 and the implication (vi) ⇒ (iv) of Theorem 4.6. 5 The comp letion of the pro of of the main r esult In this section, we pro v e seve ral theorems needed to complete the pro of of Th eorem 4.6. F or the sak e of con v enience, we firs t rephrase the d efinition of Martin-L¨ of T -randomness of a real n umber as follo ws. W e denote b y I th e set { ( n, q , r ) ∈ N + × Q × Q | q < r } . A su b set D of I is called a r ational M artin-L¨ of T -test if D is an r .e. set and ∀ n ∈ N + X ( q , r ) ∈ D ( n ) ( r − q ) T ≤ 2 − n , where D ( n ) =  ( q , r )   ( n, q , r ) ∈ D  . W e can then sh o w the follo wing lemma, which rephrases the definition of the Martin-L¨ of T -rand omness of a r eal n um b er to giv e it more fl exibilit y . Lemma 5.1. F or eve ry α ∈ R , α is Martin-L¨ of T -r andom if and only if for every r ational Martin- L¨ of T -test D , ther e e xi sts n ∈ N + such that, for every q , r ∈ Q , if ( q , r ) ∈ D ( n ) then α / ∈ [ q , r ] , wher e [ q , r ] = { x ∈ R | q ≤ x ≤ r } . Pr o of. First, w e sho w the “if ” part b y sho win g its con trap osition. Sup p ose that α is not Martin-L¨ of T -random. T hen there exists a Martin-L¨ of T -test C suc h that ∀ n ∈ N + ∃ k ∈ N + α k ∈ C n . (2) W e define a set D ⊂ I by D = { ( n, 0 .s + ⌊ α ⌋ , 0 .s + 2 −| s | + ⌊ α ⌋ ) | s ∈ C n } . Since C is an r.e. set, D is also an r.e. set. W e also s ee that, for eac h n ∈ N + , X ( q , r ) ∈ D ( n ) ( r − q ) T = X s ∈ C n 2 − T | s | ≤ 2 − n . 8 Th us, D is a rational Martin-L¨ of T -test. On the other hand , note that β ∈ [0 .β k + ⌊ β ⌋ , 0 .β k + 2 −| β k | + ⌊ β ⌋ ] f or ev ery β ∈ R and ev ery k ∈ N + . It follo ws from (2) th at, for ev ery n ∈ N + , there exist q , r ∈ Q suc h th at ( q , r ) ∈ D ( n ) and α ∈ [ q , r ]. T h is completes the pro of of the “if ” part. Next, w e show the “only if ” part b y sho wing its contraposition. Sup p ose that there exists a rational Martin-L¨ of T -test D such that ∀ n ∈ N + ∃ q , r ∈ Q [ ( q , r ) ∈ D ( n ) & α ∈ [ q , r ] ] . (3) In the case of α ∈ Q , α is not Martin-L¨ of T -random, obviously . This can b e sho wn as follo w s. W e choose an y one m ∈ N + with T m ≥ 1. W e then defin e a set C ⊂ N + × { 0 , 1 } ∗ b y C = { ( n, α mn ) | n ∈ N + } . Recall here that α mn ∈ { 0 , 1 } ∗ denotes the fi rst mn bits of the b ase-t wo expansion of α mo d 1 with infi nitely man y ze ros. O b viously , C is an r.e. s et. W e al so see th at, for ea c h n ∈ N + , X s ∈ C n 2 − T | s | = 2 − T mn ≤ 2 − n . Therefore, C is a Martin-L¨ of T -test. On the other hand, α mn ∈ C n for ev ery n ∈ N + . Hence, α is not Martin-L¨ of T -rand om, as desired. Th us, in w hat follo ws, w e assume th at α / ∈ Q . W e c ho ose an y one n 0 ∈ N such that 2 − n 0 T < min { α − ⌊ α ⌋ , ⌊ α ⌋ + 1 − α } . W e then define a set D (0) ⊂ I b y D (0) = { ( n, q − ⌊ α ⌋ , r − ⌊ α ⌋ ) | n ∈ N + & ( n + n 0 , q , r ) ∈ D & ⌊ α ⌋ < q , r < ⌊ α ⌋ + 1 } . Ob viously , D (0) is an r .e. set. W e also see that X ( q , r ) ∈ D (0) ( n ) ( r − q ) T ≤ X ( q , r ) ∈ D ( n + n 0 ) ( r − q ) T ≤ 2 − ( n + n 0 ) ≤ 2 − n for eac h n ∈ N + . Thus, D (0) is a rational Martin-L¨ of T -test, and also D (0) ⊂ N + × (0 , 1) × (0 , 1). O n the other hand, b y the c hoice of n 0 , it is easy to see that, for every ( n, q , r ) ∈ I , if ( q , r ) ∈ D ( n + n 0 ) and α ∈ [ q , r ], then r − q ≤ 2 − n 0 /T and therefore ⌊ α ⌋ < q , r < ⌊ α ⌋ + 1. It foll o ws from (3) th at ∀ n ∈ N + ∃ q , r ∈ Q [ ( q , r ) ∈ D (0) ( n ) & α mo d 1 ∈ [ q , r ] ] . (4) F or eac h q , r ∈ Q with 0 < q < r < 1, let v ( q , r ) and w ( q , r ) b e finite binary str in gs suc h that (i) v ( q , r ) = q k and w ( q , r ) = r k for some k ∈ N + , and (ii) v ( q , r ) + 1 = w ( q , r ) wh ere v ( q , r ) and w ( q , r ) are regarded as a d y adic integ er. S uc h a pair ( v ( q , r ) , w ( q , r )) of finite bin ary strings exists uniquely sin ce 0 < q < r < 1. Then, for ev ery q , r ∈ Q w ith 0 < q < r < 1, it follo w s that (i) 2 −| v ( q, r ) | = 2 −| w ( q ,r ) | ≤ r − q , and (ii) for every β ∈ R , if β mo d 1 ∈ [ q , r ] then there exists k ∈ N + suc h that either β k = v ( q , r ) or β k = w ( q , r ). W e define a set C ⊂ N + × { 0 , 1 } ∗ b y C = [ ( n +1 ,q ,r ) ∈D (0) { ( n, v ( q , r )) , ( n , w ( q , r )) } . 9 Note that, giv en q , r ∈ Q with 0 < q < r < 1, one can compute b oth v ( q , r ) and w ( q , r ). Th us, since D (0) is an r .e. set, C is also an r .e. set. W e also s ee that, for eac h n ∈ N + , X s ∈ C n 2 − T | s | = X ( q , r ) ∈D (0) ( n +1) n 2 − T | v ( q ,r ) | + 2 − T | w ( q, r ) | o ≤ X ( q , r ) ∈D (0) ( n +1) 2( r − q ) T ≤ 2 − n . Th us, C is a Martin-L¨ of T -test. On the other hand, it is easy to see that, for eve ry ( n, q , r ) ∈ I , if ( q, r ) ∈ D (0) ( n + 1) and α mo d 1 ∈ [ q , r ], then there exists k ∈ N + suc h that either α k = v ( q , r ) or α k = w ( q , r ), and therefore ( n, α k ) ∈ C . It follo ws from (4 ) that, for ev ery n ∈ N + , there exists k ∈ N + suc h that α k ∈ C n . Thus, α is n ot Martin-L¨ of T -random. This co mpletes the pro of of the “only if ” part. Lemma 5.2 and Theorem 5.7 b elo w can b e p ro ved, based on the generaliza tion of the tec hniques used in the pro of of Theorem 2.1 of Kuˇ cera and Slaman [8] o v er p artial randomness. W e also us e Lemma 5.1 to pro v e Lemma 5.2 b elo w. Lemma 5.2. L et α b e an r.e. r e al numb er, and let { d n } b e a c omputable se quenc e of p ositive r ational numb ers such that P ∞ n =0 d n T ≤ 1 . If α is Martin-L¨ of T -r andom, then for every ε > 0 ther e exist a c omputable, incr e asing se quenc e { a n } of r ational numb ers and a r ational numb er q > 0 suc h that a n +1 − a n > q d n for every n ∈ N , a 0 > α − ε , and α = lim n →∞ a n . Pr o of. W e c ho ose an y one rational n um b er r with 2 − 1 /T ≥ r > 0. Since α is an r.e. r eal n umb er, there exists a compu table, incr easing sequ en ces { b n } of rational num b ers s uc h that b 0 > α − ε and α = lim n →∞ b n . W e construct a rational Martin-L¨ of T -test D by enumerating D ( i ) for eac h i ∈ N + as follo ws. Dur ing the en u meration of D ( i ) we sim u ltaneously construct a sequ en ce { a ( i ) n } n of rational num b ers. Initially , we set D ( i ) := ∅ and then sp ecify a ( i ) 0 b y a ( i ) 0 := b 0 . In general, whenever a ( i ) n is sp ecified as a ( i ) n := b m , w e up date D ( i ) b y D ( i ) := D ( i ) ∪ { ( i, a ( i ) n , a ( i ) n + r i d n ) } , and calculate b m +1 , b m +2 , b m +3 , · · · one b y one. During the calculation, if we fin d m 1 suc h that m 1 > m and b m 1 > a ( i ) n + r i d n , then we sp ecify a ( i ) n +1 b y a ( i ) n +1 := b m 1 and we rep eat this pro cedure for n + 1. F or the completed D through th e ab o ve pro cedu re, we see that, for ev ery i ∈ N + , X ( r 1 ,r 2 ) ∈ D ( i ) ( r 2 − r 1 ) T = X n ( r i d n ) T ≤ 2 − i X n d n T ≤ 2 − i . Here the second and third sums on n may b e finite or infi n ite. Thus, D is a rational Martin-L¨ of T -test. Since α is Martin-L¨ of T -random, th er e exists k ∈ N + suc h that, for every r 1 , r 2 ∈ Q , if ( r 1 , r 2 ) ∈ D ( k ) then α / ∈ [ r 1 , r 2 ]. It follo ws from α = lim n →∞ b n that in the ab o v e pr o cedure for en u merating D ( k ), for eve ry n ∈ N w e eve r find m 1 suc h that m 1 > m and b m 1 > a ( k ) n + r k d n . Therefore, D ( k ) is constru cted as an infinite set and also { a ( k ) n } n is constructed as an infi nite sequence of rational num b ers. Thus, we hav e a ( k ) n +1 > a ( k ) n + r k d n for all n ∈ N . Sin ce { a ( k ) n } n is a su bsequence of { b n } , it follo ws that the sequence { a ( k ) n } n is incr easing, a ( k ) 0 > α − ε , and α = lim n →∞ a ( k ) n . This completes the pro of. Theorem 5.3. Supp ose that T is c omputable. F or every r.e. r e al numb er α > 0 , if α is Martin-L¨ of T -r andom, then for every r.e. T -c onver gent r e al numb er γ > 0 ther e exist an r.e . r e al numb er β > 0 and a r ational numb er q > 0 such tha t α = β + q γ . 10 Pr o of. Supp ose that γ is an arbitrary r .e. T -con ve rgen t real num b er with γ > 0. T hen there exists a T -con v ergent compu table, increasing sequence { c n } of rational num b ers whic h con v erges to γ . Since γ > 0, without loss of generalit y we can assume that c 0 = 0. W e choose any one r ational n um b er ε > 0 such that ∞ X n =0 ( c n +1 − c n ) T ≤  1 ε  T . Suc h ε exists s in ce the s equ ence { c n } is T -con vergen t. It follo w s that ∞ X n =0 [ ε ( c n +1 − c n )] T ≤ 1 . Note that the sequence { ε ( c n +1 − c n ) } is a computable sequence of p ositiv e rational n um b ers. Thus, since α is r.e. and Ma rtin-L¨ of T -random by the assumption, it follo ws from Lemma 5.2 that there exist a computable, increasing sequence { a n } of rational n u m b ers and a rational n u m b er r > 0 suc h that a n +1 − a n > r ε ( c n +1 − c n ) for eve ry n ∈ N , a 0 > 0, an d α = lim n →∞ a n . W e then define a sequence { b n } of p ositiv e real num b ers by b n = a n +1 − a n − r ε ( c n +1 − c n ). It f ollo ws that { b n } is a computable sequence of rational n umb ers and P ∞ n =0 b n con verge s to α − a 0 − r ε ( γ − c 0 ). Thus we ha ve α = a 0 + P ∞ n =0 b n + r εγ , where a 0 + P ∞ n =0 b n is a p ositiv e r.e. real n u m b er. This completes the pr o of. Theorem 5.4. Su pp ose that T is c omputable. F or every r e al numb er α , if for every r.e. T - c onver gent r e al numb er γ > 0 ther e exist an r.e. r e al numb er β ≥ 0 and a r ational numb er q > 0 such that α = β + q γ , then ther e exist an optimal c omputer V and an r.e. r e al numb er β ≥ 0 such that α = β + Ω V ( T ) . Pr o of. First, for the optimal compu ter U , it follo ws fr om Theorem 4.3 that Ω U ( T ) is an r.e. T - con verge n t real num b er. Thus, by the assumption there exist an r.e. real num b er β ≥ 0 and a rational n umb er q > 0 su c h that α = β + q Ω U ( T ). W e c ho ose an y one n ∈ N w ith q > 2 − n/T . W e then define a partial function V : { 0 , 1 } ∗ → { 0 , 1 } ∗ b y the conditions that (i) dom V = { 0 n p | p ∈ dom U } and (ii) for ev ery p ∈ d om U , V (0 n p ) = U ( p ). Since dom V is a p refix-free set, it follo ws th at V is a computer. It is then easy to see that H V ( s ) = H U ( s )+ n f or ev ery s ∈ { 0 , 1 } ∗ . Th erefore, s in ce U is an optimal computer, V is also an optimal compu ter. It follo ws that Ω V ( T ) = 2 − n/T Ω U ( T ). Th us w e hav e α = β + ( q − 2 − n/T )Ω U ( T ) + Ω V ( T ). On the other hand, since T is computable, β + ( q − 2 − n/T )Ω U ( T ) is an r .e. real num b er. Th is completes the pro of. Theorem 5.5. Supp ose that T is c omputable. F or every r e al numb er α ∈ (0 , 1) , i f for ev e ry r.e. T - c onver gent r e al numb er γ > 0 ther e exist an r.e. r e al numb er β ≥ 0 and a r ational numb er q > 0 such that α = β + q γ , then ther e exists a universal pr ob ability m such that α = P s ∈{ 0 , 1 } ∗ m ( s ) 1 T . Pr o of. First, based on the optimal computer U w e defin e a computer V : { 0 , 1 } ∗ → { 0 , 1 } ∗ b y the conditions that (i) H V ( s ) = H ( s ) + 1 for ev ery s ∈ { 0 , 1 } ∗ and (ii) for ev ery s ∈ { 0 , 1 } ∗ and every n ∈ N , if n > H ( s ) then there exists a u nique p ∈ { 0 , 1 } ∗ suc h th at | p | = n and V ( p ) = s . The existence of such a computer V can b e easily sho wn using Theorem 3.2 of [4], based on the fact that the set { ( n, s ) ∈ N × { 0 , 1 } ∗ | n > H ( s ) } is r.e. an d X n>H ( s ) 2 − n = X s ∈{ 0 , 1 } ∗ ∞ X n = H ( s )+1 2 − n = X s ∈{ 0 , 1 } ∗ 2 − H ( s ) < 1 , 11 where the first sum is o ver all ( n, s ) ∈ N × { 0 , 1 } ∗ with n > H ( s ). It follo ws th at V is optimal and Ω V ( T ) = X s ∈{ 0 , 1 } ∗ ∞ X n = H ( s )+1 2 − n/T = 1 2 1 /T − 1 X s ∈{ 0 , 1 } ∗ 2 − H ( s ) /T . (5) By Theorem 4.3, we also see that Ω V ( T ) is an r.e. T -con v ergent real n umber. Th us, b y the assumption, there exist an r .e. real n u m b er β ≥ 0 and a rational num b er q > 0 suc h that α = β + q Ω V ( T ). W e c ho ose an y one rational n umber ε > 0 such that ε ≤ 1 − α T and ε 1 /T < q / (2 1 /T − 1). It follo ws from (5) that α = β + q 2 1 /T − 1 2 − H ( λ ) /T +  q 2 1 /T − 1 − ε 1 /T  X s 6 = λ 2 − H ( s ) /T + X s 6 = λ  ε 2 − H ( s )  1 /T . (6) Let γ b e the sum of the first, second, and third terms on the right -hand side of (6). Then , since T is compu table, γ is an r.e. real n u m b er. W e defin e a function m : { 0 , 1 } ∗ → (0 , ∞ ) by m ( s ) = γ T if s = λ ; m ( s ) = ε 2 − H ( s ) otherwise. S ince γ T < α T ≤ 1 − ε , we see that P s ∈{ 0 , 1 } ∗ m ( s ) < γ T + ε < 1. Since T is right -computable, γ T is an r.e. real num b er. Therefore, sin ce 2 − H ( s ) is a lo we r-computable semi-measure by Th eorem 2.5, m is also a lo wer-computable semi-measure. Thus, since 2 − H ( s ) is a un iv ersal p robabilit y by Theorem 2.5 again and γ T > 0, it is easy to see that m is a universal probabilit y . On the other hand , it follo w s from (6) th at α = P s ∈{ 0 , 1 } ∗ m ( s ) 1 T . Th is completes th e pro of. Theorem 5.6 b elo w is obtained by generalizing the pro ofs of Solo v ay [14 ] and Theorem 6.4 of Calude, Hertling, Khoussaino v, and W ang [1 ]. Theorem 5.6. Su pp ose that T is c omputable. F or every α ∈ (0 , 1) , if ther e exists a universal pr ob ability m such tha t α = P s ∈{ 0 , 1 } ∗ m ( s ) 1 T , then every c omputable, incr e asing se quenc e of r ational numb ers which c onver ges to α is T -unive rsal. Pr o of. Supp ose that { a n } is an arb itrary computable, increasing sequence of rational num b ers whic h conv erges to α = P s ∈{ 0 , 1 } ∗ m ( s ) 1 T . Since m is a low er-computable semi-measure and T is left-computable, there exists a total recursiv e fun ction f : N → N + suc h that, for eve ry n ∈ N , f ( n ) < f ( n + 1) and f ( n ) − 1 X k =0 m ( k ) 1 T ≥ a n . (7) Recall here that we id en tify { 0 , 1 } ∗ with N . W e then defin e a total recursive fu nction g : N → N b y g ( k ) = min { n ∈ N | k ≤ f ( n ) } . It follo ws that g ( f ( n )) = n for ev ery n ∈ N and lim k →∞ g ( k ) = ∞ . Supp ose that { b n } is an arbitrary T -con verge n t computable, increasing and conv erging sequence of rational n u mb ers. W e then choose any one d ∈ N + with P ∞ n =0 ( b n +1 − b n ) T ≤ d . W e then define a function r : N → [0 , ∞ ) b y r ( k ) = ( b g ( k +1) − b g ( k ) ) T /d . Sin ce T is computable, { b n } is T -con v ergen t, and g ( k + 1) = g ( k ) , g ( k ) + 1, we see th at r is a lo wer-computable semi-measure. Thus, since m 12 is a u niv ersal p robabilit y , there exists c ∈ N + suc h that, for ev ery k ∈ N , cm ( k ) ≥ r ( k ). I t f ollo ws from (7) that, for eac h n ∈ N , ( cd ) 1 T ( α − a n ) ≥ d 1 T ∞ X k = f ( n ) ( cm ( k )) 1 T ≥ ∞ X k = f ( n ) ( b g ( k +1) − b g ( k ) ) = β − b n , where β = lim n →∞ b n . T his completes the pr o of. Note that, using Lemma 5.2, w e can directly show that the condition (ii) implies the condition (iii) in Theorem 4.6 without assu ming the computabilit y of T ∈ (0 , 1], as follo ws. Theorem 5.7 b elo w holds for an arb itrary real num b er T ∈ (0 , 1]. Theorem 5.7. F or every r.e. r e al numb er α , if α is M artin-L¨ of T -r andom, then α is Ω ( T ) -like. Pr o of. Supp ose that β is an arbitrary r.e. T -con ve rgen t real n umb ers. Then there is a T -con vergen t computable, increasing sequence { b n } of ratio nal n u m b ers whic h con verge s to β . Since { b n } is T - con verge n t, withou t loss of generalit y w e can assume that P ∞ n =0 ( b n +1 − b n ) T ≤ 1. S ince α is r.e. and Martin-L¨ of T -random by the assump tion, it follo ws from Lemm a 5.2 th at there exist a computable, increasing sequence { a n } of rational n umb ers and a r ational num b er q > 0 s uc h that a n +1 − a n > q ( b n +1 − b n ) for ev ery n ∈ N and α = lim n →∞ a n . It is then easy to see that α − a n > q ( β − b n ) for every n ∈ N . Therefore α dominates β . This completes the pro of. 6 Some results on T -con ve rgence In this section, w e in vestig ate some pr op erties of the notion of T -conv ergence. As one of the applications of T heorem 4.6, the follo wing theorem can b e obtained fi rst. Theorem 6.1. Supp ose that T is c omputable. F or every r.e. r e al numb er α , if α is T -c onver gent, then α is T -c ompr essib le. Pr o of. Using (vi) ⇒ (iv) of Theorem 4.6, we see that H ( α n ) ≤ H ((Ω U ( T )) n ) + O (1) for ev ery r.e. T -con verge n t r eal n u m b er α . It follo ws from Theorem 2.10 (i) that α is T -compressible for ev ery r.e. T -con v ergen t real num b er α . In the case of T < 1, the con v erse of Theorem 6.1 do es not hold, as seen in th e foll o wing theorem in a sh arp er form. Theorem 6.2. Supp ose that T is c omputable and T < 1 . Then ther e exists an r.e. r e al numb er η such that (i) η is we akly Chaitin T -r andom and T -c ompr e ssib le, and (ii) η is not T -c onver g e nt. In order to pr o ve Th eorem 6.2, the follo wing lemma is useful. Lemma 6.3. (i) If { a n } is a T -c onver gent incr e asing se quenc e of r e al numb ers, then every subse qu enc e of the se quenc e { a n } is also T -c onver gent. 13 (ii) L et α b e a T -c onver gent r.e. r e al numb er. If { a n } i s a c omputable, incr e asing se quenc e of r a- tional numb ers c onver ging to α , then ther e exists a subse quenc e { a ′ n } of the se qu e nc e { a n } such that { a ′ n } is a T -c onver gent c omputable, incr e asing se quenc e of r ational numb ers c onver g i ng to α . Pr o of. (i) Let f : N → N such that f ( n ) < f ( n + 1) for all n ∈ N . Then, u s ing rep eatedly th e inequalit y ( x + y ) t ≤ x t + y t for real num b ers x, y > 0 and t ∈ (0 , 1], w e h a ve  a f ( n +1) − a f ( n )  T =   f ( n +1) − 1 X k = f ( n ) ( a k +1 − a k )   T ≤ f ( n +1) − 1 X k = f ( n ) ( a k +1 − a k ) T . It follo ws that m X n =0  a f ( n +1) − a f ( n )  T ≤ f ( m +1) − 1 X k = f (0) ( a k +1 − a k ) T . Since { a n } is T -con vergen t, w e see th at the su bsequence { a f ( n ) } of { a n } is also T -con vergen t. (ii) W e c ho ose any one T -con v ergent computable, increasing sequen ce { b n } of rational num b ers con vergi ng to α . It is then easy to sho w th at there exist total recursive functions g : N → N and h : N → N suc h that, for all n ∈ N , (i) g ( n ) < g ( n + 1), (ii) h ( n ) < h ( n + 1), and (iii) b g ( n ) < a h ( n ) < b g ( n +1) . I t follo ws from Lemma 6.3 (i) that the sub sequence { b g ( n ) } of { b n } is T -con v ergen t. Usin g th e inequalit y ( x + y ) t ≤ x t + y t for real num b ers x, y > 0 and t ∈ (0 , 1], we see that  a h ( n +1) − a h ( n )  T <  b g ( n +2) − b g ( n )  T ≤  b g ( n +2) − b g ( n +1)  T +  b g ( n +1) − b g ( n )  T . Th us, we see that the sub sequence { a h ( n ) } of { a n } is a T -con verge n t compu table, increasing se- quence of r ational n umb ers con verging to α . The pro of Th eorem 6.2 is give n as follo ws. Pr o of of The or em 6.2. W e choose an y one recur siv e en umeration p 0 , p 1 , p 2 , . . . of the r.e. set dom U , and defin e η by η = ∞ X i =0 | p i | 2 −| p i | /T . Then, since T is computable and T < 1, by T heorem 3 of T adaki [18] we see that η is an r.e. real n um b er w hic h is weakly Chaitin T -random and T-compressib le. 1 Since T is compu table, it is easy to sho w that th er e exists a computable, increasing s equ ence { a n } of rational n umb ers s u c h that n − 1 X i =0 | p i | 2 −| p i | /T < a n < n X i =0 | p i | 2 −| p i | /T (8) for all n ∈ N + . O b viously , { a n } is an in cr easing sequence of rational num b ers conv erging to η . 1 In Theorem 3 of T adaki [18], η is furthermore shown to b e Chaitin T -random, i.e., lim n →∞ H ( η n ) − T n = ∞ holds. 14 T o sho w th at η satisfies the condition (ii) of Theorem 6.2, let u s assume con trarily that η is T -con v ergen t. Then it follo ws from Lemma 6.3 (ii) that there exists a total r ecursiv e function f : N → N such that f ( n ) < f ( n + 1) for all n ∈ N , and { a f ( n ) } is a T -con v ergen t computable, increasing sequence of rational n umb ers conv erging to η . On the other h and, since T is computable, it is easy to sho w that th ere exists a computable, in cr easing sequence { b n } of rational num b er s suc h that f ( n ) X i =0 2 −| p i | /T < b n < f ( n +1) X i =0 2 −| p i | /T (9) for all n ∈ N . Obviously , { b n } is an increasing sequence of rational num b ers con vergi ng to Ω U ( T ). Since U is an optimal computer, usin g (vi) ⇒ (viii) of Th eorem 4.6, we see that there exists c ∈ N + suc h that c (Ω U ( T ) − b n ) ≥ η − a f ( n ) for all n ∈ N . It follo ws from (8) and (9) that c ∞ X i = f ( n )+1 2 −| p i | /T > ∞ X i = f ( n )+1 | p i | 2 −| p i | /T for all n ∈ N + . T herefore, we ha ve ∞ X i = f ( n )+1 ( c − | p i | )2 −| p i | /T > 0 (10) for all n ∈ N + . On the other h an d , it is easy to sho w that lim i →∞ | p i | = ∞ . T herefore, since lim n →∞ f ( n ) = ∞ , th ere exists n 0 ∈ N + suc h th at, for all i ∈ N , if i ≥ f ( n 0 ) + 1 then | p i | ≥ c . Th us, by setting n to n 0 in (10), we ha ve a con tradiction. This completes the p ro of. Let T 1 and T 2 b e arb itrary computable real num b ers w ith 0 < T 1 < T 2 < 1, and let V b e an arbitrary optimal compu ter. By Theorem 2.10 (i) and Th eorem 6.1, we see that th e r.e. real n um b er Ω V ( T 2 ) is not T 1 -con ve rgen t and therefore ev er y computable, increasing sequence { a n } of rational num b ers whic h conv erges to Ω V ( T 2 ) is not T 1 -con ve rgen t. Thus, con versely , the follo win g question natur ally arises: Is there an y computable, increasing s equ ence of r ational num b ers which con verge s to Ω V ( T 1 ) and which is n ot T 2 -con ve rgen t ? W e can answer th is question affirmativ ely in the f ollo wing f orm. Theorem 6.4. L et T 1 and T 2 b e arbitr ary c omputable r e al numb ers with 0 < T 1 < T 2 < 1 . Then ther e exist an optimal c omputer V and a c omputable, incr e asing se quenc e { a n } of r ational numb ers such that (i) Ω V ( T 1 ) = lim n →∞ a n , (ii) { a n } is T -c onver gent f or every T ∈ ( T 2 , ∞ ) , and (iii) { a n } is not T -c onver gent for every T ∈ (0 , T 2 ] . Pr o of. First, we c ho ose any one computable, increasing sequence { c n } of real n um b ers suc h that (i) { c n } conv erges to a computable real n um b er γ > 0, (ii) { c n } is T -con v ergen t for ev ery T ∈ ( T 2 , ∞ ), and (iii) { c n } is not T -con v ergen t for ev ery T ∈ (0 , T 2 ]. S uc h { c n } can b e obtained, for example, in the f ollo wing m anner. Let { c n } b e an in creasing sequence of r eal num b ers with c n = n +1 X k =1  1 k  1 T 2 . 15 Since T 2 > 0, w e first see that { c n } is T -con v ergent for ev ery T ∈ ( T 2 , ∞ ), and { c n } is n ot T - con verge n t for ev ery T ∈ (0 , T 2 ]. Since T 2 is a co mputable real n um b er with 0 < T 2 < 1, it is easy to see th at { c n } is a computable sequence of real n um b ers wh ich con ve rges to a computable r eal n um b er γ > 0. Th us, this sequence { c n } has the p rop erties (i), (ii) , and (iii ) desired ab o v e. W e c ho ose an y one rational num b er r with 0 < r < 1 /γ , and let β = r γ . Obvio usly , β is a computable r eal num b er with 0 < β < 1. Let b = 2 1 T 1 . T hen 1 < b . W e can then effectiv ely expand β to the base- b , i.e., Prop ert y 1 b elo w holds for the pair of β and b . Prop ert y 1. Ther e exists a total r e cursive function f : N + → N such that f ( k ) ≤ ⌈ b ⌉ − 1 for al l k ∈ N + and β = P ∞ k =1 f ( k ) b − k . This can b e p ossible since b oth β and b are computable. The detail is as follo ws. In the case where Prop ert y 2 below holds f or the p air of β and b , Prop ert y 1 holds, obviously . Prop ert y 2. Ther e exist m ∈ N + and a fu nc tion g : { 1 , 2 , . . . , m } → N such that g ( k ) ≤ ⌈ b ⌉ − 1 for al l k ∈ { 1 , 2 , . . . , m } and β = P m k =1 g ( k ) b − k . Th us, in what follo ws, we assu me that P r op ert y 2 d o es not hold. In this case, we construct the total recursive function f : N + → N by calculating f (1) , f (2) , f (3) , . . . , f ( m ) , . . . on e by one in this order, based on r ecur sion on stages m . W e start with stage 1 and follo w the instructions b elow. Note there that the sum P m − 1 k =1 f ( k ) b − k is regarded as 0 in the case of m = 1. A t the b eginning of stage m , assume that f (1) , f (2) , f (3) , . . . , f ( m − 1) are cal culated already . W e appro ximate the real num b er β − P m − 1 k =1 f ( k ) b − k and the ⌈ b ⌉ − 1 real num b ers b − m , 2 b − m , . . . , ( ⌈ b ⌉ − 2) b − m , ( ⌈ b ⌉ − 1) b − m b y r ational n umb ers with increasing precision. Durin g the appro ximation, if w e find l ∈ { 0 , 1 , 2 , . . . , ⌈ b ⌉− 1 } suc h that lb − m < β − m − 1 X k =1 f ( k ) b − k < ( l + 1) b − m , (11) then we set f ( m ) := l and b egin stage m + 1. W e can c h ec k that our recursion works pr op erly , as follo ws. S ince 0 < β < 1 ≤ ⌈ b ⌉ b − 1 , we see that 0 < β − P m − 1 k =1 f ( k ) b − k < ⌈ b ⌉ b − m at the b eginning of stage m = 1. Thus, in general, w e assume th at 0 < β − P m − 1 k =1 f ( k ) b − k < ⌈ b ⌉ b − m at th e b eginning of stage m . T hen, sin ce β and b are computable and Pr op ert y 2 do es not hold, we can eve n tually find l ∈ { 0 , 1 , 2 , . . . , ⌈ b ⌉ − 1 } whic h satisfies (11). Sin ce b − m ≤ ⌈ b ⌉ b − ( m +1) , w e ha ve 0 < β − P m k =1 f ( k ) b − k < ⌈ b ⌉ b − ( m +1) at the b eginning of stage m + 1. Th us, Prop ert y 1 holds in an y case. W e choose an y one L ∈ N with 2 L ≥ ⌈ b ⌉ − 1. Then P ∞ k =1 f ( k )2 − ( k + L ) ≤ P ∞ k =1 ( ⌈ b ⌉ − 1)2 − ( k + L ) ≤ 1. Hence, b y T heorem 3.2 of [4], it is easy to sho w that there exists a computer C suc h that (i) # { p | | p | = k + L & p ∈ dom C } = f ( k ) for ev ery k ∈ N + , and (ii) | p | ≥ 1 + L for ev ery p ∈ dom C . W e then defin e a partial function V : { 0 , 1 } ∗ → { 0 , 1 } ∗ b y the conditions that (i) dom V = { 0 p | p ∈ dom U } ∪ { 1 p | p ∈ dom C } , (ii) V (0 p ) = U ( p ) for all p ∈ dom U , and (iii) V (1 p ) = C ( p ) for all p ∈ dom C . Since d om V is a prefix-free set, it follo ws that V is a computer. It is then easy to c heck that H V ( s ) ≤ H U ( s ) + 1 16 for ev ery s ∈ { 0 , 1 } ∗ . Therefore, s ince U is an optimal computer, V is also an optimal computer. On the other hand, we see that Ω V ( T 1 ) = X p ∈ dom U 2 − ( | p | +1) /T 1 + X p ∈ dom C 2 − ( | p | +1) /T 1 = 2 − 1 T 1 Ω U ( T 1 ) + 2 − L +1 T 1 ∞ X k =1 f ( k )2 − k /T 1 = 2 − 1 T 1 Ω U ( T 1 ) + 2 − L +1 T 1 β . (12) Since T 1 is compu table with 0 < T 1 < 1, it f ollo ws from Theorem 4.3 that there exists a T 1 - con verge n t compu table, increasing sequence { w n } of rational num b ers which con verges to Ω U ( T 1 ). Then, sin ce T 1 is computable, it is easy to sh o w that ther e exists a computable, incr easing sequence { a n } of ratio nal num b ers such that η w n + ξ c n < a n < η w n +1 + ξ c n +1 for all n ∈ N , w h ere η = 2 − 1 T 1 and ξ = 2 − L +1 T 1 r . Obviously , b y (12) w e ha ve lim n →∞ a n = 2 − 1 T 1 Ω U ( T 1 ) + 2 − L +1 T 1 r γ = Ω V ( T 1 ). Using the inequalit y ( x + y ) t ≤ x t + y t for real n um b ers x, y > 0 and t ∈ (0 , 1], w e ha v e ( a n +1 − a n ) T < [( η w n +2 + ξ c n +2 ) − ( η w n + ξ c n )] T ≤ η T ( w n +2 − w n ) T + ξ T ( c n +2 − c n ) T ≤ η T ( w n +2 − w n +1 ) T + η T ( w n +1 − w n ) T + ξ T ( c n +2 − c n +1 ) T + ξ T ( c n +1 − c n ) T . Th us, for eac h T ∈ ( T 2 , ∞ ), sin ce b oth { w n } and { c n } are T -con v ergent, { a n } is also T -con verge n t. W e also ha ve ( c n +2 − c n +1 ) T < [ η ( w n +2 − w n +1 ) + ξ ( c n +2 − c n +1 )] T /ξ T = [( η w n +2 + ξ c n +2 ) − ( η w n +1 + ξ c n +1 )] T /ξ T < ( a n +2 − a n ) T /ξ T ≤ ( a n +2 − a n +1 ) T /ξ T + ( a n +1 − a n ) T /ξ T . Th us, f or eac h T ∈ (0 , T 2 ], sin ce { c n } is not T -conv ergen t, it is easy to see that { a n } is not T -con v ergen t also. Th is completes the p ro of. 7 Concluding remarks In this pap er, w e ha ve generalized the equiv alen t charac terizatio ns of randomn ess for a recursiv ely en u merable real o v er the n otion of partial randomness, s o that the generalized c haracterizations are all equ iv alent to the w eak Chaitin T -r andomness. As a stronger notion of p artial randomness of a real num b er α , T adaki [15, 16] introduced the notion of th e Chaitin T -randomness of α , whic h 17 is defined as the condition on α that lim n →∞ H ( α n ) − T n = ∞ . 2 Th us, fu ture w ork ma y aim at mo difying our equiv alen t c haracterizations of partial randomness so that th ey b ecome equiv alen t to the C haitin T -randomness. Ac kno wledgmen ts This wo rk w as supp orted b oth by KAKENHI, Gran t-in-Aid f or Scien tific Researc h (C ) (2054013 4) and by SCOPE (Strategic Inf orm ation and C omm u n ications R&D Promotion P rogramme) from the Ministry of In tern al Affairs and Comm unications of Japan . References [1] C. S. C alude, P . H. Hertling, B. Khoussainov, and Y. W ang, “Recursive ly en u merable reals and Chaitin Ω n umb ers,” The or et. Comput. Sci , vol. 255, pp. 125–149, 20 01. [2] C. S. Calud e, L. Staiger, and S. A. T erwijn , “On p artial ran d omness,” Annals of Pur e and Applie d L o gic , v ol. 138, pp. 20–30, 200 6. [3] C. S . Calude and M. A. Sta y , “Natural h alting pr obabilities, partial randomn ess, and zeta functions,” Inform. and Comput. , v ol. 204, pp. 1718–1739, 20 06. [4] G. J. Chaitin, “A theory of program size formally iden tical to information theory ,” J. Asso c . Comput. Mach. , v ol. 22, pp. 329–340 , 1975. [5] G. J. Chaitin, Al gorithmic Information The ory . Cam b ridge Universit y Press, Cam br idge, 198 7. [6] R. G. Do wney and J . Reimann (2007) Algorithmic randomn ess. Schola rp edia, 2(10):257 4. Av ailable at URL: http:// www.schol arpedia.o rg/article/Algorithmic_randomness [7] P . G´ acs, “On the symmetry of algo rithmic information,” Soviet Math. Dokl. , vol. 15, pp. 1477 – 1480, 1974; correction, ibid. v ol. 15 , pp. 1480, 19 74. [8] A. Kuˇ cera and T. A. Slaman, “Randomness and recursive enumerabilit y ,” SIAM J. Comput. , v ol. 31, No. 1, pp . 199–2 11, 2001 . [9] L. A. Levin, “La w s of information conserv ation (non-gro w th) and asp ects of the foundations of probabilit y theory ,” P r oblems of Inform. T r ansmission , v ol. 10, pp. 206–210, 19 74. [10] P . Martin-L¨ of, “The definition of random sequences,” Information and Contr ol , v ol. 9, pp. 602– 619, 1966. [11] M. B. P our-El and J. I. Ric h ards, Computability in Analysis and Physics . Persp ectiv es in Mathematica l Logic, Sprin ger-V erlag, Berlin, 1989 . [12] J. Reimann and F. Stephan, On h ierarc hies of randomn ess tests. Pro ceedings of the 9th Asian Logic Conferen ce, W orld S cien tific Publishing, August 16-19, 20 05, No v osibirsk , Russia. 2 The actual separation of the Chaitin T - randomness from the we ak Chaitin T - randomness is d one by Reimann and Stephan [12]. 18 [13] C.-P . Sc hn orr , “Pro cess complexit y and effectiv e random tests,” J. Comput. System Sci. , v ol. 7, pp. 376–388 , 1973. [14] R. M. Solo v a y , “Draft of a pap er (or series of pap ers) on Chaitin’s w ork ... done for the most part du r ing the p eriod of Sep t.–Dec. 1974,” unpu blished m anuscript, IBM Thomas J. W atson Researc h Cen ter, Y orkto wn Heigh ts, New Y ork, Ma y 19 75, 215 pp. [15] K. T adaki, Algorithmic information theory and fractal sets. P ro ceedings of 1999 W orkshop on Information-Based Indu ction S ciences (IBIS ’99), pp. 105–1 10, August 26-27, 1999, S yuzenji, Shizuok a, Japan. In J apanese. [16] K. T adaki, “A generalization of Ch aitin’s halting probabilit y Ω and halting self-similar sets,” Hokkaido Math. J. , vol . 31, pp. 219–253 , 2 002. Electronic V ersion Av ailable: http://a rxiv.org/ abs/nlin/0212001v1 [17] K. T adaki, “An exte nsion of Chaitin’s h alting p robabilit y Ω to a measur emen t op erator in an infinite dimensional qu an tum system,” Math. L o g. Quart. , v ol. 52, p p. 419–438 , 2006 . [18] K. T adaki, A statistical m echanical in terpr etatio n of algorithmic information theory . T o app ear in the Pro ceedings of Computability in Europ e 2008 (CiE 2008), June 15- 20, 2008, Universit y of A thens, G reece. Extended and Electronic V ersion Av ailable: http://a rxiv.org/ abs/0801.4194v1 [19] K. T adaki, The Tsallis en trop y an d the Shan n on en tr opy of a u niv ersal pr obabilit y . T o app ear in the Pr o ceedings of th e 2008 I E EE Int ernational Symp osium on Informa- tion Theory (ISIT2008), July 6-11, 2008, T oron to, Canada. Electronic V ersion Av ailable: http://a rxiv.org/ abs/0805.0154v1 [20] K. W eihr auc h, Computable Ana lysis . Spr inger-V erlag, Berlin, 20 00. [21] A. K. Zvo nkin and L. A. Levin, “The complexit y of fi n ite ob jects and the dev elopmen t of the concepts of inf ormation and randomness by m eans of the theory of algorithms,” Russian Math. Surveys , vo l. 25, no. 6, pp. 83–124, 19 70. 19