A statistical mechanical interpretation of algorithmic information theory

A statistical mec hanical in terpretation of algorithmic information theory Koh taro T adaki Researc h and Dev elopmen t Initiat iv e, Chuo Univ ersit y 1–13– 27 Kasuga, Bunkyo- ku, T oky o 112-85 51, Japan E-mail: tadaki@k c.c huo-u.ac.jp Abstract. W e deve lop a statistical mec hanical interpretation of algorithmic inf orma- tion theory b y in tro d ucing the notion of thermo dynamic quan tities, suc h as free energy , energy , sta tistical mec hanical en tropy , and sp eciﬁc heat, in to algo r ithmic information theory . W e in v estigate the prop erties of these quant ities b y means of pr ogram-size com- plexit y fr om the p oint of view of algo rithmic randomn ess. It is then disco v ered that, in the interpretatio n, the temp erature pla ys a role as the compression rate of the v alues of all these th er m o dynamic quan tities, wh ic h include the temp erature itself. Reﬂecting this self-referen tial natur e of the compression rate of the temp erature, w e obtain ﬁxed p oint theorems on compression r ate. Key wor ds : algorithmic information theory , algorithmic randomness, C h aitin’s Ω, com- pression rate, ﬁ xed p oint th eorem, statistical mec hanics, temp erature 1 In tro duction Algorithmic information theory is a framework to apply information-theoretic and probabilistic ideas to recursive f unction theory . One of the pr imary concepts of algorithmic information theory is the pr o gr am-size c omplexity (or Kolmo g or ov c omplexity ) H ( s ) of a ﬁnite binary string s , whic h is d eﬁned as the length of the shortest b inary pr ogram for the un iv ersal self-delimiting T urin g mac hine U to ou tp ut s . By th e deﬁn ition, H ( s ) can b e though t of as the information con tent of the individual ﬁ nite b in ary string s . In fact, algorithmic in formation theory has precisely the formal prop erties of classical information theory (see [4]). The conce pt of program-size complexit y pla ys a crucial role in c haracterizing th e randomness of a ﬁnite or inﬁnite binary string. In [4] Chaitin in tro duced the h alting probabilit y Ω as an example of rand om inﬁnite strin g. His Ω is deﬁn ed as the probabilit y that the unive rsal self-delimiting T uring mac hine U halts, and plays a cen tral role in the d ev elopment of algorithmic in formation theory . Th e ﬁrst n b its of the base-t wo expansion of Ω solv es the halting pr oblem for a program of size n ot greate r than n . By this prop erty , the base-t wo expansion of Ω is sho wn to b e a rand om inﬁnite binary string. In [5] Chaitin enco d ed this random p rop erty of Ω on to an exp onential Diophantine equation in the manner th at a certain prop erty of the set of the s olutions of the equation is in d istinguishable f rom coin tosses. Moreo ve r, based on this ran d om prop ert y of the equation, Chaitin derived seve ral quan titativ e versions of G¨ odel’s incompleteness theorems. In [15, 16] w e generalized Chaitin’s halting probab ility Ω to Ω D b y Ω D = X p ∈ dom U 2 − | p | D , (1) 1 so that the d egree of randomness of Ω D can b e con trolled by a real num b er D with 0 < D ≤ 1. Here, dom U denotes the set of all pr ograms p for U . As D b ecomes larger, the degree of rand omness of Ω D increases. When D = 1, Ω D b ecomes a random real num b er, i.e. , Ω 1 = Ω. The prop erties of Ω D and its relat ions to s elf-similar sets w ere studied in [15, 16]. Recen tly , Calude an d Sta y [3] p oin ted out a formal corresp ondence b et w een Ω D and a par- tition fun ction in statistical mec hanics. In statistical mec hanics, the partition function Z ( T ) at temp erature T is deﬁned b y Z ( T ) = X x ∈ X e − E x kT , where X is a complete set of energy eigenstates of a sta tistical mec hanical system and E x is the energy of an energy eigenstate x . The constan t k is calle d the Boltzmann Constan t. The partition function Z ( T ) is of p articular imp ortance in equilibrium statistical mec hanics. This is b ecause all the thermo d ynamic quantit ies of the system can b e expressed by u sing the partition fun ction Z ( T ), and the knowledge of Z ( T ) is suﬃ cien t to understand all the macroscopic prop erties of the system. Calude and Sta y [3] p oin ted out that the p artition function Z ( T ) h as the same form as Ω D b y p erformin g th e follo win g replacemen ts in Z ( T ): Replacemen ts 1.1. (i) R eplac e the c omplete set X of ener gy eigenstates x by the set dom U of al l pr o gr ams p for U . (ii) R e plac e the ener gy E x of an ener gy eigenstate x by the length | p | of a pr o gr am p . (iii) Set the Boltzm ann Constant k to 1 / ln 2 , wher e the ln denot es the natur al lo garithm. In this p ap er, insp ired b y their suggestion, w e dev elop a statistica l mec hanical in terpr etation of algorithmic information theory , where Ω D app ears as a partition function. Generally sp eaking, in ord er to giv e a statistical mechanica l int erpretation to a framew ork which lo oks unrelated to statistical mec h anics at ﬁrst glance , it is imp ortan t to identify a micro canonical ensem b le in the framew ork. Once we can do so, we can easily deve lop an equilibr ium statisti- cal mec hanics on th e framew ork according to the theoretic al devel opmen t of normal equilibrium statistica l m echanics. Here, the m icro canonical ensem b le is a certain sort of uniform probab ility distribution. In fact, in the w ork [17] w e dev elop ed a statistical mec hanical in terpr etation of the noiseless source co ding sc heme in inf orm ation theory b y identi fying a microcanonical ensem ble in the sc heme. Then, in [17] the notions in statistical mec h anics such as statistical mec hanical en tropy , temp erature, and thermal equilibrium are translated in to the con text of noiseless source co d ing. Th us, in order to dev elop a statistical mechanica l interpretation of algorithmic information theory , it is appropriate to iden tify a m icro canonica l ensem b le in the framew ork of the theory . Note, ho w ever, that algorithmic information theory is not a p h ysical th eory but a p u rely mathe- matical theory . Therefore, in order to obtain signiﬁcant resu lts for the dev elopment of algo r ithmic information theory itself, we ha ve to dev elop a statistic al mec hanical interpretatio n of algorithmic information theory in a mathematica lly r igorous manner, unlik e in norm al statistical mechanics in physic s where arguments are not necessarily mathematically r igorous. A fully rigorous mathemati- cal treatmen t of statistica l mec hanics is already dev elop ed (see [14]). At presen t, how ev er, it w ould not as yet seem to b e an easy task to mer ge algorithmic information theory with this mathematical treatmen t in a satisfactory manner. 2 On the ot her hand, if w e d o not stick to th e mathematical strictness of an argumen t, we can dev elop a statistical mec h anical int erpretation of algorithmic information theory while realiz ing a p erfect corresp ondence to n ormal statistical mechanics. In fact, in th e last part of this pap er (i.e., in Section 6) w e dev elop a statistical m ec hanical interpretation of algorithmic information theory b y making an argumen t on the same lev el of mathematical strictness as statistical mechanics in physic s. There, we iden tify a micro canonical ensem ble in algorithmic information theory in a similar manner to [17], based on the probabilit y measure whic h gives Chaitin’s Ω the meaning of the halting p robabilit y actually . In consequence, for example, the stat istical mec hanical meanin g of Ω D is clariﬁed. In the main part of this pap er, for mathematical strictness w e dev elop a statistical mec hanical in terp retation of algo rithmic in formation theory in a diﬀerent w a y from the physic al argumen t in Section 6. 1 W e int ro duce the notion of thermo d ynamic quantiti es in to algorithmic information the- ory based on Replacement s 1.1 ab o ve. Sectio n 6 p la ys a r ole in clarifying the s tatistica l mec hanical meaning of these notion and motiv ating the introd uction of them in the main part of this pap er. After the preliminary s ection on the mathematica l n otion needed in this pap er, we pro ve some results on the d egree of rand omness of real num b er s in Section 3. These results themselv es and the tec hniques used in p ro ving these results are frequ en tly u sed throughout this pap er. Then, in S ection 4 w e in tr o duce th e notion of the thermo dynamic quan tities at any giv en ﬁxed temp erature T , s uc h as p artition function, free energy , energy , statisti cal mec hanical entrop y , and sp eciﬁc heat, into algorithmic in formation theory by p erforming Replacemen ts 1.1 for the corresp ondin g thermo dynamic quantitie s in statistical mec h anics. Th ese thermo dynamic quan tities in algorithmic information theory are real n um b ers w h ic h dep end only on the temp erature T . W e pro ve that if the temp eratur e T is a compu table real num b er w ith 0 < T < 1 then , for eac h of these thermo dyn amic quan tities, the compression rate by the program-size complexit y H is equal to T . Th us, th e temp eratur e T pla ys a r ole as the compression rate of the thermo d ynamic quantitie s in this statistical mechanical inte rpretation of alg orithmic information theory . Among all thermod ynamic qu an tities in thermo dyn amics, one of the most t ypical therm o dy- namic quan tities is temp erature itself. Thus, based on the r esults of Section 4, the follo w ing question natur ally arises: Can the co mpression rate of the temp erature T b e equ al to the temp er- ature itself in the statistical mec hanical interpretatio n of alg orithmic in f ormation theory ? This question is rather self-referent ial. Ho we v er, in Section 5 w e answe r it aﬃr mativ ely by pr o ving Theorem 5.1. On e consequence of Theorem 5.1 has the follo wing f orm: F or eve ry T ∈ (0 , 1), if Ω T = P p ∈ dom U 2 − | p | T is a computable real n u m b er, then lim n →∞ H ( T n ) n = T , where T n is the ﬁrst n bits of the base-t w o expansion of T . Th is is just a ﬁxed p oin t theorem on compression rate, whic h reﬂ ects the self-referen tial nature of the qu estion. The w orks [15, 16] on Ω D migh t b e regarded as an elab oration of the tec hnique u s ed by Ch aitin [4] to pro ve that Ω is random. The mathematica l results of this pap er, which are obtained except for in Section 6, ma y be r egarded as fu rther elab orations of the tec hniqu e. Finally , in Section 6, based on a physica l and informal argumen t w e dev elop a to tal stati stical mec hanical inte rpretation of algorithmic inform ation theory which attains a p erfect corresp ondence 1 W e mak e an a rgument in a fully mathematicall y rigoro us manner in this paper e xcept fo r Section 6. An y consequence of the argumen t in Section 6 is not used in any other parts of this paper. 3 to normal statistical mec hanics. In consequen ce, w e justify th e in terp retation of Ω D as a partition function and clarify the statistical mec h anical meaning of the therm o dynamic quantitie s introd uced in to alg orithmic information theory in Secti on 4. 2 Preliminaries W e start with some notation ab out num b ers and strin gs whic h will b e used in this pap er. N = { 0 , 1 , 2 , 3 , . . . } is the set of n atural num b ers, and N + is the set of p ositiv e in tegers. Z i s the set of intege rs, and Q is the set of rational num b er s . R is th e set of real num b ers. { 0 , 1 } ∗ = { λ, 0 , 1 , 00 , 01 , 10 , 11 , 000 , 001 , 0 10 , . . . } is the set of ﬁnite binary strin gs where λ d en otes the empty string . F or any s ∈ { 0 , 1 } ∗ , | s | is the length of s . A subset S of { 0 , 1 } ∗ is cal led a pr eﬁx-fr e e set if no string in S is a p reﬁx of another string in S . { 0 , 1 } ∞ is the set of inﬁnite binary strin gs, where an in ﬁnite binary string is in ﬁnite to the righ t but ﬁnite to the left. F or an y α ∈ { 0 , 1 } ∞ , α n is the preﬁx of α of length n . F or an y partial function f , the domain of deﬁnition of f is denoted by dom f . W e w r ite “r.e.” instead of “recursiv ely en umerable.” Normally , o ( n ) denotes an y one function f : N + → R suc h that lim n →∞ f ( n ) /n = 0. On the other hand, O (1) denotes an y one f u nction g : N + → R such that there is C ∈ R with the prop erty that | g ( n ) | ≤ C for all n ∈ N + . Let T b e an arbitrary real n um b er. T mo d 1 denotes T − ⌊ T ⌋ , where ⌊ T ⌋ is the greatest intege r less than or equal to T , and T mo d ′ 1 den otes T − ⌈ T ⌉ + 1, where ⌈ T ⌉ is the sm allest intege r greater than or equ al to T . Hence, T mo d 1 ∈ [0 , 1) but T mo d ′ 1 ∈ (0 , 1]. W e identi fy a real n u m b er T with th e inﬁnite b inary string α such that 0 .α is the base-t wo expansion of T mo d 1 w ith inﬁ nitely man y zeros. Thus, T n denotes the ﬁrst n bits of the b ase-t w o expansion of T mo d 1 with in ﬁnitely man y zeros. W e say th at a real num b er T is c omputable if there exists a total recursiv e f unction f : N + → Q suc h that | T − f ( n ) | < 1 /n for all n ∈ N + . W e sa y that T is right-c omputable if there exists a total recursiv e function g : N + → Q su c h that T ≤ g ( n ) for all n ∈ N + and lim n →∞ g ( n ) = T . W e say that T is left-c omputable if − T is right-co m putable. I t is then easy to see that, for any T ∈ R , T is computable if and only if T is b oth righ t-computable and left-computable. See e.g. [11, 19] f or the detail of the treatmen t of th e computabilit y of r eal n um b ers and r eal functions on a discrete set. 2.1 Algorithmic information theory In the follo wing w e concisely review s ome deﬁnitions and results o f algorithmic information the- ory [4, 6]. A c omputer is a partial recursiv e function C : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h th at dom C is a preﬁx-free set. F o r 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 compu ter U is said to b e optimal if for eac h computer C there exists a constan t sim( C ) with the follo wing pr op ert y; if C ( p ) is deﬁned, then there is a p ′ for whic h U ( p ′ ) = C ( p ) and | p ′ | ≤ | p | + sim( C ). It is then sho w n that there exists an optimal computer. W e c ho ose any one optimal computer U as the standard one for use, and deﬁne H ( s ) as H U ( s ), whic h is referred to as the pr o gr am-size c omplexity of s , the informat ion c ontent of s , or the Kolmo gor ov c omp lexity of s [9, 10, 4]. Th us, H ( s ) has the follo w ing prop ert y: ∀ C : computer H ( s ) ≤ H C ( s ) + sim( C ) . (2) 4 It can b e sho wn that there is c ∈ N su c h that, for an y s 6 = λ , H ( s ) ≤ | s | + 2 log 2 | s | + c. (3) F or eac h s ∈ { 0 , 1 } ∗ , P ( s ) is deﬁned as P U ( p )= s 2 −| p | . Chaitin ’s halting pr ob ability Ω is deﬁned b y Ω = X p ∈ dom U 2 −| p | . F or any α ∈ { 0 , 1 } ∞ , we s a y that α is we akly Chaitin r andom if there exists c ∈ N su c h that n − c ≤ H ( α n ) for all n ∈ N + [4, 6]. Then [4] sho wed that Ω is weakly Chaitin random. F or an y α ∈ { 0 , 1 } ∞ , w e say th at α is Chaitin r andom if lim n →∞ H ( α n ) − n = ∞ [4, 6]. It is then sho w n that, for any α ∈ { 0 , 1 } ∞ , α is weakly Ch aitin r andom if and only if α is Chaitin random (see [6] for the p ro of and historical detail) . Thus Ω is Ch aitin random. The class of compu ters is equal to the class of f unctions which are computed b y self-delimiting T uring machines . A self-delimiting T urin g mac hine is a deterministic T uring mac hine whic h has t wo tap es, a program tap e and a w ork tap e. T h e program tap e is inﬁnite to the right, while the w ork tap e is inﬁnite in b oth d ir ections. An input string in { 0 , 1 } ∗ is put on the pr ogram tap e. See Chaitin [4] for the detail of self-delimiting T uring m achines. Let M b e a self-delimiting T urin g mac hine w hic h computes th e op timal compu ter U . Then P ( s ) is the probabilit y that M halts and outputs s when M starts on the pr ogram ta p e ﬁlled with an inﬁ nite binary string generate d by inﬁnitely rep eated tosses o f a fair coin. Therefore Ω = P s ∈{ 0 , 1 } ∗ P ( s ) is th e probabilit y that M just halts under the same setting. The program-size complexit y H ( s ) is originally deﬁned using the concept of p rogram-size, as stated ab o v e. Ho wev er, it is p ossib le to deﬁ n e H ( s ) without referring to suc h a concept, i.e., w e ﬁrst in tro duce a universal pr ob ability m , and then deﬁne H ( s ) as − log 2 m ( s ). A universal prob ab ility is deﬁned through th e follo wing t w o deﬁnitions [20]. Deﬁnition 2.1. F or any r : { 0 , 1 } ∗ → [0 , 1] , we say tha t r is a lower-c omp utable semi-me asur e if r satisﬁes the fol lowing two c onditions: (i) P s ∈{ 0 , 1 } ∗ r ( s ) ≤ 1 . (ii) Ther e e xi sts a total r e cursive function f : N + × { 0 , 1 } ∗ → Q such that, for e ach s ∈ { 0 , 1 } ∗ , lim n →∞ f ( n, s ) = r ( s ) and ∀ n ∈ N + f ( n, s ) ≤ r ( s ) . Deﬁnition 2.2. L et m b e a lower-c omputable semi-me asur e. We say that m is a universal pr ob a- bility if for any lower -c omputable semi-me asur e r , ther e exists a r e al numb er c > 0 such tha t, for al l s ∈ { 0 , 1 } ∗ , c r ( s ) ≤ m ( s ) . Chaitin [4] sho w ed the follo wing theorem. Theorem 2.3. Both 2 − H ( s ) and P ( s ) ar e univ ersal pr ob abilities. By Theorem 2.3, w e see that, for any univ ers al probabilit y m , H ( s ) = − log 2 m ( s ) + O (1) . (4) Th us it is p ossible to deﬁ n e H ( s ) as − log 2 m ( s ) with an y one univ ers al probability m instead of as H U ( s ). Note that the diﬀerence u p to an additiv e constan t is in essen tial to algorithmic information theory . 5 In the w orks [15, 16], w e generalized the notion of the r andomness of an in ﬁnite binary string so that th e degree of the randomness can b e c haracterized b y a real num b er D with 0 < D ≤ 1 as follo ws. Deﬁnition 2.4 (wea kly Chaitin D -random) . L et D ∈ R with D ≥ 0 , and let α ∈ { 0 , 1 } ∞ . W e say that α is we akly Chaitin D -r andom if ther e exists c ∈ R such that D n − c ≤ H ( α n ) for al l n ∈ N + . Deﬁnition 2.5 ( D -compr essible) . L et D ∈ R with D ≥ 0 , and let α ∈ { 0 , 1 } ∞ . We say that α is D -c ompr e ssib le if H ( α n ) ≤ D n + o ( n ) , which is e quivalent to lim n →∞ H ( α n ) n ≤ D . In the case of D = 1, the weak Chaitin D -randomn ess resu lts in the w eak Chaitin randomness. F or an y D ∈ [0 , 1] and any α ∈ { 0 , 1 } ∞ , if α is weakly Ch aitin D -rand om and D -compressible, then lim n →∞ H ( α n ) n = D . (5) Hereafter th e left-hand side of (5) is r eferred to as the c ompr ession r ate of an inﬁ nite b inary string α in general. Note, ho wev er, that (5 ) do es not necessarily imply that α is weakly Chaitin D -ran d om. In the wo rks [15, 16], w e generalized Chaitin’s halting probabilit y Ω to Ω D b y (1 ) f or any real n u m b er D > 0. Thus, Ω = Ω 1 . If 0 < D ≤ 1, then Ω D con verge s and 0 < Ω D < 1, since Ω D ≤ Ω < 1. Theorem 2.6 (T adaki [15, 16]) . L et D ∈ R . (i) If 0 < D ≤ 1 and D is c omputable, then Ω D is we akly Chaitin D - r andom and D -c ompr essible. (ii) If 1 < D , then Ω D diver ges to ∞ . Deﬁnition 2.7 (Chaitin D -randomn ess, T adaki [15, 16]) . L et D ∈ R with D ≥ 0 , and let α ∈ { 0 , 1 } ∞ . W e say that α is Chaitin D -r andom if lim n →∞ H ( α n ) − D n = ∞ . In the case of D = 1, the Chaitin D -rand omness r esults in the Chaitin rand omness. Ob viously , for an y D ∈ [0 , 1] and an y α ∈ { 0 , 1 } ∞ , if α is Chaitin D -rand om, then α is wea kly Ch aitin D -random. How ev er, in 20 05 Reimann and S tephan [13 ] sh o wed that, in the case of D < 1, the con verse do es not necessarily hold. T his con trasts w ith the equ iv alence b et ween the w eakly Chaitin randomness and the Chaitin rand omn ess, eac h of wh ic h corresp onds to the ca se of D = 1. In the next section, f or any computable real n umber D with 0 < D < 1, w e giv e an instance of a real n u m b er whic h is Chaitin D -random and D -compressible. 3 Chaitin D -randomness and div ergence F or eac h real n um b ers Q > 0 and D > 0, we deﬁne W ( Q, D ) b y W ( Q, D ) = X p ∈ dom U | p | Q 2 − | p | D . As the ﬁ rst result of this pap er, w e show the follo wing theorem. 6 Theorem 3.1. L et Q and D b e p ositive r e al numb ers. (i) If Q and D ar e c omputable and 0 < D < 1 , then W ( Q, D ) c onver ges to a left- c omputable r e al numb er which is Chaitin D - r andom and D -c ompr essible. (ii) If 1 ≤ D , then W ( Q, D ) diver ges to ∞ . The tec hniques used in the pr o ofs of T heorem 2.6 (see [16]) and Theorem 3.1 are frequently used throughout the rest of this p ap er as basic to ols. W e see that th e w eak Chaitin D -randomness in Theorem 2.6 is replaced by the Chaitin D -rand omness in T heorem 3.1 in exchange for the div ergence at D = 1. In order to d er ive this dive rgence we mak e us e of Theorem 3.2 (i) b elow. W e pro ve Theorem 3.2 in a m ore general form, and show that the Shannon entr opy − P s ∈{ 0 , 1 } ∗ m ( s ) log 2 m ( s ) of an arbitrary un iv ersal probabilit y m div erges to ∞ . W e sa y that a function f : N + → [0 , ∞ ) is lower-c omputable if there exists a total recursiv e function a : N + × N + → Q such th at, for eac h n ∈ N + , lim k →∞ a ( k , n ) = f ( n ) and ∀ k ∈ N + a ( k , n ) ≤ f ( n ). Theorem 3.2. L et A b e an inﬁnite r.e. subset of { 0 , 1 } ∗ and let f : N + → [0 , ∞ ) b e a lower- c omputable function such th at lim n →∞ f ( n ) = ∞ . Then the fol lowing hold. (i) P U ( p ) ∈ A f ( | p | )2 −| p | diver ges to ∞ . (ii) If ther e exists l 0 ∈ N + such tha t f ( l )2 − l is a nonincr e asing fu nction of l for al l l ≥ l 0 , then P s ∈ A f ( H ( s ))2 − H ( s ) diver ges to ∞ . Pr o of. (i) Contrarily , assume that P U ( p ) ∈ A f ( | p | )2 −| p | con verge s. Then, there exists d ∈ N + suc h that P U ( p ) ∈ A f ( | p | )2 −| p | ≤ d . W e deﬁne the fun ction r : { 0 , 1 } ∗ → [0 , ∞ ) by r ( s ) = 1 d X U ( p )= s f ( | p | )2 −| p | if s ∈ A ; r ( s ) = 0 otherwise. Then we see th at P s ∈{ 0 , 1 } ∗ r ( s ) ≤ 1 and therefore r is a low er- computable semi-measure. Since P ( s ) is a un iversal p r obabilit y , there exists c ∈ N + suc h that r ( s ) ≤ cP ( s ) for all s ∈ { 0 , 1 } ∗ . Hence we ha ve X U ( p )= s ( cd − f ( | p | )) 2 −| p | ≥ 0 (6) for all s ∈ A . On the other hand, since A is an inﬁnite set and lim n →∞ f ( n ) = ∞ , there is s 0 ∈ A suc h that f ( | p | ) > cd f or all p with U ( p ) = s 0 . Th erefore we ha v e P U ( p )= s 0 ( cd − f ( | p | ) )2 −| p | < 0. Ho we v er, this contradicts (6), and th e pro of of (i) is completed. (ii) W e ﬁr st n ote that there is n 0 ∈ N su c h that H ( s ) ≥ l 0 for all s with | s | ≥ n 0 . No w, let us assume contrarily that P s ∈ A f ( H ( s ))2 − H ( s ) con verge s. Then, there exists d ∈ N + suc h that P s ∈ A f ( H ( s ))2 − H ( s ) ≤ d . W e deﬁne the fun ction r : { 0 , 1 } ∗ → [0 , ∞ ) by r ( s ) = 1 d f ( H ( s ))2 − H ( s ) if s ∈ A and | s | ≥ n 0 ; r ( s ) = 0 otherwise. Th en we see that P s ∈{ 0 , 1 } ∗ r ( s ) ≤ 1 and therefore r is a lo w er-computable semi-measur e. Since 2 − H ( s ) is a un iv ersal probabilit y by Th eorem 2.3 , there 7 exists c ∈ N + suc h that r ( s ) ≤ c 2 − H ( s ) for all s ∈ { 0 , 1 } ∗ . Hence, if s ∈ A and | s | ≥ n 0 , then cd ≥ f ( H ( s )). On th e other hand, since A is an inﬁ n ite set and lim n →∞ f ( n ) = ∞ , there is s 0 ∈ A suc h that | s 0 | ≥ n 0 and f ( H ( s 0 )) > cd . Th u s, w e ha ve a cont radiction, and the pro of of (ii) is completed. Corollary 3.3. If m is a universal pr ob ability and A is an inﬁnite r.e. subset of { 0 , 1 } ∗ , then − P s ∈ A m ( s ) log 2 m ( s ) diver ges to ∞ . Pr o of. W e ﬁr st note that there is a real n um b er x 0 > 0 suc h that the fun ction x 2 − x of a real n u m b er x is decreasing f or x ≥ x 0 . F or this x 0 , there is n 0 ∈ N suc h that − log 2 m ( s ) ≥ x 0 for all s with | s | ≥ n 0 . On the other han d , by (4), there is c ∈ N such th at − log 2 m ( s ) ≤ H ( s ) + c for all s ∈ { 0 , 1 } ∗ . Thus, w e see that − X s ∈ A & | s |≥ n 0 m ( s ) log 2 m ( s ) ≥ X s ∈ A & | s |≥ n 0 ( H ( s ) + c )2 − H ( s ) − c = 2 − c X s ∈ A & | s |≥ n 0 H ( s )2 − H ( s ) + c 2 − c X s ∈ A & | s |≥ n 0 2 − H ( s ) . By Theorem 3.2 (ii), P s ∈ A H ( s )2 − H ( s ) div erges to ∞ . Hence, we see, b y th e inequalit y ab o ve, that − P s ∈ A m ( s ) log 2 m ( s ) also d iv erges to ∞ . By Corollary 3.3, w e see that the Shann on en tropy of an arbitrary universal p robabilit y div erges to ∞ . The pro of of Theorem 3.1 is give n as follo ws. The pr o of of The or em 3.1. Let p 1 , p 2 , p 3 , . . . b e a recursive en u meration of the r.e. set dom U . Then, for ev ery D > 0, W ( Q, D ) = lim m →∞ f W m ( Q, D ), w here f W m ( Q, D ) = m X i =1 | p i | Q 2 − | p i | D . (i) First we sho w th at W ( Q, D ) con v erges to a left-computable real num b er. Since D < 1, there is l 0 ∈ N + suc h that 1 D − Q log 2 l l ≥ 1 for all l ≥ l 0 . Then there is m 0 ∈ N + suc h that | p i | ≥ l 0 for all i > m 0 . Th u s, w e see that, for eac h i > m 0 , | p i | Q 2 − | p i | D = 2 − ( 1 D − Q log 2 | p i | | p i | ) | p i | ≤ 2 −| p i | . Hence, for ea c h m > m 0 , f W m ( Q, D ) − f W m 0 ( Q, D ) = m X i = m 0 +1 | p i | Q 2 − | p i | D ≤ m X i = m 0 2 −| p i | < Ω . Th us, s in ce { f W m ( Q, D ) } m is an increasing sequence of real n um b ers, it con v erges to a real num b er W ( Q, D ) as m → ∞ . Moreo ver, since Q and D are computable, W ( Q, D ) is sho wn to b e left- computable. 8 W e th en sh o w that W ( Q, D ) is Ch aitin D -r andom. Let α b e the inﬁ nite b inary string such that 0 .α is the base-t w o expansion of W ( Q, D ) mo d ′ 1 with inﬁn itely man y ones. Then, since Q and D are computable real n umb ers and ⌈ W ( Q, D ) ⌉ − 1 + 0 .α n < W ( Q, D ) for all n ∈ N + , there exists a partial recursive function ξ : { 0 , 1 } ∗ → N + suc h that, for all n ∈ N + , ⌈ W ( Q, D ) ⌉ − 1 + 0 .α n < f W ξ ( α n ) ( Q, D ) . It is then easy to see that W ( Q, D ) − f W ξ ( α n ) ( Q, D ) < 2 − n . It follo ws that, for all i > ξ ( α n ), | p i | Q 2 − | p i | D < 2 − n and therefore QD log 2 | p i | < | p i | − D n . Thus, giv en α n , by calculating the set  U ( p i )   i ≤ ξ ( α n )  and picking an y one ﬁnite b inary string whic h is not in this set, one can obtai n s ∈ { 0 , 1 } ∗ suc h that QD log 2 H ( s ) < H ( s ) − D n . Hence, there exists a partial recursive function Ψ : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that QD log 2 H (Ψ( α n )) < H (Ψ ( α n )) − D n. Applying this inequalit y to itself, we ha ve QD log 2 n < H (Ψ ( α n )) − D n + O (1). On the other hand, using (2 ) there is a natural num b er c Ψ suc h that H (Ψ ( α n )) < H ( α n ) + c Ψ . Therefore, we ha ve QD log 2 n < H ( α n ) − D n + O (1) . Hence, α is Chaitin T -random. It follo ws that α has inﬁ n itely man y zeros, whic h implies that W ( Q, D ) mo d 1 = W ( Q, D ) mo d ′ 1 = 0 .α and therefore ( W ( Q, D )) n = α n . Th us , W ( Q, D ) is Chaitin D -rand om. Next, w e show th at W ( Q, D ) is D -compressible. S ince Q and D are computable r eal num b ers, there exists a total recursiv e function g : N + × N + → Z su c h that, for all m, n ∈ N + ,    f W m ( Q, D ) − ⌊ W ( Q, D ) ⌋ − 2 − n g ( m, n )    < 2 − n . (7) Let d b e an y computable real n umb er with D < d < 1. Then, th e limit v alue W ( Q, d ) exists sin ce d < 1. Let β b e the inﬁnite binary string su c h that 0 .β is the base-t wo expansion of W ( Q, d ) mo d ′ 1 with inﬁn itely man y ones. Giv en n and β ⌈ Dn/d ⌉ (i.e., the ﬁrst ⌈ D n/d ⌉ bits of β ), one can ﬁnd m 0 ∈ N + suc h that ⌈ W ( Q, d ) ⌉ − 1 + 0 .β ⌈ Dn/d ⌉ < f W m 0 ( Q, d ) . This is p ossible since ⌈ W ( Q, d ) ⌉ − 1 + 0 .β ⌈ Dn/d ⌉ < W ( Q, d ) and lim m →∞ f W m ( Q, d ) = W ( Q, d ). It is then easy to see that ∞ X i = m 0 +1 | p i | Q 2 − | p i | d < 2 − D n/d . Raising b oth sides of this inequalit y to th e p o w er d/D and using the inequalit y a c + b c ≤ ( a + b ) c for real num b ers a, b > 0 and c ≥ 1, 2 − n > ∞ X i = m 0 +1 | p i | Qd/D 2 − | p i | D > ∞ X i = m 0 +1 | p i | Q 2 − | p i | D . 9 It follo ws that    W ( Q, D ) − f W m 0 ( Q, D )    < 2 − n . (8) F r om (7), (8), an d | ⌊ W ( Q, D ) ⌋ + 0 . ( W ( Q, D )) n − W ( Q, D ) | < 2 − n , it is shown that | ( W ( Q, D )) n − g ( m 0 , n ) | < 3 and therefore ( W ( Q, D )) n = g ( m 0 , n ) , g ( m 0 , n ) ± 1 , g ( m 0 , n ) ± 2 , where ( W ( Q, D )) n is r egarded as a dyadic in teger. Thus, there are still 5 p ossibilities of ( W ( Q, D )) n , so that on e n eeds only 3 b its more in order to determine ( W ( Q, D )) n . Th us, there exists a partial recursive function Φ : N + × { 0 , 1 } ∗ × { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that ∀ n ∈ N + ∃ s ∈ { 0 , 1 } ∗ | s | = 3 & Φ( n, β ⌈ Dn/d ⌉ , s ) = ( W ( Q, D )) n . It f ollo ws from (3) that H (( W ( Q, D )) n ) ≤ | β ⌈ Dn/d ⌉ | + o ( n ) ≤ D n/d + o ( n ), whic h implies that W ( Q, D ) is D/d -compressible. Since d is any computable real n umb er w ith D < d < 1, it follo w s that W ( Q, D ) is D -compressible. (ii) W e c h o ose an y one compu table real n u m b er Q ′ with Q ≥ Q ′ > 0. Then, using T heorem 3.2 (i), we can show that W ( Q ′ , 1) = P p ∈ dom U | p | Q ′ 2 −| p | div erges to ∞ . Thus, since f W m ( Q, D ) ≥ f W m ( Q ′ , 1), we see that W ( Q, D ) also div erges to ∞ , and the pro of is complete d. 4 T emp erature as a compression rate In this section we int ro duce the notion of thermo d ynamic quantitie s suc h as p artition fun ction, free energy , energy , en tropy , and sp eciﬁc heat, in to algorithmic in formation theory b y p erforming Replacemen ts 1.1 for the corresp onding thermo dyn amic quan tities in statist ical mechanics. 2 W e in vestig ate their con ve rgence and th e degree of randomn ess. F or that p urp ose, we ﬁrst choose an y one en u meration q 1 , q 2 , q 3 , . . . of the coun tably inﬁn ite set dom U as the standard one for u se throughout this section. 3 In statistical mec hanics, the p artition function Z sm ( T ) at te mp erature T is giv en b y Z sm ( T ) = X x ∈ X e − E x kT . (9) Motiv ated b y the form u la (9) and taking in to accoun t Replacement s 1.1, we introd uce the n otion of partition function in to alg orith m ic in formation theory as follo ws. Deﬁnition 4.1 (partition function) . F or e ach n ∈ N + and e ach r e al numb er T > 0 , we deﬁne Z n ( T ) by Z n ( T ) = n X i =1 2 − | q i | T . 2 F or the thermo dy namic quantities in statistical mec hanics, see Chapter 16 of [1] and Chapter 2 of [18]. T o b e precise, t he partition function is not a thermo dynamic qu antit y but a statistical mechanical q uantit y . 3 The enumeration { q i } is quite arbitrary and therefore we d o not, ever, require { q i } to b e a recursive enumeration of dom U . 10 Then, for e ach T > 0 , the p artition fu nc tion Z ( T ) is deﬁne d by Z ( T ) = lim n →∞ Z n ( T ) . Since Z ( T ) = Ω T , we restate Theorem 2.6 as in the follo wing form. Theorem 4.2 (T adaki [15, 16]) . L et T ∈ R . (i) If 0 < T ≤ 1 and T is c omputable, then Z ( T ) c onver ges to a left-c omputa ble r e al numb er which is we akly Chaitin T - r andom and T -c ompr essible. (ii) If 1 < T , then Z ( T ) diver ges to ∞ . In statistical mec hanics, the fr ee energy F sm ( T ) at te mp erature T is giv en by F sm ( T ) = − k T ln Z sm ( T ) , (10) where Z sm ( T ) is giv en b y (9). Motiv ated by the form u la (10) and taking int o accoun t Replace- men ts 1.1, w e in tro du ce the notion of fr ee energy in to algorithmic information theory as follo w s. Deﬁnition 4.3 (free energy) . F or e ach n ∈ N + and e ach r e al numb er T > 0 , we deﬁne F n ( T ) by F n ( T ) = − T log 2 Z n ( T ) . Then, for e ach T > 0 , the fr e e ener gy F ( T ) is deﬁne d by F ( T ) = lim n →∞ F n ( T ) . Theorem 4.4. L et T ∈ R . (i) If 0 < T ≤ 1 and T is c omputable, then F ( T ) c onver ges to a right-c omputable r e al numb er which is we akly Chaitin T - r andom and T -c ompr essible. (ii) If 1 < T , then F ( T ) diver ges to −∞ . Pr o of. (i) Sin ce Z ( T ) con v erges b y Theorem 4.2 (i) and Z ( T ) > 0, F ( T ) also con verges and F ( T ) = − T log 2 Z ( T ) . Note that T is a right- computable real n u m b er and − log 2 Z ( T ) > 0. Sin ce Z ( T ) is a left-computable real num b er by Theorem 4.2 (i), F ( T ) is a r igh t-computable real num b er. W e sho w that F ( T ) is w eakly Chaitin T -rand om. By the mean v alue th eorem, there exists c ∈ N + suc h that, for any A, B ∈ R , if A ≥ F ( T ) and B ≥ 1 /T then 0 ≤ Z ( T ) − 2 − AB ≤ 2 c max { A − F ( T ) , B − 1 /T } . (11) Since F ( T ) is a righ t-computable real num b er, there exists a tota l recur s iv e fu nction f : N + → Q suc h that F ( T ) ≤ f ( m ) for all m ∈ N + and lim m →∞ f ( m ) = F ( T ). S ince T is a computable real n u m b er, there exists a total recursiv e function g : N + → Q suc h th at 0 ≤ g ( n ) − 1 /T < 2 − n for all n ∈ N + . Giv en ( F ( T )) n , one can ﬁnd m 0 ∈ N + suc h that f ( m 0 ) < ⌊ F ( T ) ⌋ + 0 . ( F ( T )) n + 2 − n . 11 This is p ossib le b ecause F ( T ) < ⌊ F ( T ) ⌋ + 0 . ( F ( T )) n + 2 − n . It follo ws that 0 ≤ f ( m 0 ) − F ( T ) < 2 − n . Therefore, by (11) it is s h o wn that 0 ≤ Z ( T ) − 2 − f ( m 0 ) g ( n ) < 2 c − n . Let l n b e the ﬁrst n bits of the base-t wo expansion of 2 − f ( m 0 ) g ( n ) with inﬁn itely man y zeros. It follo ws that 0 ≤ 0 . ( Z ( T )) n − 0 .l n < Z ( T ) − 2 − f ( m 0 ) g ( n ) + 2 − n < (2 c + 1)2 − n . Hence ( Z ( T )) n = l n , l n + 1 , l n + 2 , . . . , l n + (2 c + 1) , where ( Z ( T )) n and l n are regarded as a d y adic in teger. Therefore, there are still 2 c + 2 p ossibilities of ( Z ( T )) n , so that one needs only c + 1 bits more in ord er to determine ( Z ( T )) n . Th us, there exists a partial recursive function Φ : { 0 , 1 } ∗ × { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that ∀ n ∈ N + ∃ s ∈ { 0 , 1 } ∗ | s | = c + 1 & Φ(( F ( T )) n , s ) = ( Z ( T )) n . It follo ws that there exists c Φ ∈ N + suc h that, for all n ∈ N + , H (( Z ( T )) n ) ≤ H (( F ( T )) n ) + c Φ . Hence, F ( T ) is w eakly Chaitin T -random by Theorem 4.2 (i). Next, w e sho w that F ( T ) is T -compressible. Since T is a computable real n um b er, there exists a total r ecursiv e function a : { 0 , 1 } ∗ × N + → Z su c h that, for all s ∈ { 0 , 1 } ∗ and all n ∈ N + , if 0 .s > 0 then   − T log 2 0 .s − ⌊ F ( T ) ⌋ − 2 − n a ( s, n )   < 2 − n . (12) By the mean v alue theorem, it is also sh o wn th at there is d ∈ N + suc h that, for all n ∈ N + , if 0 . ( Z ( T )) n > 0 then | − T log 2 0 . ( Z ( T )) n − F ( T ) | < 2 d − n . (13) F r om (12), (13) , and | ⌊ F ( T ) ⌋ + 0 . ( F ( T )) n − F ( T ) | < 2 − n , it is shown that, for all n ∈ N + , if 0 . ( Z ( T )) n > then | ( F ( T )) n − a (( Z ( T )) n , n ) | < 2 d + 2 and therefore ( F ( T )) n = a (( Z ( T )) n , n ) , a (( Z ( T )) n , n ) ± 1 , a (( Z ( T )) n , n ) ± 2 , . . . , a (( Z ( T )) n , n ) ± (2 d + 1) , where ( F ( T )) n is r egarded as a dya dic intege r. T herefore, there are still 2 d +1 + 3 p ossib ilities of ( F ( T )) n , so that one needs only d + 2 bits more in ord er to determine ( F ( T )) n . Th us, there exists a partial recursive function Ψ : { 0 , 1 } ∗ × { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that ∀ n ∈ N + ∃ s ∈ { 0 , 1 } ∗ | s | = d + 2 & Ψ (( Z ( T )) n , s ) = ( F ( T )) n . It follo ws that there exists c Ψ ∈ N + suc h that, for all n ∈ N + , H (( F ( T )) n ) ≤ H (( Z ( T )) n ) + c Ψ . Since Z ( T ) is T -compressible b y Th eorem 4.2 (i), F ( T ) is also T -compressible. (ii) In the case of T > 1, since lim n →∞ Z n ( T ) = ∞ by Theorem 4.2 (ii), we see that F n ( T ) div erges to −∞ as n → ∞ . 12 In statistical mec hanics, the energy E sm ( T ) at te mp erature T is giv en b y E sm ( T ) = 1 Z sm ( T ) X x ∈ X E x e − E x kT , (14) where Z sm ( T ) is giv en b y (9). Motiv ated by the form u la (14) and taking int o accoun t Replace- men ts 1.1, w e in tro du ce the notion of energy in to algorithmic information theory as follo ws. Deﬁnition 4.5 (energy) . F or e ach n ∈ N + and e ach r e al numb er T > 0 , we deﬁne E n ( T ) by E n ( T ) = 1 Z n ( T ) n X i =1 | q i | 2 − | q i | T . Then, for e ach T > 0 , the ener gy E ( T ) is deﬁne d by E ( T ) = lim n →∞ E n ( T ) . Theorem 4.6. L et T ∈ R . (i) If 0 < T < 1 and T is c omputable, then E ( T ) c onver ges to a left-c omputable r e al numb er which is Cha itin T -r andom and T -c ompr e ssible. (ii) If 1 ≤ T , then E ( T ) diver ges to ∞ . Pr o of. (i) First w e sho w that E ( T ) con v erges. By Th eorem 4.2 (i), the denominator Z n ( T ) of E n ( T ) con verges to the real n umb er Z ( T ) > 0 as n → ∞ . On the other hand , by Theorem 3.1 (i), the numerator P n i =1 | q i | 2 − | q i | T of E n ( T ) conv erges to the real n u m b er W (1 , T ) as n → ∞ . Th us , E n ( T ) conv erges to the real num b er W (1 , T ) / Z ( T ) as n → ∞ . Next, w e sho w that E ( T ) is a left-computable real num b er. L et p 1 , p 2 , p 3 , . . . b e a recursiv e en u meration of th e r.e. set dom U . F or eac h m ∈ N + , w e deﬁne f W m ( T ) and e Z m ( T ) by f W m ( T ) = m X i =1 | p i | 2 − | p i | T and e Z m ( T ) = m X i =1 2 − | p i | T , and then deﬁn e e E m ( T ) by e E m ( T ) = f W m ( T ) / e Z m ( T ). Since the n umerator and the d enominator of E n ( T ) are p ositiv e term series whic h con verge as n → ∞ , w e see that W (1 , T ) = lim m →∞ f W m ( T ), Z ( T ) = lim m →∞ e Z m ( T ), and E ( T ) = lim m →∞ e E m ( T ). W e then see that e E m +1 ( T ) − e E m ( T ) = e Z m ( T ) | p m +1 | − f W m ( T ) e Z m +1 ( T ) e Z m ( T ) 2 − | p m +1 | T . Since f W m ( T ) an d e Z m ( T ) conv erge as m → ∞ and lim m →∞ | p m | = ∞ , there exist a ∈ N + and m 1 ∈ N + suc h that, for any m ≥ m 1 , e E m +1 ( T ) − e E m ( T ) > | p m +1 | 2 − | p m +1 | T − a . (15) In particular, e E m ( T ) is an increasing function of m for all m ≥ m 1 b y the ab o ve inequalit y . Thus, since T is a computable real num b er, E ( T ) is sho wn to b e a left-computable real n u m b er. W e then sho w that E ( T ) is Chaitin T -rand om. Let α b e the inﬁnite binary string suc h that 0 .α is th e base-t wo expansion of E ( T ) mo d ′ 1 with inﬁ n itely many ones. Then, s ince T is a computable 13 real num b er and ⌈ E ( T ) ⌉ − 1 + 0 .α n < E ( T ) for all n ∈ N + , there exists a partial recursiv e function ξ : { 0 , 1 } ∗ → N + suc h that, for all n ∈ N + , ξ ( α n ) ≥ m 1 and ⌈ E ( T ) ⌉ − 1 + 0 .α n < e E ξ ( α n ) ( T ) . It is then easy to see that E ( T ) − e E ξ ( α n ) ( T ) < 2 − n . It follo ws from (15) that, for all i > ξ ( α n ), | p i | 2 − | p i | T − a < 2 − n and therefore T log 2 | p i | − T a < | p i | − T n . Th us, giv en α n , by calculating th e set  U ( p i )   i ≤ ξ ( α n )  and pic king an y one ﬁnite b inary string whic h is not in this set, one can obtain s ∈ { 0 , 1 } ∗ suc h that T log 2 H ( s ) − T a < H ( s ) − T n . Hence, there exists a partial recursive function Ψ : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that T log 2 H (Ψ( α n )) − T a < H (Ψ( α n )) − T n. Applying this inequalit y to itself, w e ha ve T log 2 n < H (Ψ( α n )) − T n + O (1). On the other h and, using (2 ) there is a natural num b er c Ψ suc h that H (Ψ ( α n )) < H ( α n ) + c Ψ . Therefore, we ha ve T log 2 n < H ( α n ) − T n + O (1) . Hence, α is Chaitin T -random. It follo ws that α has inﬁ n itely man y zeros, whic h implies that E ( T ) mo d 1 = E ( T ) mo d ′ 1 = 0 .α and ther efore ( E ( T )) n = α n . Th u s, E ( T ) is C haitin T -r andom. Next, w e s h o w that E ( T ) is T -compressible. Since T is a computable real num b er, there exists a total r ecur siv e function g : N + × N + → Z su c h that, for all m, n ∈ N + ,    e E m ( T ) − ⌊ E ( T ) ⌋ − 2 − n g ( m, n )    < 2 − n . (16) It is also sho wn that there is c ∈ N + suc h that, for all m ∈ N + ,    E ( T ) − e E m ( T )    < 2 c max n    W (1 , T ) − f W m ( T )    ,    Z ( T ) − e Z m ( T )    o . (17) Let t b e an y compu table real n u m b er with T < t < 1, Then, W (1 , t ) = lim m →∞ f W m ( t ), where f W m ( t ) = m X i =1 | p i | 2 − | p i | t . The limit v alue W (1 , t ) exists since t < 1. Let β b e the inﬁ nite binary s tr ing suc h that 0 .β is the base-t wo expansion of W (1 , t ) mo d ′ 1 with in ﬁ nitely man y ones. Giv en n and β ⌈ T n/t ⌉ (i.e., the ﬁrst ⌈ T n/t ⌉ bits of β ), one can ﬁnd m 0 ∈ N + suc h that ⌈ W (1 , t ) ⌉ − 1 + 0 .β ⌈ T n/t ⌉ < f W m 0 ( t ) . This is p ossible since t is a computable real num b er and ⌈ W (1 , t ) ⌉ − 1 + 0 .β ⌈ T n/t ⌉ < W (1 , t ). It is then easy to see that ∞ X i = m 0 +1 | p i | 2 − | p i | t < 2 − T n/t . 14 Raising b oth sides of this inequ ality to the pow er t/T and using the inequ alit y a c + b c ≤ ( a + b ) c for real num b ers a, b > 0 and c ≥ 1, 2 − n > ∞ X i = m 0 +1 | p i | t/T 2 − | p i | T > ∞ X i = m 0 +1 | p i | 2 − | p i | T and therefore 2 − n > ∞ X i = m 0 +1 2 − | p i | T . It follo ws that    W (1 , T ) − f W m 0 ( T )    < 2 − n and    Z ( T ) − e Z m 0 ( T )    < 2 − n . (18) F r om (16), (17) , (18), and | ⌊ E ( T ) ⌋ + 0 . ( E ( T )) n − E ( T ) | < 2 − n , it is shown that | ( E ( T )) n − g ( m 0 , n ) | < 2 c + 2 and therefore ( E ( T )) n = g ( m 0 , n ) , g ( m 0 , n ) ± 1 , g ( m 0 , n ) ± 2 , . . . , g ( m 0 , n ) ± (2 c + 1) , where ( E ( T )) n is regarded as a dy adic in teger. Th us, there are still 2 c +1 + 3 p ossibilities of ( E ( T )) n , so that on e n eeds only c + 2 bits more in ord er to determine ( E ( T )) n . Th us, there exists a partial recursive function Φ : N + × { 0 , 1 } ∗ × { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that ∀ n ∈ N + ∃ s ∈ { 0 , 1 } ∗ | s | = c + 2 & Φ( n, β ⌈ T n/t ⌉ , s ) = ( E ( T )) n . It follo w s f rom (3) that H (( E ( T )) n ) ≤ | β ⌈ T n/t ⌉ | + o ( n ) ≤ T n/t + o ( n ), whic h imp lies th at E ( T ) is T /t -compressible. Since t is an y computable real n umber with T < t < 1, it follo ws that E ( T ) is T -compressible. (ii) In the case of T = 1, by Theorem 3.1 (ii), the n umerator W (1 , 1) = P p ∈ dom U | p | 2 −| p | of E (1) diverges to ∞ . On the other h and, the denominator Z (1) of E (1) conv erges. Thus, E (1) div erges to ∞ . The case of T > 1 is treated as follo ws. W e n ote that lim n →∞ | q n | = ∞ . Giv en M > 0, there is n 0 ∈ N + suc h that | q i | ≥ 2 M for all i > n 0 . Sin ce lim n →∞ Z n ( T ) = ∞ b y Theorem 4.2 (ii), there is n 1 ∈ N + suc h that 1 Z n ( T ) n 0 X i =1 2 − | q i | T ≤ 1 2 15 for all n > n 1 . Thus, for ev ery n > max { n 0 , n 1 } , E n ( T ) = 1 Z n ( T ) n 0 X i =1 | q i | 2 − | q i | T + 1 Z n ( T ) n X i = n 0 +1 | q i | 2 − | q i | T > 2 M Z n ( T ) n X i = n 0 +1 2 − | q i | T = 2 M 1 − 1 Z n ( T ) n 0 X i =1 2 − | q i | T ! ≥ 2 M 1 2 = M . Hence, lim n →∞ E n ( T ) = ∞ , and th e pro of is complete d. In statistical mec hanics, the entrop y S sm ( T ) at te mp erature T is giv en b y S sm ( T ) = 1 T E sm ( T ) + k ln Z sm ( T ) , (19) where Z sm ( T ) and E sm ( T ) are giv en by (9) and (14), resp ectiv ely . Motiv ated b y the formula (19) and taking into acco u n t Replacemen ts 1.1, w e in tro d uce the notion of statistical mec han ical entrop y in to alg orithmic information theory as follo w s. Deﬁnition 4.7 (statistica l mec h anical en trop y) . F or e ach n ∈ N + and e ach r e al numb er T > 0 , we deﬁne S n ( T ) by S n ( T ) = 1 T E n ( T ) + log 2 Z n ( T ) . Then, for e ach T > 0 , the statistic al me chanic al e ntr opy S ( T ) is deﬁne d by S ( T ) = lim n →∞ S n ( T ) . Theorem 4.8. L et T ∈ R . (i) If 0 < T < 1 and T is c omputable, then S ( T ) c onver ges to a left-c omputable r e al numb er which is Cha itin T -r andom and T -c ompr e ssible. (ii) If 1 ≤ T , then S ( T ) diver ges to ∞ . Pr o of. (i) Since Z ( T ) and E ( T ) con v erge b y Th eorem 4.2 (i) and Theorem 4.6 (i), resp ectiv ely , S ( T ) also conv erges and S ( T ) = 1 T E ( T ) + log 2 Z ( T ) . Since E ( T ) is a left-computable real num b er and E ( T ) / ∈ N by T heorem 4.6 (i), there exists a total recursiv e function f : N + → Q such that 0 < f ( m ) ≤ E ( T ) and ⌊ f ( m ) ⌋ = ⌊ E ( T ) ⌋ for all m ∈ N + and lim m →∞ f ( m ) = E ( T ). Since T is a computable r eal n um b er, th ere exists a total recursiv e fun ction g : N + → Q su c h that 0 ≤ g ( m ) ≤ 1 /T f or all m ∈ N + and lim m →∞ g ( m ) = 1 /T . Since Z ( T ) is a left-computable real num b ers by Theorem 4.2 (i), there exists a total recursive function h : N + → Q such that h ( m ) ≤ log 2 Z ( T ) for all m ∈ N + and lim m →∞ h ( m ) = log 2 Z ( T ). 16 Hence, g ( m ) f ( m ) + h ( m ) ≤ S ( T ) for all m ∈ N + and lim m →∞ g ( m ) f ( m ) + h ( m ) = S ( T ). Th us , S ( T ) is a left-c omputable real n u m b er. W e then show th at S ( T ) is w eakly C haitin T -random. Let α b e the inﬁnite binary strin g suc h that 0 .α is the base-t w o expansion of S ( T ) m o d ′ 1 with in ﬁ nitely man y ones. Giv en α n , one can ﬁn d m 0 ∈ N + suc h that ⌈ S ( T ) ⌉ − 1 + 0 .α n < g ( m 0 ) f ( m 0 ) + h ( m 0 ) . This is p ossible because ⌈ S ( T ) ⌉ − 1 + 0 .α n < S ( T ) and lim m →∞ g ( m ) f ( m ) + h ( m ) = S ( T ). It is sho w n that 2 − n >  1 T E ( T ) + log 2 Z ( T )  − ( g ( m 0 ) f ( m 0 ) + h ( m 0 )) ≥ 1 T E ( T ) − g ( m 0 ) f ( m 0 ) ≥ 1 T ( E ( T ) − f ( m 0 )) ≥ E ( T ) − f ( m 0 ) . Th us, 0 ≤ E ( T ) − f ( m 0 ) < 2 − n . Let l n b e the ﬁrst n bits of the b ase-t w o expan s ion of the r ational n u m b er f ( m 0 ) − ⌊ f ( m 0 ) ⌋ with inﬁ nitely man y zeros. It follo w s that 0 ≤ 0 . ( E ( T )) n − 0 .l n < E ( T ) − f ( m 0 ) + 2 − n < 2 · 2 − n . Hence ( E ( T )) n = l n , l n + 1 , where ( E ( T )) n and l n are regarded as a dyadic in teger. Thus, there are still 2 p ossibilities of ( E ( T )) n , so that one needs only 1 bit more in ord er to determine ( E ( T )) n . Th us, there exists a partial recursive function Φ : { 0 , 1 } ∗ × { 0 , 1 } → { 0 , 1 } ∗ suc h that ∀ n ∈ N + ∃ b ∈ { 0 , 1 } Φ( α n , b ) = ( E ( T )) n . It follo ws that there exists c Φ ∈ N + suc h that, for all n ∈ N + , H (( E ( T )) n ) ≤ H ( α n ) + c Φ . Hence, α is Ch aitin T -rand om b y Theorem 4.6 (i). It follo ws that α has inﬁnitely many zeros, whic h implies that S ( T ) mo d 1 = S ( T ) mo d ′ 1 = 0 .α and th erefore ( S ( T )) n = α n . T h u s, S ( T ) is also Chaitin T -random. Next, we show that S ( T ) is T -compr essib le. Let p 1 , p 2 , p 3 , . . . b e a recursive enumeration of the r.e. set d om U . F or eac h m ∈ N + , w e deﬁne f W m ( T ) and e Z m ( T ) by f W m ( T ) = m X i =1 | p i | 2 − | p i | T and e Z m ( T ) = m X i =1 2 − | p i | T , and then d eﬁne e S m ( T ) by e S m ( T ) = 1 T f W m ( T ) e Z m ( T ) + log 2 e Z m ( T ) . 17 Since, in the deﬁnition of S n ( T ), Z n ( T ) and the n umerator and the denominator of E n ( T ) are p ositiv e term series whic h con v erge as n → ∞ , we see that W (1 , T ) = lim m →∞ f W m ( T ), Z ( T ) = lim m →∞ e Z m ( T ), and S ( T ) = lim m →∞ e S m ( T ). Since T is a computable real n umb er, there exists a to tal recursive fu nction a : N + × N + → Z suc h that, for all m, n ∈ N + ,    e S m ( T ) − ⌊ S ( T ) ⌋ − 2 − n a ( m, n )    < 2 − n . (20) It is also sho wn that there is d ∈ N + suc h that, for all m ∈ N + ,    S ( T ) − e S m ( T )    < 2 d max n    W (1 , T ) − f W m ( T )    ,    Z ( T ) − e Z m ( T )    o . (21) Based on the inequalities (20) and (21) , in the s ame manner as the pro of of the T -compressibilit y of E ( T ) in Theorem 4.6 (i), we can sho w that S ( T ) is T -compressible. (ii) In the case of T ≥ 1, since lim n →∞ E n ( T ) = ∞ by Th eorem 4.6 (ii) and log 2 Z n ( T ) is b ound ed to the b elo w b y Theorem 4.2, we see th at lim n →∞ S n ( T ) = ∞ . Finally , in statistical m ec hanics, the sp eciﬁc h eat C sm ( T ) at te mp erature T is giv en by C sm ( T ) = d dT E sm ( T ) , (22) where E sm ( T ) is give n by (14). Motiv ated by the form u la (22), w e in tro duce the notion of sp eciﬁc heat int o algorithmic in formation theory as follo ws. Deﬁnition 4.9 (sp eciﬁc heat) . F or e ach n ∈ N + and e ach r e al numb er T > 0 , we deﬁne C n ( T ) by C n ( T ) = E ′ n ( T ) , wher e E ′ n ( T ) is the derive d function of E n ( T ) . Then, for e ach T > 0 , the sp e ciﬁc he at C ( T ) is deﬁne d by C ( T ) = lim n →∞ C n ( T ) . Theorem 4.10. L et T ∈ R . (i) If 0 < T < 1 and T is c omputable, then C ( T ) c onver ges to a left-c omputable r e al numb er which is Chaitin T -r andom and T -c ompr essible, and mor e over C ( T ) = E ′ ( T ) wher e E ′ ( T ) is the derive d function of E ( T ) . (ii) If T = 1 , then C ( T ) diver ges to ∞ . Pr o of. (i) First w e s h o w that C ( T ) conv erges. Note that C n ( T ) = ln 2 T 2 ( Y n ( T ) Z n ( T ) −  W n ( T ) Z n ( T )  2 ) , where Y n ( T ) = n X i =1 | q i | 2 2 − | q i | T and W n ( T ) = n X i =1 | q i | 2 − | q i | T . 18 By Theorem 4.2 (i), Z n ( T ) con ve rges to the real n umb er Z ( T ) > 0 as n → ∞ . On the other hand, by Theorem 3.1 (i), Y n ( T ) and W n ( T ) con ve rge to the real num b ers W (2 , T ) and W (1 , T ), resp ectiv ely , as n → ∞ . Thus, C n ( T ) also conv erges to a real num b er C ( T ) as n → ∞ . Next, w e sho w that C ( T ) is a left-computable real num b er. Let p 1 , p 2 , p 3 , . . . b e a recurs iv e en u meration of th e r.e. set dom U . F or eac h m ∈ N + , w e deﬁne e Y m ( T ), f W m ( T ), and e Z m ( T ) by e Y m ( T ) = m X i =1 | p i | 2 2 − | p i | T , f W m ( T ) = m X i =1 | p i | 2 − | p i | T , an d e Z m ( T ) = m X i =1 2 − | p i | T , resp ectiv ely . Since Y n ( T ), W n ( T ), and Z n ( T ) are positive term series which con v erge as n → ∞ , w e see that W (2 , T ) = lim m →∞ e Y m ( T ), W (1 , T ) = lim m →∞ f W m ( T ), Z ( T ) = lim m →∞ e Z m ( T ), and C ( T ) = lim m →∞ e C m ( T ) wh ere e C m ( T ) = ln 2 T 2    e Y m ( T ) e Z m ( T ) − f W m ( T ) e Z m ( T ) ! 2    . W e then see th at e C m +1 ( T ) − e C m ( T ) is calc ulated as ln 2 T 2 2 − | p m +1 | T e Z m +1 ( T ) " | p m +1 | 2 − ( f W m +1 ( T ) e Z m +1 ( T ) + f W m ( T ) e Z m ( T ) ) | p m +1 | + ( f W m +1 ( T ) e Z m +1 ( T ) + f W m ( T ) e Z m ( T ) ) f W m ( T ) e Z m ( T ) − e Y m ( T ) e Z m ( T ) # . (23) Since e Y m ( T ), f W m ( T ), and e Z m ( T ) con verge as m → ∞ and lim m →∞ | p m | = ∞ , ther e exists m 1 ∈ N + suc h that, for an y m ≥ m 1 , e C m +1 ( T ) − e C m ( T ) > | p m +1 | 2 − | p m +1 | T . (24) In particular, e C m ( T ) is an increasing function of m for all m ≥ m 1 b y the ab o ve inequalit y . Thus, since T is a computable real num b er, C ( T ) is sh o wn to b e a left-computable r eal num b er. Based on the compu tabilit y of T and the inequalit y (24), in the same manner as the proof of Theorem 4.6 (i) w e can sho w th at C ( T ) is Chaitin T -random. Next, w e sho w that C ( T ) is T -compressible. Since T is a computable real n um b er, there exists a total r ecur siv e function g : N + × N + → Z su c h that, for all m, n ∈ N + ,    e C m ( T ) − ⌊ C ( T ) ⌋ − 2 − n g ( m, n )    < 2 − n . (25) It is also sho wn that there is c ∈ N + suc h that, for all m ∈ N + ,    C ( T ) − e C m ( T )    < 2 c max n    Y ( T ) − e Y m ( T )    ,    W ( T ) − f W m ( T )    ,    Z ( T ) − e Z m ( T )    o . (26) Let t b e an y compu table real n u m b er with T < t < 1. Then, W (2 , t ) = lim m →∞ e Y m ( t ), where e Y m ( t ) = m X i =1 | p i | 2 2 − | p i | t . 19 The limit v alue W (2 , t ) exists since t < 1. Let β b e the inﬁ nite binary s tr ing suc h that 0 .β is the base-t wo expansion of W (2 , t ) mo d ′ 1 with in ﬁ nitely man y ones. Giv en n and β ⌈ T n/t ⌉ (i.e., the ﬁrst ⌈ T n/t ⌉ bits of β ), one can ﬁnd m 0 ∈ N + suc h that ⌈ W (2 , t ) ⌉ − 1 + 0 .β ⌈ T n/t ⌉ < e Y m 0 ( t ) . This is p ossible since t is a computable real num b er and ⌈ W (2 , t ) ⌉ − 1 + 0 .β ⌈ T n/t ⌉ < W (2 , t ). It is then easy to see that ∞ X i = m 0 +1 | p i | 2 2 − | p i | t < 2 − T n/t . Raising b oth sides of this inequ ality to the pow er t/T and using the inequ alit y a c + b c ≤ ( a + b ) c for real num b ers a, b > 0 and c ≥ 1, 2 − n > ∞ X i = m 0 +1 | p i | 2 t/T 2 − | p i | T > ∞ X i = m 0 +1 | p i | 2 2 − | p i | T and therefore 2 − n > ∞ X i = m 0 +1 | p i | 2 − | p i | T and 2 − n > ∞ X i = m 0 +1 2 − | p i | T . It follo ws that max n    W (2 , T ) − e Y m 0 ( T )    ,    W (1 , T ) − f W m 0 ( T )    ,    Z ( T ) − e Z m 0 ( T )    o < 2 − n . (27) F r om (25), (26) , (27), and | ⌊ C ( T ) ⌋ + 0 . ( C ( T )) n − C ( T ) | < 2 − n , it is shown that | ( C ( T )) n − g ( m 0 , n ) | < 2 c + 2 and therefore ( C ( T )) n = g ( m 0 , n ) , g ( m 0 , n ) ± 1 , g ( m 0 , n ) ± 2 , . . . , g ( m 0 , n ) ± (2 c + 1) , where ( C ( T )) n is regarded as a dy adic in teger. Thus, there are still 2 c +1 + 3 p ossibilities of ( C ( T )) n , so that on e n eeds only c + 2 bits more in ord er to determine ( C ( T )) n . Th us, there exists a partial recursive function Φ : N + × { 0 , 1 } ∗ × { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that ∀ n ∈ N + ∃ s ∈ { 0 , 1 } ∗ | s | = c + 2 & Φ( n, β ⌈ T n/t ⌉ , s ) = ( C ( T )) n . It follo w s f rom (3) that H (( C ( T )) n ) ≤ | β ⌈ T n/t ⌉ | + o ( n ) ≤ T n/t + o ( n ), whic h implies that C ( T ) is T /t -compressible. Since t is any compu table real n um b er with T < t < 1, it follo ws that C ( T ) is T -compressible. By ev aluating C n +1 ( x ) − C n ( x ) for al l x ∈ (0 , 1) lik e (2 3), w e ca n show that C n ( x ) conv erges uniformly in the wider sense on (0 , 1) to C ( x ) as n → ∞ . Hence, w e hav e C ( T ) = E ′ ( T ). 20 (ii) It can b e sho wn that C n (1) = ln 2 2 1 Z n (1) 2 n X i =1 n X j =1 ( | q i | − | q j | ) 2 2 −| q i | 2 −| q j | . By Theorem 3.2 (i), P n j =1 ( | q 1 | − | q j | ) 2 2 −| q j | div erges to ∞ as n → ∞ . Therefore P n i =1 P n j =1 ( | q i | − | q j | ) 2 2 −| q i | 2 −| q j | also div erges to ∞ as n → ∞ . On th e other hand, by Theorem 4.2 (i), Z n (1) con verge s as n → ∞ . Th u s, C (1) div erges to ∞ , and the pro of is completed. Th us, the theorems in this s ection sho w that the temp eratur e T p la ys a role as th e compression rate for all the thermo dyn amic quan tities in tro d uced in to algorithmic information theory in this section. These theorems also sho w that the v alues of the thermo dynamic qu an tities: p artition function, free energy , energy , and statistical mechanical en tropy div erge in th e case of T > 1. This phenomenon migh t b e regarded as some s ort of ph ase transition in statistical mec hanics. 4 5 Fixed p oin t theorems on compression rate In this section, w e pro v e the follo w in g th eorem. Theorem 5.1 (ﬁxed p oint theorem on compression rate) . F or every T ∈ (0 , 1) , if Z ( T ) is a c omputable r e al numb er, then the fol lowing hold: (i) T is right-c omputable and not left-c omputable. (ii) T is we akly Chaitin T -r andom and T -c ompr essib le. (iii) lim n →∞ H ( T n ) /n = T . Theorem 5.1 follo ws immediately from Th eorem 5.3, Theorem 5.5, and Th eorem 5.6 b elo w. F r om a pu rely m athematical p oint of view, T heorem 5.1 is just a ﬁxed p oint theorem on compr ession rate, w here the computabilit y of th e v alue Z ( T ) giv es a su ﬃcien t condition for T to b e a ﬁxed p oin t on compression rate. Note that Z ( T ) is a monotonically increasing con tin uous function on (0 , 1). In fact, [15, 16] sh o wed that Z ( T ) is a function of class C ∞ on (0 , 1). Thus, s in ce computable real n u m b ers are d ense in R , we ha ve the follo win g corollary of T heorem 5.1. Corollary 5.2. The set { T ∈ (0 , 1) | lim n →∞ H ( T n ) /n = T } is dense in [0 , 1] . F r om the p oin t of view of th e statistical m echanica l interpretatio n in tro duced in the p revious section, Theorem 5.1 sho w s that the co mpression rate of temp erature is equal to the temp eratur e itself. Thus, Theorem 5.1 further conﬁrms the role of temp erature as the compression rate, whic h is observ ed in the previous section. As a ﬁr st s tep to prov e Theorem 5.1, we prov e the follo wing theorem whic h give s the w eak Chaitin T -rand omness of T in T h eorem 5.1. Theorem 5.3. F or every T ∈ (0 , 1) , if Z ( T ) is a right-c omputable r e al numb er, then T is we akly Chaitin T -r andom. 4 It is still open whether C ( T ) diverge s or not in the case of T > 1. 21 Pr o of. Let p 1 , p 2 , p 3 , . . . b e a recursive en umeration of the r.e. set dom U . F or eac h k ∈ N + , we deﬁne a fun ction e Z k : (0 , 1) → R by e Z k ( x ) = k X i =1 2 − | p i | x . Then, lim k →∞ e Z k ( x ) = Z ( x ) for ev ery x ∈ (0 , 1). On the other hand , since Z ( T ) is right - computable, there exists a total recursiv e function g : N + → Q suc h that Z ( T ) ≤ g ( m ) f or all m ∈ N + , and lim m →∞ g ( m ) = Z ( T ). W e c ho ose an y one real num b er t with T < t < 1. F or eac h i ∈ N + , using the mean v alue theorem we see that 2 − | p i | x − 2 − | p i | T < ln 2 T 2 | p i | 2 − | p i | t ( x − T ) for all x ∈ ( T , t ). W e c ho ose an y one c ∈ N with W (1 , t ) ln 2 /T 2 ≤ 2 c . Here, the limit v alue W (1 , t ) exists by T heorem 3.1 (i), since 0 < t < 1. Then, it follo w s that e Z k ( x ) − e Z k ( T ) < 2 c ( x − T ) (28) for all k ∈ N + and x ∈ ( T , t ). W e c ho ose an y one n 0 ∈ N + suc h that T < 0 .T n + 2 − n < t for all n ≥ n 0 . Suc h n 0 exists s ince T < t and lim n →∞ 0 .T n + 2 − n = T . Giv en T n with n ≥ n 0 , one can ﬁnd k 0 , m 0 ∈ N + suc h that g ( m 0 ) < e Z k 0 (0 .T n + 2 − n ) . This is p ossible from Z ( T ) < Z (0 .T n + 2 − n ), lim k →∞ e Z k (0 .T n + 2 − n ) = Z (0 .T n + 2 − n ), and the prop erties of g . It follo ws from Z ( T ) ≤ g ( m 0 ) and (28) th at ∞ X i = k 0 +1 2 − | p i | T = Z ( T ) − e Z k 0 ( T ) < e Z k 0 (0 .T n + 2 − n ) − e Z k 0 ( T ) < 2 c − n . Hence, for ev ery i > k 0 , 2 − | p i | T < 2 c − n and therefore T n − T c < | p i | . Th us, b y ca lculating the set { U ( p i )   i ≤ k 0 } and pic king an y one ﬁn ite bin ary str in g wh ic h is not in this set, one can then obtain an s ∈ { 0 , 1 } ∗ suc h that T n − T c < H ( s ). Hence, there exists a partial recurs iv e fu n ction Ψ : { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that T n − T c < H (Ψ( T n )) for all n ≥ n 0 . Using (2), there is c Ψ ∈ N + suc h that H (Ψ( T n )) < H ( T n ) + c Ψ for all n ≥ n 0 . T herefore, T n − T c − c Ψ < H ( T n ) for all n ≥ n 0 . I t follo ws that T is wea k ly Chaitin T -random. Remark 5.4. B y elab or ating The or em 4.2 (i), we c an se e that the left-c omputability of T r e- sults in the we ak Chaitin T -r andomness of Z ( T ) . On th e other hand, by The or em 5.3, the right- c omputability of Z ( T ) r esults in the we ak Chaitin T -r andomness of T . We c an inte gr ate these two extr emes into the fol lowing form: F or every T ∈ (0 , 1] , ther e exists c ∈ N + such that, for ev ery n ∈ N + , T n − c ≤ H ( T n , ( Z ( T )) n ) , (29) 22 wher e H ( s, t ) is deﬁne d as H ( < s , t > ) with any one c omputable b i je ction < s, t > fr om ( s, t ) ∈ { 0 , 1 } ∗ × { 0 , 1 } ∗ to { 0 , 1 } ∗ (se e [4] for the detail of the notion of H ( s, t ) ). In fact, if T is left- c omputable, then we c an show that H (( Z ( T )) n ) = H ( T n , ( Z ( T )) n ) + O (1) , and ther efor e the in- e quality (2 9) r esults in the we ak Chaitin T -r andomness of Z ( T ) . On the other hand, if Z ( T ) is right-c omputable, then we c an show that H ( T n ) = H ( T n , ( Z ( T )) n ) + O (1) , and ther efor e the ine quality (29) r e sults in the we ak Chaitin T -r andomness of T . Note, how ever, that the ine quality (29) is not ne c e ssarily tight exc ept for these two extr emes, that is, th e fol lowing ine q u ality do es not hold: F or every T ∈ (0 , 1] , H ( T n , ( Z ( T )) n ) ≤ T n + o ( n ) , (30) wher e o ( n ) dep ends on T in addition to n . T o se e th is, c ontr arily assume that the ine quality (30) holds. Then, by setting T to Chaitin ’s Ω , we have H (Ω n ) ≤ H (Ω n , ( Z (Ω)) n ) + O (1) ≤ Ω n + o ( n ) . Sinc e Ω < 1 , this c ontr adicts th e fact that Ω is we akly Chaitin r andom. Thus, the ine quality (30 ) do es not hold. The follo wing Th eorem 5.5 and Theorem 5. 6 giv e the T -compressibilit y of T in Th eorem 5.1 together. Theorem 5.5. F or e very T ∈ (0 , 1) , if Z ( T ) is a right-c omp utable r e al numb er, then T is also a right-c omputable r e al numb er. Pr o of. Let p 1 , p 2 , p 3 , . . . b e a recursive en umeration of the r.e. set dom U . F or eac h k ∈ N + , we deﬁne a fun ction e Z k : (0 , 1) → R by e Z k ( x ) = k X i =1 2 − | p i | x . Then, lim k →∞ e Z k ( x ) = Z ( x ) f or every x ∈ (0 , 1). Since Z ( T ) is right- computable, there exists a total recursive fun ction g : N + → Q suc h that Z ( T ) ≤ g ( m ) for all m ∈ N + , and lim m →∞ g ( m ) = Z ( T ). Since Z ( x ) is an increasing function of x , we see that, for ev ery x ∈ Q with 0 < x < 1, T < x if and only if there are m, k ∈ N + suc h that g ( m ) < e Z k ( x ). Th us, T is right-c omputable. This is b ecause th e set { ( m, n ) | m ∈ Z & n ∈ N + & T < m/n } is r.e. if and on ly if T is righ t-computable. The con verse of Theorem 5.5 do es not hold. T o see this, consider an arb itrary computable real n um b er T ∈ (0 , 1). Then, obvio usly T is righ t-computable. On the other hand , Z ( T ) is left- computable and w eakly Chaitin T -random b y Theorem 4.2 (i). T hus, Z ( T ) is not righ t-computable. Theorem 5.6. F or every T ∈ (0 , 1) , i f Z ( T ) is a left-c omputable r e al numb er and T is a right- c omputable r e al numb er, then T is T -c ompr essible. Pr o of. Let p 1 , p 2 , p 3 , . . . b e a recursive en umeration of the r.e. set dom U . F or eac h k ∈ N + , we deﬁne a fun ction e Z k : (0 , 1) → R by e Z k ( x ) = k X i =1 2 − | p i | x . 23 Then, lim k →∞ e Z k ( x ) = Z ( x ) for every x ∈ (0 , 1). F or eac h i ∈ N + , using the mean v alue theorem w e see that 2 − | p 1 | t − 2 − | p 1 | T > (ln 2) | p 1 | 2 − | p 1 | T ( t − T ) for all t ∈ ( T , 1). W e c h o ose an y one c ∈ N + suc h that (ln 2) | p 1 | 2 − | p 1 | T ≥ 2 − c . Then, it follo ws that e Z k ( t ) − e Z k ( T ) > 2 − c ( t − T ) (31) for all k ∈ N + and t ∈ ( T , 1). Since T is a right- computable real n umb er with T < 1, there exists a total recur s iv e fu nction f : N + → Q suc h th at T < f ( l ) < 1 for all l ∈ N + , and lim l →∞ f ( l ) = T . On the other hand, since Z ( T ) is left-computable, there exists a total recursive function g : N + → Q such that g ( m ) ≤ Z ( T ) for all m ∈ N + , and lim m →∞ g ( m ) = Z ( T ). Let β b e the in ﬁnite b inary sequ ence su c h that 0 .β is the base-t w o expansion of Z (1) (i.e., Ch aitin’s Ω). Giv en n and β ⌈ T n ⌉ (i.e., the ﬁrst ⌈ T n ⌉ bits of β ), one can ﬁnd k 0 ∈ N + suc h that 0 .β ⌈ T n ⌉ < k 0 X i =1 2 −| p i | . It is then easy to see that ∞ X i = k 0 +1 2 −| p i | < 2 − T n . Using the inequ alit y a d + b d ≤ ( a + b ) d for real num b ers a, b > 0 an d d ≥ 1, it follo w s that Z ( T ) − e Z k 0 ( T ) < 2 − n . (32) Note that e Z k 0 ( T ) < e Z k 0 ( f ( l )) for all l ∈ N + , and lim l →∞ e Z k 0 ( f ( l )) = e Z k 0 ( T ). T h u s, since e Z k 0 ( T ) < Z ( T ), one can then ﬁn d l 0 , m 0 ∈ N + suc h that e Z k 0 ( f ( l 0 )) < g ( m 0 ) . It follo ws from (32) and (31) that 2 − n > g ( m 0 ) − e Z k 0 ( T ) > e Z k 0 ( f ( l 0 )) − e Z k 0 ( T ) > 2 − c ( f ( l 0 ) − T ) . Th us, 0 < f ( l 0 ) − T < 2 c − n . Let t n b e the ﬁrst n bits of the base-t w o expansion of the ratio n al n u m b er f ( l 0 ) with inﬁn itely many zeros. Then, | f ( l 0 ) − 0 .t n | < 2 − n . It follo ws from | T − 0 .T n | < 2 − n that | 0 .T n − 0 .t n | < (2 c + 2)2 − n . Hence T n = t n , t n ± 1 , t n ± 2 , . . . , t n ± (2 c + 1) , where T n and t n are regarded as a d y adic integ er. Thus, there are still 2 c +1 + 3 p ossibilities of T n , so that on e n eeds only c + 2 bits more in ord er to determine T n . Th us, there exists a partial recursive function Φ : N + × { 0 , 1 } ∗ × { 0 , 1 } ∗ → { 0 , 1 } ∗ suc h that ∀ n ∈ N + ∃ s ∈ { 0 , 1 } ∗ | s | = c + 2 & Φ ( n, β ⌈ T n ⌉ , s ) = T n . It follo ws fr om (3) that H ( T n ) ≤ | β ⌈ T n ⌉ | + o ( n ) ≤ T n + o ( n ), whic h implies that T is T -compressible. 24 In a similar manner to the pro of of Theorem 5.1, w e can prov e another v ersion of a ﬁxed p oin t theorem on compr ession rate as follo ws. Here, the w eak Chaitin T -randomness is replaced by the Chaitin T -rand omness. Theorem 5.7 (ﬁxed p oin t theorem on compr ession rate I I) . L et Q b e a c omputable r e al numb er with Q > 0 . F or every T ∈ (0 , 1) , if W ( Q, T ) is a c omputable r e al numb er, then the fol lowing hold: (i) T is right-c omputable and not left-c omputable. (ii) T is Chaitin T -r ando m and T -c ompr essible. Remark 5.8. The c omputability of Z ( T ) in the pr emise of The or em 5.1 c an b e r eplac e d by the c om- putability of F ( T ) . On the other hand, the c omputability of W ( Q, T ) in the pr emise of The or em 5.7 c an b e r eplac e d by the c omputability of E ( T ) or S ( T ) . 6 T otal statistical mec hanical in terpretation of algorithmic infor- mation theory: Ph ysical and informal argumen t In what follo ws, based on a physical argumen t we d ev elop a total statistica l mec hanical int erpreta- tion of algo rithmic information th eory whic h attains a p erfect corresp ondence to norm al statistica l mec hanics. In consequence, we ju s tify the in terp r etation of Ω D as a partition function and clarify the statistical mec hanical meaning of the thermo dyn amic quan tities in tr o duced into algorithmic information theory in Section 4. In the w ork [17], we dev elop ed a statistical mec hanical inte rpre- tation of the noiseless sou r ce co din g sc h eme b ased on an absolutely optimal instan taneous co de b y iden tifyin g a micro canonical ensem ble in the sc heme. In a similar manner to [17] w e d ev elop a statistica l mec h anical in terpretation of algo r ithmic information theory in what f ollo ws. Th is can b e p ossible b ecause the set dom U is preﬁx-fr ee and therefore the action of the op timal compu ter U can b e regarded as an instan taneous co de whic h is extended ov er an in ﬁnite set. Note that, in what follo ws , we do not stic k to the mathematical strictness of the argumen t and w e make an argumen t on the same lev el of mathematical strictness as statistica l mec hanics in p h ysics. W e start with some r eviews of statistical mec hanics. In statistical mec hanics we consid er a quant um system S total whic h consists in a large num b er of iden tical quant um subsystems. Let N b e the n um b er of suc h su bsystems. F or example, N ∼ 10 22 for 1 cm 3 of a gas at ro om temp erature. W e assume here that eac h quantum su bsystem can b e distinguishable fr om others. Th u s, we d eal with quan tum particles wh ic h ob ey Maxwe ll-Boltz mann statistics and n ot Bose-Einstein statistic s or F ermi-Dirac statistics. Und er this assum p tion, we can iden tify the i th quan tum subsystem S i for eac h i = 1 , . . . , N . In qu an tum mec hanics, any q u an tu m system is d escrib ed b y a qu an tu m state completely . In statisti cal mec hanics, among all quan tum states, energy eigenstates are of particular imp ortance. An y energy eigenstate of eac h sub system S i can b e sp eciﬁed by a num b er n = 1 , 2 , 3 , . . . , called a quantum numb er , where the sub system in the energy eigenstate sp eciﬁed by n has the energy E n . Then, an y energy eigenstate of the system S total can b e sp eciﬁed b y an N -tuple ( n 1 , n 2 , . . . , n N ) of quan tum n um b ers. If the state of the system S total is the energy eigenstate sp eciﬁed b y ( n 1 , n 2 , . . . , n N ), then the state of eac h subsystem S i is the energy eige nstate sp eciﬁed by n i and th e system S total has the energy E n 1 + E n 2 + · · · + E n N . 25 Then, the fundamenta l p ostulate of statistical m echanics, calle d the principle of e qual pr ob ability , is stated as follo ws. The Principle of Equal Probabilit y: If th e energy of the system S total is kno wn to ha ve a constan t v alue in the range b et w een E and E + δ E , wher e δ E is the indeterminacy in measuremen t of the energy of the sy s tem S total , then the system S total is equ ally lik ely to b e in any energy eigenstate sp eciﬁed by ( n 1 , n 2 , . . . , n N ) such that E ≤ E n 1 + E n 2 + · · · + E n N ≤ E + δ E . Let Θ( E , N ) b e th e total num b er of energy eigenstates of S total sp eciﬁed b y ( n 1 , n 2 , . . . , n N ) such that E ≤ E n 1 + E n 2 + · · · + E n N ≤ E + δ E . The ab o v e p ostulate states that an y energy eigenstate of S total whose energy lies betw een E and E + δ E o ccurs with the probabilit y 1 / Θ( E , N ). This uniform distrib ution of energy eigenstates wh ose energy lie b et w een E and E + δ E is called a micr o c anonic al ensemble . I n statistica l mec hanics, the e ntr opy S ( E , N ) of the system S total is then deﬁned by S ( E , N ) = k ln Θ( E , N ) , where k is a p ositiv e constan t, called the Boltzmann Constant , and the ln denotes the natural logarithm. T he a v erage energy ε p er one sub system is giv en by E / N . In a n ormal case where ε has a ﬁnite v alue, the en tr op y S ( E , N ) is prop ortional to N . O n the other hand, the indeterminacy δ E of the energy con tr ibutes to S ( E , N ) through th e term k ln δ E , whic h can b e ignored compared to N unless δ E is too small. T h us the magnitude of the indeterminacy δ E of the energy do es not matter to the v alue of the en tropy S ( E , N ) unless it is to o small. Th e temp e r atur e T ( E , N ) of the system S total is deﬁned b y 1 T ( E , N ) = ∂ S ∂ E ( E , N ) . Th us the temp erature is a fun ction of E and N . No w we give a statistical mechanica l interpretati on to algorithmic inf ormation theory . As considered in [4], thin k of the optimal computer U as decodin g equip men t at the receiving end of a noiseless binary comm unication c hannel. Regard its programs (i.e., ﬁ nite b in ary strings in dom U ) as co dew ords and regard the r esult of th e computation by U , wh ic h is a ﬁnite binary string, as a deco ded “sym b ol.” Since dom U is a preﬁx-free set, such co dew ords form wh at is called an “instan taneous co d e,” so that successiv e symbols s ent throu gh the channel in the form of concatenati on of co dew ord s can b e separated. F or establishing the statistical mec hanical in terpretation of algo rithmic information theory , w e assume that the inﬁnite bin ary string s en t through the channel is generated b y inﬁnitely rep eated tosses of a fair coin. Under this assump tion, the success probability of deco ding one sym b ol is equal to Chaitin’s halting probabilit y Ω, and the pr obabilit y of getting a ﬁnite binary strin g s as the ﬁ rst deco ded symbol is equal to P ( s ). Hereafter the inﬁn ite binary strin g sen t through the channel is referred to as the channel inﬁnite string . F or eac h r ∈ { 0 , 1 } ∗ , let Q ( r ) b e the probabilit y that the c hann el inﬁnite string has the preﬁx r . It follo w s that Q ( r ) = 2 −| r | . Thus, the c h annel inﬁn ite string is th e random v ariable d ra wn acc ording to Leb esgue measure on { 0 , 1 } ∞ . Let N b e a large num b er, sa y N ∼ 10 22 . W e relate algorithmic information theory to the statistica l m echanics review ed ab ov e in the follo wing manner. Among all inﬁnite binary strings, consider inﬁ nite bin ary strings of the form p 1 p 2 · · · p N α with p 1 , p 2 , . . . , p N ∈ dom U and α ∈ { 0 , 1 } ∞ . F or eac h i , the i th slot fed by p i corresp onds to the i th quantum subsystem S i . On the other hand , the ord ered sequence of th e 1st slot, the 2nd slot, . . . , and the N th slot corresp on d s to 26 the qu an tum system S total . W e relate a co deword p ∈ dom U to an energy eigenstate of a subsystem, and relate a co dewo rd length | p | to an energy E n of the energy eigenstate of th e subsystem. Th en, a ﬁnite binary s tring p 1 · · · p N corresp onds to an energy eigenstate of S total sp eciﬁed by ( n 1 , . . . , n N ). Th us, | p 1 | + · · · + | p N | = | p 1 · · · p N | corresp onds to the energy E n 1 + · · · + E n N of the energy eige nstate of S total . W e d eﬁne a subs et C ( L, N ) of { 0 , 1 } ∗ as the set of all ﬁn ite binary strings of the form p 1 · · · p N with p i ∈ dom U whose total length | p 1 · · · p N | lie b et ween L and L + δ L . Then, Θ( L, N ) is d eﬁned as the cardinalit y of C ( L, N ). Therefore, Θ ( L, N ) is th e total num b er of a ll concatenatio ns of N co dew ord s whose total length lie b et we en L and L + δ L . W e can see that if p 1 · · · p N ∈ C ( L, N ), then 2 − ( L + δL ) ≤ Q ( p 1 · · · p N ) ≤ 2 − L . T hus, all concatenatio n s p 1 · · · p N ∈ C ( L, N ) of N co dewo rds o ccur in a preﬁx of the c hannel inﬁ nite string with the same probabilit y 2 − L . Note h ere that we care nothing ab out the magnitude of δ L , as in th e case of statistical mec hanics. Thus, the follo wing principle, called the principle of e qual c onditio nal pr ob ability , h olds. The Principle of Equal Conditional Probabilit y: Giv en that a concatenation of N co dewo rds of total length L o ccurs in a pr eﬁx of the channel inﬁnite string, all such concate nations occur with the same pr obabilit y 1 / Θ( L, N ). W e introdu ce a m icro canonica l ensem ble into algorithmic information theory in this m anner. Th us, w e can dev elop a certain s ort of statistic al mec hanics on algorithmic information th eory . Note that, in statistical mechanics, the principle of equal pr ob ab ility is just a conjecture wh ic h is not yet pro ved completely in a realistic physical system. On th e other hand, in our statist ical mec hanical in terp retation of algorithmic information theory , th e p rinciple of equal co nditional p robabilit y is automatica lly satisﬁed. The statistic al me chanic al entr opy S ( L, N ) is deﬁned by S ( L, N ) = log 2 Θ( L, N ) . (33) The temp er atur e T ( L, N ) is then deﬁn ed b y 1 T ( L, N ) = ∂ S ∂ L ( L, N ) . (34) Th us, the temp erature is a function of L and N . According to the theoretical develo p men t of equ ilibrium statistical mec hanics, 5 w e can introdu ce a canonical ensem ble in to alg orithmic in formation theory in the f ollo wing manner. W e in vesti gate the probabilit y distr ib ution of the left-most codeword p 1 of the c hannel inﬁn ite string, giv en that a concatenation of N co dewords of total length L o ccurs in a preﬁx of the c hann el in ﬁ nite string. F or eac h p ∈ { 0 , 1 } ∗ , let R ( p ) b e the probabilit y that the left-most co dew ord of the channel inﬁnite string is p , giv en that a concatenation o f N cod ew ords of tot al length L occurs in a p reﬁx of the c hann el inﬁnite s tring. Based on the principle of equal conditional p robabilit y , it can b e sh o wn that R ( p ) = Θ( L − | p | , N − 1) Θ( L, N ) . F r om the general deﬁnition (33) of statistical mec hanical en tropy , we ha ve R ( p ) = 2 S ( L −| p | ,N − 1) − S ( L,N ) . (35) 5 W e follo w th e argument of S ection 16-1 of Callen [1 ] in particular. 27 Let E ( L, N ) b e the exp ected length of the left-most codeword of the c hannel inﬁ nite string, giv en that a concatenati on of N co dewords of total length L occurs in a preﬁx of the c han n el inﬁnite string. Then, the follo wing equ alit y is exp ected to h old: S ( L, N ) = S ( E ( L, N ) , 1) + S ( L − E ( L, N ) , N − 1) . (36) Here, the term S ( L, N ) in the left-hand sid e denotes the statistical mec hanical en trop y of the whole concatenati on of N cod ew ord s of total length L . On the other hand, the ﬁrst term S ( E ( L, N ) , 1) in the righ t-hand side denotes the statistical mec hanical entrop y of the left-most cod ew ord of the concatenati on of N codewo rds of total length L while the second term S ( L − E ( L, N ) , N − 1) in the righ t-hand side denotes the statistica l mec hanical en tropy of the r emaining N − 1 co dewords of the concatenati on of N co dew ords of total length L . Thus, the equalit y (36) represents the additivit y of the statistical mec hanical entrop y . W e assume here that the equalit y (36) holds. By expandin g S ( L − | p | , N − 1) aroun d the equilibr iu m p oint L − E ( L, N ), w e ha v e S ( L − | p | , N − 1) = S ( L − E ( L, N ) + E ( L, N ) − | p | , N − 1) = S ( L − E ( L, N ) , N − 1) + ∂ S ∂ L ( L − E ( L, N ) , N − 1)( E ( L, N ) − | p | ) . (37) Here, we ignore the higher order terms than the ﬁrst order. Sin ce N ≫ 1 and L ≫ E ( L, N ), using the deﬁn ition (34) of temp erature w e ha v e ∂ S ∂ L ( L − E ( L, N ) , N − 1) = ∂ S ∂ L ( L, N ) = 1 T ( L, N ) . (38) Hence, by (37) and (38), w e ha v e S ( L − | p | , N − 1) = S ( L − E ( L, N ) , N − 1) + 1 T ( L, N ) ( E ( L, N ) − | p | ) . (39) Th us, using (35), (36), and (39) , w e obtain R ( p ) = 2 E ( L,N ) − T ( L,N ) S ( E ( L,N ) , 1) T ( L,N ) 2 − | p | T ( L,N ) . Then, according to statistical mec h an ics we deﬁne the fr e e ener gy F ( L, N ) of the left-most co d ew ord of the concatenation of N co dew ords of total length L by F ( L, N ) = E ( L, N ) − T ( L, N ) S ( E ( L, N ) , 1) . (40) It follo ws that R ( p ) = 2 F ( L,N ) T ( L,N ) 2 − | p | T ( L,N ) . (41) Using P p ∈ dom U R ( p ) = 1, we can sho w that, for an y p ∈ dom U , R ( p ) = 1 Z ( T ( L, N )) 2 − | p | T ( L,N ) , (42) where Z ( T ) is deﬁn ed b y Z ( T ) = X p ∈ dom U 2 − | p | T ( T > 0) . (43) 28 Z ( T ) is called the p artition fu nc tion (of the left-most co d ew ord of the c h annel in ﬁnite string). Th us, in our statistica l mec hanical inte rpretation of algorithmic information theory , the partition function Z ( T ) has exactly the same form as Ω D . The distribution in the form of R ( p ) is called a c anonic al ensemble in statistica l mec hanics. Then, usin g (41) and (42), F ( L, N ) is calculate d as F ( L, N ) = F ( T ( L, N )) , (44) where F ( T ) is deﬁned b y F ( T ) = − T log 2 Z ( T ) ( T > 0) . (4 5) On the other hand, from the d eﬁnition of R ( p ), E ( N , L ) is calculat ed as E ( N , L ) = X p ∈ dom U | p | R ( p ) . Th us, we hav e E ( L, N ) = E ( T ( L, N )) , (46) where E ( T ) is deﬁned by E ( T ) = 1 Z ( T ) X p ∈ dom U | p | 2 − | p | T ( T > 0) . (47) Then, using (40), (4 4), (45), and (46), the statistical mechanica l en tr op y S ( E ( L, N ) , 1) of th e left-most co deword of the concatenation of N cod ew ord s of total length L is calculated as S ( E ( L, N ) , 1) = S ( T ( L, N )) , where S ( T ) is d eﬁned by S ( T ) = 1 T E ( T ) + log 2 Z ( T ) ( T > 0) . (48) Note that th e statistical mec hanical en tropy S ( E ( L, N ) , 1) coincides with the Shanno n entr opy − X p ∈ dom U R ( p ) log 2 R ( p ) of the distr ibution R ( p ). Finally , the sp e ciﬁc he at C ( T ) of the left-most cod ew ord of th e channel inﬁn ite string is d eﬁ ned b y C ( T ) = E ′ ( T ) ( T > 0) , (49) where E ′ ( T ) is th e d eriv ed function of E ( T ). Th us, a s tatistica l mec hanical interpretation of algo rithmic information theory can b e estab- lished, based on a physical argum ent. W e can c h ec k that th e form u las in this argument: the partition fu nction (43), the fr ee energy (45), the exp ected length of the left-most co dewo rd (47) , the statistica l mec hanical en tropy (48), and the sp eciﬁc heat (49) corresp ond to the deﬁnitions in Section 4: Deﬁnition 4.1, Deﬁnition 4.3, Deﬁnition 4.5, Deﬁnition 4.7, and Deﬁnition 4.9, re- sp ectiv ely . Thus, the statistic al mec hanical meaning of the notion of ther m o dynamic qu an tities in tro duced in Section 4 in to algorithmic information theory is cla riﬁed b y this argum en t. 29 7 Concluding remarks In this p ap er, w e ha ve d ev elop ed a statistica l mec hanical in terpr etation of algorithmic information theory by introdu cing the notion of thermo dynamic quan tities in to algorithmic information th eory and in v estigating their prop erties from the p oint of view of algorithmic randomness. As a result, w e hav e disco v ered that, in the interpretation, the temp erature p la ys a role as the compression rate of all these thermo dyn amic quan tities, whic h include the temp erature itself. Th u s, in particular, w e ha ve obtained ﬁxed p oint theorems on compression rate, which r eﬂ ect this self-referenti al nature of the compression rate of the temp erature. In the last part of this pap er, w e ha ve also d evelo p ed a total statistical mechanica l interpretation of algorithmic information theory , whic h realizes a p erfect corresp ondence to normal statistical mec hanics and motiv es the ab o ve introdu ction of the thermo dyn amic quantiti es int o algorithmic information theory . Ho wev er, th e argumen t used in the total statisti cal mec hanical in terp retation is on the same lev el of mathematical strictness as statistica l mec h anics. Thus, we try to mak e the argum en t a mathematically rigorous form in a future study . 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A statistical mechanical interpretation of algorithmic information theory

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