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A Hierarchy of Imperative Languages for the Feasible Classes DTIMEF (nk) and for the Superexponential Classes DTIMEF (kn)

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A Hierarchy of Imperative Languages for the Feasible Classes DTIMEF (nk) and for the Superexponential Classes DTIMEF (kn)
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  A Hierarchy of Imperative Languages for theFeasible Classes DTIMEF ( n k )  and for theSuperexponential Classes DTIMEF ( k n ) Salvatore CaporasoUniversit`a di BariDipartimento di InformaticaVia Orabona, I-70125 BariItalycaporaso@di.uniba.itEmanuele CovinoUniversit`a di BariDipartimento di InformaticaVia Orabona, I-70125 BariItalycovino@di  uniba it Paolo GissiUniversit`a di BariDipartimento di InformaticaVia Orabona, I-70125 BariItaly gissi di uniba it Giovanni PaniUniversit`a di BariDipartimento di InformaticaVia Orabona, I-70125 BariItaly pani di uniba it  Abstract:  An imperative programming language is defined by closure of a free word-algebra of de/con-structorsunder two new operators (simultaneous safe recurrence and constructive diagonalization). By assigning ordinalsto its programs a transfinite hierarchy of imperative languages is introduced which singles-out the feasible classesDTIMEF ( n k )  and the superexponential classes DTIMEF ( k n ) . Key–Words:  Implicit computational complexity, Computational complexity, Elementary functions, Superexponen-tial classes 1 Statement of the result. Assume defined a scheme of   inessential substitu-tion  ( isbst ), such that classes like  DTIMEF ( n k )  areclosed under this scheme, together with a scheme of  safe recursion  (safe in the sense of [1, 14]) Let us saythat f   isdefinedby constructivediagonalizationintheenumerating function e if we have f  ( n ) =  { e ( n ) } ( n ) ,where { m } isKleene’snotationforthefunctioncodedby  m . Define a hierarchy  T   α  by T   1  is a characterization of   DTIMEF ( n ) ; T   α +1  is the closure under  isbst  of the class of allfunctions obtained by at most one application of (our)safe recurrence scheme to functions in  T   α ; T   λ  is the closure under  isbst  of the class of allfunctions obtained by at most one application of con-structive diagonalization in a given enumerator  e  ∈T   λ 1  such that  { e ( n ) } ∈ T   λ n .We then have (writing  clps  ( α,m )  for the result of re-placing  ω  by  m  in Cantor normal form (CNF) for  α ) DTIMEF ( n clps ( α,n ) )  ⊆ T   α ⊆  DTIMEF (( n + 4) clps ( α,n +4) ) .  (1)Moreover,  T   k  =  DTIMEF ( n k )  for every finite  k . Forexample ( c +1 n  stands for  n c n ) DTIMEF ( n n )  ⊆ T   ω  ⊆  DTIMEF (( n + 4) n +4 );  β<ω  T   β   =  PTIMEF ; DTIMEF ( k n )  ⊆  β<ω k  T   β   ⊆  DTIMEF ( k ( n + 4));  β< 0  T   β   =  E  . Under a more heavy syntax, the  spread   4 in (1) can bereduced to 1 (see Note 17). 2 Diagonalization and other defini-tion schemes a,b,a 1 ,...  are  digits  of the ternary alphabet  T  = { 0 , 1 , 2 }  and  p,...,s,...,w,... are numerals over  T ;  is the empty word.  u,w,u 1 ,... are variables definedon T + . Wedenoteby x,y,z  variablespreviouslyusedas, respectively,  auxiliary variables, parameters, re-cursion variables . This convention is tantamount tothe use of a semicolon ( z ; x,y ) to separate safe andunsafe arguments by Simmons [14] and Bellantoni & Proceedings of the 6th WSEAS International Conference on Applied Computer Science, Hangzhou, China, April 15-17, 2007 315  Cook [ ? ]). When we write  f  ( u 1 ,...,u n )  we alwaysassume that some of the indicated variables may beabsent. Definition 1  Given a word  s  in which  n  zeroes oc-cur,  ( s ) i  (1  ≤  i  ≤  n  + 1)  is the  i -th rightmostword (possibly empty) in the alphabet  { 1 , 2 }  whichoccurs between two zeroes of   s  (0 replaced by    when i  = 1 ,n  + 1 ); if   i > n  + 1  then  ( s ) i  does not exist(cf. Schwichtenberg [13] and Sect. 6 for this use of ternary words to represent tuples). Given  i  ≥  0  and a  = 1 , 2 ,1. the  constructor   c ai ( s )  adds a digit  a  at the rightof   ( s ) i ;2. the  destructor  d i ( s )  erases the right-most digit of  ( s ) i  if any;all these constructors and destructors leave  s  un-changed if   ( s ) i  doesn’t exist;3. the construct  case i [ f,g,h ]( s )  returns  f  ( s )  (re-spectively  g ( s ) ) if the rightmost digit of   ( s ) i  is 1(respectively 2); it returns  h ( s )  if   ( s ) i  does notexist or is   .For example  d 2 (010) =  d 2 (00) = 00;  d 1 (1) =  ;  c 11 (0) = 01 . Definition 2  f   =  sr ( g,h )  is defined by  (this pa- per’s) safe recursion  in the  basis function  g ( x,y )  andin the  step function  h ( x,y,z )  if for all  s,t,r  we have   f  ( s,t, ) =  g ( s,t ) f  ( s,t,ra ) =  h ( f  ( s,t,r ) ,t,ra ) . An  iteration  is a sr in which parameter and recursionvariable are both absent in the step function. Definition 3  f   =  cdiag ( e )  is defined by  (construc-tive) diagonalization  in the  enumerator   e  if for all s,t,r , we have f  ( s,t,r ) =  { e ( r ) } ( s,t,r ) . Definition 4  1.  f   =  asg ( s,u,g )  is the result of the assignment   of   s  to variable  u  in  g ;2.  f   =  idt x ( g )  is the result of the  identification  of  x  as  y  in  g  (thus,  f  ( y,z ) =  g ( y,y,z ) ); similarly, f   =  idt z ( g )  is the result of the  identification  of  z  as  y  in  g  (thus,  f  ( x,y ) =  g ( x,y,y ) ).3.  f   =  sbst ( u,h,g )  is defined by  substitution  in  g and  h  if it is obtained by substitution of   g  for  u in  h .An essential point is that identification of   z  as  x  is notallowed and, therefore, the step function cannot as-sign the recursion variable with the previous value of the function being defined by safe recursion. (Accord-ing to Bellantoni&Cook’s terminology, in this way  z keeps  safe ). Definition 5  (1) Class  T   0  is the closure under  asg,idt x ,  idt z  and  sbst  of   c ai ,  d i ,  case i ;(2) Class  T   1  is the closure under  asg, idt x , idt z  and  sbst  of functions obtained by at most oneiteration from functions in  T   0 .In other words, this class is the indicated closure of allfunctions  f  ( x,z ) =  h | z | ( x )  for some  h  ∈ T   0 . Definition 6  The  number of components  #( f  )  of  f   ∈T   1  is  max { i | d i  or  c ai  or  case i  occurs in  f  } . Example 7  Define (by iteration of   c 10 ) function  g 1  ∈T   1  such that  g 1 ( x,y ) =  x 1 | y | . Define further   f  n +1 ( s,t, ) =  sf  n +1 ( s,t,ra ) =  g n ( f  n +1 ( s,t,r ) ,t )  ( sr ) g n +1 ( x,y ) =  f  n +1 ( x,y,y ) ( idt z ) By induction one shows that  | f  n +1 ( s,ta,rb ) |  =  | s |  + | t | n | r |  and, therefore,  | g n ( ,ta ) |  =  | t | n . Assume de-fined a function  e  ∈ T   1  such that  e ( r ) =   g | r |  . Bysetting  f  ω  := cdiag ( e ) , we then have | f  ω ( ,t,ta ) |  =  g | t | ( ,ta ) =  | t | | t | . Lemma 8  DTIMEF ( n ) =  T   1 . Proof.  1. To show that every function in  T   1  can becomputed in linear time by a  TM  with input and out-put on its first tape, let  g  ∈ T   0  and  f  ( s,t ) =  g | t | ( s ) be given. A  TM  M  f   with  m  := #( f  ) + 1  tapes canbe defined which, by input  s  on tape 1: (a) copies ( s )  j  (  j  ≤  m )  on tape  j +1 ; (b) computes g  in constanttime; (c) after  | t |  repetitions collates back the contentsof tapes  2 ,...,m + 1  into tape 1.2. For every  m -tapes  TM  M   define in  T   0  a function nxt M   which uses two components for the part at theright (read in reverse order) and the part at the leftof the observed symbol of each tape, and the last oneof its  2 m  + 1  components for the internal state. Thebehaviour of   M   by input  s  for  | t |  steps can then besimulated by a function  linsim M  ( x,z )  ∈ T   1 , de-fined by iteration of   nxt M  . Let  M   by input  s  on itsfirst tape stop operating within  c | s |  steps. Since  T   1  isclosed under substitution, we may define sim M  ( s ) := linsim M  ( s, times  c ( s )) , where  times  c ( s ) = 1 c | s | isobtained from function  g 1  of Ex. 7 by assigning    to  x and by some  sbst ’s. Proceedings of the 6th WSEAS International Conference on Applied Computer Science, Hangzhou, China, April 15-17, 2007 316  3 The hierarchy Definition 9  We adopt the following assignment of fundamental sequences to all  λ <  0 λ n  =  n  if   n  ≤  1  and  λ < ω 2 ω µ n if CNF for  λ  is  ω µ ω α n  if CNF for  λ  is  ω α +1 µ + ( ω α ) n  if CNF for  λ  is  µ + ω α . Comment.  Assignment  S   of last definition differsfrom the traditional standard assignment  S  ∗ (see [11,p. 78]) at line 1 (adopted to simplify the proof of Lemma 16 — cf. Case 2 of the induction); and at line3, where we multiply by  n  instead of   n +1  in order tocope with the fact that, as we shall see,  n  nested  sr ’sgrow like  | t | n , not as  | t | n +1 . The slow hierarchy  G α ,when defined with respect to S   (cf.  § 1.1), is obviouslydominated by the one defined with respect to  S  ∗ . Onthe other hand the former hierarchy is not collapsingsince the Bachmann property ( λ n  < λ n +1  for all  λ and  n ) keeps holding for  S   (see [11, Th. 3.6]). Definition 10  f   =  isbst ( u,h,g )  is defined by inessential substitution  in  g  and  h  if it is defined bysubstitution in  g  and  h  and  h  ∈ T   1 .We may now define the following hierarchy of classes of functions. Definition 11  (a)  T   α +1  ( α >  0)  is the closure under asg, idt x ,  idt z ,  isbst  of the class of all func-tions obtained by at most one application of  sr to  T   α ;(b)  T   λ  is the closure under  asg, idt x ,  idt z , isbst  of the functions of the form  cdiag ( e ) , for e  ∈ T   λ 1 , and such that  { e ( r ) } ∈ T   λ | r | . Notation 12  B α ( n ) := max(2 ,n ) clps ( α,n ) By induction on  α  one sees that for all  n  ≥  2  wehave  B α ( n ) =  n G α ( n ) . Theorem 13 1.  For all  ω  ≤  α <  0  we have DTIMEF ( B α ( n ))  ⊆ T   α  ⊆ DTIMEF ( B α ( n + 4)) . 2.  For all finite  k ,  dtime ( n k ) =  T   k . Proof.  1.  By proof of Lemma 8, there is a function sim  M  ( s,t )  ∈ T   1  returning the instantaneous descrip-tion of the  TM  M   after  | t |  steps. By Lemma 20, forall  α <  0  we can define in  T   α  a function  g α  which computes in unary  B α ( | t | ) . The first inclusion thenfollows by  isbst  of   g α ( s )  for  t  in  sim .The second inclusion and part  2.  follow byLemma 16, in which an interpreter is defined, insteadof mere simulation, in order to handle diagonaliza-tion. 4 Proofs 4.1 Codes All expressions introduced throughout this papermay be thought of as transcriptions of a Pol-ish prefix language over a  united alphabet   U  = { 0 , 1 , 2 , isbst,sr,cdiag,itrt  ,... } . Codes arebuilt-up by juxtaposition from the codes for the let-ters of   U , unique parsing being ensured by the  arity associated tacitly with each such letter. Definition 14  The code   L   for the  i -th letter  L of   U  is  2 i +1 1 . Let us write   E  1 ,...,E  n   for  E  1  ...  E  n  . If the arity of   L  ∈  U  is  n  then  L,E  1 ,...,E  n   codes the expression  LE  1 ...E  n . Example 15  If   f  ( x,z )  ∈ T   1  is the  | z | -th iterate of function  e  ∈ T   0 ,then its code is   itrt ,e  . The codefor functions  g n +1  of Ex.7 is   idt z ,  sr ,x,g n  ,where   g 1   =   idt z ,  itrt,c 10  . 4.2 Simulation by TM’s Lemma 16  For all  α  ≥  ω  we have T   α  ⊆ DTIMEF ( B α ( n  + 4)) ; for all finite  m , T   m  ⊆ DTIMEF ( B m ( n )) . Proof.  We first associate each integer  d  with an inter-preter  INT  d . By input   f   ,s,t,r  it returns  f  ( s,t,r ) provided that the following  d-condition  holds: (a) | s |  +  | t |  +  | r | ≤  d  or (b) no  cdiag  occurs in  f   and #( g )  ≤  d , for each  g  ∈ T   1  used in the construction of  f  .When the  d -condition holds, since by Def. 5 thenumber of zeroes doesn’t increase during the compu-tation of   f  , the number of tapes needed to store theparts of the arguments which can be modified by  f   ina number of tapes which depends only on  d . In the fi-nal part of this proof, we will reduce the family  INT  d to a single interpreter using only two tapes. Definition of the interpreter  We have to avoid thewaste of time resulting from moving back and forththe value of the function being recursed upon from thestorage which are reserved to the data to that contain-ing the current results. To this purpose, the interpreter INT  d  (see Fig. 1) uses the following stacks:(a) T  x ,T  y ,T  z , to store the values of  x,y,z  during thecomputation;  T  x consists of   d  tapes, one for each of the modifiable parts of the value assigned to  x ;(b)  T  u , to store the value of the principal variable of the current enumeration or recursion;(c)  T  f  , to store (the codes for) some sub-functions of  f  ;the initial contents of  T  x ,T  y ,T  z ,T  f   are, respectively,the input values  s,t,r , and   f   ;  T  u is initially empty. Proceedings of the 6th WSEAS International Conference on Applied Computer Science, Hangzhou, China, April 15-17, 2007 317  At the end of the computation  INT   collates the com-ponents of the result into the first of tapes  T  x . INT  d  repeats, until  T  f   is not empty, the followingcycle- it pops a function  k  from the top of   T  f  , and un-neststhe outermost sub-function  j  of   k ;- according to the form of   j , it carries-out a differentaction on the stacks;- if the form of   j  is  itrt ( g )  with  g  ∈ T   0 , it calls aninterpreter  ITRT   for  T   1  which simulates  g  on  T  x for | t |  times, where  t  is the top record of   T  y ;- in all other cases, it pushes into T  f   an information of the form  jMRKk , where MRK   is a mark informingabout the outermost scheme used to define  j . Time complexity  We first show that for all  f,s,t,r respecting the  d -condition  f   ∈ T   α  implies INT  d (  f   ,s,t,r ) =  f  ( s,t,r ) within time | s |  +  | f  | B α ( | t |  +  | r |  + 1 α ) where  1 α  = 0  if   α < ω  and  1 α  = 1  otherwise. Theresult follows, since every function  f   ∈ T   α  is thencomputed in  DTIMEF ( B α ( n  + 1 α ))  by the sequencecomposition of the constant-time  TM  writing the codefor  f   with  INT  d .Define  m  :=  | s | , n  :=  | t |  +  | r | ;  X   :=   f   ;  c  := | X  | . We show that, for all  f   ∈ T   α ,  INT  d  moveswithin  m  +  cB α ( n  + 1 α )  steps from an istantaneousdescription of the form T  f   =  ZX  ;  T  x =  s 0 s ;  T  y =  t 0 t ;  T  z =  r 0 r ;  T  u =  q, to a new istantaneous description of the form T  f   =  Z  ;  T  x =  s 0 ( { X  } ( s,t,r ));  T  y =  t 0 t ; T  z =  r 0 r ;  T  u =  q. Induction on  α  and on the construction of   f  . Basis. α  = 1 . We have  1 α  = 0 . The complexity of   ITRT   isobviously  < m + cn .Step. Case 1.  f   = sr ( g,h ) . We have  α  =  β   +1 ; let  r be the word  a | r | ...a 1 . By the induction on  α ,  INT  d needs time  ≤  m  +  | g | B β  ( n  + 1 α )  to produce theistantaneous description T  f   =  ZX RC  ;  T  x =  s 0  g ( s,t,a 1 );  T  y =  t 0 t ; T  z =  r 0 ra 1 ;  T  u =  qr. If   | r |  >  1  then  INT  d  puts  T  f   :=  Z X RC    h   and T  z :=  r 0 ra 2 a 1 , and calls itself in order to compute h  and the next value of   f  . Again by the inductionon  α  we have that  INT  d  needs time  ≤ | g ( s,t,a 1 ) |  + | h | B β  ( n  + 1 α )  to produce an istantaneous descrip-tion of the form T  f   =  ZX RC  ;  T  x =  s 0  ( h ( g ( s,t,a 1 ) ,t,a 2 a 1 )); T  y =  t 0 t ;  T  z =  r 0 ra 2 a 1 ;  T  u =  qr. After  | r | −  1  simulations of   h  we obtain the promisedistantaneous description within an overall time m +  | r | max( | g | , | h | ) B β  ( n + 1 α )  ≤ m +  | r | cB β  ( n + 1 α )  ≤  m + cB α ( n + 1 α ) , where, since  α  ≥  2 , in these evaluations we maycompensate the quadratic amount of time needed tocopy r  and its digits with the difference between c and max( | g | , | h | ) .Case 2.  f   = cdiag ( h )  ∈ T   λ . We have  h  ∈T   λ 1 ,  1 α  = 1 , and (recall that  λ 1  = 1  when  λ  ≤  ω 2 ) B λ 1 ( n +1) · B λ n ( n +1)  ≤  B λ n +1 ( n +1) =  B λ ( n +1) . (2) INT  d  computes  h ( r ) , understands from the mark  DG  that the result is the code for the function to becomputed, and, accordingly, pushes it into  T  f  .To compute  h ( r )  and  { h ( r ) } ( s,t,r )  the inter-preter  INT  d  needs, by the induction on  α , time ≤  m + | h | B λ (1) ( | r | +1)+ |{ h ( r ) }| B λ ( | r | ) ( n +1)  ≤ (by eq. (2))  m + | h | B λ ( n +1)  ≤  m + cB λ ( n +1) . INT  ( X,s,t,r ) := T  f   :=  X  ;  T  x :=  s ;  T  y :=  t ;  T  z :=  r ; while  T  f   not empty  do  A  :=  last record(s) of   T  f  ; case A  =  ISBST  ( X,g,h )  then X   :=  g h ; push  g X   in  T  f  A  =  X   :=  x then copy last record of   T  x into  T  X A  =  REN  ( X,Y,h )  then push  h  in  T  f  ; copy last record of   T  X into  T  Y  A  =  DIAG ( h )  then push  DG h  into  T  f  ; copy last record of   T  x into  T  u A  =  DG  then pop  T  f  ; pop last record of   T  x and push it into  T  f  ;pop last record from  T  u and push it into  T  x A  =  SREC  ( g,h )  then push  A RC g  into  T  f  ; copy last record of   T  z into  T  u push last digit of   T  u into  T  z A  =  SREC  ( g,h )  RC   thenif   T  u =  T  z then  pop  T  f  ; pop  T  u ; pop  T  z else  push  h  into  T  f  ;pop last digit of   T  u and push it into  T  z A  =  ITRT  ( g ) then call  ITRT  .end  case ; end  while . Fig. 1 Reduction to  2  tapes  Let  e  be an enumerator suchthat for all  d  there is  r  such that  #( { e ( r ) } )  > d . Proceedings of the 6th WSEAS International Conference on Applied Computer Science, Hangzhou, China, April 15-17, 2007 318  If such an  e  occurs in the computation of   f  , then f  ( s,t,r )  is simulated by a  TM  INT  c (  f   ,s,t,r ) ( c  = | s |  +  | t |  +  | r | )  which obviously depends on the ar-guments for  f  . However, we know from [7] thata  k -tapes  TM  with time bound  T  ( n )  can be simu-lated by a 2-tapes  TM  INT   in time  kT  ( n )log( T  ( n )) .Hence we can define a single 2-tapes  TM  INT   whichbehaves like the  INT  c ’s. For  α  ≥  ω , runtime for INT   is  T  ( n ) =  nB α ( n  + 1)log( B α ( n  + 1))  ≤ nB α ( n + 2)log( n + 1)  ≤  B α ( n + 4) . Note 17  We see from the proof above that the spread4 in the statement of last lemma is reduced to 1 if wehave  #( g )  ≤  d  for some  d  and for all  g  involvedin the computation of   f  . This can be obtained byadding some cumbrous syntactic clauses to Defini-tions 1 and 3. 4.3 Simulation of TM’s Lemma 18  Given(a) a function  {  p }  such that, for a numerical function F  ( n ) , we have  |{  p } ( s,t ) |  =  | s |  + F  ( | t | ) ;(b) the function  h  in  T   1  defined as follows   h ( x, ) =  xh ( x,ya ) =   idt z ,  sr ,x, { h ( x,y ) } we have that for all  t,q  :  |{ h (  p,q  ) } ( ,t ) |  = F  ( | t | ) | t | | q | . Example 19  For  p 1  :=  c 11  , we have  |{  p 1 } ( s,t ) |  = | s |  + 1 . Last lemma (for  F  ( n ) = 1 )says that  |{ h (  p 1 ,q  ) } ( ,t ) |  =  | t | | q | . Hence cdiag ( asg (  p 1 ,x,h ))  is the  f  ω  of Ex.7, which  com- putes in unary  B ω . Define further   k (  p, ) =  pk (  p,qa ) =   cdiag ,  asg ,k (  p,q  ) ,x,h ( x,y )  By last lemma and induction on  | q  |  we may provethat, if   p ω  codes  f  ω , we have  |{ k (  p ω ,q  ) } ( ,t ) |  = B ω ·| q | ( | t | ) . Hence,  cdiag ( k (  p ω ,y ))  computes inunary  B ω 2 .Notice that all these enumerators are obtained by asafe recursion which adds at each step a constant in-formation to its previous value, and, therefore, theybelong to  T   1 . A uniform way to build codes of func-tions  {  p α }  computing in unary  B α  for every  α  ≤  ω ω can be defined similarly. Proof of Lemma  18. Note that the first occurrence of  x in “  idt z ,  sr ,x, { h ( x,y ) } ” is the code for a vari-able, while the second is an argument (cf. Ex.15).Function  h (  p,q  )  yields the codes for  q   nestings of  idt z ’s and  sr ’s over  {  p } . We show by induction on | q  |  that we have |{ h (  p,q  ) } ( s,t ) |  =  | s |  + F  ( | t | ) | t | | q | .  (3)Basis.  q   =   . We have |{ h (  p, ) } ( s,t ) |  =  |{  p } ( s,t ) | by the definition of   h =  | s |  + F  ( | t | ) | t | 0 by the hypothesis on  {  p } . Step. Define  e  := sr ( x, { h (  p,q  ) } ) . We show by in-duction on  | r |  that we have | e ( s,t,r ) |  =  | s |  + F  ( | t | ) | t | | q | | r | .  (4) | e ( s,t,rb ) |  =  |{ h (  p,q  ) } ( e ( s,t,r ) ,t ) | by definition of   e =  | e ( s,t,r ) |  + F  ( | t | ) | t | | q | by (3), that is by the induction on  q  =  | s |  + F  ( | t | ) | t | | q | | r |  +  | F  ( | t | ) | t | | q | by (4), that is by the induction on  | r | =  | s |  + F  ( | t | ) | t | | q | ( | r |  + 1) . The result follows from (4) since we have |{ h (  p,qa ) } ( s,t ) |  =  | e ( s,t,t ) |  =  | s |  + F  ( | t | ) | t | | q | +1 . Lemma 20  1. For all  α <  0  there exists  g α  ∈ T   α such that  | g α ( s ) |  =  B α ( | s | ) .2. Every  TM  whose runtime is bounded above by B α  can be simulated in  T   α . Proof.  1. Case 1.  α  =  β   + 1 . Define  g α  = { h (  g β   , 1) } , where  g β   is granted by the inductionand  h  is given by last lemma.Case 2.  α  =  ω β  . Define in  T   β   by  isbst  e α  = { h (  c 11  ,g β  ( y )) } , where  g β   ∈ T   β   is given by the in-duction. The induction and Lemma 18 give |{ e α ( s ) } ( s ) |  =  | s | B β ( | s | ) . Since  β   ≤  ( ω β  ) 1 , we may define g α ( s ) := cdiag ( e α )  in  T   α . By the inductionand last lemma | g α ( s ) |  =  |{ e α ( s ) } ( s ) |  =  | s | B β ( | s | ) . The result follows since  clps ( ω β  ,n ) =  n clps ( α,n ) im-plies  B ω β ( n ) =  n B β ( n ) .Case 3.  α  =  λ  +  ω β  ( λ  ≥  ω β  ) . Define in  T   β   by isbst  e α ( s ) :=  { h (  g λ  ,g β  ( y )) } . Since  ω β  ≤  α 1 , Proceedings of the 6th WSEAS International Conference on Applied Computer Science, Hangzhou, China, April 15-17, 2007 319
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