  
  [1X3 [33X[0;0YLogged Rewriting Systems[133X[101X
  
  [33X[0;0YA [13Xlogged rewrite system[113X is associated with a group presentation. Each [13Xlogged
  rewrite rule[113X contains, in addition to the standard rewrite rule, a record or
  [13Xlog  component[113X which expresses the rule in terms of the original relators of
  the  group.  We  represent  such  a rule by a triple [10X[ u, [L1,L2,..,Lk], v][110X,
  where  [10X[u,v][110X  is  a  rewrite  rule  and [22XL_i = [n_i,w_i][122X where [22Xn_i[122X is a group
  relator  and  [22Xw_i[122X  is  a  word. These three components obey the identity [22Xu =
  n_1^w_1 ... n_k^w_k v[122X.[133X
  
  [33X[0;0YRules  of the form [22Xg^+g^- -> id[122X apply to the monoid presentation, but not to
  the group presentation, so are given an empty logged component.[133X
  
  
  [1X3.1 [33X[0;0YLogged Knuth-Bendix Completion[133X[101X
  
  [33X[0;0YThe  functions  in  this  section  are  the  logged versions of those in the
  previous chapter.[133X
  
  [1X3.1-1 LoggedOnePassKB[101X
  
  [33X[1;0Y[29X[2XLoggedOnePassKB[102X( [3Xgrp[103X, [3Xloggedrules[103X ) [32X operation[133X
  
  [33X[0;0YGiven a logged rewrite system for the group [10Xgrp[110X, this function finds all the
  rules  that  would  be  added to complete the rewrite system of [10XOnePassKB[110X in
  [14X2.2-3[114X,  and  also  the logs which relate the new rules to the originals. The
  result  of  applying this function to [10Xloggedrules[110X is to add new logged rules
  to the system without changing the monoid it defines.[133X
  
  [33X[0;0YIn the example, we first convert the presentation for [10Xq8[110X into an initial set
  of logged rules, and then apply one pass of Knuth-Bendix.[133X
  
  [33X[0;0YThe function returns a two-element list of lists. The first element is a set
  of  logged  rules  and  the  second element is an empty list of r-sequences,
  which will be explained later.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xl0 := ListWithIdenticalEntries( 8, 0 );;[127X[104X
    [4X[25Xgap>[125X [27Xfor j in [1..8] do [127X[104X
    [4X[25X>[125X [27X       r := r0[j];[127X[104X
    [4X[25X>[125X [27X       if ( j<5 ) then[127X[104X
    [4X[25X>[125X [27X          l0[j] := [ r[1], [ [j,id] ], r[2] ];[127X[104X
    [4X[25X>[125X [27X       else[127X[104X
    [4X[25X>[125X [27X          l0[j] := [ r[1], [ ], r[2] ];[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xl0;[127X[104X
    [4X[28X[ [ q8_M1^4, [ [ 1, <identity ...>] ], <identity. ..> ], [128X[104X
    [4X[28X  [ q8_M2^4, [ [ 2, <identity ...>] ], <identity ...> ], [128X[104X
    [4X[28X  [ q8_M1*q8_M2*q8_M1*q8_M4, [ [ 3, <identity ...> ] ], <identity ...> ],   [128X[104X
    [4X[28X  [ q8_M1^2*q8_M2^2, [ [ 4, <identity ...> ] ], <identity ...> ], [128X[104X
    [4X[28X  [ q8_M1*q8_M3, [ ], <identity ...> ], [ q8_M2*q8_M4, [ ], <identity ...> ], [128X[104X
    [4X[28X  [ q8_M3*q8_M1, [ ], <identity ...> ], [ q8_M4*q8_M2, [ ], <identity ...> ] ] [128X[104X
    [4X[25Xgap>[125X [27Xl1 := LoggedOnePassKB( q8, l0 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( l1[1] ); [127X[104X
    [4X[28X21[128X[104X
    [4X[25Xgap>[125X [27Xl1[1][16];[127X[104X
    [4X[28X[ q8_M2^2, [ [ -4, <identity ...> ], [ 2, q8_M3^2 ] ], q8_M1^2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the  length  [22X21[122X of [10Xl1[110X is, as expected, the same as that of [10Xr1[110X in
  [14X2.2-3[114X.  If  we  write  [22Xa^+,b^+,a^-,b^-[122X  for  [10XM1,M2,M3,M4[110X  and label the four
  original   relators   as  [22Xq=a^+4,r=b^+4,s=a^+b^+a^+b^-,t=a^+2b^+2[122X  then  the
  sixteenth  identity  (for  example)  says  that [22Xb^+2 = (t^-1r^a^-2})a^+2[122X. To
  verify this, we may expand the right-hand side as follows:[133X
  
  
  [24X[33X[0;6Y(b^{-2}a^{-2}).a^{+2}(b^{+4})a^{-2}.a^{+2}                               ~=~
  b^{-2}(a^{+2}a^{-2})b^{+4}(a^{-2}a^{+2}) ~=~ b^{-2}b^{+4} ~=~ b^{+2}.[133X
  
  [124X
  
  [1X3.1-2 LoggedKnuthBendix[101X
  
  [33X[1;0Y[29X[2XLoggedKnuthBendix[102X( [3Xgrp[103X, [3Xloggedrules[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XLoggedRewriteReduce[102X( [3Xgrp[103X, [3Xloggedrules[103X ) [32X operation[133X
  
  [33X[0;0YThe  function  [10XLoggedRewriteReduce[110X  removes  unnecessary rules from a logged
  rewrite system. It works on the same principle as [10XRewriteReduce[110X in [14X2.2-3[114X.[133X
  
  [33X[0;0YThe  function [10XLoggedKnuthBendix[110X repeatedly applies functions [10XLoggedOnePassKB[110X
  and [10XLoggedRewriteReduce[110X until no new rules are added and no unnecessary ones
  are included. The output is a reduced complete logged rewrite system.[133X
  
  [33X[0;0YAs  a further example, consider the second rule in [10Xl2[110X which shows how [22Xb^+a^+[122X
  reduces  to  [22Xa^+b^-[122X.  For  this  rule  [10X[u,L,v][110X  we  will  verify  that  [22Xu  =
  n_1^w_1n_2^w_2n_3^w_3  v[122X,  as  in the introduction to this chapter. The rule
  is:[133X
  
  [4X[32X[104X
    [4X[104X
    [4X[ q8_M2*q8_M1, [ [3,q8_M1], [-1,<identity...>], [4,q8_M3] ], q8_M1*q8_M4 ]. [104X
    [4X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  relators  [22X3  ≡  s,  4  ≡  t[122X  and  the  inverse  relator  [22X-1  ≡ q^-1[122X are
  [22Xa^+b^+a^+b^-,  a^+2b^+2[122X  and [22Xa^-4[122X, and these are conjugated by [22X(a^-)^-1=a^+,
  (a^+)^-1=a^-[122X and the identity respectively. So the second and third parts of
  the rule expand to:[133X
  
  
  [24X[33X[0;6Y(a^-(a^+b^+a^+b^-)a^+)(a^{-4})(a^+(a^{+2}b^{+2})a^-)a^+b^-               ~=~
  (a^-a^+)b^+a^+(b^-(a^+a^{-4}a^{+3})b^+)(b^+(a^-a^+)b^-)[133X
  
  [124X
  
  [33X[0;0Ywhich reduces to the first part of the rule, [22Xb^+a^+[122X.[133X
  
  [33X[0;0YIn version 2.41 of this package the function [10XLoggedOnePassKB[110X was modified to
  return  a  two-element  list  of lists. The first element is a set of logged
  rules (as has been returned by earlier versions) while the second element is
  a  list  of  [22X51[122X  identities.  See  chapter  [14X6[114X  for  more  information  about
  identities.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xl11 := LoggedRewriteReduce( q8, l1[1] );;[127X[104X
    [4X[25Xgap>[125X [27XPerform( l11, Display );[127X[104X
    [4X[28X[ q8_M1*q8_M3, [  ], <identity ...> ][128X[104X
    [4X[28X[ q8_M2^2, [ [ -4, <identity ...> ], [ 2, q8_M3^2 ] ], q8_M1^2 ][128X[104X
    [4X[28X[ q8_M2*q8_M4, [  ], <identity ...> ][128X[104X
    [4X[28X[ q8_M3*q8_M1, [  ], <identity ...> ][128X[104X
    [4X[28X[ q8_M4*q8_M2, [  ], <identity ...> ][128X[104X
    [4X[28X[ q8_M1^3, [ [ 1, <identity ...> ] ], q8_M3 ][128X[104X
    [4X[28X[ q8_M1^2*q8_M2, [ [ 4, <identity ...> ] ], q8_M4 ][128X[104X
    [4X[28X[ q8_M1*q8_M2*q8_M1, [ [ 3, <identity ...> ] ], q8_M2 ][128X[104X
    [4X[28X[ q8_M2*q8_M1*q8_M4, [ [ 3, q8_M1 ] ], q8_M3 ] [128X[104X
    [4X[25Xgap>[125X [27XLength( l11 );[127X[104X
    [4X[28X9[128X[104X
    [4X[25Xgap>[125X [27Xl2 := LoggedKnuthBendix( q8, l11 );;[127X[104X
    [4X[25Xgap>[125X [27Xl2[1]; [127X[104X
    [4X[28X[ [ q8_M1*q8_M3, [  ], <identity ...> ], [128X[104X
    [4X[28X  [ q8_M2*q8_M1, [ [ 3, q8_M1 ], [ -1, <identity ...> ], [ 4, q8_M3 ] ], [128X[104X
    [4X[28X      q8_M1*q8_M4 ], [128X[104X
    [4X[28X  [ q8_M2^2, [ [ -4, <identity ...> ], [ 2, q8_M3^2 ] ], q8_M1^2 ], [128X[104X
    [4X[28X  [ q8_M2*q8_M3, [ [ -3, <identity ...> ] ], q8_M1*q8_M2 ], [128X[104X
    [4X[28X  [ q8_M2*q8_M4, [  ], <identity ...> ], [ q8_M3*q8_M1, [  ], <identity ...> ][128X[104X
    [4X[28X    , [ q8_M3*q8_M2, [ [ -1, <identity ...> ], [ 4, q8_M3 ] ], q8_M1*q8_M4 ], [128X[104X
    [4X[28X  [ q8_M3^2, [ [ -1, <identity ...> ] ], q8_M1^2 ], [128X[104X
    [4X[28X  [ q8_M3*q8_M4, [128X[104X
    [4X[28X      [ [ -1, <identity ...> ], [ -2, q8_M3^2 ], [ 4, <identity ...> ], [128X[104X
    [4X[28X          [ 3, q8_M1*q8_M4 ], [ -3, <identity ...> ] ], q8_M1*q8_M2 ], [128X[104X
    [4X[28X  [ q8_M4*q8_M1, [ [ -4, <identity ...> ], [ 3, q8_M3 ] ], q8_M1*q8_M2 ], [128X[104X
    [4X[28X  [ q8_M4*q8_M2, [  ], <identity ...> ], [128X[104X
    [4X[28X  [ q8_M4*q8_M3, [ [ -3, q8_M1*q8_M2 ] ], q8_M1*q8_M4 ], [128X[104X
    [4X[28X  [ q8_M4^2, [ [ -4, <identity ...> ] ], q8_M1^2 ], [128X[104X
    [4X[28X  [ q8_M1^3, [ [ 1, <identity ...> ] ], q8_M3 ], [128X[104X
    [4X[28X  [ q8_M1^2*q8_M2, [ [ 4, <identity ...> ] ], q8_M4 ], [128X[104X
    [4X[28X  [ q8_M1^2*q8_M4, [ [ -4, q8_M3^2 ], [ 1, <identity ...> ] ], q8_M2 ] ][128X[104X
    [4X[25Xgap>[125X [27XLength( l2[1] );[127X[104X
    [4X[28X16[128X[104X
    [4X[25Xgap>[125X [27XLength( l2[2] );[127X[104X
    [4X[28X51[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YLogged reduction of a word[133X[101X
  
  [1X3.2-1 LoggedReduceWordKB[101X
  
  [33X[1;0Y[29X[2XLoggedReduceWordKB[102X( [3Xword[103X, [3Xloggedrules[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XLoggedOnePassReduceWord[102X( [3Xword[103X, [3Xloggedrules[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XShorterLoggedRule[102X( [3Xlogrule1[103X, [3Xlogrule2[103X ) [32X operation[133X
  
  [33X[0;0YGiven    a    word    and    a   logged   rewrite   system,   the   function
  [10XLoggedOnePassReduceWord[110X  makes  one  reduction  pass  of  the word (possibly
  involving  several  reductions)  (as  does  [10XOnePassReduceWord[110X  in [14X2.2-2[114X) and
  records this, using the log part of the rule(s) used and the position in the
  original word of the replaced part.[133X
  
  [33X[0;0YThe  function  [10XLoggedReduceWordKB[110X repeatedly applies [10XOnePassLoggedReduceWord[110X
  until  the  word  can  no  longer  be reduced. Each step of the reduction is
  logged,  showing  how  the  original  word  can be expressed in terms of the
  original relators and the irreducible word. When [10Xloggedrules[110X is complete the
  reduced  word is a unique normal form for that group element. The log of the
  reduction depends on the order in which the rules are applied.[133X
  
  [33X[0;0YThe  function  [10XShorterLoggedrule[110X  decides  whether one logged rule is better
  than  another,  using  the  same  criteria  as  [10XShorterRule[110X in [14X2.2-3[114X. In the
  example  we  perform  logged reductions of [22Xw_0 = a^9b^9[122X corresponding to the
  ordinary reductions performed in the previous chapter (section [14X2.2-2[114X).[133X
  
  [33X[0;0YIn  order  to  clarify  the  following  output, note that, in the log below,
  [22Xb^9a^9[122X  reduces to [22Xb^5a^5[122X in [10Xlw1[110X and to [22Xba[122X in the first [10Xlw2[110X. These expand to
  the initial [10Xw0[110X using the given logged parts as follows:[133X
  
  
  [24X[33X[0;6Y(b^{+9}a^{+4}b^{-9})(b^{+4})b^{+5}a^{+5}      ~=~      b^{+9}a^{+9}      ~=~
  (b^{+9}a^{+4}b^{-9})(b^{+4})(b^{+5}a^{+4}b^{-5})(b^{+4})b^+a^+\,.[133X
  
  [124X
  
  [33X[0;0YThe corresponding expansion of the final [10Xlw2[110X is too lengthy to include here.
  (It's  hard to believe that the logged part of this identity is the simplest
  possible.  Further  investigation is needed to determine whether or not this
  logged part can be simplified.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xw0; [127X[104X
    [4X[28Xq8_M2^9*q8_M1^9[128X[104X
    [4X[25Xgap>[125X [27Xlw1 := LoggedOnePassReduceWord( w0, l0 );[127X[104X
    [4X[28X[ [ [ 1, q8_M2^-9 ], [ 2, <identity ...> ] ], q8_M2^5*q8_M1^5 ][128X[104X
    [4X[25Xgap>[125X [27Xlw2 := LoggedReduceWordKB( w0, l0 ); [127X[104X
    [4X[28X[ [ [ 1, q8_M2^-9 ], [ 2, <identity ...> ], [ 1, q8_M2^-5 ], [128X[104X
    [4X[28X      [ 2, <identity ...> ] ], q8_M2*q8_M1 ][128X[104X
    [4X[25Xgap>[125X [27Xlw2 := LoggedReduceWordKB( w0, l2[1] ); [127X[104X
    [4X[28X[ [ [ 3, q8_M1*q8_M2^-8 ], [ -1, q8_M2^-8 ], [ 4, q8_M3*q8_M2^-8 ], [128X[104X
    [4X[28X      [ -4, <identity ...> ], [ 2, q8_M3^2 ], [128X[104X
    [4X[28X      [ -4, q8_M1^-1*q8_M2^-6*q8_M1^-2 ], [128X[104X
    [4X[28X      [ 3, q8_M3*q8_M1^-1*q8_M2^-6*q8_M1^-2 ], [128X[104X
    [4X[28X      [ 1, q8_M2^-1*q8_M1^-2*q8_M2^-6*q8_M1^-2 ], [ 4, <identity ...> ], [128X[104X
    [4X[28X      [ 3, q8_M1*q8_M2^-4*q8_M4^-1 ], [ -1, q8_M2^-4*q8_M4^-1 ], [128X[104X
    [4X[28X      [ 4, q8_M3*q8_M2^-4*q8_M4^-1 ], [ -4, q8_M4^-1 ], [128X[104X
    [4X[28X      [ 2, q8_M3^2*q8_M4^-1 ], [128X[104X
    [4X[28X      [ -3, q8_M1^-1*q8_M4^-1*q8_M1^-1*q8_M2^-2*q8_M1^-2*q8_M4^-1 ], [128X[104X
    [4X[28X      [ -4, <identity ...> ], [ 3, q8_M3 ], [128X[104X
    [4X[28X      [ 1, q8_M2^-1*q8_M1^-2*q8_M4^-1*q8_M1^-1*q8_M2^-1*(q8_M2^-1*q8_M1^-1)^2 [128X[104X
    [4X[28X         ], [ 4, q8_M4^-1*q8_M1^-1*q8_M2^-1*(q8_M2^-1*q8_M1^-1)^2 ], [128X[104X
    [4X[28X      [ 3, <identity ...> ], [ -1, q8_M1^-1 ], [ 4, q8_M3*q8_M1^-1 ], [128X[104X
    [4X[28X      [ -4, q8_M4^-1*q8_M1^-2 ], [ 2, q8_M3^2*q8_M4^-1*q8_M1^-2 ], [128X[104X
    [4X[28X      [ -4, q8_M1^-2 ], [ 3, q8_M3*q8_M1^-2 ], [128X[104X
    [4X[28X      [ -4, q8_M1^-2*q8_M2^-1*q8_M1^-3 ], [ 1, <identity ...> ], [128X[104X
    [4X[28X      [ 3, q8_M1*q8_M3^-1 ], [ -1, q8_M3^-1 ], [ 4, <identity ...> ], [128X[104X
    [4X[28X      [ -4, <identity ...> ], [ 3, q8_M3 ], [ 3, <identity ...> ], [128X[104X
    [4X[28X      [ -1, q8_M1^-1 ], [ 4, q8_M3*q8_M1^-1 ], [ -4, q8_M1^-2 ], [128X[104X
    [4X[28X      [ 3, q8_M3*q8_M1^-2 ], [ 1, <identity ...> ], [ -1, <identity ...> ], [128X[104X
    [4X[28X      [ 4, q8_M3 ] ], q8_M1*q8_M4 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X3.2-2 LoggedRewritingSystemFpGroup[101X
  
  [33X[1;0Y[29X[2XLoggedRewritingSystemFpGroup[102X( [3Xgrp[103X ) [32X attribute[133X
  
  [33X[0;0YGiven   a  group  presentation,  the  function  [10XLoggedRewritingSystemFpGroup[110X
  determines a logged rewrite system based on the relators. The initial logged
  rewrite system associated with a group presentation consists of two types of
  rule.  These  are  logged  versions  of  the two types of rule in the monoid
  presentation.  Corresponding to the j-th relator [10Xrel[110X of the group there is a
  logged  rule  [10X[rel,[[j,id]],id][110X.  For each inverse relator there is a logged
  rule  [10X[  gen*inv,  [],  id ][110X. The function then attempts a completion of the
  logged  rewrite  system. The rules in the final system are partially ordered
  by the function [10XShorterLoggedRule[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xlrws := LoggedRewritingSystemFpGroup( q8 );;[127X[104X
    [4X[25Xgap>[125X [27XPerform( lrws, Display ); [127X[104X
    [4X[28X[ q8_M4*q8_M2, [  ], <identity ...> ][128X[104X
    [4X[28X[ q8_M3*q8_M1, [  ], <identity ...> ][128X[104X
    [4X[28X[ q8_M2*q8_M4, [  ], <identity ...> ][128X[104X
    [4X[28X[ q8_M1*q8_M3, [  ], <identity ...> ][128X[104X
    [4X[28X[ q8_M1^2*q8_M4, [ [ -8, q8_M3^2 ], [ 5, <identity ...> ] ], q8_M2 ][128X[104X
    [4X[28X[ q8_M1^2*q8_M2, [ [ 8, <identity ...> ] ], q8_M4 ][128X[104X
    [4X[28X[ q8_M1^3, [ [ 5, <identity ...> ] ], q8_M3 ][128X[104X
    [4X[28X[ q8_M4^2, [ [ -8, <identity ...> ] ], q8_M1^2 ][128X[104X
    [4X[28X[ q8_M4*q8_M3, [ [ -7, q8_M1*q8_M2 ] ], q8_M1*q8_M4 ][128X[104X
    [4X[28X[ q8_M4*q8_M1, [ [ -8, <identity ...> ], [ 7, q8_M3 ] ], q8_M1*q8_M2 ][128X[104X
    [4X[28X[ q8_M3*q8_M4, [128X[104X
    [4X[28X  [ [ -5, <identity ...> ], [ -6, q8_M3^2 ], [ 8, <identity ...> ], [128X[104X
    [4X[28X      [ 7, q8_M1*q8_M4 ], [ -7, <identity ...> ] ], q8_M1*q8_M2 ][128X[104X
    [4X[28X[ q8_M3^2, [ [ -5, <identity ...> ] ], q8_M1^2 ][128X[104X
    [4X[28X[ q8_M3*q8_M2, [ [ -5, <identity ...> ], [ 8, q8_M3 ] ], q8_M1*q8_M4 ][128X[104X
    [4X[28X[ q8_M2*q8_M3, [ [ -7, <identity ...> ] ], q8_M1*q8_M2 ][128X[104X
    [4X[28X[ q8_M2^2, [ [ -8, <identity ...> ], [ 6, q8_M3^2 ] ], q8_M1^2 ][128X[104X
    [4X[28X[ q8_M2*q8_M1, [ [ 7, q8_M1 ], [ -5, <identity ...> ], [ 8, q8_M3 ] ],[128X[104X
    [4X[28X  q8_M1*q8_M4 ][128X[104X
    [4X[25Xgap>[125X [27XLength( lrws );[127X[104X
    [4X[28X16[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YConsider  now  the  two-generator abelian group [22XT[122X considered in the previous
  chapter  ([14X2.2-1[114X). Using the alternative ordering on the monoid generators, [10X[
  T_M1[110X[22X=a^+[122X,  [10XT_M2[110X[22X=a^-[122X,  [10XT_M3[110X[22X=b^+[122X, [10XT_M4[110X[22X=b^-[122X [10X][110X, we obtain the following set of [22X8[122X
  logged rules. The last of these may be checked as follows:[133X
  
  
  [24X[33X[0;6Y(b^+a^+(b^-a^-b^+a^+)a^-b^-)a^+b^+                                       ~=~
  b^+a^+(b^-(a^-(b^+(a^+a^-)b^-)a^+)b^+)[133X
  
  [124X
  
  [33X[0;0Yand is a logged version of the rule [22Xb^+a^+ -> a^+b^+[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XlrwsT := LoggedRewritingSystemFpGroup( T );[127X[104X
    [4X[28X[ [ T_M4*T_M3, [  ], <identity ...> ], [ T_M3*T_M4, [  ], <identity ...> ], [128X[104X
    [4X[28X  [ T_M2*T_M1, [  ], <identity ...> ], [ T_M1*T_M2, [  ], <identity ...> ], [128X[104X
    [4X[28X  [ T_M4*T_M2, [ [ -5, <identity ...> ] ], T_M2*T_M4 ], [128X[104X
    [4X[28X  [ T_M4*T_M1, [ [ 5, T_M2 ] ], T_M1*T_M4 ], [128X[104X
    [4X[28X  [ T_M3*T_M2, [ [ 5, T_M4 ] ], T_M2*T_M3 ], [128X[104X
    [4X[28X  [ T_M3*T_M1, [ [ -5, T_M2*T_M4 ] ], T_M1*T_M3 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
