  
  [1X3 [33X[0;0YFunctionally recursive machines[133X[101X
  
  [33X[0;0YAt  the  core  of  this  package  are  [13Xfunctionally  recursive machines[113X. The
  internals  of  specific machines will be described later, but each machine [22XM[122X
  has  an  associated [13Xalphabet[113X [22XX[122X, a [13Xset of states[113X [22XQ[122X, and a [13Xtransition function[113X
  [22Xϕ:Q  ×  X  ->  X × Q[122X. An [13Xelement[113X, as we will see in Section [14X4[114X, is given by a
  machine and an initial state [22Xq∈ Q[122X.[133X
  
  [33X[0;0YThe element [22X(M,q)[122X defines a transformation on the set [22XX^*[122X of strings (finite
  or  infinite)  over  the  alphabet [22XX[122X, as follows: the empty string is always
  fixed.  Given  a  string  [22Xx_1x_2... x_n[122X with [22Xnge0[122X, compute [22Xϕ(q,x_1)=(y_1,r)[122X;
  then  compute  the  action  of  [22X(M,r)[122X on the string [22Xx_2... x_n[122X, and call the
  result  [22Xy_2...  y_n[122X.  Then  the  action  of  [22X(M,q)[122X  on  [22Xx_1x_2... x_n[122X yields
  [22Xy_1y_2... y_n[122X.[133X
  
  [33X[0;0YThis  can  be understood more formally as follows. The transition function [22Xϕ[122X
  induces  a  map [22Xϕ^n:Q× X^n -> X^n × Q[122X, defined by successively applying [22Xϕ[122X to
  move the [22XQ[122X from the left to the right. Similarly, [22Xϕ[122X can be extended to a map
  [22XQ^m× X^n -> X^n × Q^m[122X.[133X
  
  [33X[0;0YWe  see  that  the action on finite strings preserves their length, and also
  preserves prefixes: if [22X(M,q)[122X maps [22Xx_1... x_n[122X to [22Xy_1... y_m[122X, then necessarily
  [22Xm=n[122X; and if [22Xk<n[122X then [22XT[122X maps [22Xx_1... x_k[122X to [22Xy_1... y_k[122X.[133X
  
  [33X[0;0YThe  strings  over  the  alphabet [22XX[122X can be naturally organised into a rooted
  tree.  The  root represents the empty string, and the strings [22Xx_1... x_n[122X and
  [22Xx_1... x_n+1[122X are connected by an edge, for all [22Xx_i∈ X[122X.[133X
  
  
  [1X3.1 [33X[0;0YTypes of machines[133X[101X
  
  [33X[0;0YMachines  must  be  accessible to computation; therefore it is reasonable to
  assume that their alphabet [22XX[122X is finite.[133X
  
  [33X[0;0YIf the stateset [22XQ[122X is also finite, the machine is called a [13XMealy machine[113X, and
  its transition function [22Xϕ[122X can be stored as a table.[133X
  
  [33X[0;0YMore general machines are obtained if one allows the stateset [22XQ[122X to be a free
  group/semigroup/monoid  generated  by  a  finite  set  [22XS[122X, and the transition
  function  [22Xϕ[122X  to be specified on [22XS× X[122X. Its values are then extended naturally
  by composition.[133X
  
  [33X[0;0YMachines store their transitions (second coordinate of [22Xϕ[122X), and their output,
  (first  coordinate  of  [22Xϕ[122X)  in  a  matrix  indexed  by  state and letter. In
  particular, [10XPermList(output[state])[110X gives the action on the first level.[133X
  
  [33X[0;0YBecause   of  the  way  [5XGAP[105X  handles  permutations  and  transformations,  a
  permutation  is never equal to a transformation, even though both can answer
  [9Xtrue[109X  to [22XIsOne[122X. Therefore, [5XFR[105X stores the output as a list, which can be then
  accessed  (e.g.  in commands such as [10XActivityPerm[110X and [10XActivityTransformation[110X
  either  as a permutation or as a transformation. The command [10XActivity[110X itself
  will return a permutation if possible, and otherwise a transformation.[133X
  
  
  [1X3.2 [33X[0;0YProducts of machines[133X[101X
  
  [33X[0;0YMachines  can  be  combined in different manners. If two machines act on the
  same  alphabet,  then  their  [13Xsum[113X and [13Xproduct[113X are defined as machines acting
  again  on  the  same alphabet; the statesets are the free products (which is
  also  their sum, in the category of semigroups) of the respective statesets.
  The  sum or product of machines has a stateset of highest possible category,
  i.e.  is  a group unless some argument is a monoid, and a monoid unless some
  argument  is  a semigroup. The transition and output functions are the union
  of the respective functions of their arguments.[133X
  
  [33X[0;0YIf  a non-empty collection of machines have same stateset, then their [13Xtensor
  sum[113X  and [13Xtensor product[113X are machines again with same stateset; the alphabets
  on  which  the  machines  act are the disjoint union, respectively cartesian
  product,  of  the  arguments' alphabets. The transition and output functions
  are  again  the  union  or  composition of the respective functions of their
  arguments.[133X
  
  [33X[0;0YThe  [13Xdirect  sum[113X  and  [13Xdirect product[113X of a collection of machines are always
  defined. Its stateset is generated by the union of the arguments' statesets,
  as  for a sum or a product; its alphabet is the disjoint union, respectively
  cartesian  product  of  its  arguments'  alphabets,  as  for a tensor sum or
  product.  The  transition  and  output  functions are again the union of the
  respective functions of their arguments.[133X
  
  
  [1X3.3 [33X[0;0YCreators for [10XFRMachine[110X[101X[1Xs[133X[101X
  
  [1X3.3-1 FRMachineNC[101X
  
  [29X[2XFRMachineNC[102X( [3Xfam[103X, [3Xfree[103X, [3Xtransitions[103X, [3Xoutputs[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA new FR machine.[133X
  
  [33X[0;0YThis  function  constructs a new FR machine, belonging to family [3Xfam[103X. It has
  stateset  the free group/semigroup/monoid [3Xfree[103X, and transitions described by
  [3Xstates[103X and [3Xoutputs[103X.[133X
  
  [33X[0;0Y[3Xtransitions[103X  is  a list of lists; [3Xtransitions[103X[[3Xs[103X][[3Xx[103X] is a word in [3Xfree[103X, which
  is the state reached by the machine when started in state [3Xs[103X and fed input [3Xx[103X.[133X
  
  [33X[0;0Y[3Xoutputs[103X is also a list of lists; [3Xoutputs[103X[[3Xs[103X][[3Xx[103X] is the output produced by the
  machine is in state [3Xs[103X and inputs [3Xx[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := FreeGroup(2);[127X[104X
    [4X[28X<free group on the generators [ f1, f2 ]>[128X[104X
    [4X[25Xgap>[125X [27Xm := FRMachineNC(FRMFamily([1,2]),f,[[One(f),f.1],[One(f),f.2^-1]],[127X[104X
    [4X[28X                      [[2,1],[1,2]]);[128X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1, f2 ] )>[128X[104X
  [4X[32X[104X
  
  [1X3.3-2 FRMachine[101X
  
  [29X[2XFRMachine[102X( [[3Xnames[103X, ][3Xtransitions[103X, [3Xoutputs[103X ) [32X operation
  [29X[2XFRMachine[102X( [3Xfree[103X, [3Xtransitions[103X, [3Xoutputs[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA new FR machine.[133X
  
  [33X[0;0YThis  function  constructs  a  new  FR  machine.  It  has  stateset  a  free
  group/semigroup/monoid, and structure described by [3Xtransitions[103X and [3Xoutputs[103X.[133X
  
  [33X[0;0YIf  there  is  an argument [3Xfree[103X, it is the free group/monoid/semigroup to be
  used  as  stateset.  Otherwise,  the  stateset  will  be  guessed  from  the
  [3Xtransitions[103X  and  [3Xoutputs[103X;  it  will  be  a  free  group  if  all states are
  invertible,  and a monoid otherwise. [3Xnames[103X is then an optional list, with at
  position  [3Xs[103X  a  string naming generator [3Xs[103X of the stateset. If [3Xnames[103X contains
  too few entries, they are completed by the names [3X__1,__2,...[103X.[133X
  
  [33X[0;0Y[3Xtransitions[103X  is  a list of lists; [10Xtransitions[s][x][110X is either an associative
  word, or a list of integers describing the state reached by the machine when
  started  in  state [3Xs[103X and fed input [3Xx[103X. Positive integers indicate a generator
  index,  negative integers its inverse, the empty list in the identity state,
  and  lists  of  length  greater than one indicate a product of states. If an
  entry  is  an  FR  element,  then its machine is incorporated into the newly
  constructed one.[133X
  
  [33X[0;0Y[3Xoutputs[103X   is   a   list;   at  position  [3Xs[103X  it  contains  a  permutation,  a
  transformation,  or  a  list  of  integers (the images of a transformation),
  describing  the  activity  of  state  [3Xs[103X.  If  all states are invertible, the
  outputs   are   all   converted   to  permutations,  while  if  there  is  a
  non-invertible state then the outputs are all converted to transformations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["tau","mu"],[[[],[1]],[[],[-2]]],[(1,2),(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ tau, mu ] )>[128X[104X
    [4X[25Xgap>[125X [27Xm=n;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDisplay(n);[127X[104X
    [4X[28X     |     1         2[128X[104X
    [4X[28X-----+--------+---------+[128X[104X
    [4X[28X tau | <id>,2     tau,1[128X[104X
    [4X[28X  mu | <id>,2   mu^-1,1[128X[104X
    [4X[28X-----+--------+---------+[128X[104X
    [4X[25Xgap>[125X [27Xm := FRMachine([[[],[FRElement(n,1)]]],[()]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1, f2, f3 ] )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(m);[127X[104X
    [4X[28X    |     1         2[128X[104X
    [4X[28X----+--------+---------+[128X[104X
    [4X[28X f1 | <id>,1      f2,2[128X[104X
    [4X[28X f2 | <id>,2      f2,1[128X[104X
    [4X[28X f3 | <id>,2   f1^-1,1[128X[104X
    [4X[28X----+--------+---------+[128X[104X
    [4X[25Xgap>[125X [27Xf := FreeGroup(2);[127X[104X
    [4X[28X<free group on the generators [ f1, f2 ]>[128X[104X
    [4X[25Xgap>[125X [27Xp := FRMachine(f,[[One(f),f.1],[One(f),f.2^-1],[(1,2),(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1, f2 ] )>[128X[104X
    [4X[25Xgap>[125X [27Xn=p;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-3 UnderlyingFRMachine[101X
  
  [29X[2XUnderlyingFRMachine[102X( [3Xobj[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YAn FR machine underlying [3Xobj[103X.[133X
  
  [33X[0;0YFR  elements,  FR  groups etc. often have an underlying FR machine, which is
  returned by this command.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := FRMachine(["a","b"],[[[],[2]],[[],[1]]],[(1,2),()]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ a, b ] )>[128X[104X
    [4X[25Xgap>[125X [27Xa := FRElement(m,1); b := FRElement(m,2);[127X[104X
    [4X[28X<2|a>[128X[104X
    [4X[28X<2|b>[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingFRMachine(a)=m;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-4 AsGroupFRMachine[101X
  
  [29X[2XAsGroupFRMachine[102X( [3Xm[103X ) [32X attribute
  [29X[2XAsMonoidFRMachine[102X( [3Xm[103X ) [32X attribute
  [29X[2XAsSemigroupFRMachine[102X( [3Xm[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YAn FR machine isomorphic to [3Xm[103X, on a free group/monoid/semigroup.[133X
  
  [33X[0;0YThis  function constructs, from the FR machine [3Xm[103X, an isomorphic FR machine [10Xn[110X
  with    a   free   group/monoid/semigroup   as   stateset.   The   attribute
  [10XCorrespondence(n)[110X is a mapping (homomorphism or list) from the stateset of [3Xm[103X
  to the stateset of [10Xn[110X.[133X
  
  [33X[0;0Y[3Xm[103X  can  be an arbitrary FR machine, or can be an free group/monoid/semigroup
  endomorphism. It is then converted to an FR machine on a 1-letter alphabet.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := FreeSemigroup(1);;[127X[104X
    [4X[25Xgap>[125X [27Xsm := FRMachine(s,[[GeneratorsOfSemigroup(s)[1],[127X[104X
    [4X[28X                         GeneratorsOfSemigroup(s)[1]^2]],[(1,2)]);[128X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Semigroup( [ s1 ] )>[128X[104X
    [4X[25Xgap>[125X [27Xm := FreeMonoid(1);;[127X[104X
    [4X[25Xgap>[125X [27Xmm := FRMachine(m,[[One(m),GeneratorsOfMonoid(m)[1]^2]],[(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Monoid( [ m1 ], ... )>[128X[104X
    [4X[25Xgap>[125X [27Xg := FreeGroup(1);;[127X[104X
    [4X[25Xgap>[125X [27Xgm := FRMachine(g,[[One(g),GeneratorsOfGroup(g)[1]^-2]],[(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1 ] )>[128X[104X
    [4X[25Xgap>[125X [27XAsGroupFRMachine(sm); Display(last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1 ] )>[128X[104X
    [4X[28X    |   1        2[128X[104X
    [4X[28X----+------+--------+[128X[104X
    [4X[28X f1 | f1,2   f1^2,1[128X[104X
    [4X[28X----+------+--------+[128X[104X
    [4X[25Xgap>[125X [27XCorrespondence(last);[127X[104X
    [4X[28XMappingByFunction( <free semigroup on the generators[128X[104X
    [4X[28X[ s1 ]>, <free group on the generators [ f1 ]>, function( w ) ... end )[128X[104X
    [4X[25Xgap>[125X [27XAsGroupFRMachine(mm); Display(last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1 ] )>[128X[104X
    [4X[28X    |     1        2[128X[104X
    [4X[28X----+--------+--------+[128X[104X
    [4X[28X f1 | <id>,2   f1^2,1[128X[104X
    [4X[28X----+--------+--------+[128X[104X
    [4X[25Xgap>[125X [27XAsGroupFRMachine(gm); Display(last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1 ] )>[128X[104X
    [4X[28X    |     1         2[128X[104X
    [4X[28X----+--------+---------+[128X[104X
    [4X[28X f1 | <id>,2   f1^-2,1[128X[104X
    [4X[28X----+--------+---------+[128X[104X
    [4X[25Xgap>[125X [27XAsMonoidFRMachine(sm); Display(last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Monoid( [ m1 ], ... )>[128X[104X
    [4X[28X    |   1        2[128X[104X
    [4X[28X----+------+--------+[128X[104X
    [4X[28X m1 | m1,2   m1^2,1[128X[104X
    [4X[28X  ----+------+--------+[128X[104X
    [4X[25Xgap>[125X [27XAsMonoidFRMachine(mm); Display(last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Monoid( [ m1 ], ... )>[128X[104X
    [4X[28X    |     1        2[128X[104X
    [4X[28X----+--------+--------+[128X[104X
    [4X[28X m1 | <id>,2   m1^2,1[128X[104X
    [4X[28X----+--------+--------+[128X[104X
    [4X[25Xgap>[125X [27XAsMonoidFRMachine(gm); Display(last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Monoid( [ m1, m2 ], ... )>[128X[104X
    [4X[28X    |     1        2[128X[104X
    [4X[28X----+--------+--------+[128X[104X
    [4X[28X m1 | <id>,2   m2^2,1[128X[104X
    [4X[28X m2 | m1^2,2   <id>,1[128X[104X
    [4X[28X----+--------+--------+[128X[104X
    [4X[25Xgap>[125X [27XAsSemigroupFRMachine(sm); Display(last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Semigroup( [ s1 ] )>[128X[104X
    [4X[28X    |   1        2[128X[104X
    [4X[28X----+------+--------+[128X[104X
    [4X[28X s1 | s1,2   s1^2,1[128X[104X
    [4X[28X----+------+--------+[128X[104X
    [4X[25Xgap>[125X [27XAsSemigroupFRMachine(mm); Display(last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Semigroup( [ s1, s2 ] )>[128X[104X
    [4X[28X    |   1        2[128X[104X
    [4X[28X----+------+--------+[128X[104X
    [4X[28X s1 | s2,2   s1^2,1[128X[104X
    [4X[28X s2 | s2,1     s2,2[128X[104X
    [4X[28X----+------+--------+[128X[104X
    [4X[25Xgap>[125X [27XAsSemigroupFRMachine(gm); Display(last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Semigroup( [ s1, s2, s3 ] )>[128X[104X
    [4X[28X    |     1        2[128X[104X
    [4X[28X----+--------+--------+[128X[104X
    [4X[28X s1 |   s3,2   s2^2,1[128X[104X
    [4X[28X s2 | s1^2,2     s3,1[128X[104X
    [4X[28X s3 |   s3,1     s3,2[128X[104X
    [4X[28X----+--------+--------+[128X[104X
    [4X[28Xgap>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(GuptaSidkiMachines(3));[127X[104X
    [4X[28X   |  1     2     3[128X[104X
    [4X[28X---+-----+-----+-----+[128X[104X
    [4X[28X a | a,1   a,2   a,3[128X[104X
    [4X[28X b | a,2   a,3   a,1[128X[104X
    [4X[28X c | a,3   a,1   a,2[128X[104X
    [4X[28X d | b,1   c,2   d,3[128X[104X
    [4X[28X---+-----+-----+-----+[128X[104X
    [4X[25Xgap>[125X [27XAsGroupFRMachine(GuptaSidkiMachines(3));[127X[104X
    [4X[28X<FR machine with alphabet [ 1 .. 3 ] on Group( [ f1, f2 ] )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X    |     1         2        3[128X[104X
    [4X[28X----+--------+---------+--------+[128X[104X
    [4X[28X f1 | <id>,2    <id>,3   <id>,1[128X[104X
    [4X[28X f2 |   f1,1   f1^-1,2     f2,3[128X[104X
    [4X[28X----+--------+---------+--------+[128X[104X
    [4X[25Xgap>[125X [27XCorrespondence(last);[127X[104X
    [4X[28X[ <identity ...>, f1, f1^-1, f2 ][128X[104X
    [4X[25Xgap>[125X [27XAsGroupFRMachine(GroupHomomorphism(g,g,[g.1],[g.1^3]));[127X[104X
    [4X[28X<FR machine with alphabet [ 1 ] on Group( [ f1 ] )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X G  |     1[128X[104X
    [4X[28X----+--------+[128X[104X
    [4X[28X f1 | f1^3,1[128X[104X
    [4X[28X----+--------+[128X[104X
  [4X[32X[104X
  
  [1X3.3-5 AsGroupFRMachine[101X
  
  [29X[2XAsGroupFRMachine[102X( [3Xf[103X ) [32X attribute
  [29X[2XAsMonoidFRMachine[102X( [3Xf[103X ) [32X attribute
  [29X[2XAsSemigroupFRMachine[102X( [3Xf[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YAn FR machine.[133X
  
  [33X[0;0YThis  function creates an FR machine on a 1-letter alphabet, that represents
  the  endomorphism  [3Xf[103X.  It is specially useful when combined with products of
  machines; indeed the usual product of machines corresponds to composition of
  endomorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := FreeGroup(2);;[127X[104X
    [4X[25Xgap>[125X [27Xh := GroupHomomorphismByImages(f,f,[f.1,f.2],[f.2,f.1*f.2]);[127X[104X
    [4X[28X[ f1, f2 ] -> [ f2, f1*f2 ][128X[104X
    [4X[25Xgap>[125X [27Xm := AsGroupFRMachine(h);[127X[104X
    [4X[28X<FR machine with alphabet [ 1 ] on Group( [ f1, f2 ] )>[128X[104X
    [4X[25Xgap>[125X [27Xmm := TensorProduct(m,m);[127X[104X
    [4X[28X<FR machine with alphabet [ 1 ] on Group( [ f1, f2 ] )>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(mm);[127X[104X
    [4X[28X G  |         1[128X[104X
    [4X[28X----+------------+[128X[104X
    [4X[28X f1 |    f1*f2,1[128X[104X
    [4X[28X f2 | f2*f1*f2,1[128X[104X
    [4X[28X----+------------+[128X[104X
  [4X[32X[104X
  
  
  [1X3.4 [33X[0;0YAttributes for [10XFRMachine[110X[101X[1Xs[133X[101X
  
  [1X3.4-1 StateSet[101X
  
  [29X[2XStateSet[102X( [3Xm[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe set of states associated with [3Xm[103X.[133X
  
  [33X[0;0YThis  function  returns  the  stateset of [3Xm[103X. It can be either a list (if the
  machine  is  of  Mealy type), or a free group/semigroup/monoid (in all other
  cases).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["tau","mu"],[[[],[1]],[[],[-2]]],[(1,2),(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ tau, mu ] )>[128X[104X
    [4X[25Xgap>[125X [27XStateSet(n);[127X[104X
    [4X[28X<free group on the generators [ tau, mu ]>[128X[104X
    [4X[25Xgap>[125X [27XStateSet(AsMealyMachine(n));[127X[104X
    [4X[28X[ 1 .. 4 ][128X[104X
  [4X[32X[104X
  
  [1X3.4-2 GeneratorsOfFRMachine[101X
  
  [29X[2XGeneratorsOfFRMachine[102X( [3Xm[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe generating set of the stateset of [3Xm[103X.[133X
  
  [33X[0;0YThis  function  returns  the  generating set of the stateset of [3Xm[103X. If [3Xm[103X is a
  Mealy machine, it returs the stateset.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["tau","mu"],[[[],[1]],[[],[-2]]],[(1,2),(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ tau, mu ] )>[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfFRMachine(n);[127X[104X
    [4X[28X[ tau, mu ][128X[104X
  [4X[32X[104X
  
  [1X3.4-3 Output[101X
  
  [29X[2XOutput[102X( [3Xm[103X ) [32X operation
  [29X[2XOutput[102X( [3Xm[103X, [3Xs[103X ) [32X operation
  [29X[2XOutput[102X( [3Xm[103X, [3Xs[103X, [3Xx[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA transformation of [3Xm[103X's alphabet.[133X
  
  [33X[0;0YIn the first form, this function returns the output of [3Xm[103X.[133X
  
  [33X[0;0YIn the second form, this function returns the transformation of [3Xm[103X's alphabet
  associated  with  state  [3Xs[103X.  This  transformation  is  returned as a list of
  images.[133X
  
  [33X[0;0Y[3Xs[103X  is  also  allowed  to  be  a list, in which case it is interpreted as the
  corresponding product of states.[133X
  
  [33X[0;0YIn the third form, the result is actually the image of [3Xx[103X under [10XOutput(m,s)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["a","b"],[[[],[2]],[[],[1]]],[(1,2),()]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ a, b ] )>[128X[104X
    [4X[25Xgap>[125X [27XOutput(n,[1,2]);[127X[104X
    [4X[28X[2,1][128X[104X
    [4X[25Xgap>[125X [27XOutput(n,Product(GeneratorsOfFRMachine(n)));[127X[104X
    [4X[28X[2,1][128X[104X
  [4X[32X[104X
  
  [1X3.4-4 Transition[101X
  
  [29X[2XTransition[102X( [3Xm[103X, [3Xs[103X, [3Xi[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YAn element of [3Xm[103X's stateset.[133X
  
  [33X[0;0YThis function returns the state reached by [3Xm[103X when started in state [3Xs[103X and fed
  input  [3Xi[103X.  This  input  may  be an alphabet letter or a sequence of alphabet
  letters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["a","b"],[[[],[2]],[[],[1]]],[(1,2),()]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ a, b ] )>[128X[104X
    [4X[25Xgap>[125X [27XTransition(n,[2,1],2);[127X[104X
    [4X[28Xa*b[128X[104X
    [4X[25Xgap>[125X [27XTransition(n,Product(GeneratorsOfFRMachine(n))^2,1);[127X[104X
    [4X[28Xa*b[128X[104X
  [4X[32X[104X
  
  [1X3.4-5 Transitions[101X
  
  [29X[2XTransitions[102X( [3Xm[103X, [3Xs[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA list of elements of [3Xm[103X's stateset.[133X
  
  [33X[0;0YThis  function  returns  the states reached by [3Xm[103X when started in state [3Xs[103X and
  fed  inputs  from the alphabet. The state may be expressed as a word or as a
  list of states.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["a","b"],[[[],[2]],[[],[1]]],[(1,2),()]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ a, b ] )>[128X[104X
    [4X[25Xgap>[125X [27XTransitions(n,[2,1]);[127X[104X
    [4X[28X[ <identity ...>, a*b ][128X[104X
    [4X[25Xgap>[125X [27XTransitions(n,Product(GeneratorsOfFRMachine(n))^2);[127X[104X
    [4X[28X[ a*b, b*a ][128X[104X
  [4X[32X[104X
  
  [1X3.4-6 WreathRecursion[101X
  
  [29X[2XWreathRecursion[102X( [3Xm[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YA function on the stateset of [3Xm[103X.[133X
  
  [33X[0;0YThis  function  returns  a  function  on  [3Xm[103X's  stateset.  This  function, on
  receiving  state  [3Xq[103X  as  input,  returns  a  list. Its first entry is a list
  indexed  by  [3Xm[103X's  alphabet, with in position [3Xx[103X the state [3Xm[103X would be in if it
  received  input  [3Xx[103X  when  in  state  [3Xq[103X.  The second entry is the list of the
  permutation of [3Xm[103X's alphabet induced by [3Xq[103X.[133X
  
  [33X[0;0Y[3XWreathRecursion(machine)(q)[1][a][103X  is  equal  to [3XTransition(machine,q,a)[103X and
  [3XWreathRecursion(machine)(q)[2][103X is equal to [3XOutput(machine,q)[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["a","b"],[[[],[2]],[[],[1]]],[(1,2),()]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ a, b ] )>[128X[104X
    [4X[25Xgap>[125X [27XWreathRecursion(n)(GeneratorsOfFRMachine(n)[1]);[127X[104X
    [4X[28X[ [ <identity ...>, b ], [2,1] ][128X[104X
    [4X[25Xgap>[125X [27XWreathRecursion(n)(GeneratorsOfFRMachine(n)[2]);[127X[104X
    [4X[28X[ [ <identity ...>, a ], [1,2] ][128X[104X
  [4X[32X[104X
  
  
  [1X3.5 [33X[0;0YOperations for [10XFRMachine[110X[101X[1Xs[133X[101X
  
  [1X3.5-1 StructuralGroup[101X
  
  [29X[2XStructuralGroup[102X( [3Xm[103X ) [32X operation
  [29X[2XStructuralMonoid[102X( [3Xm[103X ) [32X operation
  [29X[2XStructuralSemigroup[102X( [3Xm[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA   finitely   presented   group/monoid/semigroup   capturing  the
            structure of [3Xm[103X.[133X
  
  [33X[0;0YThis  function  returns  a  finitely  presented group/monoid/semigroup, with
  generators    the    union    of   the   [2XAlphabetOfFRObject[102X   ([14X10.1-3[114X)   and
  [2XGeneratorsOfFRMachine[102X  ([14X3.4-2[114X)  of  [3Xm[103X,  and  relations  all [22Xqa'=aq'[122X whenever
  [22Xϕ(q,a)=(a',q')[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["a","b","c"],[[[2],[3]],[[3],[2]],[[1],[1]]],[(1,2),(1,2),()]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ a, b, c ] )>[128X[104X
    [4X[25Xgap>[125X [27XStructuralGroup(n);[127X[104X
    [4X[28X<fp group on the generators [ a, b, c, 1, 2 ]>[128X[104X
    [4X[25Xgap>[125X [27XRelatorsOfFpGroup(last);[127X[104X
    [4X[28X[ a*2*b^-1*1^-1, a*1*c^-1*2^-1, b*2*c^-1*1^-1,[128X[104X
    [4X[28X  b*1*b^-1*2^-1, c*1*a^-1*1^-1, c*2*a^-1*2^-1 ][128X[104X
    [4X[25Xgap>[125X [27XSimplifiedFpGroup(last2);[127X[104X
    [4X[28X<fp group on the generators [ a, 1 ]>[128X[104X
    [4X[25Xgap>[125X [27XRelatorsOfFpGroup(last);[127X[104X
    [4X[28X[ 1^-1*a^2*1^4*a^-2*1^-1*a*1^-2*a^-1, 1*a*1^-1*a*1^2*a^-1*1*a^-2*1^-3*a,[128X[104X
    [4X[28X  1^-1*a^2*1^2*a^-1*1^-1*a*1^2*a^-2*1^-2 ][128X[104X
  [4X[32X[104X
  
  [1X3.5-2 \+[101X
  
  [29X[2X\+[102X( [3Xm1[103X, [3Xm2[103X ) [32X method
  [6XReturns:[106X  [33X[0;10YA new FR machine, in the same family as its arguments.[133X
  
  [33X[0;0YThis  function  returns  a  new FR machine [10Xr[110X, with stateset generated by the
  union  of  the  statesets  of  its  arguments.  The arguments [3Xm1[103X and [3Xm2[103X must
  operate  on  the same alphabet. If the stateset of [3Xm1[103X is free on [22Xn_1[122X letters
  and  the  stateset  of [3Xm2[103X is free on [22Xn_2[122X letters, then the stateset of their
  sum  is  free  on  [22Xn_1+n_2[122X  letters, with the first [22Xn_1[122X identified with [3Xm1[103X's
  states and the next [22Xn_2[122X with [3Xm2[103X's.[133X
  
  [33X[0;0YThe transition and output functions are naturally extended to the sum.[133X
  
  [33X[0;0YThe arguments may be free group, semigroup or monoid machines. The sum is in
  the weakest containing category: it is a group machine if both arguments are
  group  machines; a monoid if both are either group of monoid machines; and a
  semigroup machine otherwise.[133X
  
  [33X[0;0YThe  maps  from  the  stateset  of  [3Xm1[103X  and  [3Xm2[103X  to the stateset of [10Xr[110X can be
  recovered    as    [10XCorrespondence(r)[1][110X    and   [10XCorrespondence(r)[2][110X;   see
  [2XCorrespondence[102X ([14X3.5-12[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtau := FRMachine([[[],[1]]],[(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1 ] )>[128X[104X
    [4X[25Xgap>[125X [27Xmu := FRMachine([[[],[-1]]],[(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1 ] )>[128X[104X
    [4X[25Xgap>[125X [27Xsum := tau+mu;; Display(sum);[127X[104X
    [4X[28X     |     1          2[128X[104X
    [4X[28X-----+--------+----------+[128X[104X
    [4X[28X f11 | <id>,2      f11,1[128X[104X
    [4X[28X f12 | <id>,2   f12^-1,1[128X[104X
    [4X[28X-----+--------+----------+[128X[104X
    [4X[25Xgap>[125X [27XCorrespondence(sum)[1];[127X[104X
    [4X[28X[ f1 ] -> [ f11 ][128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfFRMachine(tau)[1]^last;[127X[104X
    [4X[28Xf11[128X[104X
  [4X[32X[104X
  
  [1X3.5-3 \*[101X
  
  [29X[2X\*[102X( [3Xmachine1[103X, [3Xmachine2[103X ) [32X method
  [6XReturns:[106X  [33X[0;10YA new FR machine, in the same family as its arguments.[133X
  
  [33X[0;0YThe  product  of two FR machines coincides with their sum, since the natural
  free  object  mapping  to  the  product of the statesets is generated by the
  union of the statesets. See therefore [2X\+[102X ([14X3.5-2[114X).[133X
  
  [1X3.5-4 TensorSumOp[101X
  
  [29X[2XTensorSumOp[102X( [3XFR_machines[103X, [3Xmachine[103X ) [32X method
  [6XReturns:[106X  [33X[0;10YA  new  FR  machine  on  the  disjoint  union  of  the  arguments'
            alphabets.[133X
  
  [33X[0;0YThe  tensor  sum  of  FR  machines  with  same stateset is defined as the FR
  machine  acting  on  the disjoint union of the alphabets; if these alphabets
  are  [10X[1..n1][110X up to [10X[1..nk][110X, then the alphabet of their sum is [10X[1..n1+...+nk][110X
  and the transition functions are similarly concatenated.[133X
  
  [33X[0;0YThe  first  argument  is  a list; the second argument is any element of that
  list, and is used only to improve the method selection algorithm.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := TensorSum(AddingMachine(2),AddingMachine(3),AddingMachine(4));[127X[104X
    [4X[28XAddingMachine(2)(+)AddingMachine(3)(+)AddingMachine(4)[128X[104X
    [4X[25Xgap>[125X [27XDisplay(m);[127X[104X
    [4X[28X   |  1     2     3     4     5     6     7     8     9[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+-----+-----+-----+[128X[104X
    [4X[28X a | a,1   a,2   a,3   a,4   a,5   a,6   a,7   a,8   a,9[128X[104X
    [4X[28X b | a,2   b,1   a,4   a,5   b,3   a,7   a,8   a,9   b,6[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+-----+-----+-----+[128X[104X
  [4X[32X[104X
  
  [1X3.5-5 TensorProductOp[101X
  
  [29X[2XTensorProductOp[102X( [3XFR[103X, [3Xmachines[103X, [3Xmachine[103X ) [32X method
  [6XReturns:[106X  [33X[0;10YA  new  FR  machine  on  the  cartesian  product of the arguments'
            alphabets.[133X
  
  [33X[0;0YThe  tensor  product  of FR machines with same stateset is defined as the FR
  machine  acting  on  the  cartesian product of the alphabets. The transition
  function  and  output  function  behave as if a single letter, in the tensor
  product's  alphabet,  were a word (read from left to right) in the machines'
  alphabets.[133X
  
  [33X[0;0YThe  first  argument  is  a list; the second argument is any element of that
  list, and is used only to improve the method selection algorithm.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := TensorProduct(AddingMachine(2),AddingMachine(3));[127X[104X
    [4X[28XAddingMachine(2)(*)AddingMachine(3)[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X   |  1     2     3     4     5     6[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+[128X[104X
    [4X[28X a | a,1   a,2   a,3   a,4   a,5   a,6[128X[104X
    [4X[28X b | a,4   a,5   a,6   a,2   a,3   b,1[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+[128X[104X
  [4X[32X[104X
  
  [1X3.5-6 DirectSumOp[101X
  
  [29X[2XDirectSumOp[102X( [3XFR[103X, [3Xmachines[103X, [3Xmachine[103X ) [32X method
  [6XReturns:[106X  [33X[0;10YA  new  FR  machine  on  the  disjoint  union  of  the  arguments'
            alphabets.[133X
  
  [33X[0;0YThe  direct  sum  of  FR machines is defined as the FR machine with stateset
  generated  by  the  disjoint  union of the statesets, acting on the disjoint
  union  of  the alphabets; if these alphabets are [10X[1..n1][110X up to [10X[1..nk][110X, then
  the  alphabet  of  their sum is [10X[1..n1+...+nk][110X and the output and transition
  functions are similarly concatenated.[133X
  
  [33X[0;0YThe  first  argument  is  a list; the second argument is any element of that
  list, and is used only to improve the method selection algorithm.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := DirectSum(AddingMachine(2),AddingMachine(3),AddingMachine(4));[127X[104X
    [4X[28XAddingMachine(2)#AddingMachine(3)#AddingMachine(4)[128X[104X
    [4X[25Xgap>[125X [27XDisplay(m);[127X[104X
    [4X[28X   |  1     2     3     4     5     6     7     8     9[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+-----+-----+-----+[128X[104X
    [4X[28X a | a,1   a,2   a,3   a,4   a,5   a,6   a,7   a,8   a,9[128X[104X
    [4X[28X b | a,2   b,1   b,3   b,4   b,5   b,6   b,7   b,8   b,9[128X[104X
    [4X[28X c | c,1   c,2   a,3   a,4   a,5   c,6   c,7   c,8   c,9[128X[104X
    [4X[28X d | d,1   d,2   a,4   a,5   b,3   d,6   d,7   d,8   d,9[128X[104X
    [4X[28X e | e,1   e,2   e,3   e,4   e,5   a,6   a,7   a,8   a,9[128X[104X
    [4X[28X f | f,1   f,2   f,3   f,4   f,5   a,7   a,8   a,9   b,6[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+-----+-----+-----+[128X[104X
  [4X[32X[104X
  
  [1X3.5-7 DirectProductOp[101X
  
  [29X[2XDirectProductOp[102X( [3XFR[103X, [3Xmachines[103X, [3Xmachine[103X ) [32X method
  [6XReturns:[106X  [33X[0;10YA  new  FR  machine  on  the  cartesian  product of the arguments'
            alphabets.[133X
  
  [33X[0;0YThe direct product of FR machines is defined as the FR machine with stateset
  generated  by  the  product  of  the statesets, acting on the product of the
  alphabets;  if  these alphabets are [10X[1..n1][110X up to [10X[1..nk][110X, then the alphabet
  of  their  product is [10X[1..n1*...*nk][110X and the output and transition functions
  act component-wise.[133X
  
  [33X[0;0YThe  first  argument  is  a list; the second argument is any element of that
  list, and is used only to improve the method selection algorithm.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := DirectProduct(AddingMachine(2),AddingMachine(3));[127X[104X
    [4X[28XAddingMachine(2)xAddingMachine(3)[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X   |  1     2     3     4     5     6[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+[128X[104X
    [4X[28X a | a,1   a,2   a,3   a,4   a,5   a,6[128X[104X
    [4X[28X b | a,2   a,3   b,1   a,5   a,6   b,4[128X[104X
    [4X[28X c | a,4   a,5   a,6   c,1   c,2   c,3[128X[104X
    [4X[28X d | a,5   a,6   b,4   c,2   c,3   d,1[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+[128X[104X
  [4X[32X[104X
  
  [1X3.5-8 TreeWreathProduct[101X
  
  [29X[2XTreeWreathProduct[102X( [3Xm[103X, [3Xn[103X, [3Xx0[103X, [3Xy0[103X ) [32X method
  [6XReturns:[106X  [33X[0;10YA  new  FR  machine  on  the  cartesian  product of the arguments'
            alphabets.[133X
  
  [33X[0;0YThe  [13Xtree-wreath  product[113X  of  two  FR  machines  is a machine acting on the
  product  of  its arguments' alphabets [22XX,Y[122X, in such a way that many images of
  the first machine's states under conjugation by the second commute.[133X
  
  [33X[0;0YIt  is  introduced  (in  lesser  generality,  and  with small variations) in
  [Sid05],  and  may  be  described  as  follows:  one takes two copies of the
  stateset  of  [3Xm[103X,  one copy of the stateset of [3Xn[103X, and, if necessary, an extra
  identity state.[133X
  
  [33X[0;0YThe  first  copy  of  [3Xm[103X  fixes  the  alphabet  [22XX×  Y[122X;  its state [22Xtilde s[122X has
  transitions  to  the  identity except [22Xtilde s[122X at [22X(x0,y0)[122X and [22Xs[122X at [22X(*,y0)[122X for
  any  other  [22X*[122X.  The  second  copy of [3Xm[103X is also trivial except that, on input
  [22X(x,y0)[122X,  its  state  [22Xs[122X  goes  to  state  [22Xs'[122X  with  output [22X(x',y0)[122X whenever [22Xs[122X
  originally  went,  on  input  [22Xx[122X,  to state [22Xs'[122X with output [22Xx'[122X. This copy of [3Xm[103X
  therefore  acts  only  in  the  [22XX[122X  direction,  on the subtree [22X(X×{y0})^∞[122X, on
  subtrees below vertices of the form [22X(x0,y0)^t(x,y0)[122X.[133X
  
  [33X[0;0YA  state  [22Xt[122X  in the copy of [3Xn[103X maps the input [22X(x,y)[122X to [22X(x,y')[122X and proceeds to
  state  [22Xt'[122X  if [22Xy=y0[122X, and to the identity state otherwise, when on input [22Xy[122X the
  original machine mapped state [22Xt[122X to output [22Xt'[122X and output [22Xy'[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm := TreeWreathProduct(AddingMachine(2),AddingMachine(3),1,1);[127X[104X
    [4X[28XAddingMachine(2)~AddingMachine(3)[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X   |  1     2     3     4     5     6[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+[128X[104X
    [4X[28X a | c,2   c,3   a,1   c,5   c,6   c,4[128X[104X
    [4X[28X b | c,4   c,2   c,3   b,1   c,5   c,6[128X[104X
    [4X[28X c | c,1   c,2   c,3   c,4   c,5   c,6[128X[104X
    [4X[28X d | d,1   c,2   c,3   b,4   c,5   c,6[128X[104X
    [4X[28X---+-----+-----+-----+-----+-----+-----+[128X[104X
  [4X[32X[104X
  
  [1X3.5-9 SubFRMachine[101X
  
  [29X[2XSubFRMachine[102X( [3Xmachine1[103X, [3Xmachine2[103X ) [32X operation
  [29X[2XSubFRMachine[102X( [3Xmachine1[103X, [3Xf[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YEither  [9Xfail[109X  or  an  embedding  of  the states of [3Xmachine2[103X in the
            states of [3Xmachine1[103X.[133X
  
  [33X[0;0YIn  its  first  form, this function attempts to locate a copy of [3Xmachine2[103X in
  [3Xmachine1[103X.  If  is  succeeds,  it returns a homomorphism from the stateset of
  [3Xmachine2[103X into the stateset of [3Xmachine1[103X; otherwise it returns [9Xfail[109X.[133X
  
  [33X[0;0YIn  its  second  form,  this  function  attempts to construct a machine with
  stateset  the  source  of  [3Xf[103X,  that  could  be identified as a submachine of
  [3Xmachine1[103X via [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["tau","mu"],[[[],[1]],[[],[-2]]],[(1,2),(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ tau, mu ] )>[128X[104X
    [4X[25Xgap>[125X [27Xtauinv := FRMachine([[[1],[]]],[(1,2)]);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1 ] )>[128X[104X
    [4X[25Xgap>[125X [27XSubFRMachine(n,tauinv);[127X[104X
    [4X[28X[ f1 ] -> [ tau^-1 ][128X[104X
    [4X[25Xgap>[125X [27XSubFRMachine(n,last);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ f1 ] )>[128X[104X
  [4X[32X[104X
  
  [1X3.5-10 ChangeFRMachineBasis[101X
  
  [29X[2XChangeFRMachineBasis[102X( [3Xm[103X[, [3Xl[103X][, [3Xp[103X] ) [32X attribute
  [6XReturns:[106X  [33X[0;10YAn equivalent FR machine, in a new basis.[133X
  
  [33X[0;0YThis  function constructs a new group FR machine, given a group FR machine [3Xm[103X
  and,  optionally,  a  list  of  states  [3Xl[103X  (as  elements  of the free object
  [10XStateSet(m)[110X)   and   a   permutation  [3Xp[103X,  which  defaults  to  the  identity
  permutation.[133X
  
  [33X[0;0YThe  new  machine  has  the  following  transitions: if alphabet letter [10Xa[110X is
  mapped  to  [10Xb[110X by state [10Xs[110X in [3Xm[103X, leading to state [10Xt[110X, then, in the new machine,
  the  input  letter  [10Xa^p[110X  is  mapped  to  [10Xb^p[110X  by  state  [10Xs[110X, leading to state
  [10Xl[a]^-1*t*l[b][110X.[133X
  
  [33X[0;0YThe  group generated by the new machine is isomorphic to the group generated
  by  [3Xm[103X.  This command amounts to a change of basis of the associated bimodule
  (see  [Nek05,  Section  2.2]). It amounts to conjugation by the automorphism
  [10Xc=FRElement("c",[l[1]*c,...,l[n]*c],[()],1)[110X.[133X
  
  [33X[0;0YIf  the  second  argument  is absent, this command attempts to choose a list
  that makes many entries of the recursion trivial.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["tau","mu"],[[[],[1]],[[],[-2]]],[(1,2),(1,2)]);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay(n);[127X[104X
    [4X[28X G   |     1         2[128X[104X
    [4X[28X-----+--------+---------+[128X[104X
    [4X[28X tau | <id>,2     tau,1[128X[104X
    [4X[28X  mu | <id>,2   mu^-1,1[128X[104X
    [4X[28X-----+--------+---------+[128X[104X
    [4X[25Xgap>[125X [27Xnt := ChangeFRMachineBasis(n,GeneratorsOfFRMachine(n){[1,1]});;[127X[104X
    [4X[25Xgap>[125X [27XDisplay(nt);[127X[104X
    [4X[28X G   |     1                    2[128X[104X
    [4X[28X-----+--------+--------------------+[128X[104X
    [4X[28X tau | <id>,2                tau,1[128X[104X
    [4X[28X  mu | <id>,2   tau^-1*mu^-1*tau,1[128X[104X
    [4X[28X-----+--------+--------------------+[128X[104X
  [4X[32X[104X
  
  [1X3.5-11 Minimized[101X
  
  [29X[2XMinimized[102X( [3Xm[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA minimized machine equivalent to [3Xm[103X.[133X
  
  [33X[0;0YThis  function  attempts  to construct a machine equivalent to [3Xm[103X, but with a
  stateset  of  smaller  rank.  Identical generators are collapsed to a single
  generator  of  the  stateset; if [3Xm[103X is a group or monoid machine then trivial
  generators  are  removed;  if  [3Xm[103X  is  a  group machine then mutually inverse
  generators  are  grouped.  This  function  sets  as [10XCorrespondence(result)[110X a
  mapping  between  the  stateset  of  [3Xm[103X  and  the stateset of the result; see
  [2XCorrespondence[102X ([14X3.5-12[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn := FRMachine(["tau","mu"],[[[],[1]],[[],[-2]]],[(1,2),(1,2)]);;[127X[104X
    [4X[25Xgap>[125X [27Xm := FRMachine(["tauinv"],[[[1],[]]],[(1,2)]);;[127X[104X
    [4X[25Xgap>[125X [27Xsum := n+m+n;[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ tau1, mu1, tauinv1, tau2, mu2 ] )>[128X[104X
    [4X[25Xgap>[125X [27Xmin := Minimized(sum);[127X[104X
    [4X[28X<FR machine with alphabet [ 1, 2 ] on Group( [ tau1, mu1 ] )>[128X[104X
    [4X[25Xgap>[125X [27XCorrespondence(min);[127X[104X
    [4X[28X[ tau1, mu1, tauinv1, tau2, mu2 ] -> [ tau1, mu1, tau1^-1, tau1, mu1 ][128X[104X
  [4X[32X[104X
  
  [1X3.5-12 Correspondence[101X
  
  [29X[2XCorrespondence[102X( [3Xm[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YA mapping between statesets of FR machines.[133X
  
  [33X[0;0YIf  a  machine  [3Xm[103X was created as a minimized group/monoid/semigroup machine,
  then  [10XCorrespondence(m)[110X  is  a  mapping between the stateset of the original
  machine and the stateset of [3Xm[103X. See [2XMinimized[102X ([14X3.5-11[114X) for an example.[133X
  
  [33X[0;0YIf  [3Xm[103X  was created as a minimized Mealy machine, then [10XCorrespondence(m)[110X is a
  list identifying, for each state of the original machine, a state of the new
  machine. If the original state is inaccessible, the corresponding list entry
  is unbound. See [2XMinimized[102X ([14X5.2-2[114X) for an example.[133X
  
  [33X[0;0YIf  [3Xm[103X was created using [2XAsGroupFRMachine[102X ([14X3.3-4[114X), [2XAsMonoidFRMachine[102X ([14X3.3-4[114X),
  [2XAsSemigroupFRMachine[102X    ([14X3.3-4[114X),    or    [2XAsMealyMachine[102X    ([14X5.2-18[114X),   then
  [10XCorrespondence(m)[110X is a list or a homomorphism identifying for each generator
  of  the  original machine a generator, or word in the generators, of the new
  machine. It is a list if either the original or the final machine is a Mealy
  machine, and a homomorphism in other cases.[133X
  
  [33X[0;0YIf  [3Xm[103X  was  created  as  a  sum  of  two  machines,  then  [3Xm[103X  has  a mapping
  [10XCorrespondence(m)[i][110X  between  the  stateset  of  machine  [10Xi=1,2[110X and its own
  stateset. See [2X\+[102X ([14X3.5-2[114X) for an example.[133X
  
