  
  [1XC [33X[0;0YInverse automata and subgroups of the free group[133X[101X
  
  [33X[0;0YInverse  automata  with a single initial/accepting state are in a one to one
  correspondence  with finitely generated subgroups of the free group over the
  alphabet  of  the automaton. See [MSW01], [KM02] for details, as well as for
  concepts such as [13Xflower automaton[113X and [13XStallings foldings[113X.[133X
  
  
  [1XC.1 [33X[0;0YFrom subgroups to inverse automata[133X[101X
  
  [33X[0;0YA  finitely  generated  subgroup of a finitely generated free group is given
  through  a  list whose first element is the number of generators of the free
  group and the remaining elements are the generators of the subgroup. The set
  of generators of the free group of rank [22Xn[122X consists on the [22Xn[122X first letters of
  the  set  [22X{a,b,c,d,e,f,g}[122X. In particular, free groups of rank greater than [22X8[122X
  must  not  be  considered. A formal inverse of a generator is represented by
  the corresponding capital letter.[133X
  
  [33X[0;0YA  generator  of  the  subgroup  may be given through a string of letters or
  through a list of positive integers as should be clear from the example that
  follows.[133X
  
  [33X[0;0YFor example, [10X[2,"abA","bbabAB"][110X stands for the subgroup of the free group of
  rank  2  on the generators [22Xaba^-1[122X and [22Xbbaba^-1b^-1[122X. The same subgroup may be
  given  as  [10X[2,[1,2,3],[2,2,1,2,3,4]][110X.  The  number [22Xn+j[122X represents the formal
  inverse   of   the   generator  represented  by  [22Xj[122X.  One  can  go  from  one
  representation to another, using the following functions.[133X
  
  [1XC.1-1 GeneratorsToListRepresentation[101X
  
  [33X[1;0Y[29X[2XGeneratorsToListRepresentation[102X( [3XL[103X ) [32X function[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=[2,"abA","bbabAB"];;[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsToListRepresentation(L);[127X[104X
    [4X[28X[ 2, [ 1, 2, 3 ], [ 2, 2, 1, 2, 3, 4 ] ][128X[104X
  [4X[32X[104X
  
  [1XC.1-2 ListToGeneratorsRepresentation[101X
  
  [33X[1;0Y[29X[2XListToGeneratorsRepresentation[102X( [3XK[103X ) [32X function[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XK:=[2,[1,2,3],[2,2,1,2,3,4]];;[127X[104X
    [4X[25Xgap>[125X [27XListToGeneratorsRepresentation(K);[127X[104X
    [4X[28X[ 2, "abA", "bbabAB" ][128X[104X
  [4X[32X[104X
  
  [1XC.1-3 FlowerAutomaton[101X
  
  [33X[1;0Y[29X[2XFlowerAutomaton[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YThe  argument  [10XL[110X  is  a  subgroup of the free group given through any of the
  representations described above. Returns the flower automaton.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XW:=[2,"bbbAB","abAbA"];;[127X[104X
    [4X[25Xgap>[125X [27XA:=FlowerAutomaton(W);[127X[104X
    [4X[28X< non deterministic automaton on 2 letters with 9 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(A);[127X[104X
    [4X[28X   |  1          2       3       4    5       6       7    8       9[128X[104X
    [4X[28X---------------------------------------------------------------------[128X[104X
    [4X[28X a | [ 6, 9 ]                        [ 4 ]                [ 7 ][128X[104X
    [4X[28X b | [ 2, 5 ]   [ 3 ]   [ 4 ]                [ 7 ]        [ 9 ][128X[104X
    [4X[28XInitial state:   [ 1 ][128X[104X
    [4X[28XAccepting state: [ 1 ][128X[104X
  [4X[32X[104X
  
  [1XC.1-4 FoldFlowerAutomaton[101X
  
  [33X[1;0Y[29X[2XFoldFlowerAutomaton[102X( [3XA[103X ) [32X function[133X
  
  [33X[0;0YMakes  identifications  on  the flower automaton [10XA[110X. In the literature, these
  identifications are called Stallings foldings.[133X
  
  [33X[0;0Y(This  function  may  have  [10Xtrue[110X  as  a second argument. WARNING: the second
  argument should only be used when facilities to draw automata are available.
  In that case, one may visualize the identifications that are taking place.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB := FoldFlowerAutomaton(A);[127X[104X
    [4X[28X< deterministic automaton on 2 letters with 7 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(B);[127X[104X
    [4X[28X   |  1  2  3  4  5  6  7[128X[104X
    [4X[28X--------------------------[128X[104X
    [4X[28X a |  5  4              6[128X[104X
    [4X[28X b |  2  3  4     6     5[128X[104X
    [4X[28XInitial state:   [ 1 ][128X[104X
    [4X[28XAccepting state: [ 1 ][128X[104X
  [4X[32X[104X
  
  [1XC.1-5 SubgroupGenToInvAut[101X
  
  [33X[1;0Y[29X[2XSubgroupGenToInvAut[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YReturns the inverse automaton corresponding to the subgroup given by [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=[2,"bbbAB","AbAbA"];;[127X[104X
    [4X[25Xgap>[125X [27XSubgroupGenToInvAut(L);[127X[104X
    [4X[28X< deterministic automaton on 2 letters with 8 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X   |  1  2  3  4  5  6  7  8[128X[104X
    [4X[28X-----------------------------[128X[104X
    [4X[28X a |  8  4        1     6[128X[104X
    [4X[28X b |  2  3  4     6     8[128X[104X
    [4X[28XInitial state:   [ 1 ][128X[104X
    [4X[28XAccepting state: [ 1 ][128X[104X
  [4X[32X[104X
  
  
  [1XC.2 [33X[0;0YFrom inverse automata to subgroups[133X[101X
  
  [33X[0;0YGiven  an  inverse  automaton with a single initial/accepting state, one can
  find  a  set  of  generators  for the subgroup represented by the automaton.
  Moreover,  using  a  geodesic  tree,  one  can find a Nielsen reduced set of
  generators [KM02].[133X
  
  [1XC.2-1 GeodesicTreeOfInverseAutomaton[101X
  
  [33X[1;0Y[29X[2XGeodesicTreeOfInverseAutomaton[102X( [3XA[103X ) [32X function[133X
  
  [33X[0;0YReturns  a subautomaton of [3XA[103Xwhose underlying graph is a geodesic tree of the
  underlying graph of the inverse automaton [10XA[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:=Automaton("det",4,2,[ [ 3, 4, 0, 0 ], [ 2, 3, 4, 0 ] ],[ 1 ],[ 1 ]);;[127X[104X
    [4X[25Xgap>[125X [27XG := GeodesicTreeOfInverseAutomaton(A);[127X[104X
    [4X[28X< deterministic automaton on 2 letters with 4 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(G);[127X[104X
    [4X[28X   |  1  2  3  4[128X[104X
    [4X[28X-----------------[128X[104X
    [4X[28X a |  3[128X[104X
    [4X[28X b |  2     4[128X[104X
    [4X[28XInitial state:   [ 1 ][128X[104X
    [4X[28XAccepting state: [ 1 ][128X[104X
  [4X[32X[104X
  
  [1XC.2-2 InverseAutomatonToGenerators[101X
  
  [33X[1;0Y[29X[2XInverseAutomatonToGenerators[102X( [3XA[103X ) [32X function[133X
  
  [33X[0;0YReturns  a  set of generators (given trough the representation above) of the
  subgroup of the free group corresponding to the automaton [10XA[110X given.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNW := InverseAutomatonToGenerators(A);[127X[104X
    [4X[28X[ 2, "baBA", "bbA" ][128X[104X
  [4X[32X[104X
  
