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#############################################################################
##
#W frmachine.gi Laurent Bartholdi
##
#Y Copyright (C) 2006-2013, Laurent Bartholdi
##
#############################################################################
##
## This file implements the category of functionally recursive machines.
##
#############################################################################
############################################################################
##
#O AlphabetOfFRObject
#V FR_FAMILIES
#O FRMFamily(alphabet)
##
InstallMethod(AlphabetOfFRObject, "(FR) for an FR object",
[IsFRObject],
function(M)
local a;
a := FamilyObj(M)!.alphabet;
IsRange(a);
return a;
end);
INSTALLPRINTERS@(IsFRObject);
InstallMethod(FRMFamily, "(FR) for an alphabet",
[IsListOrCollection],
function(d)
local i;
for i in FR_FAMILIES do
if i[1] = d then return i[2]; fi;
od;
i := NewFamily(Concatenation("FRMachine(",String(d),")"), IsFRMachine);
i!.standard := Size(d)<2^28 and d=[1..Size(d)];
if i!.standard then
i!.alphabet := [1..Size(d)];
else
i!.alphabet := d;
i!.a2n := x->Position(Enumerator(d),x);
i!.n2a := x->Enumerator(d)[x];
fi;
ConvertToRangeRep(i!.alphabet);
MakeImmutable(i!.alphabet);
Add(FR_FAMILIES,[d,i]);
return i;
end);
InstallMethod(IsGroupFRMachine, [IsFRMachine], ReturnFalse);
InstallMethod(IsMonoidFRMachine, [IsFRMachine], ReturnFalse);
InstallMethod(IsSemigroupFRMachine, [IsFRMachine], ReturnFalse);
#############################################################################
#############################################################################
##
#O FRMachine(Transitions, Output)
#O FRMachine(Names, Transitions, Output)
#O FRMachineNC(Family, [Semi]Group, Transitions, Output)
#O FRMachine([Semi]Group, Transitions, Output)
##
InstallOtherMethod(FRMachineNC, "(FR) for a family, a free group, a list of transitions and a list of outputs",
[IsFamily, IsGroup, IsList, IsList],
function(fam,free,transitions,output)
local M, F;
F := FamilyObj(Representative(free));
M := Objectify(NewType(fam, IsGroupFRMachine and IsFRMachineStdRep),
rec(free := free,
pack := l->AssocWordByLetterRep(F,l),
transitions := Immutable(transitions),
output := Immutable(output)));
SetIsInvertible(M, true);
return M;
end);
InstallOtherMethod(FRMachineNC, "(FR) for a family, a free semigroup, a list of transitions and a list of outputs",
[IsFamily, IsFreeSemigroup, IsList, IsList],
function(fam,free,transitions,output)
local M, F;
F := FamilyObj(Representative(free));
M := Objectify(NewType(fam, IsSemigroupFRMachine and IsFRMachineStdRep),
rec(free := free,
pack := l->AssocWordByLetterRep(F,l),
transitions := Immutable(transitions),
output := Immutable(output)));
return M;
end);
InstallOtherMethod(FRMachineNC, "(FR) for a family, a free monoid, a list of transitions and a list of outputs",
[IsFamily, IsFreeMonoid, IsList, IsList],
function(fam,free,transitions,output)
local M, F, T;
F := FamilyObj(Representative(free));
M := Objectify(NewType(fam, IsMonoidFRMachine and IsFRMachineStdRep),
rec(free := free,
pack := l->AssocWordByLetterRep(F,l),
transitions := Immutable(transitions),
output := Immutable(output)));
return M;
end);
BindGlobal("COPYFRMACHINE@", function(m)
return Objectify(NewType(FamilyObj(m), First([IsGroupFRMachine,IsMonoidFRMachine,IsSemigroupFRMachine],p->Tester(p)(m) and p(m)) and IsFRMachineStdRep),
rec(free := m!.free,
pack := m!.pack,
transitions := m!.transitions,
output := m!.output));
end);
BindGlobal("ANY2OUT@", function(x,n)
if IsList(x) then
return x;
elif IsTransformation(x) then
return ListTransformation(x,n);
elif IsPerm(x) then
return ListPerm(x,n);
fi;
end);
BindGlobal("CHECKLENGTHSCONTENTS@", function(t, transitions, output)
# check validity of arguments;
# unpack FR elements contained in the transitions;
# set t.F
local i, j, k, x, e;
if Length(transitions)<>Length(output) then
Error("<Transitions> and <Output> must have the same length\n");
return fail;
fi;
if not ForAll(transitions, IsList) or
ForAny(transitions, r->Length(r)<>Length(transitions[1])) then
Error("All rows of <Transitions> must be lists of the same length\n");
return fail;
fi;
t.F := FRMFamily([1..Length(transitions[1])]);
t.transitions := StructuralCopy(transitions);
t.output := ShallowCopy(output);
for x in t.transitions do for x in x do if IsList(x) then
i := 1; while i <= Length(x) do
if IsFRElement(x[i]) then
if IsMealyElement(x[i]) then
e := AsSemigroupFRElement(x[i]);
else
e := x[i];
fi;
k := Length(t.transitions);
for j in UnderlyingFRMachine(e)!.transitions do
Add(t.transitions,List(j,w->List(LetterRepAssocWord(w),i->i+SignInt(i)*k)));
od;
Append(t.output, UnderlyingFRMachine(e)!.output);
Remove(x,i); i := i-1;
for j in LetterRepAssocWord(InitialState(e)) do
i := i+1;
Add(x,j+SignInt(j)*k,i);
od;
elif IsInt(x[i]) and AbsInt(x[i]) in [1..Length(t.transitions)] then;
else
Error("Entry ",i," of <Transitions> is not in the state set\n");
return fail;
fi;
i := i+1;
od;
elif not IsAssocWord(x) then
Error("Transitions must be associative words or lists");
fi; od; od;
# clean up t.output, set t.invertible
t.invertible := true;
for i in [1..Length(t.output)] do
t.output[i] := ANY2OUT@(t.output[i],Length(t.F!.alphabet));
if Set(t.output[i])<>t.F!.alphabet then
t.invertible := false;
fi;
od;
for i in t.output do
if not IsSubset(t.F!.alphabet,i) then
Error("Entry ",i," of <Output> is not in alphabet ",t.F!.alphabet,"\n");
return fail;
fi;
od;
end);
InstallMethod(FRMachine, "(FR) for a list of transitions and a list of outputs",
[IsList, IsList],
function(transitions, output)
local G, elG, t;
t := rec();
CHECKLENGTHSCONTENTS@(t, transitions, output);
if t.invertible then
G := FreeGroup(Length(t.transitions));
else
G := FreeMonoid(Length(t.transitions));
fi;
elG := FamilyObj(Representative(G));
return FRMachineNC(t.F, G, List(t.transitions,t->List(t,w->AssocWordByLetterRep(elG,w))),t.output);
end);
InstallMethod(FRMachine, "(FR) for a list of names, a list of transitions and a list of outputs",
[IsList, IsList, IsList],
function(names, transitions, output)
local G, elG, t, n;
t := rec();
if not ForAll(names,IsString) then
Error("<names> should be a list of strings, and not ", names,"\n");
fi;
CHECKLENGTHSCONTENTS@(t, transitions, output);
if Length(names)>Length(t.transitions) then
Error("Too many names supplied to FRMachine()\n");
elif Length(names)<Length(t.transitions) then
n := Concatenation(names,List([1..Length(t.transitions)-Length(names)],i->Concatenation("__",String(i))));
else
n := names;
fi;
if t.invertible then
G := FreeGroup(n);
else
G := FreeMonoid(n);
fi;
elG := FamilyObj(Representative(G));
return FRMachineNC(t.F, G, List(t.transitions,t->List(t,w->AssocWordByLetterRep(elG,w))),t.output);
end);
InstallMethod(FRMachine, "(FR) for a free [semi]group, a list of transitions and a list of outputs",
[IsSemigroup, IsList, IsList],
function(free,transitions,output)
local t, elfree, r, i;
t := rec();
CHECKLENGTHSCONTENTS@(t, transitions, output);
if IsGroup(free) and not t.invertible then
Error("Outputs must be invertible in group FR machine: ",t.output);
fi;
elfree := FamilyObj(Representative(free));
for r in t.transitions do for i in [1..Length(r)] do
if IsList(r[i]) then r[i] := AssocWordByLetterRep(elfree,r[i]); fi;
od; od;
r := FRMachineNC(t.F,free,t.transitions,t.output);
if Length(t.transitions)<>Length(GeneratorsOfFRMachine(r)) then
Error("<Transition> and <Output> should have same length as ",free,"'s rank\n");
fi;
return r;
end);
#############################################################################
##
#A GeneratorsOfFRMachine(FRMachine)
##
InstallMethod(GeneratorsOfFRMachine, "(FR) for a group FR machine",
[IsGroupFRMachine],
M->GeneratorsOfGroup(M!.free));
InstallMethod(GeneratorsOfFRMachine, "(FR) for a semigroup FR machine",
[IsSemigroupFRMachine],
M->GeneratorsOfSemigroup(M!.free));
InstallMethod(GeneratorsOfFRMachine, "(FR) for a monoid FR machine",
[IsMonoidFRMachine],
M->GeneratorsOfMonoid(M!.free));
InstallMethod(StateSet, "(FR) for an FR machine",
[IsFRMachine and IsFRMachineStdRep],
M->M!.free);
#############################################################################
#############################################################################
##
#M ViewObj(FRMachine)
#M String(FRMachine)
#M Display(FRMachine)
##
InstallMethod(ViewString, "(FR) for an FR machine",
[IsFRMachine and IsFRMachineStdRep],
function(M)
return CONCAT@("<FR machine with alphabet ", AlphabetOfFRObject(M), " on ", StateSet(M), ">");
end);
InstallMethod(String, "(FR) for an FR machine",
[IsFRMachine and IsFRMachineStdRep],
function(M)
return CONCAT@("FRMachine(...,",M!.output,")");
end);
BindGlobal("DISPLAYFRMACHINE@", function(M)
local a, i, j, g, alen, slen, glen, ablank, sblank, gblank, arule, grule, srule, StringId, s;
a := AlphabetOfFRObject(M);
g := GeneratorsOfFRMachine(M);
s := "";
StringId := function(arg)
local s;
s := CallFuncList(String,arg);
if s="<identity ...>" then
s := "<id>"; if Length(arg)=2 then s := String(s,arg[2]); fi;
fi;
return s;
end;
alen := LogInt(Maximum(a),10)+3;
ablank := ListWithIdenticalEntries(alen,' ');
arule := ListWithIdenticalEntries(alen,'-');
if g=[] then
glen := 2;
slen := List(a,i->1);
else
glen := Maximum(List(g,t->Length(StringId(t))))+1;
slen := List(a,i->Maximum(List(g,t->Length(StringId(Transition(M,t,i)))))+1);
fi;
gblank := ListWithIdenticalEntries(glen,' ');
grule := ListWithIdenticalEntries(glen,'-');
sblank := List(a,i->ListWithIdenticalEntries(slen[i],' '));
srule := List(a,i->ListWithIdenticalEntries(slen[i],'-'));
if IsGroupFRMachine(M) then
s := " G";
elif IsMonoidFRMachine(M) then
s := " M";
else s := " S"; fi;
APPEND@(s,gblank{[3..glen]}," |");
for i in [1..Length(a)] do APPEND@(s,sblank[i],String(a[i],-alen)," "); od;
APPEND@(s,"\n");
APPEND@(s,grule,"-+");
for i in [1..Length(a)] do APPEND@(s,srule[i],arule,"+"); od;
APPEND@(s,"\n");
for i in [1..Length(g)] do
APPEND@(s,StringId(g[i],glen)," |");
for j in [1..Length(a)] do
APPEND@(s,StringId(M!.transitions[i][j],slen[j]),",",String(M!.output[i][j],-alen));
od;
APPEND@(s,"\n");
od;
APPEND@(s,grule,"-+");
for i in [1..Length(a)] do APPEND@(s,srule[i],arule,"+"); od;
APPEND@(s,"\n");
return s;
end);
InstallMethod(DisplayString, "(FR) for an FR machine",
[IsFRMachine and IsFRMachineStdRep],
DISPLAYFRMACHINE@);
INSTALLPRINTERS@(IsFRMachine);
#############################################################################
##
#M One(FRMachine)
##
InstallMethod(OneOp, "(FR) for an FR machine",
[IsFRMachine],
M->FRMachine([ListWithIdenticalEntries(Size(AlphabetOfFRObject(M)),[1])],[()]));
#############################################################################
#############################################################################
##
#M Zero(FRMachine)
##
InstallOtherMethod(ZeroOp, "(FR) for an FR machine",
[IsFRMachine],
M->FRMachineNC(FamilyObj(M),FreeGroup(0),[],[]));
#############################################################################
#############################################################################
##
#M InverseOp(FRMachine)
##
InstallTrueMethod(IsInvertible, IsGroupFRMachine);
BindGlobal("ISINVERTIBLE@", function(l)
return Set(l)=[1..Length(l)];
end);
BindGlobal("INVERSE@", function(l) # inverse of transformation, given as list
local r;
r := [];
r{l} := [1..Length(l)];
return r;
end);
BindGlobal("ISONE@", function(l) # identity mapping, given as list
return l=[1..Length(l)];
end);
BindGlobal("PREIMAGE@", Position); # preimage of point under transformation
InstallMethod(InverseOp, "(FR) for a group FR machine",
[IsGroupFRMachine],
function(M)
local N;
N := FRMachineNC(FamilyObj(M), M!.free,
List([1..Length(M!.transitions)], i->M!.transitions[i]{M!.output[i]}),
List(M!.output, INVERSE@));
SetInverse(M,N);
SetInverse(N,M);
return N;
end);
InstallMethod(IsReversible, "(FR) for a group FR machine",
[IsGroupFRMachine],
function(M)
local a, hom;
for a in AlphabetOfFRObject(M) do
hom := GroupHomomorphismByImages(StateSet(M),StateSet(M),GeneratorsOfFRMachine(M),M!.transitions{[1..Length(M!.transitions)]}[a]);
if not IsBijective(hom) then
return false;
fi;
od;
return true;
end);
#############################################################################
#############################################################################
##
#M Products
##
BindGlobal("SET_NAME@", function(args,sep,obj)
local i, s, n;
for i in args do if not HasName(i) then return; fi; od;
s := ShallowCopy(Name(args[1]));
for i in [2..Length(args)] do Append(s,sep); Append(s,Name(args[i])); od;
SetName(obj,s);
end);
BindGlobal("MAKENAMESUNIQUE@", function(sgen)
local i, j, nonunique;
nonunique := Set(Filtered(Collected(Concatenation(sgen)),x->x[2]>1),x->x[1]);
RemoveSet(nonunique,"<identity ...>");
for i in [1..Length(sgen)] do
for j in [1..Length(sgen[i])] do
if sgen[i][j] in nonunique then
sgen[i][j] := Concatenation(sgen[i][j],".",String(i));
fi;
od;
od;
end);
BindGlobal("LARGESTDENOMINATOR@", function(arg)
# returns homomorphisms from all its arguments' free stateset to
# a free object of highest structure (group > monoid > semigroup).
# the last entry in the returned list is a list of appropriate generators
# for each argument's free object.
local c, d, f, i, iso, states, gen, sgen, subgen, shift;
d := Length(arg);
states := List(arg,x->x!.free);
c := List(states,IdentityMapping);
gen := List(arg,GeneratorsOfFRMachine);
sgen := List(gen,x->List(x,String));
if ForAll(states,IsGroup) then
MAKENAMESUNIQUE@(sgen);
f := FreeGroup(Concatenation(sgen));
c := [];
shift := 0;
for i in [1..d] do
Add(c,GroupHomomorphismByImages(states[i],f,
gen[i],GeneratorsOfGroup(f){shift+[1..Length(gen[i])]}));
shift := shift + Length(gen[i]);
od;
elif ForAll(states,IsMonoid) then
c := List(states,IdentityMapping);
for i in [1..d] do
if IsGroup(states[i]) then
c[i] := IsomorphismFpMonoidInversesFirst(states[i]);
gen[i] := GeneratorsOfMonoid(states[i]);
sgen[i] := List(GeneratorsOfMonoid(Range(IsomorphismFpMonoidInversesFirst(FreeGroup(sgen[i])))),String);
c[i] := c[i]*MappingByFunction(Range(c[i]),FreeMonoidOfFpMonoid(Range(c[i])),UnderlyingElement);
fi;
od;
MAKENAMESUNIQUE@(sgen);
f := FreeMonoid(Concatenation(sgen));
shift := 0;
for i in [1..d] do
c[i] := c[i]*MagmaHomomorphismByImagesNC(Range(c[i]),f,
GeneratorsOfMonoid(f){shift+[1..Length(gen[i])]});
shift := shift + Length(gen[i]);
od;
else
c := List(states,IdentityMapping);
for i in [1..d] do
if IsGroup(states[i]) then
c[i] := IsomorphismFpSemigroup(states[i]);
gen[i] := GeneratorsOfSemigroup(states[i]);
sgen[i] := List(GeneratorsOfSemigroup(Range(IsomorphismFpSemigroup(FreeGroup(sgen[i])))),String);
c[i] := c[i]*MappingByFunction(Range(c[i]),FreeSemigroupOfFpSemigroup(Range(c[i])),UnderlyingElement);
elif IsMonoid(states[i]) then
gen[i] := GeneratorsOfSemigroup(states[i]);
sgen[i] := Concatenation(["<identity ...>"],sgen[i]);
c[i] := states[i]/[];
iso := IsomorphismFpSemigroup(c[i]);
c[i] := NaturalHomomorphismByGenerators(states[i],c[i])*iso*MappingByFunction(Range(iso),FreeSemigroupOfFpSemigroup(Range(iso)),UnderlyingElement);
fi;
od;
MAKENAMESUNIQUE@(sgen);
f := FreeSemigroup(Concatenation(sgen));
shift := 0;
for i in [1..d] do
c[i] := c[i]*MagmaHomomorphismByImagesNC(Range(c[i]),f,
GeneratorsOfSemigroup(f){shift+[1..Length(gen[i])]});
shift := shift + Length(gen[i]);
od;
fi;
Add(c,gen);
return c;
end);
BindGlobal("FRMSUM@", function(arg)
local c, gen, trans, out, i, j, sum;
c := CallFuncList(LARGESTDENOMINATOR@,arg);
gen := Remove(c);
trans := [];
out := [];
for i in [1..Length(arg)] do
for j in gen[i] do
Add(trans,List(AlphabetOfFRObject(arg[i]),a->Transition(arg[i],j,a)^c[i]));
Add(out,Output(arg[i],j));
od;
od;
sum := FRMachineNC(FamilyObj(arg[1]),Range(c[1]),trans,out);
SetCorrespondence(sum,c);
SET_NAME@(arg,"+",sum);
return sum;
end);
BindGlobal("FRMMINSUM@", function(left,right)
local sum, r;
sum := FRMSUM@(left,right);
r := Minimized(sum);
r!.Correspondence := List(Correspondence(sum),x->x*Correspondence(r));
return r;
end);
InstallMethod(\+, "(FR) for two FR machines",
IsIdenticalObj,
[IsFRMachine and IsFRMachineStdRep, IsFRMachine and IsFRMachineStdRep],
FRMSUM@);
InstallMethod(\*, "(FR) for two FR machines",
IsIdenticalObj,
[IsFRMachine and IsFRMachineStdRep, IsFRMachine and IsFRMachineStdRep],
FRMSUM@);
InstallMethod(TensorSumOp, "(FR) for two FR machines",
[IsList, IsFRMachine and IsFRMachineStdRep],
function(M, N)
local trans, out, t, o, i, j, x, d, s;
while ForAny(M,x->x!.free<>N!.free) do
Error("All machines should have same stateset");
od;
trans := [];
out := [];
trans := [];
out := [];
for i in [1..Length(N!.output)] do
t := [];
o := [];
d := 0;
for j in [1..Length(M)] do
Append(t,M[j]!.transitions[i]);
Append(o,M[j]!.output[i]+d);
d := d+Size(AlphabetOfFRObject(M[j]));
od;
Add(trans,t);
Add(out,o);
od;
x := FRMachineNC(FRMFamily([1..d]),N!.free,trans,out);
SET_NAME@(M,"(+)",x);
return x;
end);
InstallMethod(TensorProductOp, "(FR) for two FR machines",
[IsList, IsFRMachine and IsFRMachineStdRep],
function(M, N)
local trans, out, t, o, i, j, x, a, b, alphabet, s;
while ForAny(M,x->x!.free<>N!.free) do
Error("All machines should have same stateset");
od;
alphabet := Cartesian(List(M,AlphabetOfFRObject));
trans := [];
out := [];
for i in [1..Length(N!.output)] do
t := [];
o := [];
for a in alphabet do
b := [];
s := i;
for j in [1..Length(M)] do
Add(b,Output(M[j],s,a[j]));
s := Transition(M[j],s,a[j]);
od;
Add(o,Position(alphabet,b));
Add(t,s);
od;
Add(trans,t);
Add(out,o);
od;
x := FRMachineNC(FRMFamily([1..Length(alphabet)]),N!.free,trans,out);
SET_NAME@(M,"(*)",x);
return x;
end);
InstallMethod(DirectSumOp, "(FR) for two FR machines",
[IsList, IsFRMachine and IsFRMachineStdRep],
function(M, N)
local c, gen, trans, out, t, o, i, j, d, alph, shift, sum;
c := CallFuncList(LARGESTDENOMINATOR@,M);
gen := Remove(c);
d := 0;
alph := []; shift := [];
for i in [1..Length(M)] do
Add(alph,AlphabetOfFRObject(M[i]));
Add(shift, [d+1..d+Length(alph[i])]);
d := d+Length(alph[i]);
od;
trans := [];
out := [];
for i in [1..Length(M)] do
for j in gen[i] do
t := ListWithIdenticalEntries(d,j^c[i]);
t{shift[i]} := List(alph[i],a->Transition(M[i],j,a)^c[i]);
o := [1..d];
o{shift[i]} := shift[i]{Output(M[i],j)};
Add(trans,t);
Add(out,o);
od;
od;
sum := FRMachineNC(FRMFamily([1..d]),Range(c[1]),trans,out);
SetCorrespondence(sum,c);
SET_NAME@(M,"#",sum);
return sum;
end);
InstallMethod(DirectProductOp, "(FR) for two FR machines",
[IsList, IsFRMachine and IsFRMachineStdRep],
function(M, N)
local c, gen, trans, out, t, o, i, j, a, b, product, alphabet;
c := CallFuncList(LARGESTDENOMINATOR@,M);
gen := Remove(c);
alphabet := Cartesian(List(M,AlphabetOfFRObject));
trans := [];
out := [];
for i in [1..Length(M)] do
for j in gen[i] do
t := [];
o := [];
for a in alphabet do
b := ShallowCopy(a);
b[i] := Output(M[i],j,a[i]);
Add(o,Position(alphabet,b));
Add(t,Transition(M[i],j,a[i])^c[i]);
od;
Add(trans,t);
Add(out,o);
od;
od;
product := FRMachineNC(FRMFamily([1..Length(alphabet)]),Range(c[1]),trans,out);
SetCorrespondence(product,c);
SET_NAME@(M,"x",product);
return product;
end);
InstallMethod(TreeWreathProduct, "for two FR machines",
[IsFRMachine and IsFRMachineStdRep, IsFRMachine and IsFRMachineStdRep,
IsObject, IsObject],
function(g,h,x0,y0)
local c, gen, m, trans, out, t, o, one, i, j, alphabet;
alphabet := Cartesian(AlphabetOfFRObject(g),AlphabetOfFRObject(h));
while not [x0,y0] in alphabet do
Error("(x0,y0) must be in the product of the machines' alphabets");
od;
c := LARGESTDENOMINATOR@(g,h,g,Zero(g));
gen := Remove(c);
if gen[4]=[] then
one := One(Range(c[1]));
else
one := gen[4][1]^c[4];
fi;
trans := [];
out := [];
for i in [1..Length(gen[1])] do
t := [];
o := [];
for j in alphabet do
if j=[x0,y0] then
Add(t,gen[1][i]^c[1]);
elif j[2]=y0 then
Add(t,gen[3][i]^c[3]);
else
Add(t,one);
fi;
Add(o,Position(alphabet,j));
od;
Add(trans,t);
Add(out,o);
od;
for i in [1..Length(gen[2])] do
t := [];
o := [];
for j in alphabet do
if j[1]=x0 then
Add(t,Transition(h,gen[2][i],j[2])^c[2]);
else
Add(t,one);
fi;
Add(o,Position(alphabet,[j[1],Output(h,gen[2][i],j[2])]));
od;
Add(trans,t);
Add(out,o);
od;
for i in [1..Length(gen[3])] do
t := [];
o := [];
for j in alphabet do
if j[2]=y0 then
Add(t,Transition(g,gen[3][i],j[1])^c[3]);
Add(o,Position(alphabet,[Output(g,gen[3][i],j[1]),y0]));
else
Add(t,one);
Add(o,Position(alphabet,j));
fi;
od;
Add(trans,t);
Add(out,o);
od;
Add(trans,ListWithIdenticalEntries(Length(alphabet),one));
Add(out,[1..Length(alphabet)]);
m := Minimized(FRMachineNC(FRMFamily([1..Length(alphabet)]),Range(c[1]),trans,out));
m!.Correspondence := List(c{[1..2]},x->x*Correspondence(m));
SET_NAME@([g,h],"~",m);
return m;
end);
#############################################################################
#############################################################################
##
#M \=(FRMachine, FRMachine)
##
InstallMethod(\=, "(FR) for two FR machines",
IsIdenticalObj,
[IsFRMachine and IsFRMachineStdRep, IsFRMachine and IsFRMachineStdRep],
function(left, right)
local i, j;
if left!.output <> right!.output then
return false;
elif Length(left!.transitions) <> Length(right!.transitions) then
return false;
fi;
for i in [1..Length(left!.transitions)] do
for j in AlphabetOfFRObject(left) do
if LetterRepAssocWord(left!.transitions[i][j])<>LetterRepAssocWord(right!.transitions[i][j]) then return false; fi;
od;
od;
return true;
end);
#############################################################################
#############################################################################
##
#M \<(FRMachine, FRMachine)
##
InstallMethod(\<, "(FR) for two FR machines",
IsIdenticalObj,
[IsFRMachine and IsFRMachineStdRep, IsFRMachine and IsFRMachineStdRep],
function(left, right)
local i, j, wl, wr;
if left!.output <> right!.output then
return left!.output < right!.output;
elif Length(left!.transitions) <> Length(right!.transitions) then
return Length(left!.transitions) < Length(right!.transitions);
fi;
for i in [1..Length(left!.transitions)] do
for j in AlphabetOfFRObject(left) do
wl := LetterRepAssocWord(left!.transitions[i][j]);
wr := LetterRepAssocWord(right!.transitions[i][j]);
if wl<>wr then return wl<wr; fi;
od;
od;
return false; # they're equal
end);
#############################################################################
#############################################################################
##
#A WreathRecursion(FRMachine)
#O Output(Machine, State)
#O Transition(Machine, State, Letter)
##
InstallMethod(Output, "(FR) for an FR machine",
[IsGroupFRMachine and IsFRMachineStdRep],
function(M)
local image;
image := List(M!.output,PermList);
return GroupHomomorphismByImages(StateSet(M),Group(image),image);
end);
InstallMethod(Output, "(FR) for an FR machine",
[IsMonoidFRMachine and IsFRMachineStdRep],
function(M)
local image;
image := List(M!.output,TransformationList);
return SemigroupHomomorphismByImagesNC(StateSet(M),Monoid(image),image);
end);
InstallMethod(Output, "(FR) for an FR machine",
[IsSemigroupFRMachine and IsFRMachineStdRep],
function(M)
local image;
image := List(M!.output,TransformationList);
return SemigroupHomomorphismByImagesNC(StateSet(M),Semigroup(image),image);
end);
InstallMethod(Output, "(FR) for an FR machine and a state expressed as an integer",
[IsFRMachine and IsFRMachineStdRep, IsInt],
function(M, i)
if i > 0 then
return M!.output[i];
elif i = 0 then
return AlphabetOfFRObject(M);
else
return INVERSE@(M!.output[-i]);
fi;
end);
InstallMethod(Output, "(FR) for an FR machine and a state expressed as a word",
[IsFRMachine and IsFRMachineStdRep, IsAssocWord],
function(M, w)
local perm, i;
perm := AlphabetOfFRObject(M);
for i in LetterRepAssocWord(w) do
if i > 0 then
perm := M!.output[i]{perm};
else
perm := INVERSE@(M!.output[-i]){perm};
fi;
od;
return perm;
end);
InstallMethod(Output, "(FR) for an FR machine and a state expressed as a list",
[IsFRMachine, IsList],
function(M, l)
local perm, i;
perm := AlphabetOfFRObject(M);
for i in l do
perm := Output(M,i){perm};
od;
return perm;
end);
InstallMethod(Output, "(FR) for an FR machine, a state and a letter",
[IsFRMachine, IsObject, IsObject],
function(M, s, a)
return Output(M,s)[a];
end);
InstallMethod(Transition, "(FR) for an FR machine, a state expressed as an integer, and an input",
[IsFRMachine and IsFRMachineStdRep, IsInt, IsPosInt],
function(M, i, p)
if i > 0 then
return M!.transitions[i][p];
else
return M!.transitions[-i][PREIMAGE@(M!.output[-i],p)];
fi;
end);
InstallMethod(Transitions, "(FR) for an FR machine and a state expressed as an integer",
[IsFRMachine and IsFRMachineStdRep, IsInt],
function(M, i)
if i > 0 then
return M!.transitions[i];
else
return M!.transitions[-i]{INVERSE@(M!.output[-i])};
fi;
end);
BindGlobal("FRMTRANSITION@", function(M,l,p)
local w, i;
if IsMonoid(M!.free) then
w := One(M!.free);
else
w := fail;
fi;
for i in l do
if i > 0 then
if w=fail then
w := M!.transitions[i][p];
else
w := w*M!.transitions[i][p];
fi;
p := M!.output[i][p];
else
p := PREIMAGE@(M!.output[-i],p);
if w=fail then
w := M!.transitions[-i][p]^-1;
else
w := w/M!.transitions[-i][p];
fi;
fi;
od;
return w;
end);
InstallMethod(Transition, "(FR) for an FR machine, a state expressed as a list, and an input",
[IsFRMachine and IsFRMachineStdRep, IsList, IsPosInt],
FRMTRANSITION@);
InstallMethod(Transition, "(FR) for an FR machine, a state expressed as a word, and an input",
[IsFRMachine and IsFRMachineStdRep, IsAssocWord, IsPosInt],
function(M, v, p)
return FRMTRANSITION@(M,LetterRepAssocWord(v),p);
end);
InstallMethod(Transition, "(FR) for an FR machine, a state, and a list of letters",
[IsFRMachine, IsObject, IsList],
function(M, s, l)
local i, t;
t := s;
for i in l do t := Transition(M, t, i); od;
return t;
end);
InstallMethod(Transitions, "(FR) for an FR machine and a state expressed as a list",
[IsFRMachine and IsFRMachineStdRep, IsList],
function(M,w)
return WreathRecursion(M)(w)[1];
end);
InstallMethod(Transitions, "(FR) for an FR machine, a state expressed as a word, and an input",
[IsFRMachine and IsFRMachineStdRep, IsAssocWord],
function(M, w)
return WreathRecursion(M)(w)[1];
end);
InstallMethod(WreathRecursion, "(FR) for an FR machine",
[IsFRMachine and IsFRMachineStdRep],
function(M)
local output, alphabet, ones, transitions, onetrans;
output := M!.output;
alphabet := AlphabetOfFRObject(M);
if IsMonoid(M!.free) then
ones := List(alphabet,x->One(M!.free));
else
ones := List(alphabet,x->fail);
fi;
onetrans := AlphabetOfFRObject(M);
transitions := M!.transitions;
return function(w)
local vector, perm, i, j;
vector := ShallowCopy(ones);
perm := onetrans;
if IsAssocWord(w) then w := LetterRepAssocWord(w); fi;
for i in w do
if i > 0 then
if vector[1]=fail then
vector := ShallowCopy(transitions[i]);
else
for j in alphabet do
vector[j] := vector[j]*transitions[i][perm[j]];
od;
fi;
perm := output[i]{perm};
else
perm := INVERSE@(output[-i]){perm};
if vector[1]=fail then
vector := List(transitions[-i]{perm},Inverse);
else
for j in alphabet do
vector[j] := vector[j]/transitions[-i][perm[j]];
od;
fi;
fi;
od;
return [vector,perm];
end;
end);
InstallMethod(VirtualEndomorphism, "(FR) for a group FR machine and a vertex",
[IsGroupFRMachine,IsObject],
function(M,v)
local G, H;
G := StateSet(M);
H := Stabilizer(G,v,function(w,g) return w^FRElement(M,g); end);
return GroupHomomorphismByImages(H,G,GeneratorsOfGroup(H),List(GeneratorsOfGroup(H),x->Transition(M,x,v)));
end);
#############################################################################
#############################################################################
##
#A FRMachineRWS
##
InstallMethod(FRMachineRWS, "(FR) for an FR machine",
[IsFRMachine and IsFRMachineStdRep],
function(M)
local output, alphabet, transitions, inverse, iso, fpmonoid, gens, mgens, rws;
if IsGroupFRMachine(M) then
iso := IsomorphismFpMonoidInversesFirst(M!.free);
mgens := List(GeneratorsOfMonoid(Range(iso)),x->PreImage(iso,x));
rws := rec(rws := KnuthBendixRewritingSystem(Range(iso)),
letterrep := w->LetterRepAssocWord(UnderlyingElement(w^iso)),
letterunrep := w->Product(mgens{w},One(M!.free)));
gens := List(GeneratorsOfMonoid(Range(iso)),
w->FRElement(M,PreImagesRepresentativeNC(iso,w)));
inverse := List(gens,w->rws.letterrep(InitialState(w)^-1)[1]);
rws.cyclicallyreduce := function(w)
local i, j;
i := 1; j := Length(w);
while i<j and w[i]=inverse[w[j]] do
i := i+1; j := j-1;
od;
if i=1 then return w; else return w{[i..j]}; fi;
end;
else
rws := rec(rws := KnuthBendixRewritingSystem(M!.free/[]),
letterrep := LetterRepAssocWord);
if IsMonoidFRMachine(M) then
rws.letterunrep := w->Product(GeneratorsOfMonoid(M!.free){w},One(M!.free));
else
rws.letterunrep := w->Product(GeneratorsOfSemigroup(M!.free){w});
fi;
gens := List(GeneratorsOfFRMachine(M),w->FRElement(M,w));
fi;
output := List(gens,Output);
alphabet := AlphabetOfFRObject(M);
transitions := List(gens,w->List(alphabet,
a->rws.letterrep(Transition(w,a))));
if ValueOption("fr_maxlen")<>fail then
rws.maxlen := ValueOption("fr_maxlen");
else
rws.maxlen := 5; # do not add rules longer than that -- too slow
fi;
rws.modified := true; # whether the true rules rws.tzrules and
# the temporary rules rws.rws!.tzrules are in sync
rws.pi := function(w)
local vector, perm, i, j;
vector := List(alphabet,x->[]);
perm := alphabet;
for i in w do
for j in alphabet do
Append(vector[j],transitions[i][perm[j]]);
od;
perm := output[i]{perm};
od;
return [vector,perm];
end;
rws.reduce := w->ReduceLetterRepWordsRewSys(rws.rws!.tzrules,w);
rws.addsgrule := function(l,r,short)
local ll, lr;
ll := Length(l); lr := Length(r);
if short and (ll>rws.maxlen or lr>rws.maxlen) then return; fi;
rws.modified := true;
if ll>lr or (ll=lr and l>r) then
Info(InfoFR,3,"# Added rule ",l," -> ",r);
AddRuleReduced(rws.rws,[ShallowCopy(l),ShallowCopy(r)]);
else
Info(InfoFR,3,"# Added rule ",r," -> ",l);
AddRuleReduced(rws.rws,[ShallowCopy(r),ShallowCopy(l)]);
fi;
end;
if IsGroupFRMachine(M) then
rws.addgprule := function(w,short)
local i, l, ll, left, right;
l := Length(w);
if short and l>2*rws.maxlen then return; fi;
Info(InfoFR,3,"# Adding group rule ",w);
rws.addsgrule(w,[],short);
rws.addsgrule(inverse{w{[l,l-1..1]}},[],short);
ll := QuoInt(l,2);
left := w{[1..ll]};
right := inverse{w{[l,l-1..ll+1]}};
for i in [1..l] do
rws.addsgrule(left,right,short);
Add(left,inverse[Remove(right)]);
if IsOddInt(l) then
rws.addsgrule(left,right,short);
fi;
Add(right,inverse[Remove(left,1)],1);
od;
end;
fi;
rws.commit := function()
if rws.modified then
Info(InfoFR,3,"# Committed rules ",rws.rws!.tzrules);
rws.tzrules := StructuralCopy(rws.rws!.tzrules);
rws.modified := false;
fi;
end;
rws.restart := function()
if rws.modified then
Info(InfoFR,3,"# Restarting with fresh rules ",rws.tzrules);
rws.rws!.tzrules := StructuralCopy(rws.tzrules);
rws.modified := false;
fi;
end;
rws.commit();
return rws;
end);
InstallGlobalFunction(NewFRMachineRWS, # "(FR) will restart with fresh rules",
function(M)
local rws;
rws := FRMachineRWS(M);
rws.restart();
return rws;
end);
#############################################################################
############################################################################
##
#O States(FRMachine)
##
InstallMethod(States, "(FR) for an FR machine and an element",
[IsFRMachine, IsMultiplicativeElement],
function(M,x)
return States(M,[x]);
end);
InstallOtherMethod(States, "(FR) for an FR machine and a list of elements",
[IsFRMachine, IsMultiplicativeElementCollection],
function(M,L)
local states, i, x, stateset;
states := ShallowCopy(L);
stateset := Set(states);
i := 1;
while i <= Length(states) do
for x in Transitions(M,states[i]) do
if not x in stateset then
Add(states,x);
AddSet(stateset,x);
fi;
od;
i := i+1;
if RemInt(i,100)=0 then
Info(InfoFR, 2, "The states contain at least ", states);
fi;
od;
return states;
end);
InstallMethod(FixedStates, "(FR) for an FR machine and an element",
[IsFRMachine, IsMultiplicativeElement],
function(M,x)
return FixedStates(M,[x]);
end);
InstallMethod(FixedStates, "(FR) for a list of FR elements",
[IsFRMachine, IsMultiplicativeElementCollection],
function(M,L)
local states, alphabet, i, x, addstates, stateset;
states := [];
stateset := [];
alphabet := AlphabetOfFRObject(M);
addstates := function(x)
local i, o, t;
o := Output(M,x);
t := Transitions(M,x);
for i in [1..Length(alphabet)] do
if o[i]=alphabet[i] and not t[i] in stateset then
Add(states,t[i]);
AddSet(stateset,t[i]);
fi;
od;
end;
for x in L do addstates(x); od;
i := 1;
while i <= Length(states) do
addstates(states[i]);
i := i+1;
if RemInt(i,100)=0 then
Info(InfoFR, 2, "The fixed states contain at least ", states);
fi;
od;
return states;
end);
InstallMethod(LimitStates, "(FR) for an FR machine and element",
[IsFRMachine,IsMultiplicativeElement],
function(M,x)
return LimitStates(M,[x]);
end);
InstallMethod(LimitStates, "(FR) for an FR machine and a list of elements",
[IsFRMachine,IsMultiplicativeElementCollection],
function(M,L)
local s, d, S, oldS;
s := Set(States(M,L));
d := List(s,w->BlistList([1..Length(s)],List(DecompositionOfFRElement(w)[1],x->Position(s,x))));
S := BlistList([1..Length(s)],[1..Length(s)]);
repeat
oldS := S;
S := UnionBlist(ListBlist(d,S));
until oldS=S;
return ListBlist(s,S);
end);
InstallMethod(CoverNucleus, "(FR) for an FR machine",
[IsFRMachine],
function(M)
local s, news, olds, gens, g, h;
gens := Set(GeneratorsOfFRMachine(M));
news := gens;
s := [];
while true do
olds := ShallowCopy(s);
UniteSet(s,LimitStates(M,news));
if Length(s)=Length(olds) then
return s;
fi;
news := [];
for g in Difference(s,olds) do
if g in FixedStates(g) and Order(g)=infinity then
return fail;
fi;
for h in gens do AddSet(news,g*h); od;
od;
Info(InfoFR, 2, "Nucleus: The nucleus contains at least ",s);
od;
end);
#############################################################################
##
#M StructuralGroup(FRMachine)
#M StructuralSemigroup(FRMachine)
#M StructuralMonoid(FRMachine)
##
InstallMethod(StructuralGroup, "(FR) for a group FR machine",
[IsGroupFRMachine],
function(M)
local ggens, wgens, f, fggens, fwgens, phi;
ggens := GeneratorsOfFRMachine(M);
wgens := [1..Size(AlphabetOfFRObject(M))];
f := FreeGroup(Concatenation(List(ggens,String),List(wgens,String)));
fggens := GeneratorsOfGroup(f){[1..Length(ggens)]};
fwgens := GeneratorsOfGroup(f){[Length(ggens)+1..Length(ggens)+Length(wgens)]};
phi := GroupHomomorphismByImagesNC(StateSet(M),f,ggens,fggens);
return f / List(Cartesian([1..Length(ggens)],wgens),
p->fggens[p[1]]*fwgens[Output(M,p[1],p[2])]/Transition(M,p[1],p[2])^phi/fwgens[p[2]]);
end);
InstallMethod(StructuralMonoid, "(FR) for a monoid FR machine",
[IsMonoidFRMachine],
function(M)
local ggens, wgens, f, fggens, fwgens;
ggens := GeneratorsOfFRMachine(M);
wgens := [1..Size(AlphabetOfFRObject(M))];
f := FreeMonoid(Concatenation(List(ggens,String),List(wgens,String)));
fggens := GeneratorsOfMonoid(f){[1..Length(ggens)]};
fwgens := GeneratorsOfMonoid(f){[Length(ggens)+1..Length(ggens)+Length(wgens)]};
return f / List(Cartesian([1..Length(ggens)],wgens),
p->[fggens[p[1]]*fwgens[Output(M,p[1],p[2])],fwgens[p[2]]*MappedWord(Transition(M,p[1],p[2]),ggens,fggens)]);
end);
InstallMethod(StructuralSemigroup, "(FR) for a semigroup FR machine",
[IsSemigroupFRMachine],
function(M)
local ggens, wgens, f, fggens, fwgens;
ggens := GeneratorsOfFRMachine(M);
wgens := [1..Size(AlphabetOfFRObject(M))];
f := FreeSemigroup(Concatenation(List(ggens,String),List(wgens,String)));
fggens := GeneratorsOfSemigroup(f){[1..Length(ggens)]};
fwgens := GeneratorsOfSemigroup(f){[Length(ggens)+1..Length(ggens)+Length(wgens)]};
return f / List(Cartesian([1..Length(ggens)],wgens),
p->[fggens[p[1]]*fwgens[Output(M,p[1],p[2])],fwgens[p[2]]*MappedWord(Transition(M,p[1],p[2]),ggens,fggens)]);
end);
#############################################################################
#############################################################################
##
#M AsGroupFRMachine
#M AsMonoidFRMachine
#M AsSemigroupFRMachine
##
InstallMethod(AsGroupFRMachine, "(FR) for a group FR machine",
[IsGroupFRMachine],
function(M)
SetCorrespondence(M,IdentityMapping(M!.free));
return M;
end);
BindGlobal("ASGROUPFRMACHINE@", function(M)
local f, N, h, s;
if not ForAll(M!.output,ISINVERTIBLE@) then return fail; fi;
s := GeneratorsOfFRMachine(M);
f := FreeGroup(Length(s));
h := MagmaHomomorphismByImagesNC(M!.free,f,GeneratorsOfGroup(f));
N := FRMachineNC(FamilyObj(M),f,List(M!.transitions,r->List(r,w->w^h)),M!.output);
SetCorrespondence(N,h);
return N;
end);
InstallMethod(AsGroupFRMachine, "(FR) for a monoid FR machine",
[IsMonoidFRMachine],
ASGROUPFRMACHINE@);
InstallMethod(AsGroupFRMachine, "(FR) for a semigroup FR machine",
[IsSemigroupFRMachine],
ASGROUPFRMACHINE@);
InstallMethod(AsMonoidFRMachine, "(FR) for a group FR machine",
[IsGroupFRMachine],
function(M)
local f, N, h, sM, sN, sNinv, trans, out, o, i;
sM := GeneratorsOfGroup(M!.free);
f := FreeMonoid(2*Length(sM));
sN := GeneratorsOfMonoid(f){[1..Length(sM)]};
sNinv := GeneratorsOfMonoid(f){[Length(sM)+1..2*Length(sM)]};
h := MagmaHomomorphismByFunctionNC(M!.free,f,function(w)
local r, i;
r := [];
for i in LetterRepAssocWord(w) do
if i>0 then Add(r,i); else Add(r,Length(sM)-i); fi;
od;
return AssocWordByLetterRep(FamilyObj(sN[1]),r);
end);
trans := List(M!.transitions,r->List(r,w->w^h));
out := ShallowCopy(M!.output);
for i in [1..Length(M!.transitions)] do
o := INVERSE@(M!.output[i]);
Add(trans,List(M!.transitions[i],w->(w^-1)^h){o});
Add(out,o);
od;
N := FRMachineNC(FamilyObj(M),f,trans,out);
SetCorrespondence(N,h);
return N;
end);
InstallMethod(AsMonoidFRMachine, "(FR) for a monoid FR machine",
[IsMonoidFRMachine],
function(M)
SetCorrespondence(M,IdentityMapping(M!.free));
return M;
end);
InstallMethod(AsMonoidFRMachine, "(FR) for a semigroup FR machine",
[IsSemigroupFRMachine],
function(M)
local f, N, h, s;
s := GeneratorsOfSemigroup(M!.free);
f := FreeMonoid(Length(s));
h := MagmaHomomorphismByImagesNC(M!.free,f,GeneratorsOfMonoid(f));
N := FRMachineNC(FamilyObj(M),f,List(M!.transitions,r->List(r,w->w^h)),M!.output);
SetCorrespondence(N,h);
return N;
end);
InstallMethod(AsSemigroupFRMachine, "(FR) for a group FR machine",
[IsGroupFRMachine],
function(M)
local f, N, h, sM, sN, sNinv, one, trans, out, o, i;
sM := GeneratorsOfGroup(M!.free);
f := FreeSemigroup(2*Length(sM)+1);
sN := GeneratorsOfSemigroup(f){[1..Length(sM)]};
sNinv := GeneratorsOfSemigroup(f){[Length(sM)+1..2*Length(sM)]};
one := GeneratorsOfSemigroup(f)[2*Length(sM)+1];
h := MagmaHomomorphismByFunctionNC(M!.free,f,function(w)
local r, i;
r := [];
if IsOne(w) then
return one;
else
for i in LetterRepAssocWord(w) do
if i>0 then Add(r,i); else Add(r,Length(sM)-i); fi;
od;
return AssocWordByLetterRep(FamilyObj(one),r);
fi;
end);
trans := List(M!.transitions,r->List(r,w->w^h));
out := ShallowCopy(M!.output);
for i in [1..Length(M!.transitions)] do
o := INVERSE@(M!.output[i]);
Add(trans,List(M!.transitions[i],w->(w^-1)^h){o});
Add(out,o);
od;
Add(trans,List(AlphabetOfFRObject(M),a->one)); # add an identity state
Add(out,AlphabetOfFRObject(M));
N := FRMachineNC(FamilyObj(M),f,trans,out);
SetCorrespondence(N,h);
return N;
end);
InstallMethod(AsSemigroupFRMachine, "(FR) for a monoid FR machine",
[IsMonoidFRMachine],
function(M)
local f, N, h, sM, sN, one, trans, out, i;
sM := GeneratorsOfMonoid(M!.free);
f := FreeSemigroup(Length(sM)+1);
sN := GeneratorsOfSemigroup(f){[1..Length(sM)]};
one := GeneratorsOfSemigroup(f)[Length(sM)+1];
h := MagmaHomomorphismByFunctionNC(M!.free,f,w->MAPPEDWORD@(w,sN,one));
trans := List(M!.transitions,r->List(r,w->w^h));
out := ShallowCopy(M!.output);
Add(trans,List(AlphabetOfFRObject(M),a->one));
Add(out,AlphabetOfFRObject(M));
N := FRMachineNC(FamilyObj(M),f,trans,out);
SetCorrespondence(N,h);
return N;
end);
InstallMethod(AsSemigroupFRMachine, "(FR) for a semigroup FR machine",
[IsSemigroupFRMachine],
function(M)
SetCorrespondence(M,IdentityMapping(M!.free));
return M;
end);
BindGlobal("HOM2MACHINE@", function(f,tester,g)
local s;
s := Source(f);
if not tester(s) or s<>Range(f) then
return fail;
fi;
return FRMachineNC(FRMFamily([1]),s,List(g(s),x->[x^f]),List(g(s),x->[1]));
end);
InstallMethod(AsGroupFRMachine, "(FR) for a group homomorphism",
[IsGroupHomomorphism],
f->HOM2MACHINE@(f,IsFreeGroup,GeneratorsOfGroup));
InstallMethod(AsMonoidFRMachine, "(FR) for a monoid homomorphism",
[IsMagmaHomomorphism],
f->HOM2MACHINE@(f,IsFreeMonoid,GeneratorsOfMonoid));
InstallMethod(AsSemigroupFRMachine, "(FR) for a semigroup homomorphism",
[IsMagmaHomomorphism],
f->HOM2MACHINE@(f,IsFreeSemigroup,GeneratorsOfSemigroup));
#############################################################################
#############################################################################
##
#M Minimized(FRMachine)
##
BindGlobal("MINIMIZERWS_MAKERULES@", function(rws,p)
# p is a tuple [generators,inverses,isone?,rules]
# this command recomputes the rules
local i, l;
p[4] := [];
if p[3] then
for i in p[1] do
Add(p[4],[[i],[]]);
od;
else
l := p[1][Length(p[1])];
for i in [1..Length(p[1])-1] do
Add(p[4],[[p[1][i]],[l]]);
od;
if l in p[2] then
Add(p[4],[[l,l],[]]);
fi;
fi;
end);
BindGlobal("MINIMIZERWS@", function(M)
local rws, gens, h, i, j, si, p, part, newpart, changed;
rws := NewFRMachineRWS(M);
if IsBound(rws.partition) then
return rws;
fi;
if IsSemigroupFRMachine(M) then
gens := GeneratorsOfSemigroup(M!.free);
else
gens := GeneratorsOfMonoid(M!.free);
fi;
gens := Filtered(gens,x->rws.letterrep(x)=rws.reduce(rws.letterrep(x)));
i := List(gens,x->Output(M,x));
si := Set(i);
part := List(si,x->[[],[],HasIsBuiltFromMonoid(rws.rws) and IsBuiltFromMonoid(rws.rws) and ISONE@(x)]);
for j in [1..Length(i)] do
p := Position(si,i[j]);
Add(part[p][1],rws.letterrep(gens[j])[1]);
if IsGroupFRMachine(M) then
Add(part[p][2],rws.letterrep(gens[j]^-1)[1]);
else
Add(part[p][2],fail);
fi;
od;
for p in part do
SortParallel(p[1],p[2]);
MINIMIZERWS_MAKERULES@(rws,p);
od;
changed := true;
while changed do
#Info(InfoFR,1,"New parts: ",part);
changed := false;
rws.rws!.tzrules := Concatenation(rws.tzrules,Concatenation(List(part,p->p[4])));
for h in [1..Length(part)] do
i := List(part[h][1],x->List(rws.pi([x])[1],rws.reduce));
si := Set(i);
if Length(si)>1 then
changed := true;
newpart := List(si,x->[[],[],part[h][3] and ForAll(x,IsEmpty)]);
for j in [1..Length(i)] do
p := Position(si,i[j]);
Add(newpart[p][1],part[h][1][j]);
Add(newpart[p][2],part[h][2][j]);
od;
for p in newpart do
MINIMIZERWS_MAKERULES@(rws,p);
od;
Append(part,newpart);
part[h] := Remove(part);
break;
elif part[h][3] and not ForAll(si[1],IsEmpty) then
changed := true;
part[h][3] := false;
MINIMIZERWS_MAKERULES@(rws,part[h]);
break;
fi;
od;
od;
for p in part do
p[3] := p[3] and ForAll(p[1],x->ForAll(rws.pi([x])[1],x->rws.reduce(x)=[]));
MINIMIZERWS_MAKERULES@(rws,p);
od;
rws.rws!.tzrules := Concatenation(rws.tzrules,Concatenation(List(part,p->p[4])));
rws.modified := true;
rws.commit();
rws.partition := part;
return rws;
end);
InstallMethod(Minimized, "(FR) for a group/monoid/semigroup FR machine",
[IsFRMachine and IsFRMachineStdRep],
function(M)
local rws, gens, gensimg, i, ri, red, free, freegens, one, out, trans, map;
rws := MINIMIZERWS@(M);
gens := GeneratorsOfFRMachine(M);
i := List(gens,rws.letterrep);
red := Filtered(i,x->rws.reduce(x)=x);
if i=red then
M := COPYFRMACHINE@(M);
SetCorrespondence(M,IdentityMapping(M!.free));
return M;
fi;
if IsGroupFRMachine(M) then
free := FreeGroup(Length(red));
freegens := GeneratorsOfGroup(free);
one := One(free);
elif IsMonoidFRMachine(M) then
free := FreeMonoid(Length(red));
freegens := GeneratorsOfMonoid(free);
one := One(free);
elif IsSemigroupFRMachine(M) then
free := FreeSemigroup(Length(red));
freegens := GeneratorsOfSemigroup(free);
one := fail;
fi;
gensimg := [];
for i in gens do
ri := rws.reduce(rws.letterrep(i));
if ri=[] then
Add(gensimg,One(free));
elif ri in red then
Add(gensimg,freegens[Position(red,ri)]);
else
Add(gensimg,freegens[Position(red,rws.reduce(rws.letterrep(i^-1)))]^-1);
fi;
od;
map := MagmaHomomorphismByImagesNC(M!.free,free,gensimg);
i := List(red,rws.pi);
out := List(i,p->p[2]);
trans := [];
for i in i do
Add(trans,List(i[1],w->rws.letterunrep(rws.reduce(w))^map));
od;
i := FRMachineNC(FamilyObj(M),free,trans,out);
SetCorrespondence(i,map);
return i;
end);
#############################################################################
#############################################################################
##
#M SubFRMachine(FRMachine,FRMachine)
##
InstallMethod(SubFRMachine, "(FR) for two group/monoid/semigroup FR machines",
[IsFRMachine and IsFRMachineStdRep,IsFRMachine and IsFRMachineStdRep],
function(M,N)
local rws, S, Mgens, Ngens, Mletter, Nred;
if AlphabetOfFRObject(M)<>AlphabetOfFRObject(N) then
return fail;
elif IsIdenticalObj(M,N) then
return IdentityMapping(M!.free);
elif M=N then
if IsGroupFRMachine(M) then
return GroupHomomorphismByImages(N!.free,M!.free,GeneratorsOfGroup(N!.free),GeneratorsOfGroup(M!.free));
else
return MagmaHomomorphismByImagesNC(N!.free,M!.free,GeneratorsOfFRMachine(M));
fi;
fi;
if (IsGroupFRMachine(N) and not IsGroupFRMachine(M)) or (IsMonoidFRMachine(N) and IsSemigroupFRMachine(M)) then
return fail;
fi;
S := FRMMINSUM@(N,M);
rws := MINIMIZERWS@(S);
Mgens := GeneratorsOfSemigroup(M!.free);
Ngens := GeneratorsOfFRMachine(N);
Mletter := List(Mgens,x->rws.letterrep(x^Correspondence(S)[2]));
Nred := List(Ngens,x->rws.reduce(rws.letterrep(x^Correspondence(S)[1])));
if IsSubset(Mletter,Nred) then
Mgens := List(Nred,x->Mgens[Position(Mletter,x)]);
return MagmaHomomorphismByImagesNC(N!.free,M!.free,Mgens);
else
return fail;
fi;
end);
InstallMethod(SubFRMachine, "(FR) for a machine and a homomorphism",
[IsFRMachine and IsFRMachineStdRep, IsMapping],
function(M,f)
local S, trans, out, i, pi, x;
S := StateSet(M);
while S<>Range(f) do
Error("SubFRMachine: range and stateset must be the same\n");
od;
while not IsFreeGroup(Source(f)) do
Error("SubFRMachine: source must be a free group\n");
od;
pi := WreathRecursion(M);
trans := [];
out := [];
for i in GeneratorsOfGroup(Source(f)) do
x := pi(i^f);
x[1] := List(x[1],g->PreImagesRepresentativeNC(f,g));
if fail in x[1] then return fail; fi;
Add(trans,x[1]);
Add(out,x[2]);
od;
x := FRMachineNC(FamilyObj(M),Source(f),trans,out);
if HasAddingElement(M) then
i := PreImagesRepresentativeNC(f,InitialState(AddingElement(M)));
if i<>fail then
SetAddingElement(x,FRElement(x,i));
fi;
fi;
return x;
end);
#############################################################################
################################################################
# change basis of FR machine
BindGlobal("CHANGEFRMACHINEBASIS@", function(M,l,p)
local trans, i, d, newM;
d := Size(AlphabetOfFRObject(M));
while Length(l)<>d or not ForAll(l,x->x in StateSet(M)) do
Error("Invalid base change ",l,"\n");
od;
while LargestMovedPoint(p)>d do
Error("Invalid permutation ",p,"\n");
od;
trans := [];
for i in [1..Length(M!.transitions)] do
Add(trans,Permuted(List(AlphabetOfFRObject(M),a->l[a]^-1*M!.transitions[i][a]*l[M!.output[i][a]]),p));
od;
newM := FRMachineNC(FamilyObj(M),StateSet(M),trans,List(M!.output,r->ListPerm(PermList(r)^p,d)));
return newM;
end);
InstallMethod(ChangeFRMachineBasis, "(FR) for a group FR machine and a list",
[IsGroupFRMachine, IsCollection],
function(M,l)
return ChangeFRMachineBasis(M,l,());
end);
InstallMethod(ChangeFRMachineBasis, "(FR) for a group FR machine and a permutation",
[IsGroupFRMachine, IsPerm],
function(M,p)
return ChangeFRMachineBasis(M,List(AlphabetOfFRObject(M),x->One(StateSet(M))),p);
end);
InstallMethod(ChangeFRMachineBasis, "(FR) for a group FR machine, a list and a permutation",
[IsGroupFRMachine, IsCollection, IsPerm],
CHANGEFRMACHINEBASIS@);
InstallMethod(ChangeFRMachineBasis, "(FR) for an FR machine",
[IsGroupFRMachine],
function(M)
local cycles, basis, s, t, u, v;
# gather all permutation cycles
cycles := [];
for s in GeneratorsOfFRMachine(M) do
for t in Cycles(PermList(Output(M,s)),AlphabetOfFRObject(M)) do
if Length(t)>1 then
Add(cycles,[s,t]);
fi;
od;
od;
basis := [];
while cycles<>[] do
# first cycle connected to the partial basis
t := First([1..Length(cycles)],i->Number(cycles[i][2],i->IsBound(basis[i]))>0);
if t=fail then
# set up an anchor on the cycle
basis[First(AlphabetOfFRObject(M),i->not IsBound(basis[i]))] := One(StateSet(M));
continue;
fi;
t := Remove(cycles,t);
# anchor on the cycle
s := First(t[2],i->IsBound(basis[i]));
u := s;
repeat
v := Output(M,t[1],u);
if not IsBound(basis[v]) then
basis[v] := LeftQuotient(Transition(M,t[1],u),basis[u]);
fi;
u := v;
until u=s;
od;
return ChangeFRMachineBasis(M,basis,());
end);
BindGlobal("RIGHTACTMACHINE@", function(M,f)
local S;
S := StateSet(M);
if S<>Source(f) or S<>Range(f) then
Error("\*: source, range and stateset must be the same\n");
fi;
return FRMachineNC(FamilyObj(M),S,List(M!.transitions,r->List(r,x->x^f)),M!.output);
end);
InstallMethod(\*, "(FR) for an FR machine and a mapping",
[IsFRMachine and IsFRMachineStdRep, IsMapping],
RIGHTACTMACHINE@);
BindGlobal("LEFTACTMACHINE@", function(f,M)
local S, trans, out, i, pi, x;
S := StateSet(M);
if S<>Source(f) or S<>Range(f) then
Error("\*: source, range and stateset must be the same\n");
fi;
pi := WreathRecursion(M);
trans := [];
out := [];
for i in [1..Length(M!.output)] do
x := pi(GeneratorsOfFRMachine(M)[i]^f);
Add(trans,x[1]);
Add(out,x[2]);
od;
return FRMachineNC(FamilyObj(M),S,trans,out);
end);
InstallMethod(\*, "(FR) for a mapping and an FR machine",
[IsMapping, IsFRMachine and IsFRMachineStdRep],
LEFTACTMACHINE@);
BindGlobal("CONJACTMACHINE@", function(M,f)
local S, newS, trans, out, i, pi, x, finv;
S := StateSet(M);
if S<>Source(f) then
Error("\^: source and stateset must be the same\n");
fi;
newS := Range(f);
pi := WreathRecursion(M);
trans := [];
out := [];
finv := InverseGeneralMapping(f);
if finv=fail then return fail; fi;
for i in GeneratorsOfGroup(newS) do
x := pi(ImagesRepresentative(finv,i));
Add(trans,List(x[1],x->ImagesRepresentative(f,x)));
Add(out,x[2]);
od;
return FRMachineNC(FamilyObj(M),newS,trans,out);
end);
InstallMethod(\^, "(FR) for a group FR machine and a mapping",
[IsFRMachine and IsFRMachineStdRep, IsMapping],
CONJACTMACHINE@);
################################################################
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