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|
#############################################################################
##
#W dt.g GAP library Wolfgang Merkwitz
##
##
#Y Copyright (C) 1996, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file initializes Deep Thought.
##
## Deep Thought deals with trees. A tree < tree > is a concatenation of
## several nodes where each node is a 5-tuple of immediate integers. If
## < tree > is an atom it contains only one node, thus it is itself a
## 5-tuple. If < tree > is not an atom we obtain its list representation by
##
## < tree > := topnode(<tree>) concat left(<tree>) concat right(<tree>) .
##
## Let us denote the i-th node of <tree> by (<tree>, i) and the tree rooted
## at (<tree>, i) by tree(<tree>, i). Let <a> be tree(<tree>, i)
## The first entry of (<tree>, i) is pos(a),
## and the second entry is num(a). The third entry of (<tree>, i) gives a
## mark.(<tree>, i)[3] = 1 means that (<tree>, i) is marked,
## (<tree>, i)[3] = 0 means that (<tree>, i) is not marked. The fourth entry
## of (<tree>, i) contains the number of knodes of tree(<tree>, i). The
## fifth entry of (<tree>, i) finally contains a boundary for
## pos( tree(<tree>, i) ). (<tree>, i)[5] <= 0 means that
## pos( tree(<tree>, i) ) is unbounded. If tree(<tree>, i) is an atom we
## already know that pos( tree(<tree>, i) ) is unbound. Thus we then can
## use the fifth component of (<tree>, i) to store the side. In this case
## (<tree>, i)[5] = -1 means that tree(<tree>, i) is an atom from the
## right hand word, and (<tree>, i)[5] = -2 means that tree(<tree>, i) is
## an atom from the left hand word.
##
## A second important data structure deep thought deals with is a deep
## thought monomial. A deep thought monomial g_<tree> is a product of
## binomial coefficients with a coefficient c. Deep thought monomials
## are represented in this implementation by formula
## vectors, which are lists of integers. The first entry of a formula
## vector is 0, to distinguish formula vectors from trees. The second
## entry is the coefficient c, and the third and fourth entries are
## num( left(tree) ) and num( right(tree) ). The remaining part of the
## formula vector is a concatenation of pairs of integers. A pair (i, j)
## with i > 0 represents binomial(x_i, j). A pair (0, j) represents
## binomial(y_gen, j) when word*gen^power is calculated.
##
## Finally deep thought has to deal with pseudorepresentatives. A
## pseudorepresentative <a> is stored in list of length 4. The first entry
## stores left( <a> ), the second entry contains right( <a> ), the third
## entry contains num( <a> ) and the last entry finally gives a boundary
## for pos( <b> ) for all trees <b> which are represented by <a>.
##
#############################################################################
##
#F Dt_Mkavec(<pr>)
##
## 'Dt_Mkavec' returns the avec for the pc-presentation <pr>.
##
BindGlobal( "Dt_Mkavec", function(pr)
local i,j,vec;
vec := [];
vec[Length(pr)] := 1;
for i in [Length(pr)-1,Length(pr)-2..1] do
j := Length(pr);
while j >= 1 do
if j = vec[i+1] then
vec[i] := j;
j := 0;
else
j := j-1;
if j < i and IsBound(pr[i][j]) then
vec[i] := j+1;
j := 0;
fi;
if j > i and IsBound(pr[j][i]) then
vec[i] := j+1;
j := 0;
fi;
fi;
od;
od;
for i in [1..Length(pr)] do
if vec[i] < i+1 then
vec[i] := i+1;
fi;
od;
return vec;
end );
#############################################################################
##
#F Dt_IsEqualMonomial(<vec1>, <vec2>) . . . . . . test if <vec1> and <vec2>
## represent the same monomial
##
## 'Dt_IsEqualMonomial' returns "true" if <vec1> and <vec2> represent the
## same monomial, and "false" otherwise.
BindGlobal( "Dt_IsEqualMonomial", function(vec1,vec2)
local i,j;
if Length(vec1) <> Length(vec2) then
return false;
fi;
# Since the first four entries of a formula vector doesn't contain
# any information about the monomial it represents, it suffices to
# compare the remaining entries.
for i in [5..Length(vec1)] do
if not vec1[i] = vec2[i] then
return false;
fi;
od;
return true;
end );
#############################################################################
##
#F Dt_Sort2(<vector>) . . . . . . . . . . . . . . . . sort a formula vector
##
## 'Dt_Sort2' sorts the pairs of integers in the formula vector <vector>
## representing the binomial coefficients such that
## <vector>[5] < <vector>[7] < .. < vector[m-1], where m is the length
## of <vector>. This is done for a easier comparison of formula vectors.
##
BindGlobal( "Dt_Sort2", function(vector)
local i,list1,list2;
list1 := vector{[5,7..Length(vector)-1]};
list2 := vector{[6,8..Length(vector)]};
SortParallel(list1,list2);
for i in [1..Length(list1)] do
vector[ 2*i+3 ] := list1[i];
vector[ 2*i+4 ] := list2[i];
od;
end );
#############################################################################
##
#F Dt_AddVecToList(<evlist>,<evlistvec>,<formvec>,<pr>)
##
## 'Dt_AddVecToList' adds the formula vector <formvec> to the list <evlist>,
## computes the corresponding coefficient vector and adds the latter to
## the list <evlistvec>.
##
BindGlobal( "Dt_AddVecToList", function(evlist, evlistvec, formvec, pr)
local i,j,k;
Add(evlist, formvec);
k := [];
for i in [1..Length(pr)] do
k[i] := 0;
od;
j := pr[ formvec[3] ][ formvec[4] ];
# the coefficient that the monomial represented by <formvec> has
# in each polynomial f_l is obtained by multiplying <formvec>[2]
# with the exponent which the group generator g_l has in the
# in the word representing the commutator of g_(formvec[3]) and
# g_(formvec[4]) in the presentation <pr>.
for i in [3,5..Length(j)-1] do
k[ j[i] ] := formvec[2]*j[i+1];
od;
Add(evlistvec, k);
end );
#############################################################################
##
#F Dt_add( <pol>, <pols>, <pr> )
##
## 'Dt_add' adds the deep thought monomial <pol> to the list of polynomials
## <pols>, such that afterwards <pols> represents a simplified polynomial.
##
BindGlobal( "Dt_add", function(pol, pols, pr)
local i,j,k,rel, pos, flag;
# first sort the deep thought monomial <pol> to compare it with the
# monomials contained in <pols>.
Dt_Sort2(pol);
# then look which component of <pols> contains <pol> in case that
# <pol> is contained in <pols>.
pos := DT_evaluation(pol);
if not IsBound( pols[pos] ) then
# create the component <pols>[<pos>] and add <pol> to it
pols[pos] := rec( evlist := [], evlistvec := [] );
Dt_AddVecToList( pols[pos].evlist, pols[pos].evlistvec, pol, pr );
return;
fi;
flag := 0;
for k in [1..Length( pols[pos].evlist ) ] do
# look for <pol> in <pols>[<pos>] and if <pol> is contained in
# <pols>[<pos>] then adjust the corresponding coefficient vector.
if Dt_IsEqualMonomial( pol, pols[pos].evlist[k] ) then
rel := pr[ pol[3] ][ pol[4] ];
for j in [3,5..Length(rel)-1] do
pols[pos].evlistvec[k][ rel[j] ] :=
pols[pos].evlistvec[k][ rel[j] ] + pol[2]*rel[j+1];
od;
flag := 1;
break;
fi;
od;
if flag = 0 then
# <pol> is not contained in <pols>[<pos>] so add it to <pols>[<pos>]
Dt_AddVecToList(pols[pos].evlist, pols[pos].evlistvec, pol, pr);
fi;
end );
#############################################################################
##
#F Dt_Convert(<sortedpols>)
##
## 'Dt_Convert' converts the list of formula vectors <sortedpols>. Before
## applying <Dt_Convert> <sortedpols> is a list of records with the
## components <evlist> and <evlistvec> where <evlist> contains deep thought
## monomials and <evlistvec> contains the corresponding coefficient vectors.
## <Dt_Convert> merges the <evlist>-compondents of the records contained
## in <sortedpols> into one component <evlist> and the <evlistvec>-coponents
## into one component <evlistvec>.
##
BindGlobal( "Dt_Convert", function(sortedpols)
local k,res;
if Length(sortedpols) = 0 then
return 0;
fi;
res := rec(evlist := [],
evlistvec :=[]);
for k in sortedpols do
Append(res.evlist, k.evlist);
Append(res.evlistvec, k.evlistvec);
od;
return res;
end );
#############################################################################
##
#F Dt_ConvertCoeffVecs(<evlistvec>)
##
## 'Dt_ConvertCoeffVecs' converts the coefficient vectors in the list
## <evlistvec>. Before applying <Dt_ConvertCoeffVecs>, an entry
## <evlistvec>[i][j] = k means that the deep thought monomial <evlist>[i]
## occurs in the polynomial f_j with coefficient k. After applying
## <Dt_ConvertCoeffVecs> a pair [j, k] occuring in <eclistvec>[i] means that
## <evlist>[i] occurs in f_j with coefficient k.
##
BindGlobal( "Dt_ConvertCoeffVecs", function(evlistvec, pr)
local i,j,res;
for i in [1..Length(evlistvec)] do
res := [];
for j in [1..Length(evlistvec[i])] do
if evlistvec[i][j] <> 0 then
Append(res, [j, evlistvec[i][j] ]);
fi;
od;
evlistvec[i] := res;
od;
end );
#############################################################################
##
#F CalcOrder( <word>, <dtrws> )
##
## CalcOrder computes the order of the word <word> in the group determined
## by the rewriting system <dtrws>
##
CalcOrder := function(word, dtrws)
local gcd, m, pcp;
if Length(word) = 0 then
return 1;
fi;
if not IsBound(dtrws![PC_EXPONENTS][ word[1] ]) then
return 0;
fi;
gcd := Gcd(dtrws![PC_EXPONENTS][ word[1] ], word[2]);
m := QuoInt( dtrws![PC_EXPONENTS][ word[1] ], gcd);
gcd := DTPower(word, m, dtrws);
return m*CalcOrder(gcd, dtrws);
end;
MakeReadOnlyGlobal( "CalcOrder" );
#############################################################################
##
#F CompleteOrdersOfRws( <dtrws> )
##
## CompleteOrdersOfRws computes the orders of the generators of the
## deep thought rewriting system <dtrws>
##
BindGlobal( "CompleteOrdersOfRws", function(dtrws)
local i,j;
dtrws![PC_ORDERS] := [];
for i in [dtrws![PC_NUMBER_OF_GENERATORS],dtrws![PC_NUMBER_OF_GENERATORS]-1..1]
do
# Print("determining order of generator ",i,"\n");
if not IsBound( dtrws![PC_EXPONENTS][i] ) then
j := 0;
elif not IsBound( dtrws![PC_POWERS][i] ) then
j := dtrws![PC_EXPONENTS][i];
else
j := dtrws![PC_EXPONENTS][i]*CalcOrder(dtrws![PC_POWERS][i], dtrws);
fi;
if j <> 0 then
dtrws![PC_ORDERS][i] := j;
fi;
od;
end );
#############################################################################
##
#F Dt_RemoveHoles( <list> )
##
## Dt_RemoveHoles removes all empty entries from <list>
## It is similar to Compacted(), but works in-place
##
BindGlobal( "Dt_RemoveHoles", function( list )
local skip, i;
skip := 0;
i := 1;
while i <= Length(list) do
while not IsBound(list[i]) do
skip := skip + 1;
i := i+1;
od;
list[i-skip] := list[i];
i := i+1;
od;
for i in [Length(list)-skip+1..Length(list)] do
Unbind(list[i]);
od;
end );
#############################################################################
##
#F ReduceCoefficientsOfRws( <dtrws> )
##
## ReduceCoefficientsOfRws reduces all coefficients of each deep thought
## polynomial f_l modulo the order of the l-th generator.
##
BindGlobal( "ReduceCoefficientsOfRws", function(dtrws)
local i,j,k,l, pseudoreps;
pseudoreps := dtrws![PC_DEEP_THOUGHT_POLS];
i := 1;
while IsRecord(pseudoreps[i]) do
for j in [1..Length(pseudoreps[i].evlistvec)] do
for k in [2,4..Length(pseudoreps[i].evlistvec[j])] do
if IsBound( dtrws![PC_ORDERS][ pseudoreps[i].evlistvec[j][k-1] ] )
and (pseudoreps[i].evlistvec[j][k] > 0 or
pseudoreps[i].evlistvec[j][k] <
-dtrws![PC_ORDERS][ pseudoreps[i].evlistvec[j][k-1] ]/2)
then
pseudoreps[i].evlistvec[j][k] :=
pseudoreps[i].evlistvec[j][k] mod
dtrws![PC_ORDERS][ pseudoreps[i].evlistvec[j][k-1] ];
fi;
od;
DTCompress( pseudoreps[i].evlistvec[j] );
if Length( pseudoreps[i].evlistvec[j] ) = 0 then
Unbind( pseudoreps[i].evlistvec[j] );
Unbind( pseudoreps[i].evlist[j] );
fi;
od;
Dt_RemoveHoles( pseudoreps[i].evlistvec );
Dt_RemoveHoles( pseudoreps[i].evlist );
i := i+1;
od;
end );
#############################################################################
##
## Dt_GetMax( <tree>, <number>, <pr> )
##
## Dt_GetMax returns the maximal value for pos(tree) if num(tree) = <number>.
##
BindGlobal( "Dt_GetMax", function(tree, number, pr)
local rel, max, position;
if Length(tree) = 5 then
return tree[5];
else
if Length(tree) = 4 then
if Length(tree[1]) = 4 then
if Length(tree[2]) = 4 then
rel := pr[ tree[1][3] ][ tree[2][3] ];
else
rel := pr[ tree[1][3] ][ tree[2][2] ];
fi;
else
if Length(tree[2]) = 4 then
rel := pr[ tree[1][2] ][ tree[2][3] ];
else
rel := pr[ tree[1][2] ][ tree[2][2] ];
fi;
fi;
else
rel := pr[ tree[7] ][ tree[ 5*(tree[9]+1)+2 ] ];
fi;
position := Position(rel, number) + 1;
if rel[position] < 0 or rel[position] > 100 then
return 0;
else
return rel[position];
fi;
fi;
end );
#############################################################################
##
#F Dt_GetNumRight( <tree> )
##
## Dt_GetNumRight returns num( right( tree ) ).
##
BindGlobal( "Dt_GetNumRight", function(tree)
if Length(tree) <> 4 then
return tree[ 5*(tree[9]+1)+2 ];
fi;
if Length(tree[2]) <> 4 then
return tree[2][2];
fi;
return tree[2][3];
end );
###########################################################################
##
#F Calcrepsn(<n>, <avec>, <pr>, <max>
##
## 'Calcrepsn' returns the polynomials f_{n1}1,..,f_{nm} which have to be
## evaluated when computing word*g_n^(y_n). Here m denotes the composition
## length of the nilpotent group G given by the presentation <pr>. This is
## done by first calculating a complete sytem of <n>-pseudorepresentatives
## for the presentation <pr> with bondary <max>. Then this sytem is used
## to get the required polynomials
##
## If g_n is in the center of the group determined by the presentation <pr>
## then there don't exist any representatives exept for the atoms and
## finally 0 will be returned.
##
BindGlobal( "Calcrepsn", function(n, avec, pr, max)
local i,j,k,l, # loop variables
x,y,z,a,b,c, # trees
reps, # list of pseudorepresentatives
pols, # stores the dt polynomials
boundary, # boundary for loop
hilf,
pos,
start,
max1, max2; # maximal values for pos(x) and pos(y)
reps:=[];
pols := [];
for i in [n..Length(pr)] do
# initialize reps[i] to contain representatives for the atoms
if i <> n then
reps[i] := [ [1,i,0,1,-2] ];
else
reps[i] := [ [1,i,0,1,-1] ];
fi;
od;
# first compute the pseudorepresentatives which are also represenatives
for i in [n..max] do
if i < avec[n] then
boundary := i-1;
else
boundary := avec[n]-1;
fi;
# to get the representatives of the non-atoms loop over j and k
# and determine the representatives for all trees <z> with
# num(<z>) = i, num( left(<z>) ) = j, num( right(<z>) ) = k
# Since for all 1 <= l <= m the group generated by
# {g_(avec[l]),..,g_m} is in the center of the group generated
# by {g_l,..,g_m} it suffices to loop over all
# j <= min(i-1, avec[n]-1). Also it is sufficient only to loop over
# k while avec[k] is bigger than j.
for j in [n+1..boundary] do
k := n;
while k <= j-1 and avec[k] > j do
if IsBound(pr[j][k]) and pr[j][k][3] = i then
if k = n then
start := 1;
else
start := 2;
fi;
for x in [start..Length(reps[j])] do
for y in reps[k] do
if Length(reps[j][x]) = 5
or k >= reps[j][x][ 5*(reps[j][x][9]+1)+2 ]
then
max1 := Dt_GetMax(reps[j][x], j, pr);
max2 := Dt_GetMax(y, k, pr);
z := [1,i,0, reps[j][x][4]+y[4]+1, 0];
Append(z,reps[j][x]);
Append(z,y);
z[7] := j;
z[10] := max1;
z[ 5*(z[9]+1)+2 ] := k;
z[ 5*(z[9]+1)+5 ] := max2;
UnmarkTree(z);
# now get all representatives <z'> with
# left(<z'>) = left(<z>) ( = <x> ) and
# right(<z'>)=right(<z>) ( = <y> ) and
# num(<z'>) = o where o is an integer
# contained in pr[j][k].
FindNewReps(z, reps, pr, avec[n]-1);
fi;
od;
od;
fi;
k := k+1;
od;
od;
od;
# now get the "real" pseudorepresentatives
for i in [max+1..Length(pr)] do
if i < avec[n] then
boundary := i-1;
else
boundary := avec[n]-1;
fi;
for j in [n+1..boundary] do
k := n;
while k <= j-1 and avec[k] > j do
if IsBound(pr[j][k]) and pr[j][k][3] = i then
if k = n then
start := 1;
else
start := 2;
fi;
for x in [start..Length(reps[j])] do
for y in reps[k] do
if Length(reps[j][x]) = 5
or k >= Dt_GetNumRight(reps[j][x]) then
# since reps[j] and reps[k] may contain
# pseudorepresentatives which are trees
# as well as "real" pseudorepresentatives
# it is necessary to take several cases into
# consideration.
max1 := Dt_GetMax(reps[j][x], j, pr);
max2 := Dt_GetMax(y, k, pr);
if Length(reps[j][x]) <> 4 then
if reps[j][x][2] <> j then
# we have to ensure that
# num( <reps>[j][x] ) = j when we
# construct a new pseudorepresentative
# out of it.
a := ShallowCopy(reps[j][x]);
a[2] := j;
else
a := reps[j][x];
fi;
a[5] := max1;
else
if reps[j][x][3] <> j then
# we have to ensure that
# num( <reps>[j][x] ) = j when we
# construct a new pseudorepresentative
# out of it.
a := ShallowCopy(reps[j][x]);
a[3] := j;
else
a := reps[j][x];
fi;
a[4] := max1;
fi;
if Length(y) <> 4 then
if y[2] <> k then
# we have to ensure that num(<y>) = k
# when we construct a new
# pseudorepresentative out of it.
b := ShallowCopy(y);
b[2] := k;
else
b := y;
fi;
b[5] := max2;
else
if y[3] <> k then
# we have to ensure that num(<y>) = k
# when we construct a new
# pseudorepresentative out of it.
b := ShallowCopy(y);
b[3] := k;
else
b := y;
fi;
b[4] := max2;
fi;
# now finally construct the
# pseudorepresentative and add it to
# reps
z := [a, b, i, 0];
if i >= avec[n] then
Add(reps[i], z);
else
l := 3;
while l <= Length(pr[j][k]) and
pr[j][k][l] < avec[n] do
Add(reps[ pr[j][k][l] ], z);
l := l+2;
od;
fi;
fi;
od;
od;
fi;
k := k+1;
od;
od;
od;
# now use the pseudorepresentatives to get the desired polynomials
for i in [n..Length(pr)] do
for j in [2..Length(reps[i])] do
# the first case: reps[i][j] is a "real" pseudorepresentative
if Length(reps[i][j]) = 4 then
if reps[i][j][3] = i then
GetPols(reps[i][j], pr, pols);
fi;
# the second case: reps[i][j] is a tree
elif reps[i][j][1] <> 0 then
if reps[i][j][2] = i then
UnmarkTree(reps[i][j]);
hilf := MakeFormulaVector(reps[i][j], pr);
Dt_add(hilf, pols, pr);
fi;
# the third case: reps[i][j] is a deep thought monomial
else
Dt_add(reps[i][j], pols, pr);
fi;
od;
Unbind(reps[i]);
od;
# finally convert the polynomials to the final state
pols := Dt_Convert(pols);
if pols <> 0 then
Dt_ConvertCoeffVecs(pols.evlistvec, pr);
fi;
return(pols);
end );
#############################################################################
##
#F Calcreps2( <pr> ) . . . . . . . . . . compute the Deep-Thought-polynomials
##
## 'Calcreps2' returns the polynomials which have to be evaluated when
## computing word*g_n^(y_n) for all <dtbound> <= n <= m where m is the
## number of generators in the given presentation <pr>.
##
BindGlobal( "Calcreps2", function(pr, max, dtbound)
local i,reps,avec,max2, max1;
reps := [];
avec := Dt_Mkavec(pr);
if max >= Length(pr) then
max1 := Length(pr);
else
max1 := max;
fi;
for i in [dtbound..Length(pr)] do
if i >= max1 then
max1 := Length(pr);
fi;
reps[i] := Calcrepsn(i, avec, pr, max1);
od;
max2 := 1;
for i in [1..Length(reps)] do
if IsRecord(reps[i]) then
max2 := i;
fi;
od;
for i in [1..max2] do
if not IsRecord(reps[i]) then
reps[i] := 1;
fi;
od;
return reps;
end );
#############################################################################
##
#E dt.g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
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