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##
#W stbcrand.gi GAP library Ákos Seress
##
##
#Y Copyright (C) 1994, 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 contains the functions for a random Schreier-Sims algorithm
## with verification.
##
#############################################################################
##
#F StabChainRandomPermGroup( <gens>, <id>, <opts> ) . random Schreier-Sims
##
## The method consists of 2 phases: a heuristic construction and
## either a deterministic or two random checking phases.
## In the random checking phases, we take random elements of
## created set, multiply by random subproduct of generators, and sift down
## by dividing with the appropriate coset representatives. The stabchain is
## correct when all siftees are (). In the first checking phase, all
## computations are carried out with words in strong generators and siftees
## are checked by plugging in random points in words; in the second checking
## phase, permutations are multiplied.
##
## During the construction, we create new records for stabilizers:
## S.aux = set of strong generators for S, temporarily created during the
## construction. On the other hand, S.generators contains a strong
## generating set for S which would have been created by the
## deterministic method
## S.treegen = elements of S.aux used in S.transversal
## S.treegeninv = inverses of S.treegen; also used in S.transversal,
## and they are also elements of S.aux
## S.treedepth = depth of Schreier tree of S
## S.diam = sum of treedepths in stabilizer chain of S
##
## After the stabilizer chain is ready, the extra records are deleted.
## Transversals are rebuilt using the generators in the .generators records.
##
InstallGlobalFunction( StabChainRandomPermGroup, function( gens, id, options)
local S, # stabilizer chain
degree, # degree of S
givenbase,# list of points from which first base points should come
correct, # boolean; true if a correct base is given
size, # size of <G> as constructed
order, # size of G if given in input
limit, # upper bound on Size(G) given in input
orbits, # list of orbits of G
orbits2, # list of orbits of G
k, # number of pairs of generators checked
param, # list of parameters guiding number of repetitions
# in random constructions
where, # list indicating which orbit contains points in domain
basesize, # list; i^th entry = number of base points in orbits[i]
i,j,
ready, # boolean; true if stabilizer chain ready
warning, # used at warning if given and computed size differ
new, # list of permutations to be added to stab. chain
result, # output of checking phase; nontrivial if stabilizer
# chain is incorrect
base, # ordering of domain from which base points are taken
missing, # if a correct base was provided by input, missing
# contains those points of it which are not in
# constructed base
T; # a stabilizer chain containing the usual records
S:= rec( generators := ShallowCopy( gens ), identity := id );
if options.random = 1000 then
#case of deterministic computation with known size
k := 1;
else
k:=First([1..14],x->(3/5)^x<1-options.random/1000);
fi;
degree := LargestMovedPoint( S.generators );
if IsBound( options.knownBase) and
Length(options.knownBase)<4+LogInt(degree,10) then
param:=[k,4,0,0,0,0];
else
param:=[QuoInt(k,2),4,QuoInt(k+1,2),4,50,5];
options:=ShallowCopy(options);
Unbind(options.knownBase);
fi;
if options.random <= 200 then
param[2] := 2;
param[4] := 2;
fi;
#param[1] = number of pairs of random subproducts from generators in
# first checking phase
#param[2] = (number of random elements from created set)/S.diam
# in first checking phase
#param[3] = number of pairs of random subproducts from generators in
# second checking phase
#param[4] = (number of random elements from created set)/S.diam
# in second checking phase
#param[5] = maximum size of orbits in which we evaluate words on all
# points of orbit
#param[6] = minimum number of random points from orbit to plug in to check
# whether given word is identity on orbit
# prepare input of construction
if IsBound(options.base) then
givenbase := options.base;
else
givenbase := [];
fi;
if IsBound(options.size) then
order := options.size;
warning := 0;
limit := 0;
else
order := 0;
if IsBound(options.limit) then
limit := options.limit;
else
limit := 0;
fi;
fi;
if IsBound( options.knownBase ) then
correct := true;
else
correct := false;
fi;
if correct then
# if correct base was given as input, no need for orbit information
base:=Concatenation(givenbase,Difference(options.knownBase,givenbase));
missing := Set(options.knownBase);
basesize := [];
where := [];
orbits := [];
else
# create ordering of domain used in choosing base points and
# compute orbit information
base:=Concatenation(givenbase,Difference([1..degree],givenbase));
missing:=[];
orbits2:=OrbitsPerms(S.generators,[1..degree]);
#throw away one-element orbits
orbits:=[];
j:=0;
for i in [1..Length(orbits2)] do
if Length(orbits2[i]) >1 then
j:=j+1; orbits[j]:= orbits2[i];
fi;
od;
basesize:=[];
where:=[];
for i in [1..Length(orbits)] do
basesize[i]:=0;
for j in [1..Length(orbits[i])] do
where[orbits[i][j]]:=i;
od;
od;
# temporary solution to speed up of handling of lots of small orbits
# until compiler
if Length(orbits) > degree/40 then
param[1] := 0;
param[3] := k;
fi;
fi;
ready:=false;
new:=S.generators;
while not ready do
SCRMakeStabStrong
(S,new,param,orbits,where,basesize,base,correct,missing,true);
# last parameter of input is true if function called for original G
# in recursive calls on stabilizers, it is false
# reason: on top level,
# we do not want to add anything to generating set
# start checking
size := 1; T := S;
while Length( T.generators ) <> 0 do
size := size * Length( T.orbit );
T := T.stabilizer;
od;
if size = order or size = limit then
ready := true;
elif size < order then
# we have an incorrect stabilizer chain
# repeat checking until a new element is discovered
result := id;
if options.random = 1000 then
if correct then
result := SCRStrongGenTest(S,[1,10/S.diam,0,0,0,0],orbits,
basesize,base,correct,missing);
else
result := SCRStrongGenTest2(S,[0,0,1,10/S.diam,0,0]);
fi;
if result = id then
T := SCRRestoredRecord(S);
result := VerifySGS(T,missing,correct);
fi;
if result = id then
Print("Warning, computed and given size differ","\n");
ready := true;
S := T;
else
if not IsPerm(result) then
repeat
result := SCRStrongGenTest2(S,[0,0,1,10/S.diam,0,0]);
until result <> id;
fi;
new := [result];
fi;
else
warning := 0;
if correct then
# if correct base was provided, it is enough to check
# images of base points to check whether a word is trivial
# no need for second checking phase
while result = id do
warning := warning + 1;
if warning > 5 then
Print("Warning, computed and given size differ","\n");
fi;
result := SCRStrongGenTest(S,param,orbits,
basesize,base,correct,missing);
od;
else
while result = id do
warning := warning + 1;
if warning > 5 then
Print("Warning, computed and given size differ","\n");
fi;
result:=SCRStrongGenTest(S,param,orbits,
basesize,base,correct,missing);
if result = id then
# Print("entering SGT2","\n");
result:=SCRStrongGenTest2(S,param);
fi;
od;
fi; # correct or not
new:=[result];
fi; # end of random checking or not, when size is known
else
#no information or only upper bound about Size(S)
if options.random = 1000 then
if correct then
result := SCRStrongGenTest(S,[1,10/S.diam,0,0,0,0],orbits,
basesize,base,correct,missing);
else
result := SCRStrongGenTest2(S,[0,0,1,10/S.diam,0,0]);
fi;
if result = id then
T := SCRRestoredRecord(S);
result := VerifySGS(T,missing,correct);
fi;
if result = id then
S := T;
ready := true;
else
if not IsPerm(result) then
repeat
result := SCRStrongGenTest2(S,[0,0,1,10/S.diam,0,0]);
until result <> id;
fi;
new := [result];
fi;
else
# Print("entering SGT", "\n");
result:=SCRStrongGenTest(S,param,orbits,basesize,
base,correct,missing);
if result <> id then
new:=[result];
elif correct then
# no need for second checking phase
ready:=true;
else
# Print("entering SGT2","\n");
result:=SCRStrongGenTest2(S,param);
if result = id then
ready:=true;
else
new:=[result];
fi;
fi;
fi; # random checking or not, when size is not known
fi; # size known
od;
#restore usual record elements
if not IsBound(S.labels) then
S := SCRRestoredRecord(S);
fi;
return S;
end );
#############################################################################
##
#F SCRMakeStabStrong( ... ) . . . . . . . . . . . . . . . . . . . . . local
##
## heuristic stabilizer chain construction, with one random subproduct on
## each level, and one or two (defined by mlimit) random cosets to make
## Schreier generators
##
InstallGlobalFunction( SCRMakeStabStrong,
function ( S, new, param, orbits, where, basesize, base,
correct, missing, top )
local x,m,j,l, # loop variables
ran1, # random permutation
string, # random 0-1 string
w, # random subproduct of generators
len, # number of generators of S
mlimit, # number of random elements to be tested
coset, # word representing coset of S
residue, # first component: remainder of Schreier generator
# after factorization; second component > 0
# if factorization unsuccesful
jlimit, # number of random points to plug into residue[1]
ran, # index of random point in an orbit of S
g, # permutation to be added to S.stabilizer
gen, # permutations used in S.transversal
inv, # their inverses
firstmove; # first point of base moved by an element of new
if new <> [] then
firstmove := First( base, x->ForAny( new, gen->x^gen<>x ) );
# if necessary add a new stabilizer to the stabchain
if not IsBound(S.stabilizer) then
S.orbit := [firstmove];
S.transversal := [];
S.transversal[S.orbit[1]] := S.identity;
S.generators := [];
S.treegen := [];
S.treegeninv := [];
S.stabilizer := rec();
S.stabilizer.identity := S.identity;
S.stabilizer.aux := [];
S.stabilizer.generators := [];
S.stabilizer.diam := 0;
if not correct then
basesize[where[S.orbit[1]]]
:= basesize[where[S.orbit[1]]] + 1;
fi;
missing := Difference( missing, [ firstmove ] );
else
if Position(base,firstmove) < Position(base,S.orbit[1]) then
S.stabilizer := ShallowCopy(S);
S.orbit := [firstmove];
S.transversal := [];
S.transversal[S.orbit[1]] := S.identity;
S.generators := ShallowCopy(S.stabilizer.generators);
S.treegen := [];
S.treegeninv := [];
if not correct then
basesize[where[S.orbit[1]]]
:= basesize[where[S.orbit[1]]] + 1;
fi;
missing := Difference( missing, [ firstmove ] );
fi;
fi;
# on top level, we want to keep the original generators
if not top or Length(S.generators) = 0 then
for j in new do
StretchImportantSLPElement(j);
od;
Append(S.generators,new);
fi;
#construct orbit of basepoint
SCRSchTree(S,new);
#check whether new elements are really new in the system
while new <> [] do
g := SCRSift( S, new[Length(new)] );
Unbind( new[Length(new)] );
if g <> S.identity then
SCRMakeStabStrong(S.stabilizer,[g],param,orbits,
where,basesize,base,correct,missing,false);
S.diam:=S.treedepth+S.stabilizer.diam;
S.aux:=Concatenation(S.treegen,
S.treegeninv,S.stabilizer.aux);
fi;
od;
fi;
#check random Schreier generators
gen := Concatenation(S.treegen,S.treegeninv,[S.identity]);
inv := Concatenation(S.treegeninv,S.treegen,[S.identity]);
len := Length(S.aux);
#in case of more than one generator for S, form a random subproduct
#otherwise, use the generator
if len > 1 then
ran1 := SCRRandomPerm(len);
string := SCRRandomString(len);
w := S.identity;
for x in [1..len] do
w := w*(S.aux[x^ran1]^string[x]);
od;
else
w:=S.aux[1];
fi;
# take random coset(s)
mlimit:=1;
m:=0;
while m < mlimit do
m := m+1;
ran := Random([1..Length(S.orbit)]);
coset := CosetRepAsWord(S.orbit[1],S.orbit[ran],S.transversal);
coset := InverseAsWord(coset,gen,inv);
if w <> S.identity then
# form Schreier generator and factorize
Add(coset,w);
residue := SiftAsWord(S,coset);
# check whether factorization is succesful
if residue[2] > 0 then
# factorization is unsuccesful; use remainder for
# construction in stabilizer
g := Product(residue[1]);
SCRMakeStabStrong(S.stabilizer,[g],param,orbits,where,
basesize,base,correct,missing,false);
S.diam := S.treedepth+S.stabilizer.diam;
S.aux := Concatenation(S.treegen,S.treegeninv,
S.stabilizer.aux);
# get out of current loop
m := 0;
elif correct then
# enough to check images of points in given base
l := 0;
while l < Length(missing) do
l := l+1;
if ImageInWord(missing[l],residue[1]) <> missing[l] then
# factorization is unsuccesful;
# use remainder for construction in stabilizer
g := Product(residue[1]);
SCRMakeStabStrong(S.stabilizer,[g],param,
orbits,where,basesize,
base,correct,missing,false);
S.diam := S.treedepth+S.stabilizer.diam;
S.aux := Concatenation(S.treegen,
S.treegeninv,S.stabilizer.aux);
# get out of current loop
m := 0;
l := Length(missing);
fi;
od;
else
l:=0;
while l < Length(orbits) do
l:=l+1;
if Length(orbits[l]) > param[5] then
# in large orbits, plug in random points
j:=0;
jlimit:=Maximum(param[6],basesize[l]);
while j < jlimit do
j:=j+1;
ran:=Random([1..Length(orbits[l])]);
if ImageInWord(orbits[l][ran],residue[1])
<> orbits[l][ran]
then
# factorization is unsuccesful;
# use remainder for construction in stabilizer
g := Product(residue[1]);
SCRMakeStabStrong(S.stabilizer,[g],param,
orbits,where,basesize,
base,correct,missing,false);
S.diam := S.treedepth+S.stabilizer.diam;
S.aux := Concatenation(S.treegen,S.treegeninv,
S.stabilizer.aux);
# get out of current loop
m := 0;
j := jlimit;
l := Length(orbits);
fi;
od; #j loop
else
# in small orbits, check images of all points
j := 0;
while j < Length(orbits[l]) do
j := j+1;
if ImageInWord(orbits[l][j],residue[1])
<> orbits[l][j]
then
# factorization is unsuccesful;
# use remainder for construction in stabilizer
g := Product(residue[1]);
SCRMakeStabStrong(S.stabilizer,[g],param,
orbits,where,basesize,
base,correct,missing,false);
S.diam := S.treedepth+S.stabilizer.diam;
S.aux := Concatenation(S.treegen,S.treegeninv,
S.stabilizer.aux);
# get out of current loop
m := 0;
j := Length(orbits[l]);
l := Length(orbits);
fi;
od; #j loop
fi;
od; #l loop
fi;
fi;
od; #m loop
end );
#############################################################################
##
#F SCRStrongGenTest( ... ) . . . . . . . . . . . . . . . . . . . . . . local
##
## tests whether product of a random element of S and a random subproduct of
## the strong generators of S is in S. Computations are carried out with
## words in generators representing group elements; random points are
## plugged in to test whether a word represents the identity. If SGS for S
## not complete, returns a permutation not in S.
##
InstallGlobalFunction( SCRStrongGenTest,
function ( S, param, orbits, basesize,
base, correct, missing)
local x,i,k,m,j,l, # loop variables
ran1,ran2, # random permutations
string, # random 0-1 string
w, # list containing random subproducts of generators
len, # number of generators of S
len2, # length of short random subproduct
mlimit, # number of random elements to be tested
ranword, # random element of S as a word in generators
residue, # first component: remainder of ranword
# after factorization; second component > 0
# if factorization unsuccesful
jlimit, # number of random points to plug into residue[1]
ran, # index of random point in an orbit of S
g; # product of residue[1]
k := 0;
while k < param[1] do
k := k+1;
len := Length(S.aux);
#in case of large S.aux, form random subproducts
#otherwise, try all of them
if len > 2*param[3] then
ran1 := SCRRandomPerm(len);
ran2 := SCRRandomPerm(len);
len2 := Random([1 .. QuoInt(len,2)]);
string := SCRRandomString(len+len2);
# we form two random subproducts:
# w[1] in a random ordering of all generators
# w[2] in a random ordering of a random subset of them
w:=[];
w[1] := S.identity;
for x in [1 .. len] do
w[1] := w[1]*(S.aux[x^ran1]^string[x]);
od;
w[2] := S.identity;
for x in [1 .. len2] do
w[2] := w[2]*(S.aux[x^ran2]^string[x+len]);
od;
else
# take next two generators of S (unless only one is left)
w := [];
w[1] := S.aux[2*k-1];
if len > 2*k-1 then
w[2] := S.aux[2*k];
else
w[2] := S.identity;
fi;
fi;
# take random elements of S as words
m := 0;
mlimit := param[2]*S.diam;
while m < mlimit do
m:=m+1;
ranword := RandomElmAsWord(S);
i := 0;
while i < 2 do
i := i+1;
if w[i] <> S.identity then
Append(ranword,[w[i]]);
residue := SiftAsWord(S,ranword);
if residue[2]>0 then
# factorization is unsuccesful;
# remainder is witness that SGS for S is not complete
g := Product(residue[1]);
# Print("k=",k," i=",i," m=",m," mlimit=",mlimit,"\n");
return g;
elif correct then
# enough to check whether base points are fixed
l:=0;
while l < Length(missing) do
l:=l+1;
if ImageInWord(missing[l],residue[1])
<> missing[l]
then
# remainder is not in S
g := Product(residue[1]);
return g;
fi;
od;
else
# plug in points from each orbit to check whether
# action on orbit is trivial
l:=0;
while l < Length(orbits) do
l:=l+1;
if Length(orbits[l]) > param[5] then
# on large orbits, plug in random points
j := 0;
jlimit := Maximum(param[6],basesize[l]);
while j < jlimit do
j := j+1;
ran := Random([1..Length(orbits[l])]);
if ImageInWord(orbits[l][ran],residue[1])
<> orbits[l][ran]
then
#remainder is not in S
g := Product(residue[1]);
return g;
fi;
od; #j loop
else
# on small orbits, plug in all points
j := 0;
while j < Length(orbits[l]) do
j := j+1;
if ImageInWord( orbits[l][j],residue[1] )
<> orbits[l][j]
then
# remainder is not in S
g:=Product(residue[1]);
return g;
fi;
od; #j loop
fi;
od; #l loop
fi;
fi;
od; #i loop
od; #m loop
if len <= 2*k then
#finished making Schr. generators with all in S.aux
k := param[1];
fi;
od; #k loop
return S.identity;
end );
#############################################################################
##
#F SCRSift( <S>, <g> ) . . . . . . . . . . . . . . . . . . . . . . . . local
##
## tries to factor g as product of cosetreps in S; returns remainder
##
InstallGlobalFunction( SCRSift, function ( S, g )
local stb, # the stabilizer of S we currently work with
bpt; # first point of stb.orbit
stb := S;
while IsBound( stb.stabilizer ) do
bpt := stb.orbit[1];
if IsBound( stb.transversal[bpt^g] ) then
while bpt <> bpt^g do
g := g*stb.transversal[bpt^g];
od;
stb := stb.stabilizer;
else
#current g witnesses that input was not in S
return g;
fi;
od;
return g;
end );
#############################################################################
##
#F SCRStrongGenTest2( <S>, <param> ) . . . . . . . . . . . . . . . . . local
##
## tests whether product of a random element of S and a random subproduct of
## the strong generators of S is in S. Computations are carried out with
## complete permutations.
##
InstallGlobalFunction( SCRStrongGenTest2, function ( S, param )
local x,i,k,m, # loop variables
ran1,ran2, # random permutations
string, # random 0-1 string
w, # list containing random subproducts of generators
len, # number of generators of S
len2, # length of short random subproduct
mlimit, # number of random elements to be tested
ranelement, # random element of S
T, p,
residue; # remainder of ranelement after factorization
k := 0;
while k < param[3] do
k := k+1;
len := Length(S.aux);
#in case of large S.aux, form random subproducts
#otherwise, try all of them
if len > 2*param[3] then
ran1 := SCRRandomPerm(len);
ran2 := SCRRandomPerm(len);
len2 := Random([1 .. QuoInt(len,2)]);
string := SCRRandomString(len+len2);
# we form two random subproducts:
# w[1] in a random ordering of all generators
# w[2] in a random ordering of a random subset of them
w:=[];
w[1] := S.identity;
for x in [1 .. len] do
w[1] := w[1]*(S.aux[x^ran1]^string[x]);
od;
w[2] := S.identity;
for x in [1 .. len2] do
w[2] := w[2]*(S.aux[x^ran2]^string[x+len]);
od;
else
# take next two generators of S (unless only one is left)
w := [];
w[1] := S.aux[2*k-1];
if len > 2*k-1 then
w[2] := S.aux[2*k];
else
w[2] := S.identity;
fi;
fi;
# take random elements of S
m := 0;
mlimit := param[4]*S.diam;
while m < mlimit do
m:=m+1;
ranelement := S.identity;
T := S;
while Length( T.generators ) <> 0 do
p := Random( T.orbit );
while p <> T.orbit[ 1 ] do
ranelement := LeftQuotient( T.transversal[ p ],
ranelement );
p := p ^ T.transversal[ p ];
od;
T := T.stabilizer;
od;
i := 0;
while i < 2 do
i := i+1;
if w[i] <> S.identity then
# test whether product of ranelement and w[i] in S
ranelement := ranelement*w[i];
residue := SCRSift(S,ranelement);
if residue <> S.identity then
return residue;
fi;
fi;
od; #i loop
od; #m loop
if len <= 2*k then
#finished checking all in S.aux
k := param[3];
fi;
od; #k loop
return S.identity;
end );
#############################################################################
##
#F SCRNotice( <orb>, <transversal>, <genlist> ) . . . . . . . . . . . local
##
## checks whether orbit is closed for the action of permutations in
## genlist. If not, returns orbit point and generator witnessing.
##
InstallGlobalFunction( SCRNotice,
function ( orb, transversal, genlist )
local flag, #first component of output; true if orb is closed for
#action of genlist
i, #second component of output, index of point in orb moving out
j ; #third component, index of permutation in genlist moving orb[i]
i := 0;
flag := true;
while i < Length(orb) and flag do
i := i+1;
j := 0;
while j < Length(genlist) and flag do
j := j+1;
if not IsBound(transversal[orb[i]^genlist[j]]) then
flag := false;
fi;
od;
od;
return [flag,i,j];
end );
#############################################################################
##
#F SCRExtend( <list> ) . . . . . . . . . . . . . . . . . . . . . . . . local
##
## given a partial Schreier tree of depth d,
## SCRExtends the partial Schreier tree to depth d+1
## input, output coded in list of length 5
##
InstallGlobalFunction( SCRExtend, function ( list )
local orb, #partial orbit
transversal, #partial transversal
treegen, #list of generators
treegeninv, #inverses of elements of treegen
#both treegen, treegeninv are used in transversal
i, j, #loop variables
previous, #index, showing end of level d-1 in orb
len; #length of orb at entering routine
orb:=list[1];
transversal:=list[2];
treegen:=list[3];
treegeninv:=list[4];
previous:=list[5];
len:=Length(orb);
# for each point on level d, check whether one of the generators or
# inverses moves it out of orb. If yes, add image to orb
for i in [previous+1..len] do
for j in [1..Length(treegen)] do
if not IsBound(transversal[orb[i]^treegen[j]]) then
transversal[orb[i]^treegen[j]] := treegeninv[j];
Add(orb, orb[i]^treegen[j]);
fi;
if not IsBound(transversal[orb[i]^treegeninv[j]]) then
transversal[orb[i]^treegeninv[j]] := treegen[j];
Add(orb, orb[i]^treegeninv[j]);
fi;
od;
od;
# return Schreier tree of depth one larger
return [orb,transversal,treegen,treegeninv,len];
end );
#############################################################################
##
#F SCRSchTree( <S>, <new> ) . . . . . . . . . . . . . . . . . . . . . local
##
## creates Schreier tree for the group generated by S.generators \cup new
##
InstallGlobalFunction( SCRSchTree, function ( S, new )
local l, #output of notice
flag, #first component of output
i, #second component of output
j, #third component of output
word, #list of permutations coding a coset representative
g, #the coset representative coded by word
witness, #a permutation moving a point out of S.orbit
list; #list coding input and output of 'extend'
l := SCRNotice(S.orbit,S.transversal,new);
flag := l[1];
if flag then
#do nothing; the orbit did not change
return;
else
i := l[2];
j := l[3];
witness := new[j];
fi;
while not flag do
word := CosetRepAsWord(S.orbit[1],S.orbit[i],S.transversal);
g := Product(word);
#add a new generator to treegen which moves S.orbit[1] out of S.orbit
Add(S.treegen, g^(-1)*witness);
Add(S.treegeninv, witness^(-1)*g);
#recompute Schreier tree to new depth
S.orbit := [S.orbit[1]];
S.transversal := [];
S.transversal[S.orbit[1]] := S.identity;
S.treedepth := 0;
list := [S.orbit,S.transversal,S.treegen,S.treegeninv,0];
flag := false;
#with k generators, we build only a tree of depth 2k
while not flag and S.treedepth < 2*Length(S.treegen) do
list := SCRExtend(list);
S.orbit := list[1];
S.transversal := list[2];
if Length(S.orbit) = list[5] then
#the tree did not extend; orbit is closed for treegen
flag := true;
else
S.treedepth := S.treedepth + 1;
fi;
od;
#increased S.orbit may not be closed for all generators of S
l := SCRNotice(S.orbit,S.transversal,S.generators);
flag := l[1];
if not flag then
i := l[2];
j := l[3];
witness := S.generators[j];
fi;
od;
#update record components aux, diam
S.aux := Concatenation(S.treegen,S.treegeninv,S.stabilizer.aux);
S.diam := S.treedepth+S.stabilizer.diam;
end );
#############################################################################
##
#F SCRRandomPerm( <d> ) . . . . . . . . . . . . . . . . . . . . . . . local
##
## constructs random permutation in Sym(d)
## without creating group record of Sym(d)
##
InstallGlobalFunction( SCRRandomPerm, function ( d )
local rnd, # random permutation, result
tmp, # temporary variable for swapping
i, k; # loop variables
# use Floyd\'s algorithm
rnd := [ 1 .. d ];
for i in [ 1 .. d-1 ] do
k := Random( [ 1 .. d+1-i ] );
tmp := rnd[d+1-i];
rnd[d+1-i] := rnd[k];
rnd[k] := tmp;
od;
# return the permutation
return PermList( rnd );
end );
#############################################################################
##
#F SCRRandomString( <n> ) . . . . . . . . . . . . . . . . . . . . . . local
##
## constructs random 0-1 string of length n
## same steps as Random, but uses created random number for 28 bits
##
InstallGlobalFunction( SCRRandomString, function ( n )
local i, j, # loop variables
k, # number of 28 long substrings
rnd, # the random number which would be created by Random
string; # the random string constructed
string:=[];
k:=QuoInt(n-1,28);
for i in [0..k-1] do
# follow steps in Random to create a random number < 2^28
R_N := R_N mod 55 + 1;
R_X[R_N] := (R_X[R_N] + R_X[(R_N+30) mod 55+1]) mod 2^28;
rnd:=R_X[R_N];
# use each bit of rnd
for j in [1 .. 28] do
string[28*i+j] := rnd mod 2;
rnd := QuoInt(rnd,2);
od;
od;
# construct last <= 28 bits of string
R_N := R_N mod 55 + 1;
R_X[R_N] := (R_X[R_N] + R_X[(R_N+30) mod 55+1]) mod 2^28;
rnd:=R_X[R_N];
for j in [28*k+1 .. n] do
string[j] := rnd mod 2;
rnd := QuoInt(rnd,2);
od;
return string;
end );
#############################################################################
##
#F SCRRandomSubproduct( <list>, <id> ) . random subproduct of perms in list
##
InstallGlobalFunction( SCRRandomSubproduct, function( list, id )
local string, # 0-1 string containing the exponents of elements of list
random, # the random subproduct
i; # loop variable
string := SCRRandomString(Length(list));
random := id;
for i in [1 .. Length(list)] do
if string[i] = 1 then
random := random*list[i];
fi;
od;
return random;
end );
#############################################################################
##
#F SCRExtendRecord( <G> ) . . . . . . . . . . . . . . . . . . . . . . local
##
## defines record elements used at random stabilizer chain construction
##
InstallGlobalFunction( SCRExtendRecord, function(G)
local list, # list of stabilizer subgroups
len, # length of list
i; # loop variable
list := ListStabChain(G);
len := Length(list);
list[len].diam := 0;
list[len].aux := [];
for i in [1 .. len - 1] do
# list[len - i].real := list[len - i].generators;
if Length(list[len - i].orbit) = 1 then
# in this case, SCRSchTree will not do anything;
# we have to define records treedepth, aux, and diameter
list[len - i].treedepth := 0;
list[len - i].aux := list[len - i + 1].aux;
list[len - i].diam := list[len - i + 1].diam;
fi;
list[len - i].orbit := [ list[len - i].orbit[1] ];
list[len - i].transversal := [];
list[len - i].transversal[list[len - i].orbit[1]]
:= list[len - i].identity;
list[len - i].treegen := [];
list[len - i].treegeninv := [];
SCRSchTree( list[len - i], list[len - i].generators );
od;
end );
#############################################################################
##
#F SCRRestoredRecord( <G> ) . . . . . . . . . . . . . . . . . . . . . local
##
## restores usual group records after random stabilizer chain construction
##
InstallGlobalFunction( SCRRestoredRecord, function( G )
local sgs, T, S, l, lab, pnt,o,ind;
S := G;
sgs := [ S.identity ];
while IsBound( S.stabilizer ) do
UniteSet( sgs, S.treegen );
UniteSet( sgs, S.treegeninv );
S := S.stabilizer;
od;
T := EmptyStabChain( sgs, G.identity );
sgs := [ 2 .. Length( sgs ) ];
S := T;
while IsBound( G.stabilizer ) do
InsertTrivialStabilizer( S, G.orbit[ 1 ] );
S.genlabels := sgs;
S.generators := G.generators;
S.orbit := G.orbit;
S.transversal := G.transversal;
o:=ShallowCopy(S.orbit);
# check identity of transversal elements first: Most are
# identical and thus element comparisons are relatively
# infrequent
while Length(o)>0 do
ind:=S.transversal[o[1]];
ind:=Filtered([1..Length(o)],
i->IsIdenticalObj(S.transversal[o[i]],ind));
for l in sgs do
if S.transversal[o[1]]=S.labels[l] then
for pnt in o{ind} do # all these transv. elements are same
S.translabels[ pnt ] := l;
od;
fi;
od;
o:=o{Difference([1..Length(o)],ind)}; # the rest
od;
#was:
# (The element comparisons could be expensive)
#for pnt in S.orbit do
# if S.transversal[ pnt ] = lab then
# S.translabels[ pnt ] := l;
# fi;
#od;
sgs := Filtered( sgs, l ->
S.orbit[ 1 ] ^ S.labels[ l ] = S.orbit[ 1 ] );
S := S.stabilizer;
G := G.stabilizer;
od;
return T;
end );
#############################################################################
##
#F VerifyStabilizer( <S>, <z>, <missing>, <correct> ) . . . . verification
##
InstallGlobalFunction( VerifyStabilizer, function(S,z,missing,correct)
#z is an involution, moving first base point
# correct is boolean telling whether a base is known
# if yes, missing contains the base points which do not occur in the base of S
local pt1, # the first base point
zinv, # inverse of z
pt2, # pt1^zinv
flag, # Boolean; true if z exchanges pt1 and pt2
stab, # the stabilizer of pt1 in S
chain, # stab after base change, to bring pt2 first
stabpt2, # stabilizer of pt2 in chain
result, # output, witness perm if something is wrong
g, # generator of stabpt2
where1, # stores which orbit of stab pts of S.orbit belong to
orbit1count, # index running through orbits of stab
leaders, # list of orbit representatives of stab
i, j, l, # loop variables
residue, # result of sifting as word
where2, # boolean list to mark orbits of stabpt2 as processed
k, # point of an orbit of stab
gen, # a generator of stab
transversal, # transversal of stab, on all of its orbits
orb, # an orbit of stabpt2
img, # image of a point at computation of orb
pnt, # a point from orb
w1, w2, w3, w4, # words/permutations coding coset representatives
w1inv, # inverse of w1
schgen; # Schreier generator
pt1 := S.orbit[1];
zinv := z^(-1);
pt2 := pt1^zinv;
result := S.identity;
stab := S.stabilizer;
if pt2^zinv = pt1 then
flag := true;
result := SCRSift(stab, z^2);
else
flag := false;
fi;
# store which orbit of stab the pts of S.orbit belong to
# in each orbit, compute transversals from a representative
where1 := []; # orbits of stab
leaders := [pt2];
orbit1count := 1;
transversal := [];
transversal[pt2] := S.identity;
where1[pt2] := 1;
orb := [pt2];
j := 1;
while j <= Length( orb ) do
for gen in stab.generators do
k := orb[j] / gen;
if not IsBound( transversal[k] ) then
transversal[k] := gen;
Add( orb, k );
where1[k] := orbit1count;
fi;
od;
j := j + 1;
od;
for i in S.orbit do
if not IsBound(where1[i]) then
orbit1count := orbit1count + 1;
Add(leaders, i);
orb := [i];
where1[i] := orbit1count;
transversal[i] := S.identity;
j := 1;
while j <= Length( orb ) do
for gen in stab.generators do
k := orb[j] / gen;
if not IsBound( transversal[k] ) then
transversal[k] := gen;
Add( orb, k );
where1[k] := orbit1count;
fi;
od;
j := j + 1;
od;
fi;
od;
# check that conjugates of point stabilizers of stab are subgroups of stab
chain:=StructuralCopy(stab);
for j in [1..Length(leaders)] do
if result = S.identity then
i := leaders[Length(leaders)+1-j];
ChangeStabChain( chain, [i], false );
if i = pt2 then
w1 := z;
w1inv := zinv;
else
w1 := CosetRepAsWord(pt1, i, S.transversal);
w1 := Product(w1);
w1inv := w1^(-1);
fi;
for g in chain.stabilizer.generators do
if result = S.identity then
if correct then
residue := SiftAsWord(stab, [w1inv,g,w1]);
if residue[2] <> 0 then
result := Product(residue[1]);
else
l := 0;
while ( l < Length(missing) ) and ( result = S.identity ) do
l:=l+1;
if ImageInWord(missing[l],residue[1])
<> missing[l] then
# remainder is not in S
result := Product(residue[1]);
fi;
od;
fi;
else
result := SCRSift(stab, w1inv*g*w1);
fi;
fi;
od;
fi;
od;
if result = S.identity then
stabpt2 := chain.stabilizer;
# process orbits of stabpt2
where2:= BlistList( [1..Length(where1)],[] ) ;
for i in S.orbit do
if result = S.identity and (not where2[i]) then
orb := [i];
where2[i] := true;
for pnt in orb do
for gen in stabpt2.generators do
img := pnt^gen;
if not where2[img] then
Add( orb, img );
where2[img] := true;
fi;
od;
od;
# if we hit a new orbit of stabpt2
# and if z exchanges pt1 and pt2 then
# mark the orbit
if flag and not where2[i^z] then
orb := [i^z];
where2[i^z] := true;
for pnt in orb do
for gen in stabpt2.generators do
img := pnt^gen;
if not where2[img] then
Add( orb, img );
where2[img] := true;
fi;
od;
od;
fi;
# compute Schreier generator either as a word or perm
# if i is not a fixed point of z,
# we have to compute a Schreier generator
w1 := CosetRepAsWord(pt1, leaders[where1[i]], S.transversal);
w2 := CosetRepAsWord(leaders[where1[i]], i, transversal);
w3 := CosetRepAsWord(leaders[where1[i^z]], i^z, transversal);
if where1[i] <> where1[i^z] then
w4 := CosetRepAsWord(pt1,leaders[where1[i^z]], S.transversal);
else
w4 := w1;
fi;
schgen := (Product(w1))^(-1)*(Product(w2))^(-1);
if correct then
schgen := Concatenation([schgen,z],w3,w4);
else
schgen := schgen*z*Product(w3)*Product(w4);
fi;
# sift Schreier generator either as a word or perm
if correct then
residue := SiftAsWord(stab, schgen);
if residue[2] <> 0 then
result := Product(residue[1]);
else
l := 0;
while ( l < Length(missing) ) and ( result = S.identity ) do
l:=l+1;
if ImageInWord(missing[l],residue[1])
<> missing[l] then
# remainder is not in S
result := Product(residue[1]);
fi;
od;
fi;
else
result := SCRSift(stab, schgen);
fi;
fi; # result = S.identity and not where2[i]
od;
fi;
return result;
end );
#############################################################################
##
#F VerifySGS( <S>, <missing>, <correct> ) . . . . . . . . . . verification
##
InstallGlobalFunction( VerifySGS, function(S,missing,correct)
# correct is boolean telling whether a base is known
# if yes, missing contains the base points which do not occur in the base of S
local n, # degree of S
list, # list of subgroups in stabchain
result, # result of the test
len, # length of list
i,l, # loop variables
residue, # result of sifting as word
temp, # subgroup of S we currently work with
temp2, # temp, with possible extension on blocks
gen, # the generator whose addition we verify
longer, # generator list of temp, after gen is added
gencount, # counts the generators to be verified
set, # set of orbit of temp
orbit, # orbit of temp after adding gen
blks, # images of block when gen is added
extension, # list of length 2; first coord. is temp2, extended by
# the action on blks, second coord. is newgen
newgen, # the extension of gen
leader, # first point in orbit of temp2
block, # block containing leader
point, # another point from block
pos; # position of set in blks
n := LargestMovedPoint(S.generators);
list := ListStabChain(S);
len := Length(list);
result := S.identity;
# verify the subgroup chain from bottom up
i := 1;
while i < len and result = S.identity do
temp := ShallowCopy(list[len - i + 1]);
InsertTrivialStabilizer(temp, list[len -i].orbit[1]);
gencount := 0;
# add one generator a time
while (gencount < Length(list[len - i].generators)) and (result= S.identity ) do
gencount := gencount + 1;
gen := list[len - i].generators[gencount];
set := Set( temp.orbit );
# if adding gen did not change the fundamental orbit, just sift gen
if set = OnSets(set,gen) then
if correct then
residue := SiftAsWord(temp, [gen]);
if residue[2] <> 0 then
result := Product(residue[1]);
else
l := 0;
while ( l < Length(missing) ) and ( result = S.identity ) do
l:=l+1;
if ImageInWord(missing[l],residue[1])
<> missing[l] then
# remainder is not in S
result := Product(residue[1]);
fi;
od;
fi;
else
result := SCRSift(temp, gen);
fi;
# if the fundamental orbit increased, compute block system
# obtained when adding gen
else
if Length(set) =1 then
temp2 := temp;
newgen := gen;
# otherwise, compute the action on blocks
else
longer := Concatenation(temp.generators, [gen]);
orbit := OrbitPerms( longer, temp.orbit[ 1 ] );
blks := Blocks( GroupByGenerators( longer ), orbit, set );
if Length(blks) * Length(set) <> Length(orbit) then
result := "false0";
else
pos := Position( blks, set );
extension := ExtensionOnBlocks( temp, n, blks, [gen] );
temp2 := extension[1];
newgen := extension[2][1];
InsertTrivialStabilizer(temp2, n + pos);
fi;
fi;
# first generator in first group can be verified easily
if i=1 and Length(set) =1 then
result := newgen^(CycleLengthOp(newgen,temp2.orbit[1]));
AddGeneratorsExtendSchreierTree(temp2, [newgen]);
elif result = S.identity then
AddGeneratorsExtendSchreierTree( temp2, [ newgen ] );
blks := Blocks( GroupByGenerators(temp2.generators),
temp2.orbit);
if Length(blks) > 1 then
leader := temp2.orbit[1];
block := First(blks, x -> leader in x);
point := First(block, x -> x <> leader);
newgen := Product(CosetRepAsWord(leader ,point,
temp2.transversal));
temp2 := temp2.stabilizer;
InsertTrivialStabilizer(temp2, leader);
AddGeneratorsExtendSchreierTree(temp2, [newgen]);
result := VerifyStabilizer(temp2,newgen,missing,correct);
if leader > n then
AddGeneratorsExtendSchreierTree(temp,[RestrictedPermNC
(newgen, [1..n])]);
else
temp := temp2;
fi;
gencount := gencount - 1;
else
result := VerifyStabilizer(temp2,newgen,missing,correct);
if temp2.orbit[1] > n then
AddGeneratorsExtendSchreierTree(temp,[gen]);
fi;
fi;
if result <> S.identity and temp2.orbit[1] > n then
result := RestrictedPermNC(result, [1..n]);
fi;
fi;
fi;
od;
i := i + 1;
od;
return result;
end );
#############################################################################
##
#F ExtensionOnBlocks( <S>, <n>, <blks>, <elms> ) . . . . . . . . . extension
##
InstallGlobalFunction( ExtensionOnBlocks, function( S, n, blks, elms )
local where, j, k, hom, T, newelms;
# list which block the elements of the orbit belong to
where := [];
for j in [1..Length(blks)] do
for k in blks[j] do
where[k] := j;
od;
od;
hom := function( g )
local perm, j;
perm := [1..n];
for j in [1..Length(blks)] do
perm[n+j] := n+where[blks[j][1]^g];
od;
perm := PermList(perm);
return g * perm;
end;
T := EmptyStabChain( [ ], S.identity );
ConjugateStabChain( S, T, hom, S.identity );
# construct extensions of permutations in elms
newelms := List( elms, hom );
return [ T, newelms ];
end );
#############################################################################
##
#F ClosureRandomPermGroup( <G>, <genlist>, <options> ) make closure randomly
##
InstallGlobalFunction( ClosureRandomPermGroup,
function( G, genlist, options )
local k, # number of pairs of subproducts of generators in
# testing result
givenbase, # ordering from which initial base points should
# be chosen
gens, # generators in genlist that are not in <G>
g, # element of gens
degree, # degree of closure
orbits, # list of orbits of closure
orbits2, # list of orbits of closure
i,j, # loop variables
param, # list of parameters guiding number of repetitions
# in random constructions
where, # list indicating which orbit contains points in domain
basesize, # list; i^th entry = number of base points in orbits[i]
ready, # boolean; true if stabilizer chain ready
new, # list of permutations to be added to stab. chain
result, # output of checking phase; nontrivial if stabilizer
# chain is incorrect
base, # ordering of domain from which base points are taken
missing, # if a correct base was provided by input, missing
# contains those points of it which are not in
# constructed base
correct; # boolean; true if a correct base is given
# warning: options.base should be compatible with BaseOfGroup(G)
gens := Filtered( genlist, gen -> not(IsOne(SCRSift(G,gen))) );
if Length(gens) > 0 then
G.identity := One(gens[1]) ;
if options.random = 1000 then
#case of deterministic computation with known size
k := 1;
else
k:=First([1..14],x->(3/5)^x<1-options.random/1000);
fi;
if IsBound(options.knownBase) then
param := [k,4,0,0,0,0];
else
param := [QuoInt(k,2),4,QuoInt(k+1,2),4,50,5];
fi;
if options.random <= 200 then
param[2] := 2;
param[4] := 2;
fi;
#param[1] = number of pairs of random subproducts from generators in
# first checking phase
#param[2] = (number of random elements from created set)/S.diam
# in first checking phase
#param[3] = number of pairs of random subproducts from generators in
# second checking phase
#param[4] = (number of random elements from created set)/S.diam
# in second checking phase
#param[5] = maximum size of orbits in which we evaluate words on all
# points of orbit
#param[6] = minimum number of random points from orbit to plug in to check
# whether given word is identity on orbit
degree := LargestMovedPoint( Union( G.generators, gens ) );
# prepare input of construction
if IsBound(options.base) then
givenbase := options.base;
else
givenbase := [];
fi;
if IsBound(options.knownBase) then
correct := true;
else
correct := false;
fi;
if correct then
# if correct base was given as input,
# no need for orbit information
base := Set( givenbase );
for i in BaseStabChain(G) do
if not i in base then
Add( givenbase, i );
fi;
od;
base := Concatenation(givenbase,Difference(options.knownBase,
givenbase));
missing := Difference(options.knownBase,BaseStabChain(G));
basesize := [];
where := [];
orbits := [];
else
# create ordering of domain used in choosing base points and
# compute orbit information
base := Set( givenbase );
for i in BaseStabChain(G) do
if not i in base then
Add( givenbase, i );
fi;
od;
base := Concatenation(givenbase,Difference([1..degree],givenbase));
missing := [];
orbits2 := OrbitsPerms( Union( G.generators, gens ), [1..degree] );
#throw away one-element orbits
orbits:=[];
j:=0;
for i in [1..Length(orbits2)] do
if Length(orbits2[i]) >1 then
j:=j+1; orbits[j]:= orbits2[i];
fi;
od;
basesize:=[];
where:=[];
for i in [1..Length(orbits)] do
basesize[i]:=0;
for j in [1..Length(orbits[i])] do
where[orbits[i][j]]:=i;
od;
od;
# temporary solution to speed up of handling
# of lots of small orbits until compiler
if Length(orbits) > degree/40 then
param[1] := 0;
param[3] := k;
fi;
fi;
if not IsBound(G.aux) then
SCRExtendRecord(G);
fi;
new := gens;
#the first call of SCRMakeStabStrong has top:=false
#in order to add gens to the generating set of G;
#further calls have top:=true, in order not to add
#output of SCRStrongGenTest to generating set.
#remark: adding gens to the generating set of G before
#calling SCRMakeStabStrong gives a nasty error if first base
#point changes
for g in gens do
if not(IsOne(SCRSift(G,g))) then
SCRMakeStabStrong (G,[g],param,orbits,
where,basesize,base,correct,missing,false);
fi;
od;
ready := false;
while not ready do
if IsBound(options.limit)
and SizeStabChain(G)=options.limit
then
ready := true;
else
# we do a little random testing, to ensure heuristically a
# correct result
if correct then
result := SCRStrongGenTest(G,[1,10/G.diam,0,0,0,0],orbits,
basesize,base,correct,missing);
else
result := SCRStrongGenTest2(G,[0,0,1,10/G.diam,0,0]);
fi;
if not(IsPerm(result) and IsOne(result)) then
new := [result];
ready := false;
elif options.random = 1000 then
G.restored := SCRRestoredRecord(G);
result := VerifySGS( G.restored, missing, correct );
elif options.random > 0 then
result := SCRStrongGenTest
(G,param,orbits,basesize,base,correct,missing);
fi;
if not(IsPerm(result) and IsOne(result)) then
if not IsPerm(result) then
repeat
result := SCRStrongGenTest2(G,[0,0,1,10/G.diam,0,0]);
until not(IsPerm(result) and IsOne(result));
fi;
new := [result];
ready := false;
elif correct or options.random = 0 or options.random = 1000 then
ready := true;
else
result := SCRStrongGenTest2(G,param);
if IsPerm(result) and IsOne(result) then
ready := true;
else
new := [result];
ready := false;
fi;
fi;
if not ready then
Unbind(G.restored);
SCRMakeStabStrong (G,new,param,orbits,
where,basesize,base,correct,missing,true);
fi;
fi;
od;
if not IsBound(options.temp) or options.temp = false then
if IsBound( G.restored ) then
G := G.restored;
else
G := SCRRestoredRecord(G);
fi;
else
G.basesize := basesize;
G.correct := correct;
G.orbits := orbits;
G.missing := missing;
G.base := base;
fi;
fi; # if Length(gens) > 0
# return the closure
return G;
end );
#############################################################################
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