Quelle pbr.gi
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############################################################################
##
## elements/pbr.gi
## Copyright (C) 2015-2022 Attila Egri-Nagy
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
# This file contains an initial implementation of partitioned binary
# relations (PBRs) as defined in:
#
# Paul Martin and Volodymyr Mazorchuk, Partitioned binary relations,
# Mathematica Scandinavica, v113, n1, p. 30-52, https://arxiv.org/abs/1102.0862
# Internally a PBR is stored as the adjacency list of digraph with
# vertices [1 .. 2 * n] for some n. More precisely if <x> is a PBR, then:
#
# * <x![1]> is equal to <n>
#
# * <x![i + 1]> is the vertices adjacent to <i>
#
# The number <n> is the *degree* of <x>.
#############################################################################
# Family and type.
#
# One per degree to avoid lists with pbrs of different degrees
# belonging to IsAssociativeElementCollection.
#############################################################################
BindGlobal("TYPES_PBR", []);
BindGlobal("TYPE_PBR",
function(n)
if not IsInt(n) or n < 0 then
ErrorNoReturn("the argument is not a non-negative integer");
fi;
n := n + 1; # since the degree can be 0
if not IsBound(TYPES_PBR[n]) then
TYPES_PBR[n] := NewType(NewFamily(Concatenation("PBRFamily",
String(n - 1)),
IsPBR,
CanEasilySortElements,
CanEasilySortElements),
IsPBR and IsPositionalObjectRep);
fi;
return TYPES_PBR[n];
end);
#############################################################################
# Pickler
#############################################################################
InstallMethod(IO_Pickle, "for a PBR",
[IsFile, IsPBR],
function(file, x)
if IO_Write(file, "PABR") = fail then
return IO_Error;
fi;
return IO_Pickle(file, List([1 .. 2 * x![1] + 1], i -> x![i]));
end);
IO_Unpicklers.PABR := function(file)
local x;
x := IO_Unpickle(file);
if x = IO_Error then
return IO_Error;
fi;
return Objectify(TYPE_PBR(x![1]), x);
end;
#############################################################################
# TODO(later) the embeddings from the paper, AsPBR for a pbr (extend or
# restrict),
#############################################################################
InstallMethod(EmptyPBR, "for a pos int", [IsPosInt],
n -> PBRNC(List([1 .. n], y -> []), List([1 .. n], y -> [])));
InstallMethod(IdentityPBR, "for a pos int", [IsPosInt],
n -> PBRNC(List([1 .. n], y -> [-y]), List([1 .. n], y -> [y])));
InstallMethod(UniversalPBR, "for a pos in", [IsPosInt],
function(n)
local x;
x := Concatenation([-n .. -1], [1 .. n]);
return PBRNC(List([1 .. n], y -> x), List([1 .. n], y -> x));
end);
# The following is temporary, with the current definition of
# IsGeneratorsOfInverseSemigroup for a pbr collection, the One of any element
# in the collection does not satisfy IsGeneratorsOfInverseSemigroup, and so it
# cannot be inverted.
InstallMethod(InverseMonoidByGenerators,
[IsPBRCollection],
function(_)
ErrorNoReturn("not yet implemented");
end);
# See the comment above, this is not really correct.
InstallOtherMethod(InverseMutable, "for a PBR", [IsPBR],
function(x)
# TODO(later) change IsBlockBijection(AsBipartition(x)) to
# IsBlockBijectionPBR.
if IsPartialPermPBR(x) or
(IsBipartitionPBR(x) and IsBlockBijection(AsBipartition(x))) then
return Star(x);
fi;
return fail;
end);
# See the comment above, this is not really correct.
InstallMethod(IsGeneratorsOfInverseSemigroup, "for a pbr collection",
[IsPBRCollection],
function(coll)
return ForAll(coll, IsBipartitionPBR)
and IsGeneratorsOfInverseSemigroup(List(coll, AsBipartition));
end);
InstallMethod(StarOp, "for a pbr", [IsPBR],
function(x)
local ext;
ext := ShallowCopy(ExtRepOfObj(x) * -1);
Apply(ext, ShallowCopy);
Apply(ext[1], ShallowCopy);
Apply(ext[2], ShallowCopy);
return PBR(ext[2], ext[1]);
end);
InstallMethod(DegreeOfPBRCollection, "for a PBR collection",
[IsPBRCollection],
function(coll)
if IsPBRSemigroup(coll) then
return DegreeOfPBRSemigroup(coll);
fi;
return DegreeOfPBR(coll[1]);
end);
InstallMethod(IsBipartitionPBR, "for a pbr",
[IsPBR],
x -> IsEquivalenceBooleanMat(AsBooleanMat(x)));
InstallMethod(IsTransformationPBR, "for a pbr",
[IsPBR],
function(x)
local n, i;
n := x![1];
for i in [2 .. n + 1] do
if Length(x![i]) <> 1 or x![i][1] <= n
or not i - 1 in x![x![i][1] + 1] then
return false;
fi;
od;
for i in [n + 2 .. 2 * n + 1] do
if not ForAll(x![i], j -> j <= n and x![j + 1][1] = i - 1) then
return false;
fi;
od;
return true;
end);
InstallMethod(IsBlockBijectionPBR, "for a pbr",
[IsPBR],
x -> IsBipartitionPBR(x) and IsBlockBijection(AsBipartition(x)));
InstallMethod(IsPartialPermPBR, "for a pbr",
[IsPBR],
x -> IsBipartitionPBR(x) and IsPartialPermBipartition(AsBipartition(x)));
InstallMethod(IsPermPBR, "for a pbr",
[IsPBR],
x -> IsBipartitionPBR(x) and IsPermBipartition(AsBipartition(x)));
InstallMethod(IsDualTransformationPBR, "for a pbr",
[IsPBR],
x -> IsBipartitionPBR(x) and IsDualTransBipartition(AsBipartition(x)));
InstallMethod(NumberPBR, "for a pbr",
[IsPBR],
x -> NumberBooleanMat(AsBooleanMat(x)));
InstallMethod(PBRNumber, "for pos int and pos int",
[IsPosInt, IsPosInt],
{nr, deg} -> AsPBR(BooleanMatNumber(nr, 2 * deg)));
InstallMethod(IsEmptyPBR, "for a partition binary relation",
[IsPBR],
function(x)
local n, i;
n := x![1];
for i in [2 .. 2 * n + 1] do
if Length(x![i]) > 0 then
return false;
fi;
od;
return true;
end);
InstallMethod(IsIdentityPBR, "for a partition binary relation",
[IsPBR],
function(x)
local n, i;
n := x![1];
for i in [2 .. n + 1] do
if Length(x![i]) <> 1 or x![i][1] <> i + n - 1 then
return false;
fi;
od;
for i in [n + 2 .. 2 * n + 1] do
if Length(x![i]) <> 1 or x![i][1] <> i - n - 1 then
return false;
fi;
od;
return true;
end);
InstallMethod(IsUniversalPBR, "for a partition binary relation",
[IsPBR],
function(x)
local n, i;
n := x![1];
for i in [2 .. 2 * n + 1] do
if Length(x![i]) < 2 * n then
return false;
fi;
od;
return true;
end);
InstallMethod(AsPBR, "for a partial perm and pos int",
[IsPartialPerm, IsPosInt],
function(x, deg)
local left, right, j, i;
left := List([1 .. deg], x -> []);
right := List([1 .. deg], x -> []);
for i in [1 .. deg] do
j := i ^ x;
if j <= deg and j <> 0 then
Add(left[i], -j);
Add(right[j], i);
fi;
od;
return PBR(left, right);
end);
InstallMethod(AsPBR, "for a partial perm", [IsPartialPerm],
x -> AsPBR(x, Maximum(DegreeOfPartialPerm(x), CoDegreeOfPartialPerm(x))));
InstallMethod(AsPBR, "for a transformation and pos int",
[IsTransformation, IsPosInt],
function(x, deg)
local left, right, i;
left := List([1 .. deg], x -> []);
right := List([1 .. deg], x -> []);
for i in [1 .. deg] do
Add(left[i], -(i ^ x));
Add(right[i ^ x], i);
od;
return PBR(left, right);
end);
InstallMethod(AsPBR, "for a transformation", [IsTransformation],
x -> AsPBR(x, DegreeOfTransformation(x)));
InstallMethod(AsPBR, "for a multiplicative element",
[IsMultiplicativeElement], x -> AsPBR(AsBipartition(x)));
InstallMethod(AsPBR, "for a multiplicative element and pos int",
[IsMultiplicativeElement, IsPosInt], {x, n} -> AsPBR(AsBipartition(x, n)));
# TODO(later) The following doesn't define a monoid embedding of P_n into
# PBR_n. What is a monoid embedding from P_n to PBR_n?
InstallMethod(AsPBR, "for a bipartition",
[IsBipartition], x -> AsPBR(x, DegreeOfBipartition(x)));
InstallMethod(AsPBR, "for a bipartition and pos int",
[IsBipartition, IsPosInt],
function(x, n)
local deg, blocks, out, dom, block, i;
deg := DegreeOfBipartition(x);
blocks := ExtRepOfObj(x);
out := [[], []];
dom := Union([-n .. -1], [1 .. n]);
for block in blocks do
for i in block do
if AbsInt(i) <= n then
if i < 0 then
out[2][-i] := Intersection(block, dom);
else
out[1][i] := Intersection(block, dom);
fi;
fi;
od;
od;
for i in [deg + 1 .. n] do
Add(out[1], []);
Add(out[2], []);
od;
return PBRNC(out[1], out[2]);
end);
InstallMethod(AsPBR, "for a boolean matrix", [IsBooleanMat],
function(x)
local dim, succ;
dim := DimensionOfMatrixOverSemiring(x);
if not IsEvenInt(dim) then
ErrorNoReturn("the 1st argument (a boolean matrix)is not of ",
"even dimension");
fi;
succ := Successors(x);
return PBRNC(succ{[1 .. dim / 2]}, succ{[dim / 2 + 1 .. dim]});
end);
InstallMethod(AsPBR, "for a boolean mat and pos int",
[IsBooleanMat, IsPosInt],
function(mat, n)
local m, nbs, k, i, j;
if not IsEvenInt(n) then
ErrorNoReturn("the 2nd argument (a pos. int) is not even");
fi;
m := DimensionOfMatrixOverSemiring(mat);
if not IsEvenInt(m) then
ErrorNoReturn("the 1st argument (a boolean matrix) ",
"does not have even dimension");
fi;
nbs := [List([1 .. n / 2], x -> []),
List([1 .. n / 2], x -> [])];
if n <= m then
for i in [1 .. n / 2] do
for j in [1 .. n] do
if mat[i][j] then
Add(nbs[1][i], j);
fi;
od;
od;
for i in [n / 2 + 1 .. n] do
for j in [1 .. n] do
if mat[i][j] then
Add(nbs[2][i - n / 2], j);
fi;
od;
od;
else
k := (n - m) / 2;
for i in [1 .. m / 2] do
for j in [1 .. m / 2] do
if mat[i][j] then
Add(nbs[1][i], j);
fi;
od;
for j in [m / 2 + 1 .. m] do
if mat[i][j] then
Add(nbs[1][i], j + k);
fi;
od;
od;
for i in [m / 2 + 1 .. m] do
for j in [1 .. m / 2] do
if mat[i][j] then
Add(nbs[2][i - m / 2], j);
fi;
od;
for j in [m / 2 + 1 .. m] do
if mat[i][j] then
Add(nbs[2][i - m / 2], j + k);
fi;
od;
od;
fi;
return PBRNC(nbs[1], nbs[2]);
end);
# TODO(later) 2 arg version of this
InstallMethod(AsTransformation, "for a pbr", [IsPBR],
function(x)
local out, n, i;
if not IsTransformationPBR(x) then
ErrorNoReturn("the argument (a pbr) does not define a transformation");
fi;
out := [];
n := x![1];
for i in [2 .. n + 1] do
out[i - 1] := x![i][1] - n;
od;
return Transformation(out);
end);
# TODO(later) 2 arg version of this
InstallMethod(AsPartialPerm, "for a pbr", [IsPBR],
function(x)
if not IsPartialPermPBR(x) then
ErrorNoReturn("the argument (a pbr) does not define a partial perm");
fi;
return AsPartialPerm(AsBipartition(x));
end);
# TODO(later) 2 arg version of this
InstallMethod(AsPermutation, "for a pbr", [IsPBR],
function(x)
if not IsPermPBR(x) then
ErrorNoReturn("the argument (a pbr) does not define a permutation");
fi;
return AsPermutation(AsBipartition(x));
end);
InstallMethod(RandomPBR, "for a pos int", [IsPosInt],
function(n)
local digraph;
digraph := RandomDigraph(2 * n);
return PBRNC(OutNeighbours(digraph){[1 .. n]},
OutNeighbours(digraph){[n + 1 .. 2 * n]});
end);
InstallMethod(RandomPBR, "for a pos int", [IsPosInt, IsFloat],
function(n, p)
local digraph;
digraph := RandomDigraph(2 * n, p);
return PBRNC(OutNeighbours(digraph){[1 .. n]},
OutNeighbours(digraph){[n + 1 .. 2 * n]});
end);
InstallMethod(PBR, "for pair of dense list",
[IsDenseList, IsDenseList],
function(left, right)
local deg, i;
if Length(left) <> Length(right) then
ErrorNoReturn("the arguments (dense lists) do not have equal length");
fi;
deg := Length(left);
for i in [1 .. deg] do
if not IsHomogeneousList(left[i]) then
ErrorNoReturn("expected a homogeneous list in position ", i,
" of the 1st argument (a dense list)");
elif not IsHomogeneousList(right[i]) then
ErrorNoReturn("expected a homogeneous list in position ", i,
" of the 2nd argument (a dense list)");
elif not ForAll(left[i], j -> IsInt(j) and j <> 0
and j <= deg and j >= -deg)
or not ForAll(right[i], j -> IsInt(j) and j <> 0
and j <= deg and j >= -deg) then
# TODO(later) more informative
ErrorNoReturn("the entries in the arguments are not integers ",
"in [", -deg, " .. -1] or [1 .. ", deg, "]");
fi;
od;
return PBRNC(left, right);
end);
InstallGlobalFunction(PBRNC,
function(arg...)
local left, right, n, i, j;
arg := StructuralCopy(arg);
left := arg[1]; # things adjacent to positives
right := arg[2]; # things adjacent to negatives
n := Length(left);
for i in [1 .. n] do
for j in [1 .. Length(left[i])] do
if left[i][j] < 0 then
left[i][j] := -left[i][j] + n;
fi;
od;
left[i] := ShallowCopy(left[i]);
Sort(left[i]);
for j in [1 .. Length(right[i])] do
if right[i][j] < 0 then
right[i][j] := -right[i][j] + n;
fi;
od;
right[i] := ShallowCopy(right[i]);
Sort(right[i]);
od;
MakeImmutable(arg);
arg := Concatenation([Length(arg[1])], Concatenation(arg));
return Objectify(TYPE_PBR(arg[1]), arg);
end);
InstallMethod(DegreeOfPBR, "for a pbr",
[IsPBR], pbr -> pbr![1]);
InstallMethod(\*, "for pbrs", IsIdenticalObj,
[IsPBR, IsPBR],
function(x, y)
local n, out, x_seen, y_seen, empty, x_dfs, y_dfs, tmp, i, j;
n := x![1];
out := Concatenation([n], List([1 .. 2 * n], x -> BlistList([1 .. 2 * n],
[])));
x_seen := BlistList([1 .. 2 * n], []);
y_seen := BlistList([1 .. 2 * n], []);
empty := BlistList([1 .. 2 * n], []);
x_dfs := function(i, adj) # starting in x
local j;
if x_seen[i] then
return;
fi;
x_seen[i] := true;
for j in x![i + 1] do
if j <= n then
adj[j] := true;
else # j > n
y_dfs(j - n, adj);
fi;
od;
return;
end;
y_dfs := function(i, adj) # starting in y
local j;
if y_seen[i] then
return;
fi;
y_seen[i] := true;
for j in y![i + 1] do
if j > n then
adj[j] := true;
else # j <= n
x_dfs(j + n, adj);
fi;
od;
return;
end;
tmp := [];
for i in [1 .. n] do # find everything connected to vertex i
for j in x![i + 1] do
if j <= n then
out[i + 1][j] := true;
elif IsBound(tmp[j]) then
UNITE_BLIST(out[i + 1], tmp[j]);
else
tmp[j] := BlistList([1 .. 2 * n], []);
IntersectBlist(x_seen, empty);
IntersectBlist(y_seen, empty);
x_seen[i] := true;
y_dfs(j - n, tmp[j]);
UNITE_BLIST(out[i + 1], tmp[j]);
fi;
if SizeBlist(out[i + 1]) = 2 * n then
break;
fi;
od;
od;
for i in [n + 1 .. 2 * n] do # find everything connected to vertex i
for j in y![i + 1] do
if j > n then
out[i + 1][j] := true;
elif IsBound(tmp[j]) then
UNITE_BLIST(out[i + 1], tmp[j]);
else
tmp[j] := BlistList([1 .. 2 * n], []);
IntersectBlist(x_seen, empty);
IntersectBlist(y_seen, empty);
y_seen[i] := true;
x_dfs(j + n, tmp[j]);
UNITE_BLIST(out[i + 1], tmp[j]);
fi;
if SizeBlist(out[i + 1]) = 2 * n then
break;
fi;
od;
od;
for i in [2 .. 2 * n + 1] do
out[i] := ListBlist([1 .. 2 * n], out[i]);
od;
return Objectify(TYPE_PBR(n), out);
end);
InstallMethod(ExtRepOfObj, "for a pbr",
[IsPBR],
function(x)
local n, out, i, j, k;
n := x![1];
out := [[], []];
for i in [0, 1] do
for j in [1 + n * i .. n + n * i] do
Add(out[i + 1], []);
for k in x![j + 1] do
if k > n then
AddSet(out[i + 1][j - n * i], -(k - n));
else
AddSet(out[i + 1][j - n * i], k);
fi;
od;
od;
od;
return out;
end);
# These ViewObj and PrintObj methods are here because Print(ext[1]) produces
# nicer output than Print(ViewString(ext[1])). The ViewString and PrintString
# methods are required for use in things like Green's relations.
InstallMethod(ViewObj, "for a pbr", [IsPBR], PrintObj);
InstallMethod(PrintObj, "for a pbr", [IsPBR],
function(x)
local ext;
ext := ExtRepOfObj(x);
Print("\>\>PBR(\>\>", ext[1], "\<\<,");
if Length(String(ext[1])) > 72 or Length(String(ext[2])) > 72 then
Print("\n");
else
Print(" ");
fi;
Print("\>\>", ext[2], "\<\<\<\<)");
end);
InstallMethod(ViewString, "for a pbr", [IsPBR], PrintString);
InstallMethod(PrintString, "for a pbr",
[IsPBR],
function(x)
local ext, str;
ext := ExtRepOfObj(x);
str := Concatenation("\>\>PBR(\>\>", PrintString(ext[1]), "\<\<,");
if Length(String(ext[1])) > 72 or Length(String(ext[2])) > 72 then
Append(str, "\n");
else
Append(str, " ");
fi;
Append(str, "\>\>");
Append(str, PrintString(ext[2]));
Append(str, "\<\<\<\<)");
# print empty lists with two spaces for consistency
# see https://github.com/gap-system/gap/pull/5418
return ReplacedString(str, "[ ]", "[ ]");
end);
InstallMethod(String, "for a pbr", [IsPBR],
function(x)
local ext, str;
ext := ExtRepOfObj(x);
str := Concatenation("PBR(", String(ext[1]), ", ", String(ext[2]), ")");
# print empty lists with two spaces for consistency
# see https://github.com/gap-system/gap/pull/5418
return ReplacedString(str, "[ ]", "[ ]");
end);
InstallMethod(\=, "for pbrs", IsIdenticalObj,
[IsPBR, IsPBR],
function(x, y)
local n, i;
n := x![1];
for i in [1 .. 2 * n + 1] do
if x![i] <> y![i] then
return false;
fi;
od;
return true;
end);
InstallMethod(\<, "for pbrs", IsIdenticalObj,
[IsPBR, IsPBR],
function(x, y)
local n, i;
n := x![1];
for i in [1 .. 2 * n + 1] do
if x![i] < y![i] then
return true;
elif x![i] > y![i] then
return false;
fi;
od;
return false;
end);
InstallMethod(OneMutable, "for a pbr",
[IsPBR],
function(x)
local n, out, i;
n := x![1];
out := [n];
for i in [1 .. n] do
out[i + 1] := [i + n];
out[i + n + 1] := [i];
od;
return Objectify(TYPE_PBR(n), out);
end);
[ Dauer der Verarbeitung: 0.11 Sekunden
(vorverarbeitet)
]
|
2026-03-28
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