mirror of
https://github.com/kalmarek/Groups.jl.git
synced 2024-11-19 06:30:29 +01:00
Merge branch 'master' into enh/GroupsCore
This commit is contained in:
commit
394d9a7aac
8
.github/workflows/TagBot.yml
vendored
8
.github/workflows/TagBot.yml
vendored
@ -1,11 +1,15 @@
|
||||
name: TagBot
|
||||
on:
|
||||
schedule:
|
||||
- cron: 0 * * * *
|
||||
issue_comment:
|
||||
types:
|
||||
- created
|
||||
workflow_dispatch:
|
||||
jobs:
|
||||
TagBot:
|
||||
if: github.event_name == 'workflow_dispatch' || github.actor == 'JuliaTagBot'
|
||||
runs-on: ubuntu-latest
|
||||
steps:
|
||||
- uses: JuliaRegistries/TagBot@v1
|
||||
with:
|
||||
token: ${{ secrets.GITHUB_TOKEN }}
|
||||
ssh: ${{ secrets.DOCUMENTER_KEY }}
|
||||
|
||||
2
.github/workflows/runtests.yml
vendored
2
.github/workflows/runtests.yml
vendored
@ -14,7 +14,7 @@ jobs:
|
||||
matrix:
|
||||
version:
|
||||
- '1.3'
|
||||
- '1.5'
|
||||
- '1'
|
||||
- 'nightly'
|
||||
os:
|
||||
- ubuntu-latest
|
||||
|
||||
12
Project.toml
12
Project.toml
@ -4,26 +4,24 @@ authors = ["Marek Kaluba <kalmar@amu.edu.pl>"]
|
||||
version = "0.6.0"
|
||||
|
||||
[deps]
|
||||
AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
|
||||
GroupsCore = "d5909c97-4eac-4ecc-a3dc-fdd0858a4120"
|
||||
KnuthBendix = "c2604015-7b3d-4a30-8a26-9074551ec60a"
|
||||
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
|
||||
OrderedCollections = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
|
||||
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
|
||||
ThreadsX = "ac1d9e8a-700a-412c-b207-f0111f4b6c0d"
|
||||
|
||||
[compat]
|
||||
AbstractAlgebra = "^0.13.0, ^0.14.0, ^0.15.0"
|
||||
AbstractAlgebra = "0.15, 0.16"
|
||||
GroupsCore = "^0.3"
|
||||
KnuthBendix = "^0.2.0"
|
||||
KnuthBendix = "^0.2.1"
|
||||
OrderedCollections = "1"
|
||||
ThreadsX = "^0.1.0"
|
||||
julia = "1.3, 1.4, 1.5"
|
||||
julia = "1.3, 1.4, 1.5, 1.6"
|
||||
|
||||
[extras]
|
||||
AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
|
||||
BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
|
||||
PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
|
||||
Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
|
||||
|
||||
[targets]
|
||||
test = ["Test", "BenchmarkTools", "PermutationGroups"]
|
||||
test = ["Test", "BenchmarkTools", "AbstractAlgebra"]
|
||||
|
||||
152
README.md
152
README.md
@ -1,7 +1,153 @@
|
||||
# Groups
|
||||
[](https://travis-ci.org/kalmarek/Groups.jl)
|
||||
[](https://github.com/kalmarek/Groups.jl/actions/workflows/runtests.yml)
|
||||
[](https://codecov.io/gh/kalmarek/Groups.jl)
|
||||
|
||||
A very rudimentary implementation of finitely-presented groups (syllable representation). Relatively complete are only [automorphism groups of free groups](https://github.com/kalmarek/Groups.jl/blob/master/src/AutGroup.jl) and [wreath products](https://github.com/kalmarek/Groups.jl/blob/master/src/WreathProducts.jl) (which are not finitely-presented, but based on the standard normal form).
|
||||
An implementation of finitely-presented groups together with normalization (using Knuth-Bendix procedure).
|
||||
|
||||
Have a look into `test` directory for eample use.
|
||||
The package implements `AbstractFPGroup` with three concrete types: `FreeGroup`, `FPGroup` and `AutomorphismGroup`. Here's an example usage:
|
||||
|
||||
```julia
|
||||
julia> using Groups, GroupsCore
|
||||
|
||||
julia> A = Alphabet([:a, :A, :b, :B, :c, :C], [2, 1, 4, 3, 6, 5])
|
||||
Alphabet of Symbol:
|
||||
1. :a = (:A)⁻¹
|
||||
2. :A = (:a)⁻¹
|
||||
3. :b = (:B)⁻¹
|
||||
4. :B = (:b)⁻¹
|
||||
5. :c = (:C)⁻¹
|
||||
6. :C = (:c)⁻¹
|
||||
|
||||
julia> F = FreeGroup(A)
|
||||
free group on 3 generators
|
||||
|
||||
julia> a,b,c = gens(F)
|
||||
3-element Vector{FPGroupElement{FreeGroup{Symbol}, KnuthBendix.Word{UInt8}}}:
|
||||
a
|
||||
b
|
||||
c
|
||||
|
||||
julia> a*inv(a)
|
||||
(empty word)
|
||||
|
||||
julia> (a*b)^2
|
||||
a*b*a*b
|
||||
|
||||
julia> commutator(a, b)
|
||||
A*B*a*b
|
||||
|
||||
julia> x = a*b; y = inv(b)*a;
|
||||
|
||||
julia> x*y
|
||||
a^2
|
||||
|
||||
```
|
||||
Let's create a quotient of the free group above:
|
||||
```julia
|
||||
julia> ε = one(F);
|
||||
|
||||
julia> G = FPGroup(F, [a^2 => ε, b^3=> ε, (a*b)^7=>ε, (a*b*a*inv(b))^6 => ε, commutator(a, c) => ε, commutator(b, c) => ε ])
|
||||
┌ Warning: Maximum number of rules (100) reached. The rewriting system may not be confluent.
|
||||
│ You may retry `knuthbendix` with a larger `maxrules` kwarg.
|
||||
└ @ KnuthBendix ~/.julia/packages/KnuthBendix/i93Np/src/kbs.jl:6
|
||||
⟨a, b, c | a^2 => (empty word), b^3 => (empty word), a*b*a*b*a*b*a*b*a*b*a*b*a*b => (empty word), a*b*a*B*a*b*a*B*a*b*a*B*a*b*a*B*a*b*a*B*a*b*a*B => (empty word), A*C*a*c => (empty word), B*C*b*c => (empty word)⟩
|
||||
|
||||
```
|
||||
As you can see from the warning, the Knuth-Bendix procedure has not completed successfully. This means that we only are able to approximate the word problem in `G`, i.e. if the equality (`==`) of two group elements may return `false` even if group elements are equal. Let us try with a larger maximal number of rules in the underlying rewriting system.
|
||||
|
||||
```julia
|
||||
julia> G = FPGroup(F, [a^2 => ε, b^3=> ε, (a*b)^7=>ε, (a*b*a*inv(b))^6 => ε, commutator(a, c) => ε, commutator(b, c) => ε ], maxrules=500)
|
||||
⟨a, b, c | a^2 => (empty word), b^3 => (empty word), a*b*a*b*a*b*a*b*a*b*a*b*a*b => (empty word), a*b*a*B*a*b*a*B*a*b*a*B*a*b*a*B*a*b*a*B*a*b*a*B => (empty word), A*C*a*c => (empty word), B*C*b*c => (empty word)⟩
|
||||
|
||||
```
|
||||
This time there was no warning, i.e. Knuth-Bendix completion was successful and we may treat the equality (`==`) as true mathematical equality. Note that `G` is the direct product of `ℤ = ⟨ c ⟩` and a quotient of van Dyck `(2,3,7)`-group. Let's create a random word and reduce it as an element of `G`.
|
||||
```julia
|
||||
julia> using Random; Random.seed!(1); w = Groups.Word(rand(1:length(A), 16))
|
||||
KnuthBendix.Word{UInt16}: 4·6·1·1·1·6·5·1·5·2·3·6·2·4·2·6
|
||||
|
||||
julia> F(w) # freely reduced w
|
||||
B*C*a^4*c*A*b*C*A*B*A*C
|
||||
|
||||
julia> G(w) # w as an element of G
|
||||
B*a*b*a*B*a*C^2
|
||||
|
||||
julia> F(w) # freely reduced w
|
||||
B*C*a^4*c*A*b*C*A*B*A*C
|
||||
|
||||
julia> word(ans) # the underlying word in A
|
||||
KnuthBendix.Word{UInt8}: 4·6·1·1·1·1·5·2·3·6·2·4·2·6
|
||||
|
||||
julia> G(w) # w as an element of G
|
||||
B*a*b*a*B*a*C^2
|
||||
|
||||
julia> word(ans) # the underlying word in A
|
||||
KnuthBendix.Word{UInt8}: 4·1·3·1·4·1·6·6
|
||||
|
||||
```
|
||||
As we can see the underlying words change according to where they are reduced.
|
||||
Note that a word `w` (of type `Word <: AbstractWord`) is just a sequence of numbers -- pointers to letters of an `Alphabet`. Without the alphabet `w` has no meaning.
|
||||
|
||||
### Automorphism Groups
|
||||
|
||||
Relatively complete is the support for the automorphisms of free groups, as given by Gersten presentation:
|
||||
```julia
|
||||
julia> saut = SpecialAutomorphismGroup(F, maxrules=100)
|
||||
┌ Warning: Maximum number of rules (100) reached. The rewriting system may not be confluent.
|
||||
│ You may retry `knuthbendix` with a larger `maxrules` kwarg.
|
||||
└ @ KnuthBendix ~/.julia/packages/KnuthBendix/i93Np/src/kbs.jl:6
|
||||
automorphism group of free group on 3 generators
|
||||
|
||||
julia> S = gens(saut)
|
||||
12-element Vector{Automorphism{FreeGroup{Symbol},…}}:
|
||||
ϱ₁.₂
|
||||
ϱ₁.₃
|
||||
ϱ₂.₁
|
||||
ϱ₂.₃
|
||||
ϱ₃.₁
|
||||
ϱ₃.₂
|
||||
λ₁.₂
|
||||
λ₁.₃
|
||||
λ₂.₁
|
||||
λ₂.₃
|
||||
λ₃.₁
|
||||
λ₃.₂
|
||||
|
||||
julia> x, y, z = S[1], S[12], S[6];
|
||||
|
||||
julia> f = x*y*inv(z)
|
||||
ϱ₁.₂*λ₃.₂*ϱ₃.₂^-1
|
||||
|
||||
julia> g = inv(z)*y*x
|
||||
ϱ₃.₂^-1*ϱ₁.₂*λ₃.₂
|
||||
|
||||
julia> word(f), word(g)
|
||||
(KnuthBendix.Word{UInt8}: 1·12·18, KnuthBendix.Word{UInt8}: 18·1·12)
|
||||
|
||||
```
|
||||
Even though Knuth-Bendix did not finish successfully in automorphism groups we have another ace in our sleeve to solve the word problem: evaluation.
|
||||
Lets have a look at the images of generators under those automorphisms:
|
||||
```julia
|
||||
julia> evaluate(f) # or to be more verbose...
|
||||
(a*b, b, b*c*B)
|
||||
|
||||
julia> Groups.domain(g)
|
||||
(a, b, c)
|
||||
|
||||
julia> Groups.evaluate!(Groups.domain(g), g)
|
||||
(a*b, b, b*c*B)
|
||||
|
||||
```
|
||||
Since these automorphism map the standard generating set to the same new generating set, they should be considered as equal! And indeed they are:
|
||||
```julia
|
||||
julia> f == g
|
||||
true
|
||||
```
|
||||
This is what is happening behind the scenes:
|
||||
1. words are reduced using a rewriting system
|
||||
2. if resulting words are equal `true` is returned
|
||||
3. if they are not equal `Groups.equality_data` is computed for each argument (here: the images of generators) and the result of comparison is returned.
|
||||
|
||||
Moreover we try to amortize the cost of computing those images. That is a hash of `equality_daata` is lazily stored in each group element and used as needed. Essentially only if `true` is returned, but comparison of words returns `false` recomputation of images is needed (to guard against hash collisions).
|
||||
|
||||
----
|
||||
This package was developed for computations in [1712.07167](https://arxiv.org/abs/1712.07167) and in [1812.03456](https://arxiv.org/abs/1812.03456). If you happen to use this package please cite either of them.
|
||||
|
||||
346
src/AutGroup.jl
346
src/AutGroup.jl
@ -1,346 +0,0 @@
|
||||
export Automorphism, AutGroup, Aut, SAut
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# AutSymbol/ AutGroup / Automorphism
|
||||
#
|
||||
|
||||
struct RTransvect
|
||||
i::Int8
|
||||
j::Int8
|
||||
end
|
||||
|
||||
struct LTransvect
|
||||
i::Int8
|
||||
j::Int8
|
||||
end
|
||||
|
||||
struct FlipAut
|
||||
i::Int8
|
||||
end
|
||||
|
||||
struct PermAut
|
||||
perm::AbstractAlgebra.Generic.Perm{Int8}
|
||||
end
|
||||
|
||||
struct Identity end
|
||||
|
||||
struct AutSymbol <: GSymbol
|
||||
id::Symbol
|
||||
pow::Int8
|
||||
fn::Union{LTransvect, RTransvect, PermAut, FlipAut, Identity}
|
||||
end
|
||||
|
||||
# taken from ValidatedNumerics, under under the MIT "Expat" License:
|
||||
# https://github.com/JuliaIntervals/ValidatedNumerics.jl/blob/master/LICENSE.md
|
||||
function subscriptify(n::Integer)
|
||||
subscript_0 = Int(0x2080) # Char(0x2080) -> subscript 0
|
||||
return join([Char(subscript_0 + i) for i in reverse(digits(n))], "")
|
||||
end
|
||||
|
||||
function id_autsymbol()
|
||||
return AutSymbol(Symbol("(id)"), 0, Identity())
|
||||
end
|
||||
|
||||
function transvection_R(i::Integer, j::Integer, pow::Integer=1)
|
||||
if 0 < i < 10 && 0 < j < 10
|
||||
id = Symbol(:ϱ, subscriptify(i), subscriptify(j))
|
||||
else
|
||||
id = Symbol(:ϱ, subscriptify(i), "." ,subscriptify(j))
|
||||
end
|
||||
return AutSymbol(id, pow, RTransvect(i, j))
|
||||
end
|
||||
|
||||
function transvection_L(i::Integer, j::Integer, pow::Integer=1)
|
||||
if 0 < i < 10 && 0 < j < 10
|
||||
id = Symbol(:λ, subscriptify(i), subscriptify(j))
|
||||
else
|
||||
id = Symbol(:λ, subscriptify(i), "." ,subscriptify(j))
|
||||
end
|
||||
return AutSymbol(id, pow, LTransvect(i, j))
|
||||
end
|
||||
|
||||
function flip(i::Integer, pow::Integer=1)
|
||||
iseven(pow) && return id_autsymbol()
|
||||
id = Symbol(:ɛ, subscriptify(i))
|
||||
return AutSymbol(id, 1, FlipAut(i))
|
||||
end
|
||||
|
||||
function AutSymbol(p::AbstractAlgebra.Generic.Perm, pow::Integer=1)
|
||||
if pow != 1
|
||||
p = p^pow
|
||||
end
|
||||
|
||||
if any(p.d[i] != i for i in eachindex(p.d))
|
||||
id = Symbol(:σ, "₍", join([subscriptify(i) for i in p.d],""), "₎")
|
||||
return AutSymbol(id, 1, PermAut(p))
|
||||
end
|
||||
return id_autsymbol()
|
||||
end
|
||||
|
||||
ϱ(i::Integer, j::Integer, pow::Integer=1) = transvection_R(i, j, pow)
|
||||
λ(i::Integer, j::Integer, pow::Integer=1) = transvection_L(i, j, pow)
|
||||
ε(i::Integer, pow::Integer=1) = flip(i, pow)
|
||||
σ(v::AbstractAlgebra.Generic.Perm, pow::Integer=1) = AutSymbol(v, pow)
|
||||
|
||||
function change_pow(s::AutSymbol, n::Integer)
|
||||
iszero(n) && id_autsymbol()
|
||||
|
||||
symbol = s.fn
|
||||
if symbol isa FlipAut
|
||||
return flip(symbol.i, n)
|
||||
elseif symbol isa PermAut
|
||||
return AutSymbol(symbol.perm, n)
|
||||
elseif symbol isa RTransvect
|
||||
return transvection_R(symbol.i, symbol.j, n)
|
||||
elseif symbol isa LTransvect
|
||||
return transvection_L(symbol.i, symbol.j, n)
|
||||
elseif symbol isa Identity
|
||||
return id_autsymbol()
|
||||
else
|
||||
throw(DomainError("Unknown type of AutSymbol: $s"))
|
||||
end
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# AutGroup / Automorphism
|
||||
#
|
||||
|
||||
mutable struct AutGroup{N} <: AbstractFPGroup
|
||||
objectGroup::FreeGroup
|
||||
gens::Vector{AutSymbol}
|
||||
end
|
||||
|
||||
mutable struct Automorphism{N} <: GWord{AutSymbol}
|
||||
symbols::Vector{AutSymbol}
|
||||
modified::Bool
|
||||
savedhash::UInt
|
||||
parent::AutGroup{N}
|
||||
|
||||
function Automorphism{N}(f::Vector{AutSymbol}) where {N}
|
||||
return new{N}(f, true, zero(UInt))
|
||||
end
|
||||
end
|
||||
|
||||
Base.eltype(::Type{AutGroup{N}}) where N = Automorphism{N}
|
||||
GroupsCore.parent_type(::Type{Automorphism{N}}) where N = AutGroup{N}
|
||||
|
||||
function AutGroup(G::FreeGroup; special=false)
|
||||
S = AutSymbol[]
|
||||
n = length(gens(G))
|
||||
n == 0 && return AutGroup{n}(G, S)
|
||||
|
||||
indexing = [[i,j] for i in 1:n for j in 1:n if i≠j]
|
||||
|
||||
rmuls = [ϱ(i,j) for (i,j) in indexing]
|
||||
lmuls = [λ(i,j) for (i,j) in indexing]
|
||||
|
||||
append!(S, [rmuls; lmuls])
|
||||
|
||||
if !special
|
||||
flips = [ε(i) for i in 1:n]
|
||||
syms = [σ(p) for p in AbstractAlgebra.SymmetricGroup(Int8(n))][2:end]
|
||||
|
||||
append!(S, [flips; syms])
|
||||
end
|
||||
return AutGroup{n}(G, S)
|
||||
end
|
||||
|
||||
Aut(G::Group) = AutGroup(G)
|
||||
SAut(G::Group) = AutGroup(G, special=true)
|
||||
|
||||
Automorphism{N}(s::AutSymbol) where N = Automorphism{N}(AutSymbol[s])
|
||||
|
||||
function (G::AutGroup{N})(f::AutSymbol) where N
|
||||
g = Automorphism{N}([f])
|
||||
setparent!(g, G)
|
||||
return g
|
||||
end
|
||||
|
||||
(G::AutGroup{N})(g::Automorphism{N}) where N = (setparent!(g, G); g)
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# AutSymbol defining functions && evaluation
|
||||
# NOTE: all automorphisms operate on a tuple of FreeWords INPLACE!
|
||||
#
|
||||
|
||||
function (ϱ::RTransvect)(v, pow::Integer=1)
|
||||
rmul!(v[ϱ.i], v[ϱ.j]^pow)
|
||||
return v
|
||||
end
|
||||
|
||||
function (λ::LTransvect)(v, pow::Integer=1)
|
||||
lmul!(v[λ.i], v[λ.j]^pow)
|
||||
return v
|
||||
end
|
||||
|
||||
function (σ::PermAut)(v, pow::Integer=1)
|
||||
w = deepcopy(v)
|
||||
s = (σ.perm^pow).d
|
||||
@inbounds for k in eachindex(v)
|
||||
v[k].symbols = w[s[k]].symbols
|
||||
end
|
||||
return v
|
||||
end
|
||||
|
||||
function (ɛ::FlipAut)(v, pow::Integer=1)
|
||||
@inbounds if isodd(pow)
|
||||
v[ɛ.i].symbols = inv(v[ɛ.i]).symbols
|
||||
end
|
||||
return v
|
||||
end
|
||||
|
||||
(::Identity)(v, pow::Integer=1) = v
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Functional call overloads for evaluation of AutSymbol and Automorphism
|
||||
#
|
||||
|
||||
(s::AutSymbol)(v::NTuple{N, T}) where {N, T} = s.fn(v, s.pow)::NTuple{N, T}
|
||||
|
||||
function (f::Automorphism{N})(v::NTuple{N, T}) where {N, T}
|
||||
for s in syllables(f)
|
||||
v = s(v)::NTuple{N, T}
|
||||
end
|
||||
return v
|
||||
end
|
||||
|
||||
function domain(G::AutGroup{N}) where N
|
||||
F = G.objectGroup
|
||||
return ntuple(i->F(F.gens[i]), N)
|
||||
end
|
||||
|
||||
evaluate(f::Automorphism) = f(domain(parent(f)))
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# hashing && equality
|
||||
#
|
||||
|
||||
function hash_internal(
|
||||
g::Automorphism,
|
||||
h::UInt = 0x7d28276b01874b19; # hash(Automorphism)
|
||||
# alternatively: 0xcbf29ce484222325 from FNV-1a algorithm
|
||||
images = compute_images(g),
|
||||
prime = 0x00000100000001b3, # prime from FNV-1a algorithm
|
||||
)
|
||||
return foldl((h,x) -> hash(x, h)*prime, images, init = hash(parent(g), h))
|
||||
end
|
||||
|
||||
function compute_images(g::Automorphism)
|
||||
images = evaluate(g)
|
||||
for im in images
|
||||
reduce!(im)
|
||||
end
|
||||
return images
|
||||
end
|
||||
|
||||
function Base.:(==)(g::Automorphism{N}, h::Automorphism{N}) where N
|
||||
syllables(g) == syllables(h) && return true
|
||||
img_computed, imh_computed = false, false
|
||||
|
||||
if ismodified(g)
|
||||
img = compute_images(g) # sets modified bit
|
||||
hash(g; images=img)
|
||||
img_computed = true
|
||||
end
|
||||
|
||||
if ismodified(h)
|
||||
imh = compute_images(h) # sets modified bit
|
||||
hash(h; images=imh)
|
||||
imh_computed = true
|
||||
end
|
||||
|
||||
@assert !ismodified(g) && !ismodified(h)
|
||||
# cheap
|
||||
# if hashes differ, images must have differed as well
|
||||
hash(g) != hash(h) && return false
|
||||
|
||||
# hashes equal, hence either equal elements, or a hash conflict
|
||||
begin
|
||||
if !img_computed
|
||||
img_task = Threads.@spawn img = compute_images(g)
|
||||
# img = compute_images(g)
|
||||
end
|
||||
if !imh_computed
|
||||
imh_task = Threads.@spawn imh = compute_images(h)
|
||||
# imh = compute_images(h)
|
||||
end
|
||||
!img_computed && fetch(img_task)
|
||||
!imh_computed && fetch(imh_task)
|
||||
end
|
||||
|
||||
img != imh && @warn "hash collision in == :" g h
|
||||
return img == imh
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# String I/O
|
||||
#
|
||||
|
||||
function Base.show(io::IO, G::AutGroup)
|
||||
print(io, "Automorphism Group of $(G.objectGroup)\n")
|
||||
print(io, "Generated by $(gens(G))")
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Reduction
|
||||
#
|
||||
|
||||
getperm(s::AutSymbol) = s.fn.perm^s.pow
|
||||
|
||||
function simplifyperms!(::Type{Bool}, w::Automorphism{N}) where N
|
||||
reduced = true
|
||||
for i in 1:syllablelength(w)-1
|
||||
s, ns = syllables(w)[i], syllables(w)[i+1]
|
||||
if isone(s)
|
||||
continue
|
||||
elseif s.fn isa PermAut && ns.fn isa PermAut
|
||||
reduced = false
|
||||
setmodified!(w)
|
||||
syllables(w)[i+1] = AutSymbol(getperm(s)*getperm(ns))
|
||||
syllables(w)[i] = change_pow(s, 0)
|
||||
end
|
||||
end
|
||||
filter!(!isone, syllables(w))
|
||||
return reduced
|
||||
end
|
||||
|
||||
function reduce!(w::Automorphism)
|
||||
reduced = false
|
||||
while !reduced
|
||||
reduced = simplifyperms!(Bool, w) && freereduce!(Bool, w)
|
||||
end
|
||||
return w
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Abelianization (natural Representation to GL(N,Z))
|
||||
#
|
||||
|
||||
abelianize(A::Automorphism{N}) where N = image(A, abelianize; n=N)
|
||||
|
||||
# homomorphism definition
|
||||
abelianize(; n::Integer=1) = Matrix{Int}(I, n, n)
|
||||
abelianize(a::AutSymbol; n::Int=1) = abelianize(a.fn, n, a.pow)
|
||||
|
||||
function abelianize(a::Union{RTransvect, LTransvect}, n::Int, pow)
|
||||
x = Matrix{Int}(I, n, n)
|
||||
x[a.i,a.j] = pow
|
||||
return x
|
||||
end
|
||||
|
||||
function abelianize(a::FlipAut, n::Int, pow)
|
||||
x = Matrix{Int}(I, n, n)
|
||||
x[a.i,a.i] = -1
|
||||
return x
|
||||
end
|
||||
|
||||
abelianize(a::PermAut, n::Integer, pow) = Matrix{Int}(I, n, n)[(a.perm^pow).d, :]
|
||||
abelianize(a::Identity, n::Integer, pow) = abelianize(;n=n)
|
||||
135
src/FPGroups.jl
135
src/FPGroups.jl
@ -1,135 +0,0 @@
|
||||
###############################################################################
|
||||
#
|
||||
# FPSymbol/FPGroupElem/FPGroup definition
|
||||
#
|
||||
###############################################################################
|
||||
|
||||
struct FPSymbol <: GSymbol
|
||||
id::Symbol
|
||||
pow::Int
|
||||
end
|
||||
|
||||
FPGroupElem = GroupWord{FPSymbol}
|
||||
|
||||
mutable struct FPGroup <: AbstractFPGroup
|
||||
gens::Vector{FPSymbol}
|
||||
rels::Dict{FreeGroupElem, FreeGroupElem}
|
||||
|
||||
function FPGroup(gens::Vector{T}, rels::Dict{FreeGroupElem, FreeGroupElem}) where {T<:GSymbol}
|
||||
G = new(gens)
|
||||
G.rels = Dict(G(k) => G(v) for (k,v) in rels)
|
||||
return G
|
||||
end
|
||||
end
|
||||
|
||||
export FPGroupElem, FPGroup
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Type and parent object methods
|
||||
#
|
||||
|
||||
Base.eltype(::Type{FPGroup}) = FPGroupElem
|
||||
GroupsCore.parent_type(::Type{FPGroupElem}) = FPGroup
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# FPSymbol constructors
|
||||
#
|
||||
|
||||
FPSymbol(s::Symbol) = FPSymbol(s, 1)
|
||||
FPSymbol(s::String) = FPSymbol(Symbol(s))
|
||||
FPSymbol(s::GSymbol) = FPSymbol(s.id, s.pow)
|
||||
|
||||
FPGroup(n::Int, symbol::String="f") = FPGroup([Symbol(symbol,i) for i in 1:n])
|
||||
FPGroup(a::AbstractVector) = FPGroup([FPSymbol(i) for i in a])
|
||||
FPGroup(gens::Vector{FPSymbol}) = FPGroup(gens, Dict{FreeGroupElem, FreeGroupElem}())
|
||||
|
||||
FPGroup(H::FreeGroup) = FPGroup([FPSymbol(s) for s in H.gens])
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Parent object call overloads
|
||||
#
|
||||
|
||||
function (G::FPGroup)(w::GWord)
|
||||
if isempty(w)
|
||||
return one(G)
|
||||
end
|
||||
|
||||
@boundscheck for s in syllables(w)
|
||||
i = findfirst(g -> g.id == s.id, G.gens)
|
||||
i == 0 && throw(DomainError("Symbol $s does not belong to $G."))
|
||||
s.pow % G.gens[i].pow != 0 && throw(
|
||||
DomainError("Symbol $s doesn't belong to $G."))
|
||||
end
|
||||
|
||||
w = FPGroupElem(FPSymbol.(syllables(w)))
|
||||
setparent!(w, G)
|
||||
return reduce!(w)
|
||||
end
|
||||
|
||||
(G::FPGroup)(s::GSymbol) = G(FPGroupElem(s))
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# String I/O
|
||||
#
|
||||
|
||||
function Base.show(io::IO, G::FPGroup)
|
||||
print(io, "FPgroup on $(length(G.gens)) generators ")
|
||||
strrels = join(G.rels, ", ")
|
||||
if length(strrels) > 200
|
||||
print(io, "⟨ ", join(G.gens, ", "), " | $(length(G.rels)) relation(s) ⟩.")
|
||||
else
|
||||
print(io, "⟨ ", join(G.gens, ", "), " | ", join(G.rels, ", "), " ⟩.")
|
||||
end
|
||||
end
|
||||
|
||||
function reduce!(W::FPGroupElem)
|
||||
reduced = false
|
||||
while !reduced
|
||||
W = replace(W, parent(W).rels)
|
||||
reduced = freereduce!(Bool, W)
|
||||
end
|
||||
return W
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Misc
|
||||
#
|
||||
###############################################################################
|
||||
|
||||
freepreimage(G::FPGroup) = parent(first(keys(G.rels)))
|
||||
freepreimage(g::FPGroupElem) = freepreimage(parent(g))(syllables(g))
|
||||
|
||||
function add_rels!(G::FPGroup, newrels::Dict{FreeGroupElem,FreeGroupElem})
|
||||
for w in keys(newrels)
|
||||
haskey(G.rels, w) && continue
|
||||
G.rels[w] = newrels[w]
|
||||
end
|
||||
return G
|
||||
end
|
||||
|
||||
function Base.:/(G::FPGroup, newrels::Vector{FPGroupElem})
|
||||
for r in newrels
|
||||
parent(r) == G || throw(DomainError(
|
||||
"Can not form quotient group: $r is not an element of $G"))
|
||||
end
|
||||
H = deepcopy(G)
|
||||
F = freepreimage(H)
|
||||
newrels = Dict(freepreimage(r) => one(F) for r in newrels)
|
||||
add_rels!(H, newrels)
|
||||
return H
|
||||
end
|
||||
|
||||
function Base.:/(F::FreeGroup, rels::Vector{FreeGroupElem})
|
||||
for r in rels
|
||||
parent(r) == F || throw(DomainError(
|
||||
"Can not form quotient group: $r is not an element of $F"))
|
||||
end
|
||||
G = FPGroup(FPSymbol.(F.gens))
|
||||
G.rels = Dict(rel => one(F) for rel in unique(rels))
|
||||
return G
|
||||
end
|
||||
@ -1,73 +0,0 @@
|
||||
###############################################################################
|
||||
#
|
||||
# FreeSymbol/FreeGroupElem/FreeGroup definition
|
||||
#
|
||||
|
||||
struct FreeSymbol <: GSymbol
|
||||
id::Symbol
|
||||
pow::Int
|
||||
end
|
||||
|
||||
FreeGroupElem = GroupWord{FreeSymbol}
|
||||
|
||||
mutable struct FreeGroup <: AbstractFPGroup
|
||||
gens::Vector{FreeSymbol}
|
||||
|
||||
function FreeGroup(gens::AbstractVector{T}) where {T<:GSymbol}
|
||||
G = new(gens)
|
||||
G.gens = gens
|
||||
return G
|
||||
end
|
||||
end
|
||||
|
||||
export FreeGroupElem, FreeGroup
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Type and parent object methods
|
||||
#
|
||||
|
||||
Base.eltype(::Type{FreeGroup}) = FreeGroupElem
|
||||
GroupsCore.parent_type(::Type{FreeGroupElem}) = FreeGroup
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# FreeSymbol constructors
|
||||
#
|
||||
|
||||
FreeSymbol(s::Symbol) = FreeSymbol(s,1)
|
||||
FreeSymbol(s::AbstractString) = FreeSymbol(Symbol(s))
|
||||
FreeSymbol(s::GSymbol) = FreeSymbol(s.id, s.pow)
|
||||
|
||||
FreeGroup(n::Int, symbol::String="f") = FreeGroup([Symbol(symbol,i) for i in 1:n])
|
||||
FreeGroup(a::AbstractVector) = FreeGroup(FreeSymbol.(a))
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Parent object call overloads
|
||||
#
|
||||
|
||||
function (G::FreeGroup)(w::GroupWord{FreeSymbol})
|
||||
for s in syllables(w)
|
||||
i = findfirst(g -> g.id == s.id, G.gens)
|
||||
isnothing(i) && throw(DomainError(
|
||||
"Symbol $s does not belong to $G."))
|
||||
s.pow % G.gens[i].pow == 0 || throw(DomainError(
|
||||
"Symbol $s doesn't belong to $G."))
|
||||
end
|
||||
setparent!(w, G)
|
||||
return reduce!(w)
|
||||
end
|
||||
|
||||
(G::FreeGroup)(s::GSymbol) = G(FreeGroupElem(s))
|
||||
(G::FreeGroup)(v::AbstractVector{<:GSymbol}) = G(FreeGroupElem(FreeSymbol.(v)))
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# String I/O
|
||||
#
|
||||
|
||||
function Base.show(io::IO, G::FreeGroup)
|
||||
print(io, "Free group on $(length(G.gens)) generators: ")
|
||||
join(io, G.gens, ", ")
|
||||
end
|
||||
140
src/Groups.jl
140
src/Groups.jl
@ -1,144 +1,24 @@
|
||||
module Groups
|
||||
|
||||
using GroupsCore
|
||||
using LinearAlgebra
|
||||
using ThreadsX
|
||||
|
||||
import AbstractAlgebra
|
||||
import KnuthBendix
|
||||
import KnuthBendix: AbstractWord, Alphabet, Word
|
||||
import KnuthBendix: alphabet
|
||||
import Random
|
||||
|
||||
export gens, FreeGroup, Aut, SAut
|
||||
|
||||
include("types.jl")
|
||||
|
||||
include("FreeGroup.jl")
|
||||
include("FPGroups.jl")
|
||||
include("AutGroup.jl")
|
||||
|
||||
include("symbols.jl")
|
||||
include("words.jl")
|
||||
include("hashing.jl")
|
||||
include("freereduce.jl")
|
||||
include("arithmetic.jl")
|
||||
include("findreplace.jl")
|
||||
|
||||
module New
|
||||
import OrderedCollections: OrderedSet
|
||||
|
||||
include("new_types.jl")
|
||||
include("new_hashing.jl")
|
||||
export Alphabet, AutomorphismGroup, FreeGroup, FreeGroup, FPGroup, FPGroupElement, SpecialAutomorphismGroup
|
||||
export alphabet, evaluate, word
|
||||
|
||||
include("types.jl")
|
||||
include("hashing.jl")
|
||||
include("normalform.jl")
|
||||
include("new_autgroups.jl")
|
||||
include("autgroups.jl")
|
||||
|
||||
include("groups/sautFn.jl")
|
||||
include("groups/mcg.jl")
|
||||
|
||||
end # module New
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# String I/O
|
||||
#
|
||||
|
||||
function Base.show(io::IO, W::GWord)
|
||||
if length(W) == 0
|
||||
print(io, "(id)")
|
||||
else
|
||||
join(io, (string(s) for s in syllables(W)), "*")
|
||||
end
|
||||
end
|
||||
|
||||
function Base.show(io::IO, s::T) where {T<:GSymbol}
|
||||
if s.pow == 1
|
||||
print(io, string(s.id))
|
||||
else
|
||||
print(io, "$(s.id)^$(s.pow)")
|
||||
end
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Misc
|
||||
#
|
||||
|
||||
GroupsCore.gens(G::AbstractFPGroup) = G.(G.gens)
|
||||
|
||||
"""
|
||||
wlmetric_ball(S::AbstractVector{<:GroupElem}
|
||||
[, center=one(first(S)); radius=2, op=*])
|
||||
Compute metric ball as a list of elements of non-decreasing length, given the
|
||||
word-length metric on the group generated by `S`. The ball is centered at `center`
|
||||
(by default: the identity element). `radius` and `op` keywords specify the
|
||||
radius and multiplication operation to be used.
|
||||
"""
|
||||
function wlmetric_ball_serial(S::AbstractVector{T}; radius = 2, op = *) where {T}
|
||||
@assert radius > 0
|
||||
old = unique!([one(first(S)), S...])
|
||||
sizes = [1, length(old)]
|
||||
for i in 2:radius
|
||||
new = collect(op(o, s) for o in @view(old[sizes[end-1]:end]) for s in S)
|
||||
append!(old, new)
|
||||
resize!(new, 0)
|
||||
old = unique!(old)
|
||||
push!(sizes, length(old))
|
||||
end
|
||||
return old, sizes[2:end]
|
||||
end
|
||||
|
||||
function wlmetric_ball_thr(S::AbstractVector{T}; radius = 2, op = *) where {T}
|
||||
@assert radius > 0
|
||||
old = unique!([one(first(S)), S...])
|
||||
sizes = [1, length(old)]
|
||||
for r in 2:radius
|
||||
begin
|
||||
new =
|
||||
ThreadsX.collect(op(o, s) for o in @view(old[sizes[end-1]:end]) for s in S)
|
||||
ThreadsX.foreach(hash, new)
|
||||
end
|
||||
append!(old, new)
|
||||
resize!(new, 0)
|
||||
old = ThreadsX.unique(old)
|
||||
push!(sizes, length(old))
|
||||
end
|
||||
return old, sizes[2:end]
|
||||
end
|
||||
|
||||
function wlmetric_ball_serial(S::AbstractVector{T}, center::T; radius = 2, op = *) where {T}
|
||||
E, sizes = wlmetric_ball_serial(S, radius = radius, op = op)
|
||||
isone(center) && return E, sizes
|
||||
return c .* E, sizes
|
||||
end
|
||||
|
||||
function wlmetric_ball_thr(S::AbstractVector{T}, center::T; radius = 2, op = *) where {T}
|
||||
E, sizes = wlmetric_ball_thr(S, radius = radius, op = op)
|
||||
isone(center) && return E, sizes
|
||||
return c .* E, sizes
|
||||
end
|
||||
|
||||
function wlmetric_ball(
|
||||
S::AbstractVector{T},
|
||||
center::T = one(first(S));
|
||||
radius = 2,
|
||||
op = *,
|
||||
threading = true,
|
||||
) where {T}
|
||||
threading && return wlmetric_ball_thr(S, center, radius = radius, op = op)
|
||||
return wlmetric_ball_serial(S, center, radius = radius, op = op)
|
||||
end
|
||||
|
||||
"""
|
||||
image(w::GWord, homomorphism; kwargs...)
|
||||
Evaluate homomorphism `homomorphism` on a group word (element) `w`.
|
||||
`homomorphism` needs to implement
|
||||
> `hom(w; kwargs...)`,
|
||||
where `hom(;kwargs...)` returns the value at the identity element.
|
||||
"""
|
||||
function image(w::GWord, hom; kwargs...)
|
||||
return reduce(
|
||||
*,
|
||||
(hom(s; kwargs...) for s in syllables(w)),
|
||||
init = hom(; kwargs...),
|
||||
)
|
||||
end
|
||||
|
||||
include("wl_ball.jl")
|
||||
end # of module Groups
|
||||
|
||||
@ -1,93 +0,0 @@
|
||||
function Base.inv(W::T) where T<:GWord
|
||||
length(W) == 0 && return one(W)
|
||||
G = parent(W)
|
||||
w = T([inv(s) for s in Iterators.reverse(syllables(W))])
|
||||
return setparent!(w, G)
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# Binary operators
|
||||
#
|
||||
|
||||
function Base.push!(w::GWord{T}, s::T) where T <: GSymbol
|
||||
push!(syllables(w), s)
|
||||
return w
|
||||
end
|
||||
|
||||
function Base.pushfirst!(w::GWord{T}, s::T) where T <: GSymbol
|
||||
pushfirst!(syllables(w), s)
|
||||
return w
|
||||
end
|
||||
|
||||
function Base.append!(w::T, v::T) where T <: GWord
|
||||
append!(syllables(w), syllables(v))
|
||||
return w
|
||||
end
|
||||
|
||||
function Base.prepend!(w::T, v::T) where T <: GWord
|
||||
prepend!(syllables(w), syllables(v))
|
||||
return w
|
||||
end
|
||||
|
||||
Base.append!(w::T, v::T, others::Vararg{T,N}) where {N,T <: GWord} =
|
||||
append!(append!(w, v), others...)
|
||||
|
||||
function rmul!(out::T, x::T, y::T) where T<: GWord
|
||||
if out === x
|
||||
out = deepcopy(out)
|
||||
return freereduce!(append!(out, y))
|
||||
elseif out === y
|
||||
out = deepcopy(out)
|
||||
return freereduce!(prepend!(out, x))
|
||||
else
|
||||
slenx = syllablelength(x)
|
||||
sleny = syllablelength(y)
|
||||
resize!(syllables(out), slenx+sleny)
|
||||
syllables(out)[1:slenx] .= syllables(x)
|
||||
syllables(out)[slenx+1:slenx+sleny] .= syllables(y)
|
||||
return freereduce!(out)
|
||||
end
|
||||
end
|
||||
|
||||
rmul!(out::T, v::T) where T<:GWord = freereduce!(append!(out, v))
|
||||
lmul!(out::T, v::T) where T<:GWord = freereduce!(prepend!(out, v))
|
||||
|
||||
lmul!(out::T, x::T, y::T) where T <: GWord = rmul!(out, y, x)
|
||||
|
||||
GroupsCore.mul!(out::T, x::T, y::T) where T <: GWord = rmul!(out, x, y)
|
||||
|
||||
Base.:(*)(W::GW, Z::GW) where GW <: GWord = rmul!(deepcopy(W), W, Z)
|
||||
Base.:(*)(W::GWord, s::GSymbol) = freereduce!(push!(deepcopy(W), s))
|
||||
Base.:(*)(s::GSymbol, W::GWord) = freereduce!(pushfirst!(deepcopy(W), s))
|
||||
|
||||
function Base.power_by_squaring(W::GWord, p::Integer)
|
||||
if p < 0
|
||||
return Base.power_by_squaring(inv(W), -p)
|
||||
elseif p == 0
|
||||
return one(W)
|
||||
elseif p == 1
|
||||
return W
|
||||
elseif p == 2
|
||||
return W*W
|
||||
end
|
||||
W = deepcopy(W)
|
||||
t = trailing_zeros(p) + 1
|
||||
p >>= t
|
||||
while (t -= 1) > 0
|
||||
append!(W, W)
|
||||
end
|
||||
Z = deepcopy(W)
|
||||
while p > 0
|
||||
t = trailing_zeros(p) + 1
|
||||
p >>= t
|
||||
while (t -= 1) >= 0
|
||||
append!(W, W)
|
||||
end
|
||||
append!(Z, W)
|
||||
end
|
||||
|
||||
return freereduce!(Z)
|
||||
end
|
||||
|
||||
Base.:(^)(x::GWord, n::Integer) = Base.power_by_squaring(x,n)
|
||||
@ -84,17 +84,6 @@ Base.show(io::IO, ::Type{<:FPGroupElement{<:AutomorphismGroup{T}}}) where {T} =
|
||||
|
||||
Base.show(io::IO, A::AutomorphismGroup) = print(io, "automorphism group of ", object(A))
|
||||
|
||||
|
||||
function Base.show(io::IO, ::MIME"text/plain", a::FPGroupElement{<:AutomorphismGroup})
|
||||
println(io, " ┌ $(a):")
|
||||
im = evaluate(a)
|
||||
d = domain(a)
|
||||
for (x, imx) in zip(d, im[1:end-1])
|
||||
println(io, " │ $x ↦ $imx")
|
||||
end
|
||||
print(io, " └ $(last(d)) ↦ $(last(im))")
|
||||
end
|
||||
|
||||
## Automorphism Evaluation
|
||||
|
||||
domain(f::FPGroupElement{<:AutomorphismGroup}) = deepcopy(parent(f).domain)
|
||||
@ -116,4 +105,3 @@ function evaluate!(
|
||||
end
|
||||
|
||||
evaluate!(t::NTuple{N, T}, s::GSymbol, A, tmp=one(first(t))) where {N, T} = throw("you need to implement `evaluate!(::$(typeof(t)), ::$(typeof(s)), ::Alphabet, tmp=one(first(t)))`")
|
||||
|
||||
@ -1,183 +0,0 @@
|
||||
###############################################################################
|
||||
#
|
||||
# Replacement of symbols / sub-words
|
||||
#
|
||||
|
||||
issubsymbol(s::GSymbol, t::GSymbol) =
|
||||
s.id == t.id && (0 ≤ s.pow ≤ t.pow || 0 ≥ s.pow ≥ t.pow)
|
||||
|
||||
function issubsymbol(s::FreeSymbol, w::GWord, sindex::Integer)
|
||||
@boundscheck 1 ≤ sindex ≤ syllablelength(w) || throw(BoundsError(w, sindex))
|
||||
return issubsymbol(s, syllables(w)[sindex])
|
||||
end
|
||||
|
||||
function issubword(z::GWord, w::GWord, sindex::Integer)
|
||||
isempty(z) && return true
|
||||
@boundscheck 1 ≤ sindex ≤ syllablelength(w) || throw(BoundsError(w, sindex))
|
||||
n = syllablelength(z)
|
||||
n == 1 && return issubsymbol(first(syllables(z)), syllables(w)[sindex])
|
||||
|
||||
lastindex = sindex + n - 1
|
||||
lastindex > syllablelength(w) && return false
|
||||
|
||||
issubsymbol(first(z), syllables(w)[sindex]) || return false
|
||||
issubsymbol(syllables(z)[end], syllables(w)[lastindex]) || return false
|
||||
for (zidx, widx) in zip(2:n-1, sindex+1:lastindex-1)
|
||||
syllables(z)[zidx] == syllables(w)[widx] || return false
|
||||
end
|
||||
return true
|
||||
end
|
||||
|
||||
"""
|
||||
|
||||
Find the first syllable index k>=i such that Z < syllables(W)[k:k+syllablelength(Z)-1]
|
||||
"""
|
||||
function Base.findnext(subword::GWord, word::GWord, start::Integer)
|
||||
@boundscheck 1 ≤ start ≤ syllablelength(word) || throw(BoundsError(word, start))
|
||||
isempty(subword) && return start
|
||||
stop = syllablelength(word) - syllablelength(subword) +1
|
||||
|
||||
for idx in start:1:stop
|
||||
issubword(subword, word, idx) && return idx
|
||||
end
|
||||
return nothing
|
||||
end
|
||||
|
||||
function Base.findnext(s::FreeSymbol, word::GWord, start::Integer)
|
||||
@boundscheck 1 ≤ start ≤ syllablelength(word) || throw(BoundsError(word, start))
|
||||
isone(s) && return start
|
||||
stop = syllablelength(word)
|
||||
|
||||
for idx in start:1:stop
|
||||
issubsymbol(s, word, idx) && return idx
|
||||
end
|
||||
return nothing
|
||||
end
|
||||
|
||||
function Base.findprev(subword::GWord, word::GWord, start::Integer)
|
||||
@boundscheck 1 ≤ start ≤ syllablelength(word) || throw(BoundsError(word, start))
|
||||
isempty(subword) && return start
|
||||
stop = 1
|
||||
|
||||
for idx in start:-1:1
|
||||
issubword(subword, word, idx) && return idx
|
||||
end
|
||||
return nothing
|
||||
end
|
||||
|
||||
function Base.findprev(s::FreeSymbol, word::GWord, start::Integer)
|
||||
@boundscheck 1 ≤ start ≤ syllablelength(word) || throw(BoundsError(word, start))
|
||||
isone(s) && return start
|
||||
stop = 1
|
||||
|
||||
for idx in start:-1:stop
|
||||
issubsymbol(s, word, idx) && return idx
|
||||
end
|
||||
return nothing
|
||||
end
|
||||
|
||||
Base.findfirst(subword::GWord, word::GWord) = findnext(subword, word, 1)
|
||||
Base.findlast(subword::GWord, word::GWord) =
|
||||
findprev(subword, word, syllablelength(word)-syllablelength(subword)+1)
|
||||
|
||||
function Base.replace!(out::GW, W::GW, lhs_rhs::Pair{GS, T}; count::Integer=typemax(Int)) where
|
||||
{GS<:GSymbol, T<:GWord, GW<:GWord}
|
||||
(count == 0 || isempty(W)) && return W
|
||||
count < 0 && throw(DomainError(count, "`count` must be non-negative."))
|
||||
|
||||
lhs, rhs = lhs_rhs
|
||||
|
||||
sW = syllables(W)
|
||||
sW_idx = 1
|
||||
r = something(findnext(lhs, W, sW_idx), 0)
|
||||
|
||||
sout = syllables(out)
|
||||
resize!(sout, 0)
|
||||
sizehint!(sout, syllablelength(W))
|
||||
|
||||
c = 0
|
||||
|
||||
while !iszero(r)
|
||||
append!(sout, view(sW, sW_idx:r-1))
|
||||
a, b = divrem(sW[r].pow, lhs.pow)
|
||||
|
||||
if b != 0
|
||||
push!(sout, change_pow(sW[r], b))
|
||||
end
|
||||
|
||||
append!(sout, repeat(syllables(rhs), a))
|
||||
|
||||
sW_idx = r+1
|
||||
sW_idx > syllablelength(W) && break
|
||||
|
||||
r = something(findnext(lhs, W, sW_idx), 0)
|
||||
c += 1
|
||||
c == count && break
|
||||
end
|
||||
append!(sout, sW[sW_idx:end])
|
||||
return freereduce!(out)
|
||||
end
|
||||
|
||||
function Base.replace!(out::GW, W::GW, lhs_rhs::Pair{T, T}; count::Integer=typemax(Int)) where
|
||||
{GW<:GWord, T <: GWord}
|
||||
(count == 0 || isempty(W)) && return W
|
||||
count < 0 && throw(DomainError(count, "`count` must be non-negative."))
|
||||
|
||||
lhs, rhs = lhs_rhs
|
||||
lhs_slen = syllablelength(lhs)
|
||||
lhs_slen == 1 && return replace!(out, W, first(syllables(lhs))=>rhs; count=count)
|
||||
|
||||
sW = syllables(W)
|
||||
sW_idx = 1
|
||||
r = something(findnext(lhs, W, sW_idx), 0)
|
||||
|
||||
sout = syllables(out)
|
||||
resize!(sout, 0)
|
||||
sizehint!(sout, syllablelength(W))
|
||||
|
||||
c = 0
|
||||
|
||||
while !iszero(r)
|
||||
append!(sout, view(sW, sW_idx:r-1))
|
||||
|
||||
exp = sW[r].pow - first(syllables(lhs)).pow
|
||||
if exp != 0
|
||||
push!(sout, change_pow(sW[r], exp))
|
||||
end
|
||||
|
||||
append!(sout, syllables(rhs))
|
||||
|
||||
exp = sW[r+lhs_slen-1].pow - last(syllables(lhs)).pow
|
||||
if exp != 0
|
||||
push!(sout, change_pow(sW[r+lhs_slen-1], exp))
|
||||
end
|
||||
|
||||
sW_idx = r+lhs_slen
|
||||
sW_idx > syllablelength(W) && break
|
||||
|
||||
r = something(findnext(lhs, W, sW_idx), 0)
|
||||
c += 1
|
||||
c == count && break
|
||||
end
|
||||
|
||||
# copy the rest
|
||||
append!(sout, sW[sW_idx:end])
|
||||
return freereduce!(out)
|
||||
end
|
||||
|
||||
function Base.replace(W::GW, lhs_rhs::Pair{T, T}; count::Integer=typemax(Int)) where
|
||||
{GW<:GWord, T <: GWord}
|
||||
return replace!(one(W), W, lhs_rhs; count=count)
|
||||
end
|
||||
|
||||
function Base.replace(W::GW, subst_dict::Dict{T,T}) where {GW<:GWord, T<:GWord}
|
||||
out = W
|
||||
for toreplace in reverse!(sort!(collect(keys(subst_dict)), by=length))
|
||||
replacement = subst_dict[toreplace]
|
||||
if length(toreplace) > length(out)
|
||||
continue
|
||||
end
|
||||
out = replace(out, toreplace=>replacement)
|
||||
end
|
||||
return out
|
||||
end
|
||||
@ -1,49 +0,0 @@
|
||||
###############################################################################
|
||||
#
|
||||
# Naive reduction
|
||||
#
|
||||
|
||||
function freereduce!(::Type{Bool}, w::GWord)
|
||||
if syllablelength(w) == 1
|
||||
filter!(!isone, syllables(w))
|
||||
return syllablelength(w) == 1
|
||||
end
|
||||
|
||||
reduced = true
|
||||
@inbounds for i in 1:syllablelength(w)-1
|
||||
s, ns = syllables(w)[i], syllables(w)[i+1]
|
||||
if isone(s)
|
||||
continue
|
||||
elseif s.id === ns.id
|
||||
reduced = false
|
||||
p1 = s.pow
|
||||
p2 = ns.pow
|
||||
|
||||
syllables(w)[i+1] = change_pow(s, p1 + p2)
|
||||
syllables(w)[i] = change_pow(s, 0)
|
||||
end
|
||||
end
|
||||
if !reduced
|
||||
filter!(!isone, syllables(w))
|
||||
setmodified!(w)
|
||||
end
|
||||
return reduced
|
||||
end
|
||||
|
||||
function freereduce!(w::GWord)
|
||||
reduced = false
|
||||
while !reduced
|
||||
reduced = freereduce!(Bool, w)
|
||||
end
|
||||
return w
|
||||
end
|
||||
|
||||
reduce!(w::GWord) = freereduce!(w)
|
||||
|
||||
"""
|
||||
reduce(w::GWord)
|
||||
performs reduction/simplification of a group element (word in generators).
|
||||
The default reduction is the reduction in the free group reduction.
|
||||
More specific procedures should be dispatched on `GWord`s type parameter.
|
||||
"""
|
||||
Base.reduce(w::GWord) = reduce!(deepcopy(w))
|
||||
@ -3,7 +3,7 @@ include("gersten_relations.jl")
|
||||
|
||||
function SpecialAutomorphismGroup(F::FreeGroup; ordering = KnuthBendix.LenLex, kwargs...)
|
||||
|
||||
n = length(KnuthBendix.alphabet(F)) ÷ 2
|
||||
n = length(alphabet(F)) ÷ 2
|
||||
A, rels = gersten_relations(n, commutative = false)
|
||||
S = KnuthBendix.letters(A)[1:2(n^2-n)]
|
||||
|
||||
@ -15,6 +15,6 @@ end
|
||||
KnuthBendix.alphabet(G::AutomorphismGroup{<:FreeGroup}) = alphabet(rewriting(G))
|
||||
|
||||
function relations(G::AutomorphismGroup{<:FreeGroup})
|
||||
n = length(KnuthBendix.alphabet(object(G))) ÷ 2
|
||||
n = length(alphabet(object(G))) ÷ 2
|
||||
return last(gersten_relations(n, commutative = false))
|
||||
end
|
||||
|
||||
@ -31,12 +31,11 @@ end
|
||||
|
||||
function Base.show(io::IO, t::Transvection)
|
||||
id = if t.id === :ϱ
|
||||
"ϱ"
|
||||
'ϱ'
|
||||
else # if t.id === :λ
|
||||
"λ"
|
||||
'λ'
|
||||
end
|
||||
# print(io, id, Groups.subscriptify(t.i), ".", Groups.subscriptify(t.j))
|
||||
print(io, id, "_", t.i, ",", t.j)
|
||||
print(io, id, subscriptify(t.i), '.', subscriptify(t.j))
|
||||
t.inv && print(io, "^-1")
|
||||
end
|
||||
|
||||
|
||||
@ -1,34 +1,46 @@
|
||||
###############################################################################
|
||||
#
|
||||
# hashing, deepcopy and ==
|
||||
#
|
||||
## Hashing
|
||||
|
||||
function hash_internal(W::GWord)
|
||||
reduce!(W)
|
||||
h = hasparent(W) ? hash(parent(W)) : zero(UInt)
|
||||
return hash(syllables(W), hash(typeof(W), h))
|
||||
end
|
||||
equality_data(g::FPGroupElement) = (normalform!(g); word(g))
|
||||
|
||||
function Base.hash(W::GWord, h::UInt=UInt(0); kwargs...)
|
||||
if ismodified(W)
|
||||
savehash!(W, hash_internal(W; kwargs...))
|
||||
unsetmodified!(W)
|
||||
end
|
||||
return xor(savedhash(W), h)
|
||||
end
|
||||
bitget(h::UInt, n::Int) = Bool((h & (1 << n)) >> n)
|
||||
bitclear(h::UInt, n::Int) = h & ~(1 << n)
|
||||
bitset(h::UInt, n::Int) = h | (1 << n)
|
||||
bitset(h::UInt, v::Bool, n::Int) = v ? bitset(h, n) : bitclear(h, n)
|
||||
|
||||
# WARNING: Due to specialised (constant) hash function of GWords this one is actually necessary!
|
||||
function Base.deepcopy_internal(W::T, dict::IdDict) where T<:GWord
|
||||
G = parent(W)
|
||||
g = T(deepcopy(syllables(W)))
|
||||
setparent!(g, G)
|
||||
# We store hash of a word in field `savedhash` to use it as cheap proxy to
|
||||
# determine non-equal words. Additionally bits of `savehash` store boolean
|
||||
# information as follows
|
||||
# * `savedhash & 1` (the first bit): is word in normal form?
|
||||
# * `savedhash & 2` (the second bit): is the hash valid?
|
||||
const __BITFLAGS_MASK = ~(~(UInt(0)) << 2)
|
||||
|
||||
isnormalform(g::FPGroupElement) = bitget(g.savedhash, 0)
|
||||
_isvalidhash(g::FPGroupElement) = bitget(g.savedhash, 1)
|
||||
|
||||
_setnormalform(h::UInt, v::Bool) = bitset(h, v, 0)
|
||||
_setvalidhash(h::UInt, v::Bool) = bitset(h, v, 1)
|
||||
|
||||
_setnormalform!(g::FPGroupElement, v::Bool) = g.savedhash = _setnormalform(g.savedhash, v)
|
||||
_setvalidhash!(g::FPGroupElement, v::Bool) = g.savedhash = _setvalidhash(g.savedhash, v)
|
||||
|
||||
# To update hash use this internal method, possibly only after computing the
|
||||
# normal form of `g`:
|
||||
function _update_savedhash!(g::FPGroupElement, data)
|
||||
h = hash(data, hash(parent(g)))
|
||||
h = (h << count_ones(__BITFLAGS_MASK)) | (__BITFLAGS_MASK & g.savedhash)
|
||||
g.savedhash = _setvalidhash(h, true)
|
||||
return g
|
||||
end
|
||||
|
||||
function Base.:(==)(W::T, Z::T) where T <: GWord
|
||||
hash(W) != hash(Z) && return false # distinguishes parent and parentless words
|
||||
if hasparent(W) && hasparent(Z)
|
||||
parent(W) != parent(Z) && return false
|
||||
end
|
||||
return syllables(W) == syllables(Z)
|
||||
function Base.hash(g::FPGroupElement, h::UInt)
|
||||
_isvalidhash(g) || _update_savedhash!(g, equality_data(g))
|
||||
return hash(g.savedhash >> count_ones(__BITFLAGS_MASK), h)
|
||||
end
|
||||
|
||||
function Base.copyto!(res::FPGroupElement, g::FPGroupElement)
|
||||
@assert parent(res) === parent(g)
|
||||
resize!(word(res), length(word(g)))
|
||||
copyto!(word(res), word(g))
|
||||
res.savedhash = g.savedhash
|
||||
return res
|
||||
end
|
||||
|
||||
@ -1,46 +0,0 @@
|
||||
## Hashing
|
||||
|
||||
equality_data(g::FPGroupElement) = (normalform!(g); word(g))
|
||||
|
||||
bitget(h::UInt, n::Int) = Bool((h & (1 << n)) >> n)
|
||||
bitclear(h::UInt, n::Int) = h & ~(1 << n)
|
||||
bitset(h::UInt, n::Int) = h | (1 << n)
|
||||
bitset(h::UInt, v::Bool, n::Int) = v ? bitset(h, n) : bitclear(h, n)
|
||||
|
||||
# We store hash of a word in field `savedhash` to use it as cheap proxy to
|
||||
# determine non-equal words. Additionally bits of `savehash` store boolean
|
||||
# information as follows
|
||||
# * `savedhash & 1` (the first bit): is word in normal form?
|
||||
# * `savedhash & 2` (the second bit): is the hash valid?
|
||||
const __BITFLAGS_MASK = ~(~(UInt(0)) << 2)
|
||||
|
||||
isnormalform(g::FPGroupElement) = bitget(g.savedhash, 0)
|
||||
_isvalidhash(g::FPGroupElement) = bitget(g.savedhash, 1)
|
||||
|
||||
_setnormalform(h::UInt, v::Bool) = bitset(h, v, 0)
|
||||
_setvalidhash(h::UInt, v::Bool) = bitset(h, v, 1)
|
||||
|
||||
_setnormalform!(g::FPGroupElement, v::Bool) = g.savedhash = _setnormalform(g.savedhash, v)
|
||||
_setvalidhash!(g::FPGroupElement, v::Bool) = g.savedhash = _setvalidhash(g.savedhash, v)
|
||||
|
||||
# To update hash use this internal method, possibly only after computing the
|
||||
# normal form of `g`:
|
||||
function _update_savedhash!(g::FPGroupElement, data)
|
||||
h = hash(data, hash(parent(g)))
|
||||
h = (h << count_ones(__BITFLAGS_MASK)) | (__BITFLAGS_MASK & g.savedhash)
|
||||
g.savedhash = _setvalidhash(h, true)
|
||||
return g
|
||||
end
|
||||
|
||||
function Base.hash(g::FPGroupElement, h::UInt)
|
||||
_isvalidhash(g) || _update_savedhash!(g, equality_data(g))
|
||||
return hash(g.savedhash >> count_ones(__BITFLAGS_MASK), h)
|
||||
end
|
||||
|
||||
function Base.copyto!(res::FPGroupElement, g::FPGroupElement)
|
||||
@assert parent(res) === parent(g)
|
||||
resize!(word(res), length(word(g)))
|
||||
copyto!(word(res), word(g))
|
||||
res.savedhash = g.savedhash
|
||||
return res
|
||||
end
|
||||
@ -42,8 +42,5 @@ Defaults to the rewriting in the free group.
|
||||
"""
|
||||
@inline function normalform!(res::AbstractWord, g::FPGroupElement)
|
||||
isone(res) && isnormalform(g) && return append!(res, word(g))
|
||||
if isnormalform(g) && inv(alphabet(g), last(out)) != first(word(g))
|
||||
return append!(res, word(g))
|
||||
end
|
||||
return KnuthBendix.rewrite_from_left!(res, word(g), rewriting(parent(g)))
|
||||
end
|
||||
|
||||
@ -1,23 +0,0 @@
|
||||
change_pow(s::S, n::Integer) where S<:GSymbol = S(s.id, n)
|
||||
|
||||
function Base.iterate(s::GS, i=1) where GS<:GSymbol
|
||||
return i <= abs(s.pow) ? (change_pow(s, sign(s.pow)), i+1) : nothing
|
||||
end
|
||||
Base.size(s::GSymbol) = (abs(s.pow), )
|
||||
Base.length(s::GSymbol) = first(size(s))
|
||||
|
||||
Base.eltype(s::GS) where GS<:GSymbol = GS
|
||||
|
||||
Base.isone(s::GSymbol) = iszero(s.pow)
|
||||
Base.literal_pow(::typeof(^), s::Groups.GSymbol, ::Val{-1}) = inv(s)
|
||||
Base.inv(s::GSymbol) = change_pow(s, -s.pow)
|
||||
Base.hash(s::S, h::UInt) where S<:GSymbol = hash(s.id, hash(s.pow, hash(S, h)))
|
||||
|
||||
function Base.:(==)(s::GSymbol, t::GSymbol)
|
||||
isone(s) && isone(t) && return true
|
||||
s.pow == t.pow && s.id == t.id && return true
|
||||
return false
|
||||
end
|
||||
|
||||
Base.convert(::Type{GS}, s::GSymbol) where GS<:GSymbol = GS(s.id, s.pow)
|
||||
Base.convert(::Type{GS}, s::GS) where GS<:GSymbol = s
|
||||
238
src/types.jl
238
src/types.jl
@ -1,50 +1,210 @@
|
||||
## "Abstract" definitions
|
||||
|
||||
"""
|
||||
AbstractFPGroup
|
||||
|
||||
An Abstract type representing finitely presented groups. Every instance `` must implement
|
||||
* `KnuthBendix.alphabet(G::MyFPGroup)`
|
||||
* `rewriting(G::MyFPGroup)` : return the rewriting object which must implement
|
||||
> `KnuthBendix.rewrite_from_left!(u, v, rewriting(G))`.
|
||||
By default `alphabet(G)` is returned, which amounts to free rewriting in `G`.
|
||||
* `relations(G::MyFPGroup)` : return a set of defining relations.
|
||||
|
||||
AbstractFPGroup may also override `word_type(::Type{MyFPGroup}) = Word{UInt16}`,
|
||||
which controls the word type used for group elements. If a group has more than `255` generators you need to define e.g.
|
||||
> `word_type(::Type{MyFPGroup}) = Word{UInt16}`
|
||||
"""
|
||||
abstract type AbstractFPGroup <: GroupsCore.Group end
|
||||
|
||||
function Base.one(G::Gr) where Gr <: AbstractFPGroup
|
||||
El = eltype(G)
|
||||
id = El(eltype(El)[])
|
||||
id.parent = G
|
||||
return id
|
||||
word_type(G::AbstractFPGroup) = word_type(typeof(G))
|
||||
# the default:
|
||||
word_type(::Type{<:AbstractFPGroup}) = Word{UInt8}
|
||||
|
||||
# the default (results in free rewriting)
|
||||
rewriting(G::AbstractFPGroup) = alphabet(G)
|
||||
|
||||
Base.@propagate_inbounds function (G::AbstractFPGroup)(word::AbstractVector{<:Integer})
|
||||
@boundscheck @assert all(l -> 1 <= l <= length(KnuthBendix.alphabet(G)), word)
|
||||
return FPGroupElement(word_type(G)(word), G)
|
||||
end
|
||||
|
||||
"""
|
||||
GSymbol
|
||||
Represents a syllable. Abstract type which all group symbols of
|
||||
`AbstractFPGroups` should subtype. Each concrete subtype should implement fields:
|
||||
* `id` which is the `Symbol` representation/identification of a symbol
|
||||
* `pow` which is the (multiplicative) exponent of a symbol.
|
||||
"""
|
||||
abstract type GSymbol end
|
||||
## Group Interface
|
||||
|
||||
abstract type GWord{T<:GSymbol} <: GroupsCore.GroupElement end
|
||||
Base.one(G::AbstractFPGroup) = FPGroupElement(one(word_type(G)), G)
|
||||
|
||||
"""
|
||||
W::GroupWord{T} <: GWord{T<:GSymbol} <:GroupElem
|
||||
Basic representation of element of a finitely presented group.
|
||||
* `syllables(W)` return particular group syllables which multiplied constitute `W`
|
||||
group as a word in generators.
|
||||
* `parent(W)` return the parent group.
|
||||
Base.eltype(::Type{FPG}) where {FPG<:AbstractFPGroup} = FPGroupElement{FPG,word_type(FPG)}
|
||||
|
||||
As the reduction (inside the parent group) of word to normal form may be time
|
||||
consuming, we provide a shortcut that is useful in practice:
|
||||
`savehash!(W, h)` and `ismodified(W)` functions.
|
||||
When computing `hash(W)`, a reduction to normal form is performed and a
|
||||
persistent hash is stored inside `W`, setting `ismodified(W)` flag to `false`.
|
||||
This hash can be accessed by `savedhash(W)`.
|
||||
Future comparisons of `W` try not to perform reduction and use the stored hash as shortcut. Only when hashes collide reduction is performed. Whenever word `W` is
|
||||
changed, `ismodified(W)` returns `false` and stored hash is invalidated.
|
||||
"""
|
||||
include("iteration.jl")
|
||||
|
||||
mutable struct GroupWord{T} <: GWord{T}
|
||||
symbols::Vector{T}
|
||||
modified::Bool
|
||||
GroupsCore.ngens(G::AbstractFPGroup) = length(G.gens)
|
||||
|
||||
function GroupsCore.gens(G::AbstractFPGroup, i::Integer)
|
||||
@boundscheck 1 <= i <= GroupsCore.ngens(G)
|
||||
l = alphabet(G)[G.gens[i]]
|
||||
return FPGroupElement(word_type(G)([l]), G)
|
||||
end
|
||||
GroupsCore.gens(G::AbstractFPGroup) = [gens(G, i) for i in 1:GroupsCore.ngens(G)]
|
||||
|
||||
# TODO: ProductReplacementAlgorithm
|
||||
function Base.rand(rng::Random.AbstractRNG, rs::Random.SamplerTrivial{<:AbstractFPGroup})
|
||||
l = rand(10:100)
|
||||
G = rs[]
|
||||
nletters = length(alphabet(G))
|
||||
return FPGroupElement(word_type(G)(rand(1:nletters, l)), G)
|
||||
end
|
||||
|
||||
Base.isfinite(::AbstractFPGroup) = (@warn "using generic isfinite(::AbstractFPGroup): the returned `false` might be wrong"; false)
|
||||
|
||||
## FPGroupElement
|
||||
|
||||
mutable struct FPGroupElement{G<:AbstractFPGroup,W<:AbstractWord} <: GroupElement
|
||||
word::W
|
||||
savedhash::UInt
|
||||
parent::Group
|
||||
parent::G
|
||||
|
||||
function GroupWord{T}(symbols::AbstractVector{<:GSymbol}) where T
|
||||
return new{T}(symbols, true, zero(UInt))
|
||||
end
|
||||
GroupWord(v::AbstractVector{T}) where T<:GSymbol = GroupWord{T}(v)
|
||||
GroupWord{T}(s::GSymbol) where T<:GSymbol = GroupWord{T}(T[s])
|
||||
GroupWord(s::T) where T<:GSymbol = GroupWord{T}(s)
|
||||
FPGroupElement(word::W, G::AbstractFPGroup) where {W<:AbstractWord} =
|
||||
new{typeof(G),W}(word, UInt(0), G)
|
||||
|
||||
FPGroupElement(word::W, hash::UInt, G::AbstractFPGroup) where {W<:AbstractWord} =
|
||||
new{typeof(G),W}(word, hash, G)
|
||||
end
|
||||
|
||||
word(f::FPGroupElement) = f.word
|
||||
|
||||
#convenience
|
||||
KnuthBendix.alphabet(g::FPGroupElement) = alphabet(parent(g))
|
||||
|
||||
function Base.show(io::IO, f::FPGroupElement)
|
||||
f = normalform!(f)
|
||||
KnuthBendix.print_repr(io, word(f), alphabet(f))
|
||||
end
|
||||
|
||||
## GroupElement Interface for FPGroupElement
|
||||
|
||||
Base.parent(f::FPGroupElement) = f.parent
|
||||
GroupsCore.parent_type(::Type{<:FPGroupElement{G}}) where {G} = G
|
||||
|
||||
function Base.:(==)(g::FPGroupElement, h::FPGroupElement)
|
||||
@boundscheck @assert parent(g) === parent(h)
|
||||
normalform!(g)
|
||||
normalform!(h)
|
||||
hash(g) != hash(h) && return false
|
||||
return word(g) == word(h)
|
||||
end
|
||||
|
||||
function Base.deepcopy_internal(g::FPGroupElement, stackdict::IdDict)
|
||||
return FPGroupElement(copy(word(g)), g.savedhash, parent(g))
|
||||
end
|
||||
|
||||
Base.inv(g::FPGroupElement) = (G = parent(g); FPGroupElement(inv(alphabet(G), word(g)), G))
|
||||
|
||||
function Base.:(*)(g::FPGroupElement, h::FPGroupElement)
|
||||
@boundscheck @assert parent(g) === parent(h)
|
||||
return FPGroupElement(word(g) * word(h), parent(g))
|
||||
end
|
||||
|
||||
GroupsCore.isfiniteorder(g::FPGroupElement) = isone(g) ? true : (@warn "using generic isfiniteorder(::FPGroupElement): the returned `false` might be wrong"; false)
|
||||
|
||||
# additional methods:
|
||||
Base.isone(g::FPGroupElement) = (normalform!(g); isempty(word(g)))
|
||||
|
||||
## Free Groups
|
||||
|
||||
struct FreeGroup{T} <: AbstractFPGroup
|
||||
gens::Vector{T}
|
||||
alphabet::KnuthBendix.Alphabet{T}
|
||||
|
||||
function FreeGroup(gens, A::KnuthBendix.Alphabet) where {W}
|
||||
@assert length(gens) == length(unique(gens))
|
||||
@assert all(l -> l in KnuthBendix.letters(A), gens)
|
||||
return new{eltype(gens)}(gens, A)
|
||||
end
|
||||
end
|
||||
|
||||
function FreeGroup(A::Alphabet)
|
||||
@boundscheck @assert all(KnuthBendix.hasinverse(l, A) for l in KnuthBendix.letters(A))
|
||||
ltrs = KnuthBendix.letters(A)
|
||||
gens = Vector{eltype(ltrs)}()
|
||||
invs = Vector{eltype(ltrs)}()
|
||||
for l in ltrs
|
||||
l ∈ invs && continue
|
||||
push!(gens, l)
|
||||
push!(invs, inv(A, l))
|
||||
end
|
||||
|
||||
return FreeGroup(gens, A)
|
||||
end
|
||||
|
||||
function FreeGroup(n::Integer)
|
||||
symbols = Symbol[]
|
||||
inverses = Int[]
|
||||
sizehint!(symbols, 2n)
|
||||
sizehint!(inverses, 2n)
|
||||
for i in 1:n
|
||||
push!(symbols, Symbol(:f, i), Symbol(:F, i))
|
||||
push!(inverses, 2i, 2i-1)
|
||||
end
|
||||
return FreeGroup(symbols[1:2:2n], Alphabet(symbols, inverses))
|
||||
end
|
||||
|
||||
Base.show(io::IO, F::FreeGroup) = print(io, "free group on $(ngens(F)) generators")
|
||||
|
||||
# mandatory methods:
|
||||
KnuthBendix.alphabet(F::FreeGroup) = F.alphabet
|
||||
relations(F::FreeGroup) = Pair{eltype(F)}[]
|
||||
|
||||
# GroupsCore interface:
|
||||
# these are mathematically correct
|
||||
Base.isfinite(::FreeGroup) = false
|
||||
|
||||
GroupsCore.isfiniteorder(g::FPGroupElement{<:FreeGroup}) = isone(g) ? true : false
|
||||
|
||||
## FP Groups
|
||||
|
||||
struct FPGroup{T,R,S} <: AbstractFPGroup
|
||||
gens::Vector{T}
|
||||
relations::Vector{Pair{S,S}}
|
||||
rws::R
|
||||
end
|
||||
|
||||
KnuthBendix.alphabet(G::FPGroup) = alphabet(rewriting(G))
|
||||
rewriting(G::FPGroup) = G.rws
|
||||
|
||||
relations(G::FPGroup) = G.relations
|
||||
|
||||
function FPGroup(
|
||||
G::AbstractFPGroup,
|
||||
rels::AbstractVector{<:Pair{GEl,GEl}};
|
||||
ordering = KnuthBendix.LenLex,
|
||||
kwargs...,
|
||||
) where {GEl<:FPGroupElement}
|
||||
|
||||
O = ordering(alphabet(G))
|
||||
for (lhs, rhs) in rels
|
||||
@assert parent(lhs) === parent(rhs) === G
|
||||
end
|
||||
word_rels = [word(lhs) => word(rhs) for (lhs, rhs) in [relations(G); rels]]
|
||||
rws = KnuthBendix.RewritingSystem(word_rels, O)
|
||||
|
||||
KnuthBendix.knuthbendix!(rws; kwargs...)
|
||||
|
||||
return FPGroup(G.gens, rels, rws)
|
||||
end
|
||||
|
||||
function Base.show(io::IO, G::FPGroup)
|
||||
print(io, "⟨")
|
||||
join(io, gens(G), ", ")
|
||||
print(io, " | ")
|
||||
join(io, relations(G), ", ")
|
||||
print(io, "⟩")
|
||||
end
|
||||
|
||||
## GSymbol aka letter of alphabet
|
||||
|
||||
abstract type GSymbol end
|
||||
Base.literal_pow(::typeof(^), t::GSymbol, ::Val{-1}) = inv(t)
|
||||
|
||||
function subscriptify(n::Integer)
|
||||
subscript_0 = Int(0x2080) # Char(0x2080) -> subscript 0
|
||||
return join([Char(subscript_0 + i) for i in reverse(digits(n))], "")
|
||||
end
|
||||
|
||||
62
src/wl_ball.jl
Normal file
62
src/wl_ball.jl
Normal file
@ -0,0 +1,62 @@
|
||||
"""
|
||||
wlmetric_ball(S::AbstractVector{<:GroupElem}
|
||||
[, center=one(first(S)); radius=2, op=*])
|
||||
Compute metric ball as a list of elements of non-decreasing length, given the
|
||||
word-length metric on the group generated by `S`. The ball is centered at `center`
|
||||
(by default: the identity element). `radius` and `op` keywords specify the
|
||||
radius and multiplication operation to be used.
|
||||
"""
|
||||
function wlmetric_ball_serial(S::AbstractVector{T}; radius = 2, op = *) where {T}
|
||||
@assert radius > 0
|
||||
old = unique!([one(first(S)), S...])
|
||||
sizes = [1, length(old)]
|
||||
for i in 2:radius
|
||||
new = collect(op(o, s) for o in @view(old[sizes[end-1]:end]) for s in S)
|
||||
append!(old, new)
|
||||
resize!(new, 0)
|
||||
old = unique!(old)
|
||||
push!(sizes, length(old))
|
||||
end
|
||||
return old, sizes[2:end]
|
||||
end
|
||||
|
||||
function wlmetric_ball_thr(S::AbstractVector{T}; radius = 2, op = *) where {T}
|
||||
@assert radius > 0
|
||||
old = unique!([one(first(S)), S...])
|
||||
sizes = [1, length(old)]
|
||||
for r in 2:radius
|
||||
begin
|
||||
new =
|
||||
ThreadsX.collect(op(o, s) for o in @view(old[sizes[end-1]:end]) for s in S)
|
||||
ThreadsX.foreach(hash, new)
|
||||
end
|
||||
append!(old, new)
|
||||
resize!(new, 0)
|
||||
old = ThreadsX.unique(old)
|
||||
push!(sizes, length(old))
|
||||
end
|
||||
return old, sizes[2:end]
|
||||
end
|
||||
|
||||
function wlmetric_ball_serial(S::AbstractVector{T}, center::T; radius = 2, op = *) where {T}
|
||||
E, sizes = wlmetric_ball_serial(S, radius = radius, op = op)
|
||||
isone(center) && return E, sizes
|
||||
return c .* E, sizes
|
||||
end
|
||||
|
||||
function wlmetric_ball_thr(S::AbstractVector{T}, center::T; radius = 2, op = *) where {T}
|
||||
E, sizes = wlmetric_ball_thr(S, radius = radius, op = op)
|
||||
isone(center) && return E, sizes
|
||||
return c .* E, sizes
|
||||
end
|
||||
|
||||
function wlmetric_ball(
|
||||
S::AbstractVector{T},
|
||||
center::T = one(first(S));
|
||||
radius = 2,
|
||||
op = *,
|
||||
threading = true,
|
||||
) where {T}
|
||||
threading && return wlmetric_ball_thr(S, center, radius = radius, op = op)
|
||||
return wlmetric_ball_serial(S, center, radius = radius, op = op)
|
||||
end
|
||||
43
src/words.jl
43
src/words.jl
@ -1,43 +0,0 @@
|
||||
syllablelength(w::GWord) = length(w.symbols)
|
||||
syllables(w::GWord) = w.symbols
|
||||
ismodified(w::GWord) = w.modified
|
||||
setmodified!(w::GWord) = (w.modified = true; w)
|
||||
unsetmodified!(w::GWord) = (w.modified = false; w)
|
||||
savehash!(w::GWord, h::UInt) = (w.savedhash = h; w)
|
||||
savedhash(w::GWord) = w.savedhash
|
||||
Base.parent(w::GWord) = w.parent
|
||||
hasparent(w::GWord) = isdefined(w, :parent)
|
||||
setparent!(w::GWord, G::AbstractFPGroup) = (w.parent = G; w)
|
||||
|
||||
Base.isempty(w::GWord) = isempty(syllables(w))
|
||||
Base.isone(w::GWord) = (freereduce!(w); isempty(w))
|
||||
Base.one(w::GWord) = one(parent(w))
|
||||
|
||||
function Base.iterate(w::GWord, state=(syllable=1, pow=1))
|
||||
state.syllable > syllablelength(w) && return nothing
|
||||
next = iterate(syllables(w)[state.syllable], state.pow)
|
||||
next === nothing && return iterate(w, (syllable=state.syllable+1, pow=1))
|
||||
return first(next), (syllable=state.syllable, pow=last(next))
|
||||
end
|
||||
|
||||
Base.eltype(::Type{<:GWord{T}}) where T = T
|
||||
Base.length(w::GWord) = isempty(w) ? 0 : sum(length, syllables(w))
|
||||
Base.size(w::GWord) = (length(w),)
|
||||
Base.lastindex(w::GWord) = length(w)
|
||||
|
||||
Base.@propagate_inbounds function Base.getindex(w::GWord, i::Integer)
|
||||
csum = 0
|
||||
idx = 0
|
||||
@boundscheck 0 < i <= length(w) || throw(BoundsError(w, i))
|
||||
while csum < i
|
||||
idx += 1
|
||||
csum += length(syllables(w)[idx])
|
||||
end
|
||||
return first(syllables(w)[idx])
|
||||
end
|
||||
|
||||
Base.@propagate_inbounds Base.getindex(w::GWord, itr) = [w[i] for i in itr]
|
||||
|
||||
# no setindex! for syllable based words
|
||||
|
||||
Base.convert(::Type{GW}, s::GSymbol) where GW <: GWord = GW(s)
|
||||
@ -2,27 +2,30 @@
|
||||
|
||||
@testset "Transvections" begin
|
||||
|
||||
@test New.Transvection(:ϱ, 1, 2) isa New.GSymbol
|
||||
@test New.Transvection(:ϱ, 1, 2) isa New.Transvection
|
||||
@test New.Transvection(:λ, 1, 2) isa New.GSymbol
|
||||
@test New.Transvection(:λ, 1, 2) isa New.Transvection
|
||||
t = New.Transvection(:ϱ, 1, 2)
|
||||
@test inv(t) isa New.GSymbol
|
||||
@test inv(t) isa New.Transvection
|
||||
@test Groups.Transvection(:ϱ, 1, 2) isa Groups.GSymbol
|
||||
@test Groups.Transvection(:ϱ, 1, 2) isa Groups.Transvection
|
||||
@test Groups.Transvection(:λ, 1, 2) isa Groups.GSymbol
|
||||
@test Groups.Transvection(:λ, 1, 2) isa Groups.Transvection
|
||||
t = Groups.Transvection(:ϱ, 1, 2)
|
||||
@test inv(t) isa Groups.GSymbol
|
||||
@test inv(t) isa Groups.Transvection
|
||||
|
||||
@test t != inv(t)
|
||||
|
||||
s = New.Transvection(:ϱ, 1, 2)
|
||||
s = Groups.Transvection(:ϱ, 1, 2)
|
||||
@test t == s
|
||||
@test hash(t) == hash(s)
|
||||
|
||||
s_ = New.Transvection(:ϱ, 1, 3)
|
||||
s_ = Groups.Transvection(:ϱ, 1, 3)
|
||||
@test s_ != s
|
||||
@test hash(s_) != hash(s)
|
||||
|
||||
@test New.gersten_alphabet(3) isa Alphabet
|
||||
A = New.gersten_alphabet(3)
|
||||
@test Groups.gersten_alphabet(3) isa Alphabet
|
||||
A = Groups.gersten_alphabet(3)
|
||||
@test length(A) == 12
|
||||
|
||||
@test sprint(show, Groups.ϱ(1, 2)) == "ϱ₁.₂"
|
||||
@test sprint(show, Groups.λ(3, 2)) == "λ₃.₂"
|
||||
end
|
||||
|
||||
A4 = Alphabet(
|
||||
@ -35,18 +38,16 @@
|
||||
[ 2, 1, 4, 3, 6, 5, 8, 7,10, 9]
|
||||
)
|
||||
|
||||
F4 = New.FreeGroup([:a, :b, :c, :d], A4)
|
||||
A = New.SpecialAutomorphismGroup(F4, maxrules=1000)
|
||||
|
||||
F4 = FreeGroup([:a, :b, :c, :d], A4)
|
||||
a,b,c,d = gens(F4)
|
||||
D = ntuple(i->gens(F4, i), 4)
|
||||
|
||||
@testset "Transvection action correctness" begin
|
||||
i,j = 1,2
|
||||
r = New.Transvection(:ϱ,i,j)
|
||||
l = New.Transvection(:λ,i,j)
|
||||
r = Groups.Transvection(:ϱ,i,j)
|
||||
l = Groups.Transvection(:λ,i,j)
|
||||
|
||||
(t::New.Transvection)(v::Tuple) = New.evaluate!(v, t, A4)
|
||||
(t::Groups.Transvection)(v::Tuple) = Groups.evaluate!(v, t, A4)
|
||||
|
||||
@test r(deepcopy(D)) == (a*b, b, c, d)
|
||||
@test inv(r)(deepcopy(D)) == (a*b^-1,b, c, d)
|
||||
@ -54,45 +55,49 @@
|
||||
@test inv(l)(deepcopy(D)) == (b^-1*a,b, c, d)
|
||||
|
||||
i,j = 3,1
|
||||
r = New.Transvection(:ϱ,i,j)
|
||||
l = New.Transvection(:λ,i,j)
|
||||
r = Groups.Transvection(:ϱ,i,j)
|
||||
l = Groups.Transvection(:λ,i,j)
|
||||
@test r(deepcopy(D)) == (a, b, c*a, d)
|
||||
@test inv(r)(deepcopy(D)) == (a, b, c*a^-1,d)
|
||||
@test l(deepcopy(D)) == (a, b, a*c, d)
|
||||
@test inv(l)(deepcopy(D)) == (a, b, a^-1*c,d)
|
||||
|
||||
i,j = 4,3
|
||||
r = New.Transvection(:ϱ,i,j)
|
||||
l = New.Transvection(:λ,i,j)
|
||||
r = Groups.Transvection(:ϱ,i,j)
|
||||
l = Groups.Transvection(:λ,i,j)
|
||||
@test r(deepcopy(D)) == (a, b, c, d*c)
|
||||
@test inv(r)(deepcopy(D)) == (a, b, c, d*c^-1)
|
||||
@test l(deepcopy(D)) == (a, b, c, c*d)
|
||||
@test inv(l)(deepcopy(D)) == (a, b, c, c^-1*d)
|
||||
|
||||
i,j = 2,4
|
||||
r = New.Transvection(:ϱ,i,j)
|
||||
l = New.Transvection(:λ,i,j)
|
||||
r = Groups.Transvection(:ϱ,i,j)
|
||||
l = Groups.Transvection(:λ,i,j)
|
||||
@test r(deepcopy(D)) == (a, b*d, c, d)
|
||||
@test inv(r)(deepcopy(D)) == (a, b*d^-1,c, d)
|
||||
@test l(deepcopy(D)) == (a, d*b, c, d)
|
||||
@test inv(l)(deepcopy(D)) == (a, d^-1*b,c, d)
|
||||
end
|
||||
|
||||
@testset "AutomorphismGroup constructors" begin
|
||||
@test A isa New.AbstractFPGroup
|
||||
@test A isa New.AutomorphismGroup
|
||||
@test KnuthBendix.alphabet(A) isa Alphabet
|
||||
@test New.relations(A) isa Vector{<:Pair}
|
||||
A = SpecialAutomorphismGroup(F4, maxrules=1000)
|
||||
|
||||
@testset "AutomorphismGroup constructors" begin
|
||||
@test A isa Groups.AbstractFPGroup
|
||||
@test A isa AutomorphismGroup
|
||||
@test alphabet(A) isa Alphabet
|
||||
@test Groups.relations(A) isa Vector{<:Pair}
|
||||
@test sprint(show, A) == "automorphism group of free group on 4 generators"
|
||||
end
|
||||
|
||||
@testset "Automorphisms: hash and evaluate" begin
|
||||
@test New.domain(gens(A, 1)) == D
|
||||
@test Groups.domain(gens(A, 1)) == D
|
||||
g, h = gens(A, 1), gens(A, 8)
|
||||
|
||||
@test New.evaluate(g*h) == New.evaluate(h*g)
|
||||
@test evaluate(g*h) == evaluate(h*g)
|
||||
@test (g*h).savedhash == zero(UInt)
|
||||
|
||||
@test sprint(show, typeof(g)) == "Automorphism{FreeGroup{Symbol},…}"
|
||||
|
||||
a = g*h
|
||||
b = h*g
|
||||
@test hash(a) != zero(UInt)
|
||||
@ -106,22 +111,22 @@
|
||||
# ϱ₁₂*ϱ₂₁⁻¹*λ₁₂*ε₂ == σ₂₁₃₄
|
||||
|
||||
g = gens(A, 1)
|
||||
x1, x2, x3, x4 = New.domain(g)
|
||||
@test New.evaluate(g) == (x1*x2, x2, x3, x4)
|
||||
x1, x2, x3, x4 = Groups.domain(g)
|
||||
@test evaluate(g) == (x1*x2, x2, x3, x4)
|
||||
|
||||
g = g*inv(gens(A, 4)) # ϱ₂₁
|
||||
@test New.evaluate(g) == (x1*x2, x1^-1, x3, x4)
|
||||
@test evaluate(g) == (x1*x2, x1^-1, x3, x4)
|
||||
|
||||
g = g*gens(A, 13)
|
||||
@test New.evaluate(g) == (x2, x1^-1, x3, x4)
|
||||
@test evaluate(g) == (x2, x1^-1, x3, x4)
|
||||
end
|
||||
|
||||
@testset "Automorphisms: SAut(F₄)" begin
|
||||
N = 4
|
||||
G = New.SpecialAutomorphismGroup(New.FreeGroup(N))
|
||||
G = SpecialAutomorphismGroup(FreeGroup(N))
|
||||
|
||||
S = gens(G)
|
||||
@test S isa Vector{<:New.FPGroupElement{<:New.AutomorphismGroup{<:New.FreeGroup}}}
|
||||
@test S isa Vector{<:FPGroupElement{<:AutomorphismGroup{<:FreeGroup}}}
|
||||
|
||||
@test length(S) == 2*N*(N-1)
|
||||
@test length(unique(S)) == length(S)
|
||||
@ -141,8 +146,8 @@
|
||||
|
||||
@testset "GroupsCore conformance" begin
|
||||
test_Group_interface(A)
|
||||
g = A(rand(1:length(KnuthBendix.alphabet(A)), 10))
|
||||
h = A(rand(1:length(KnuthBendix.alphabet(A)), 10))
|
||||
g = A(rand(1:length(alphabet(A)), 10))
|
||||
h = A(rand(1:length(alphabet(A)), 10))
|
||||
|
||||
test_GroupElement_interface(g, h)
|
||||
end
|
||||
@ -152,7 +157,7 @@ end
|
||||
# using Random
|
||||
# using GroupsCore
|
||||
#
|
||||
# A = New.SpecialAutomorphismGroup(New.FreeGroup(4), maxrules=2000, ordering=KnuthBendix.RecursivePathOrder)
|
||||
# A = New.SpecialAutomorphismGroup(FreeGroup(4), maxrules=2000, ordering=KnuthBendix.RecursivePathOrder)
|
||||
#
|
||||
# # for seed in 1:1000
|
||||
# let seed = 68
|
||||
@ -163,22 +168,22 @@ end
|
||||
# @info "seed=$seed" g h
|
||||
# @time isone(g*inv(g))
|
||||
# @time isone(inv(g)*g)
|
||||
# @info "" length(New.word(New.normalform!(g*inv(g)))) length(New.word(New.normalform!(inv(g)*g)))
|
||||
# @info "" length(word(New.normalform!(g*inv(g)))) length(word(New.normalform!(inv(g)*g)))
|
||||
# a = commutator(g, h, g)
|
||||
# b = conj(inv(g), h) * conj(conj(g, h), g)
|
||||
#
|
||||
# @info length(New.word(a))
|
||||
# @info length(New.word(b))
|
||||
# @info length(word(a))
|
||||
# @info length(word(b))
|
||||
#
|
||||
# w = a*inv(b)
|
||||
# @info length(New.word(w))
|
||||
# @info length(word(w))
|
||||
# New.normalform!(w)
|
||||
# @info length(New.word(w))
|
||||
# @info length(word(w))
|
||||
#
|
||||
#
|
||||
# #
|
||||
# # @time ima = New.evaluate(a)
|
||||
# # @time imb = New.evaluate(b)
|
||||
# # @time ima = evaluate(a)
|
||||
# # @time imb = evaluate(b)
|
||||
# # @info "" a b ima imb
|
||||
# # @time a == b
|
||||
# end
|
||||
|
||||
@ -1,288 +0,0 @@
|
||||
import AbstractAlgebra.@perm_str
|
||||
|
||||
@testset "Automorphisms" begin
|
||||
|
||||
G = AbstractAlgebra.SymmetricGroup(Int8(4))
|
||||
|
||||
@testset "AutSymbol" begin
|
||||
@test_throws MethodError Groups.AutSymbol(:a)
|
||||
@test_throws MethodError Groups.AutSymbol(:a, 1)
|
||||
f = Groups.AutSymbol(:a, 1, Groups.FlipAut(2))
|
||||
@test f isa Groups.GSymbol
|
||||
@test f isa Groups.AutSymbol
|
||||
@test Groups.AutSymbol(perm"(4)") isa Groups.AutSymbol
|
||||
@test Groups.AutSymbol(perm"(1,2,3,4)") isa Groups.AutSymbol
|
||||
@test Groups.transvection_R(1,2) isa Groups.AutSymbol
|
||||
@test Groups.transvection_R(3,4) isa Groups.AutSymbol
|
||||
@test Groups.flip(3) isa Groups.AutSymbol
|
||||
|
||||
@test Groups.id_autsymbol() isa Groups.AutSymbol
|
||||
@test inv(Groups.id_autsymbol()) isa Groups.AutSymbol
|
||||
x = Groups.id_autsymbol()
|
||||
@test inv(x) == Groups.id_autsymbol()
|
||||
end
|
||||
|
||||
a,b,c,d = gens(FreeGroup(4))
|
||||
D = NTuple{4,FreeGroupElem}([a,b,c,d])
|
||||
|
||||
@testset "flip correctness" begin
|
||||
@test Groups.flip(1)(deepcopy(D)) == (a^-1, b,c,d)
|
||||
@test Groups.flip(2)(deepcopy(D)) == (a, b^-1,c,d)
|
||||
@test Groups.flip(3)(deepcopy(D)) == (a, b,c^-1,d)
|
||||
@test Groups.flip(4)(deepcopy(D)) == (a, b,c,d^-1)
|
||||
@test inv(Groups.flip(1))(deepcopy(D)) == (a^-1, b,c,d)
|
||||
@test inv(Groups.flip(2))(deepcopy(D)) == (a, b^-1,c,d)
|
||||
@test inv(Groups.flip(3))(deepcopy(D)) == (a, b,c^-1,d)
|
||||
@test inv(Groups.flip(4))(deepcopy(D)) == (a, b,c,d^-1)
|
||||
end
|
||||
|
||||
@testset "perm correctness" begin
|
||||
σ = Groups.AutSymbol(perm"(4)")
|
||||
@test σ(deepcopy(D)) == deepcopy(D)
|
||||
@test inv(σ)(deepcopy(D)) == deepcopy(D)
|
||||
|
||||
σ = Groups.AutSymbol(perm"(1,2,3,4)")
|
||||
@test σ(deepcopy(D)) == (b, c, d, a)
|
||||
@test inv(σ)(deepcopy(D)) == (d, a, b, c)
|
||||
|
||||
σ = Groups.AutSymbol(perm"(1,2)(4,3)")
|
||||
@test σ(deepcopy(D)) == (b, a, d, c)
|
||||
@test inv(σ)(deepcopy(D)) == (b, a, d, c)
|
||||
|
||||
σ = Groups.AutSymbol(perm"(1,2,3)(4)")
|
||||
@test σ(deepcopy(D)) == (b, c, a, d)
|
||||
@test inv(σ)(deepcopy(D)) == (c, a, b, d)
|
||||
end
|
||||
|
||||
@testset "rmul/transvection_R correctness" begin
|
||||
i,j = 1,2
|
||||
r = Groups.transvection_R(i,j)
|
||||
l = Groups.transvection_L(i,j)
|
||||
@test r(deepcopy(D)) == (a*b, b, c, d)
|
||||
@test inv(r)(deepcopy(D)) == (a*b^-1,b, c, d)
|
||||
@test l(deepcopy(D)) == (b*a, b, c, d)
|
||||
@test inv(l)(deepcopy(D)) == (b^-1*a,b, c, d)
|
||||
|
||||
i,j = 3,1
|
||||
r = Groups.transvection_R(i,j)
|
||||
l = Groups.transvection_L(i,j)
|
||||
@test r(deepcopy(D)) == (a, b, c*a, d)
|
||||
@test inv(r)(deepcopy(D)) == (a, b, c*a^-1,d)
|
||||
@test l(deepcopy(D)) == (a, b, a*c, d)
|
||||
@test inv(l)(deepcopy(D)) == (a, b, a^-1*c,d)
|
||||
|
||||
i,j = 4,3
|
||||
r = Groups.transvection_R(i,j)
|
||||
l = Groups.transvection_L(i,j)
|
||||
@test r(deepcopy(D)) == (a, b, c, d*c)
|
||||
@test inv(r)(deepcopy(D)) == (a, b, c, d*c^-1)
|
||||
@test l(deepcopy(D)) == (a, b, c, c*d)
|
||||
@test inv(l)(deepcopy(D)) == (a, b, c, c^-1*d)
|
||||
|
||||
i,j = 2,4
|
||||
r = Groups.transvection_R(i,j)
|
||||
l = Groups.transvection_L(i,j)
|
||||
@test r(deepcopy(D)) == (a, b*d, c, d)
|
||||
@test inv(r)(deepcopy(D)) == (a, b*d^-1,c, d)
|
||||
@test l(deepcopy(D)) == (a, d*b, c, d)
|
||||
@test inv(l)(deepcopy(D)) == (a, d^-1*b,c, d)
|
||||
end
|
||||
|
||||
@testset "AutGroup/Automorphism constructors" begin
|
||||
|
||||
f = Groups.AutSymbol(:a, 1, Groups.FlipAut(1))
|
||||
@test isa(Automorphism{3}(f), Groups.GWord)
|
||||
@test isa(Automorphism{3}(f), Automorphism)
|
||||
@test isa(AutGroup(FreeGroup(3)), GroupsCore.Group)
|
||||
@test isa(AutGroup(FreeGroup(1)), Groups.AbstractFPGroup)
|
||||
|
||||
A = AutGroup(FreeGroup(1))
|
||||
@test Groups.gens(A) isa Vector{Automorphism{1}}
|
||||
@test length(Groups.gens(A)) == 1
|
||||
@test length(Groups.gens(Aut(FreeGroup(1)))) == 1
|
||||
@test Groups.gens(A) == [A(Groups.flip(1))]
|
||||
|
||||
A = AutGroup(FreeGroup(1), special=true)
|
||||
@test isempty(Groups.gens(A))
|
||||
@test Groups.gens(SAut(FreeGroup(1))) == Automorphism{1}[]
|
||||
|
||||
A = AutGroup(FreeGroup(2))
|
||||
@test length(Groups.gens(A)) == 7
|
||||
Agens = Groups.gens(A)
|
||||
@test A(first(Agens)) isa Automorphism
|
||||
|
||||
@test A(Groups.transvection_R(1,2)) isa Automorphism
|
||||
@test A(Groups.transvection_R(1,2)) in Agens
|
||||
|
||||
@test A(Groups.transvection_R(2,1)) isa Automorphism
|
||||
@test A(Groups.transvection_R(2,1)) in Agens
|
||||
|
||||
@test A(Groups.transvection_R(1,2)) isa Automorphism
|
||||
@test A(Groups.transvection_R(1,2)) in Agens
|
||||
|
||||
@test A(Groups.transvection_R(2,1)) isa Automorphism
|
||||
@test A(Groups.transvection_R(2,1)) in Agens
|
||||
|
||||
@test A(Groups.flip(1)) isa Automorphism
|
||||
@test A(Groups.flip(1)) in Agens
|
||||
|
||||
@test A(Groups.flip(2)) isa Automorphism
|
||||
@test A(Groups.flip(2)) in Agens
|
||||
|
||||
@test A(Groups.AutSymbol(perm"(1,2)")) isa Automorphism
|
||||
@test A(Groups.AutSymbol(perm"(1,2)")) in Agens
|
||||
|
||||
@test A(Groups.id_autsymbol()) isa Automorphism
|
||||
end
|
||||
|
||||
A = AutGroup(FreeGroup(4))
|
||||
|
||||
@testset "eltary functions" begin
|
||||
|
||||
f = Groups.AutSymbol(perm"(1,2,3,4)")
|
||||
@test (Groups.change_pow(f, 2)).pow == 1
|
||||
@test (Groups.change_pow(f, -2)).pow == 1
|
||||
@test (inv(f)).pow == 1
|
||||
|
||||
g = Groups.AutSymbol(perm"(1,2)(3,4)")
|
||||
@test isa(inv(g), Groups.AutSymbol)
|
||||
|
||||
@test_throws MethodError g*f
|
||||
|
||||
@test A(g)^-1 == A(inv(g))
|
||||
|
||||
h = Groups.transvection_R(1,2)
|
||||
|
||||
@test collect(A(g)*A(h)) == [g, h]
|
||||
@test collect(A(h)^2) == [h, h]
|
||||
end
|
||||
|
||||
@testset "reductions/arithmetic" begin
|
||||
f = Groups.AutSymbol(perm"(1,2,3,4)")
|
||||
|
||||
f² = push!(A(f), f)
|
||||
@test Groups.simplifyperms!(Bool, f²) == false
|
||||
@test f²^2 == one(A)
|
||||
@test !isone(f²)
|
||||
|
||||
a = A(Groups.λ(1,2))*Groups.ε(2)
|
||||
b = Groups.ε(2)*A(inv(Groups.λ(1,2)))
|
||||
@test a*b == b*a
|
||||
@test a^3 * b^3 == one(A)
|
||||
g,h = Groups.gens(A)[[1,8]] # (g, h) = (ϱ₁₂, ϱ₃₂)
|
||||
|
||||
@test Groups.domain(A) == NTuple{4, FreeGroupElem}(gens(A.objectGroup))
|
||||
|
||||
@test (g*h)(Groups.domain(A)) == (h*g)(Groups.domain(A))
|
||||
@test (g*h).savedhash == zero(UInt)
|
||||
@test (h*g).savedhash == zero(UInt)
|
||||
a = g*h
|
||||
b = h*g
|
||||
@test hash(a) != zero(UInt)
|
||||
@test hash(b) == hash(a)
|
||||
@test a.savedhash == b.savedhash
|
||||
@test length(unique([a,b])) == 1
|
||||
@test length(unique([g*h, h*g])) == 1
|
||||
|
||||
# Not so simple arithmetic: applying starting on the left:
|
||||
# ϱ₁₂*ϱ₂₁⁻¹*λ₁₂*ε₂ == σ₂₁₃₄
|
||||
|
||||
g = A(Groups.transvection_R(1,2))
|
||||
x1, x2, x3, x4 = Groups.domain(A)
|
||||
@test g(Groups.domain(A)) == (x1*x2, x2, x3, x4)
|
||||
g = g*inv(A(Groups.transvection_R(2,1)))
|
||||
@test g(Groups.domain(A)) == (x1*x2, x1^-1, x3, x4)
|
||||
g = g*A(Groups.transvection_L(1,2))
|
||||
@test g(Groups.domain(A)) == (x2, x1^-1, x3, x4)
|
||||
g = g*A(Groups.flip(2))
|
||||
@test g(Groups.domain(A)) == (x2, x1, x3, x4)
|
||||
|
||||
@test g(Groups.domain(A)) == A(Groups.AutSymbol(perm"(1,2)(4)"))(Groups.domain(A))
|
||||
|
||||
@test g == A(Groups.AutSymbol(perm"(1,2)(4)"))
|
||||
|
||||
g_im = g(Groups.domain(A))
|
||||
@test length.(g_im) == (1,1,1,1)
|
||||
|
||||
g = A(Groups.σ(perm"(1,2)(4)"))
|
||||
h = A(Groups.σ(perm"(2,3,4)"))
|
||||
@test g*h isa Groups.Automorphism{4}
|
||||
f = g*h
|
||||
Groups
|
||||
@test Groups.syllablelength(f) == 2
|
||||
@test Groups.reduce!(f) isa Groups.Automorphism{4}
|
||||
@test Groups.syllablelength(f) == 1
|
||||
end
|
||||
|
||||
@testset "specific Aut(F4) tests" begin
|
||||
N = 4
|
||||
G = AutGroup(FreeGroup(N))
|
||||
S = G.gens
|
||||
@test isa(S, Vector{Groups.AutSymbol})
|
||||
S = [G(s) for s in unique(S)]
|
||||
@test isa(S, Vector{Automorphism{N}})
|
||||
@test S == gens(G)
|
||||
@test length(S) == 51
|
||||
S_inv = [S..., [inv(s) for s in S]...]
|
||||
@test length(unique(S_inv)) == 75
|
||||
|
||||
G = AutGroup(FreeGroup(N), special=true)
|
||||
S = gens(G)
|
||||
S_inv = [one(G), S..., [inv(s) for s in S]...]
|
||||
S_inv = unique(S_inv)
|
||||
B_2 = [i*j for (i,j) in Base.product(S_inv, S_inv)]
|
||||
@test length(B_2) == 2401
|
||||
@test length(unique(B_2)) == 1777
|
||||
end
|
||||
|
||||
@testset "abelianization homomorphism" begin
|
||||
N = 4
|
||||
G = AutGroup(FreeGroup(N))
|
||||
S = unique([gens(G); inv.(gens(G))])
|
||||
R = 3
|
||||
|
||||
@test Groups.abelianize(one(G)) isa Matrix{Int}
|
||||
@test Groups.abelianize(one(G)) == Matrix{Int}(I, N, N)
|
||||
|
||||
M = Matrix{Int}(I, N, N)
|
||||
M[1,2] = 1
|
||||
ϱ₁₂ = G(Groups.ϱ(1,2))
|
||||
λ₁₂ = G(Groups.λ(1,2))
|
||||
|
||||
@test Groups.abelianize(ϱ₁₂) == M
|
||||
@test Groups.abelianize(λ₁₂) == M
|
||||
|
||||
M[1,2] = -1
|
||||
|
||||
@test Groups.abelianize(ϱ₁₂^-1) == M
|
||||
@test Groups.abelianize(λ₁₂^-1) == M
|
||||
|
||||
@test Groups.abelianize(ϱ₁₂*λ₁₂^-1) == Matrix{Int}(I, N, N)
|
||||
@test Groups.abelianize(λ₁₂^-1*ϱ₁₂) == Matrix{Int}(I, N, N)
|
||||
|
||||
M = Matrix{Int}(I, N, N)
|
||||
M[2,2] = -1
|
||||
ε₂ = G(Groups.flip(2))
|
||||
|
||||
@test Groups.abelianize(ε₂) == M
|
||||
@test Groups.abelianize(ε₂^2) == Matrix{Int}(I, N, N)
|
||||
|
||||
M = [0 1 0 0; 0 0 0 1; 0 0 1 0; 1 0 0 0]
|
||||
|
||||
σ = G(Groups.AutSymbol(perm"(1,2,4)"))
|
||||
@test Groups.abelianize(σ) == M
|
||||
@test Groups.abelianize(σ^3) == Matrix{Int}(I, N, N)
|
||||
@test Groups.abelianize(σ)^3 == Matrix{Int}(I, N, N)
|
||||
|
||||
@test Groups.abelianize(G(Groups.id_autsymbol())) == Matrix{Int}(I, N, N)
|
||||
|
||||
function test_homomorphism(S, r)
|
||||
for elts in Iterators.product([[g for g in S] for _ in 1:r]...)
|
||||
prod(Groups.abelianize.(elts)) == Groups.abelianize(prod(elts)) || error("linear representaton test failed at $elts")
|
||||
end
|
||||
return 0
|
||||
end
|
||||
|
||||
@test test_homomorphism(S, R) == 0
|
||||
end
|
||||
end
|
||||
@ -1,18 +0,0 @@
|
||||
@testset "FPGroups definitions" begin
|
||||
F = FreeGroup(["a", "b", "c"])
|
||||
a,b,c = Groups.gens(F)
|
||||
R = [a^2, a*b*a, c*b*a]
|
||||
@test F/R isa FPGroup
|
||||
@test F isa FreeGroup
|
||||
G = F/R
|
||||
A,B,C = Groups.gens(G)
|
||||
|
||||
@test Groups.reduce!(A^2) == one(G)
|
||||
@test Groups.reduce!(A*B*A*A) == A
|
||||
@test Groups.reduce!(A*A*B*A) == A
|
||||
|
||||
@test Groups.freepreimage(G) == F
|
||||
@test Groups.freepreimage(B^2) == b^2
|
||||
|
||||
@test G/[B^2, C*B*C] isa FPGroup
|
||||
end
|
||||
@ -1,188 +0,0 @@
|
||||
@testset "Groups.FreeSymbols" begin
|
||||
s = Groups.FreeSymbol(:s)
|
||||
t = Groups.FreeSymbol(:t)
|
||||
|
||||
@testset "constructors" begin
|
||||
@test isa(Groups.FreeSymbol(:aaaaaaaaaaaaaaaa), Groups.GSymbol)
|
||||
@test Groups.FreeSymbol(:abc).pow == 1
|
||||
@test isa(s, Groups.FreeSymbol)
|
||||
@test isa(t, Groups.FreeSymbol)
|
||||
end
|
||||
@testset "eltary functions" begin
|
||||
@test length(s) == 1
|
||||
@test Groups.change_pow(s, 0) == Groups.change_pow(t, 0)
|
||||
@test length(Groups.change_pow(s, 0)) == 0
|
||||
@test inv(s).pow == -1
|
||||
@test Groups.FreeSymbol(:s, 3) == Groups.change_pow(s, 3)
|
||||
@test Groups.FreeSymbol(:s, 3) != Groups.FreeSymbol(:t, 3)
|
||||
@test Groups.change_pow(inv(s), -3) == inv(Groups.change_pow(s, 3))
|
||||
end
|
||||
@testset "powers" begin
|
||||
s⁴ = Groups.change_pow(s,4)
|
||||
@test s⁴.pow == 4
|
||||
@test Groups.change_pow(s, 4) == Groups.FreeSymbol(:s, 4)
|
||||
end
|
||||
end
|
||||
|
||||
@testset "FreeGroupSymbols manipulation" begin
|
||||
s = Groups.FreeSymbol("s")
|
||||
t = Groups.FreeSymbol(:t, -2)
|
||||
|
||||
@test isa(Groups.GroupWord(s), Groups.GWord{Groups.FreeSymbol})
|
||||
@test isa(Groups.GroupWord(s), FreeGroupElem)
|
||||
@test isa(FreeGroupElem(s), Groups.GWord)
|
||||
@test isa(convert(FreeGroupElem, s), Groups.GWord)
|
||||
@test isa(convert(FreeGroupElem, s), FreeGroupElem)
|
||||
@test isa(Vector{FreeGroupElem}([s,t]), Vector{FreeGroupElem})
|
||||
@test length(FreeGroupElem(s)) == 1
|
||||
@test length(FreeGroupElem(t)) == 2
|
||||
@test Groups.FreeSymbol(:s, 1) != Groups.FreeSymbol(:s, 2)
|
||||
@test Groups.FreeSymbol(:s, 1) != Groups.FreeSymbol(:t, 1)
|
||||
@test collect(Groups.FreeSymbol(:s, 2)) == [i for i in Groups.FreeSymbol(:s, 2)] == [s, s]
|
||||
end
|
||||
|
||||
@testset "FreeGroup" begin
|
||||
@test isa(FreeGroup(["s", "t"]), GroupsCore.Group)
|
||||
G = FreeGroup(["s", "t"])
|
||||
s, t = gens(G)
|
||||
|
||||
@testset "elements constructors" begin
|
||||
@test isa(one(G), FreeGroupElem)
|
||||
@test eltype(G.gens) == Groups.FreeSymbol
|
||||
@test length(G.gens) == 2
|
||||
@test eltype(gens(G)) == FreeGroupElem
|
||||
@test length(gens(G)) == 2
|
||||
|
||||
tt, ss = Groups.FreeSymbol(:t), Groups.FreeSymbol(:s)
|
||||
@test Groups.GroupWord([tt, inv(tt)]) isa FreeGroupElem
|
||||
|
||||
@test collect(s*t) == Groups.syllables(s*t)
|
||||
@test collect(t^2) == [tt, tt]
|
||||
ttinv = Groups.FreeSymbol(:t, -1)
|
||||
w = t^-2*s^3*t^2
|
||||
@test collect(w) == [inv(tt), inv(tt), ss, ss, ss, tt, tt]
|
||||
@test w[1] == inv(tt)
|
||||
@test w[2] == inv(tt)
|
||||
@test w[3] == ss
|
||||
@test w[3:5] == [ss, ss, ss]
|
||||
@test w[end] == tt
|
||||
|
||||
@test collect(ttinv) == [ttinv]
|
||||
|
||||
@test isone(t^0)
|
||||
@test !isone(t^1)
|
||||
end
|
||||
|
||||
@testset "internal arithmetic" begin
|
||||
|
||||
@test (s*s).symbols == (s^2).symbols
|
||||
@test hash([t^1,s^1]) == hash([t^2*inv(t),s*inv(s)*s])
|
||||
|
||||
t_symb = Groups.FreeSymbol(:t)
|
||||
tt = deepcopy(t)
|
||||
@test string(Groups.rmul!(tt, tt, inv(tt))) == "(id)"
|
||||
tt = deepcopy(t)
|
||||
@test string(Groups.lmul!(tt, tt, inv(tt))) == "(id)"
|
||||
|
||||
w = deepcopy(t)
|
||||
@test length(Groups.rmul!(w, t)) == 2
|
||||
@test length(Groups.lmul!(w, inv(t))) == 1
|
||||
w = GroupsCore.mul!(w, w, s)
|
||||
@test length(w) == 2
|
||||
@test length(Groups.lmul!(w, inv(s))) == 3
|
||||
|
||||
tt = deepcopy(t)
|
||||
push!(tt, inv(t_symb))
|
||||
@test string(tt) == "t*t^-1"
|
||||
tt = deepcopy(t)
|
||||
pushfirst!(tt, inv(t_symb))
|
||||
@test string(tt) == "t^-1*t"
|
||||
|
||||
tt = deepcopy(t)
|
||||
append!(tt, inv(t))
|
||||
@test string(tt) == "t*t^-1"
|
||||
|
||||
tt = deepcopy(t)
|
||||
prepend!(tt, inv(t))
|
||||
@test string(tt) == "t^-1*t"
|
||||
|
||||
tt = deepcopy(t)
|
||||
append!(tt, s, inv(t))
|
||||
@test string(tt) == "t*s*t^-1"
|
||||
|
||||
o = one(t)
|
||||
o_inv = inv(o)
|
||||
@test o == o_inv
|
||||
@test o !== o_inv
|
||||
Groups.rmul!(o, t)
|
||||
@test o != o_inv
|
||||
end
|
||||
|
||||
@testset "reductions" begin
|
||||
@test length(one(G).symbols) == 0
|
||||
@test length((one(G)*one(G)).symbols) == 0
|
||||
@test one(G) == one(G)*one(G)
|
||||
w = deepcopy(s)
|
||||
push!(Groups.syllables(w), (s^-1).symbols[1])
|
||||
@test Groups.reduce!(w) == one(parent(w))
|
||||
o = (t*s)^3
|
||||
@test o == t*s*t*s*t*s
|
||||
p = (t*s)^-3
|
||||
@test p == s^-1*t^-1*s^-1*t^-1*s^-1*t^-1
|
||||
@test o*p == one(parent(o*p))
|
||||
w = FreeGroupElem([o.symbols..., p.symbols...])
|
||||
w.parent = G
|
||||
@test Groups.syllables(Groups.reduce(w)) == Vector{Groups.FreeSymbol}([])
|
||||
end
|
||||
|
||||
@testset "Group operations" begin
|
||||
@test parent(s) == G
|
||||
@test parent(s) === parent(deepcopy(s))
|
||||
@test isa(s*t, FreeGroupElem)
|
||||
@test parent(s*t) == parent(s^2)
|
||||
@test s*s == s^2
|
||||
@test inv(s*s) == inv(s^2)
|
||||
@test inv(s)^2 == inv(s^2)
|
||||
@test inv(s)*inv(s) == inv(s^2)
|
||||
@test inv(s*t) == inv(t)*inv(s)
|
||||
w = s*t*s^-1
|
||||
@test inv(w) == s*t^-1*s^-1
|
||||
@test (t*s*t^-1)^10 == t*s^10*t^-1
|
||||
@test (t*s*t^-1)^-10 == t*s^-10*t^-1
|
||||
end
|
||||
|
||||
@testset "replacements" begin
|
||||
a = Groups.FreeSymbol(:a)
|
||||
b = Groups.FreeSymbol(:b)
|
||||
@test Groups.issubsymbol(a, Groups.change_pow(a,2)) == true
|
||||
@test Groups.issubsymbol(a, Groups.change_pow(a,-2)) == false
|
||||
@test Groups.issubsymbol(b, Groups.change_pow(a,-2)) == false
|
||||
@test Groups.issubsymbol(inv(b), Groups.change_pow(b,-2)) == true
|
||||
|
||||
c = s*t*s^-1*t^-1
|
||||
@test findfirst(s^-1*t^-1, c) == 3
|
||||
@test findnext(s^-1*t^-1, c*s^-1,3) == 3
|
||||
@test findnext(s^-1*t^-1, c*s^-1*t^-1, 4) == 5
|
||||
@test findfirst(c, c*t) === nothing
|
||||
|
||||
@test findlast(s^-1*t^-1, c) == 3
|
||||
@test findprev(s, s*t*s*t, 4) == 3
|
||||
@test findprev(s*t, s*t*s*t, 2) == 1
|
||||
@test findprev(Groups.FreeSymbol(:t, 2), c, 4) === nothing
|
||||
|
||||
w = s*t*s^-1
|
||||
subst = Dict{FreeGroupElem, FreeGroupElem}(w => s^1, s*t^-1 => t^4)
|
||||
@test Groups.replace(c, s*t=>one(G)) == s^-1*t^-1
|
||||
@test Groups.replace(c, w=>subst[w]) == s*t^-1
|
||||
@test Groups.replace(s*c*t^-1, w=>subst[w]) == s^2*t^-2
|
||||
@test Groups.replace(t*c*t, w=>subst[w]) == t*s
|
||||
@test Groups.replace(s*c*s*c*s, subst) == s*t^4*s*t^4*s
|
||||
|
||||
G = FreeGroup(["x", "y"])
|
||||
x,y = gens(G)
|
||||
|
||||
@test Groups.replace(x*y^9, y^2=>y) == x*y^5
|
||||
@test Groups.replace(x^3, x^2=>y) == x*y
|
||||
@test Groups.replace(y*x^3*y, x^2=>y) == y*x*y^2
|
||||
end
|
||||
end
|
||||
@ -6,7 +6,7 @@ using Groups.New
|
||||
|
||||
function wl_ball(F; radius::Integer)
|
||||
g, state = iterate(F)
|
||||
while length(New.word(g)) <= radius
|
||||
while length(word(g)) <= radius
|
||||
res = iterate(F, state)
|
||||
isnothing(res) && break
|
||||
g, state = res
|
||||
@ -20,7 +20,7 @@ end
|
||||
N = 4
|
||||
|
||||
@testset "iteration: FreeGroup" begin
|
||||
FN = New.FreeGroup(N)
|
||||
FN = FreeGroup(N)
|
||||
R = 8
|
||||
|
||||
let G = FN
|
||||
@ -46,7 +46,7 @@ end
|
||||
|
||||
@testset "iteration: SAut(F_n)" begin
|
||||
R = 4
|
||||
FN = New.FreeGroup(N)
|
||||
FN = FreeGroup(N)
|
||||
SAutFN = New.SpecialAutomorphismGroup(FN)
|
||||
|
||||
let G = SAutFN
|
||||
|
||||
@ -1,44 +1,54 @@
|
||||
@testset "FPGroups" begin
|
||||
A = Alphabet([:a, :A, :b, :B, :c, :C], [2,1,4,3,6,5])
|
||||
|
||||
F = New.FreeGroup([:a, :b, :c], A)
|
||||
@test FreeGroup(A) isa FreeGroup
|
||||
@test sprint(show, FreeGroup(A)) == "free group on 3 generators"
|
||||
|
||||
F = FreeGroup([:a, :b, :c], A)
|
||||
@test sprint(show, F) == "free group on 3 generators"
|
||||
|
||||
a,b,c = gens(F)
|
||||
@test c*b*a isa New.FPGroupElement
|
||||
@test c*b*a isa FPGroupElement
|
||||
|
||||
# quotient of F:
|
||||
G = New.FPGroup(F, [a*b=>b*a, a*c=>c*a, b*c=>c*b])
|
||||
G = FPGroup(F, [a*b=>b*a, a*c=>c*a, b*c=>c*b])
|
||||
|
||||
@test G isa New.FPGroup
|
||||
@test rand(G) isa New.FPGroupElement
|
||||
@test G isa FPGroup
|
||||
@test sprint(show, G) == "⟨a, b, c | a*b => b*a, a*c => c*a, b*c => c*b⟩"
|
||||
@test rand(G) isa FPGroupElement
|
||||
|
||||
f = a*c*b
|
||||
@test New.word(f) isa Word{UInt8}
|
||||
@test word(f) isa Word{UInt8}
|
||||
|
||||
aG,bG,cG = gens(G)
|
||||
|
||||
@test aG isa New.FPGroupElement
|
||||
@test aG isa FPGroupElement
|
||||
@test_throws AssertionError aG == a
|
||||
@test New.word(aG) == New.word(a)
|
||||
@test word(aG) == word(a)
|
||||
|
||||
g = aG*cG*bG
|
||||
|
||||
@test_throws AssertionError f == g
|
||||
@test New.word(f) == New.word(g)
|
||||
@test New.word(g) == [1, 5, 3]
|
||||
New.normalform!(g)
|
||||
@test New.word(g) == [1, 3, 5]
|
||||
@test word(f) == word(g)
|
||||
@test word(g) == [1, 5, 3]
|
||||
Groups.normalform!(g)
|
||||
@test word(g) == [1, 3, 5]
|
||||
|
||||
let g = aG*cG*bG
|
||||
# test that we normalize g before printing
|
||||
@test sprint(show, g) == "a*b*c"
|
||||
end
|
||||
|
||||
# quotient of G
|
||||
H = New.FPGroup(G, [aG^2=>cG, bG*cG=>aG], maxrules=200)
|
||||
H = FPGroup(G, [aG^2=>cG, bG*cG=>aG], maxrules=200)
|
||||
|
||||
h = H(New.word(g))
|
||||
h = H(word(g))
|
||||
|
||||
@test h isa New.FPGroupElement
|
||||
@test h isa FPGroupElement
|
||||
@test_throws AssertionError h == g
|
||||
@test_throws AssertionError h*g
|
||||
|
||||
New.normalform!(h)
|
||||
Groups.normalform!(h)
|
||||
@test h == H([5])
|
||||
|
||||
@testset "GroupsCore conformance: H" begin
|
||||
|
||||
@ -1,12 +1,12 @@
|
||||
@testset "New.FreeGroup" begin
|
||||
@testset "FreeGroup" begin
|
||||
|
||||
A3 = Alphabet([:a, :b, :c, :A, :B, :C], [4,5,6,1,2,3])
|
||||
F3 = New.FreeGroup([:a, :b, :c], A3)
|
||||
@test F3 isa New.FreeGroup
|
||||
F3 = FreeGroup([:a, :b, :c], A3)
|
||||
@test F3 isa FreeGroup
|
||||
|
||||
@test gens(F3) isa Vector
|
||||
|
||||
@test eltype(F3) <: New.FPGroupElement{<:New.FreeGroup}
|
||||
@test eltype(F3) <: FPGroupElement{<:FreeGroup}
|
||||
|
||||
w = F3([1,2,3,4])
|
||||
W = inv(w)
|
||||
@ -15,7 +15,7 @@
|
||||
|
||||
@test isone(w*W)
|
||||
|
||||
@test New.alphabet(w) == A3
|
||||
@test alphabet(w) == A3
|
||||
|
||||
@testset "generic iteration" begin
|
||||
w, s = iterate(F3)
|
||||
@ -43,7 +43,7 @@
|
||||
@testset "wl_ball" begin
|
||||
function wl_ball(F; radius::Integer)
|
||||
g, state = iterate(F)
|
||||
while length(New.word(g)) <= radius
|
||||
while length(word(g)) <= radius
|
||||
res = iterate(F, state)
|
||||
isnothing(res) && break
|
||||
g, state = res
|
||||
@ -55,11 +55,11 @@
|
||||
|
||||
E4 = wl_ball(F3, radius=4)
|
||||
@test length(E4) == 937
|
||||
@test New.word(last(E4)) == Word([6])^4
|
||||
@test word(last(E4)) == Word([6])^4
|
||||
|
||||
E8, t, _ = @timed wl_ball(F3, radius=8)
|
||||
@test length(E8) == 585937
|
||||
@test New.word(last(E8)) == Word([6])^8
|
||||
@test word(last(E8)) == Word([6])^8
|
||||
@test t/10^9 < 1
|
||||
end
|
||||
|
||||
|
||||
@ -2,43 +2,32 @@ using Test
|
||||
import AbstractAlgebra
|
||||
using Groups
|
||||
|
||||
include("symmetric.jl")
|
||||
using LinearAlgebra
|
||||
import KnuthBendix: Word
|
||||
|
||||
using GroupsCore
|
||||
include(joinpath(pathof(GroupsCore), "..", "..", "test", "conformance_test.jl"))
|
||||
|
||||
@testset "Groups" begin
|
||||
|
||||
@testset "wlmetric_ball" begin
|
||||
M = AbstractAlgebra.MatrixAlgebra(AbstractAlgebra.zz, 3)
|
||||
w = one(M); w[1,2] = 1;
|
||||
r = one(M); r[2,3] = -3;
|
||||
s = one(M); s[1,3] = 2; s[3,2] = -1;
|
||||
@testset "wlmetric_ball" begin
|
||||
M = AbstractAlgebra.MatrixAlgebra(AbstractAlgebra.zz, 3)
|
||||
w = one(M); w[1,2] = 1;
|
||||
r = one(M); r[2,3] = -3;
|
||||
s = one(M); s[1,3] = 2; s[3,2] = -1;
|
||||
|
||||
S = [w,r,s]; S = unique([S; inv.(S)]);
|
||||
_, sizes = Groups.wlmetric_ball(S, radius=4);
|
||||
@test sizes == [7, 33, 141, 561]
|
||||
_, sizes = Groups.wlmetric_ball_serial(S, radius=4);
|
||||
@test sizes == [7, 33, 141, 561]
|
||||
end
|
||||
S = [w,r,s]; S = unique([S; inv.(S)]);
|
||||
_, sizes = Groups.wlmetric_ball(S, radius=4);
|
||||
@test sizes == [7, 33, 141, 561]
|
||||
_, sizes = Groups.wlmetric_ball_serial(S, radius=4);
|
||||
@test sizes == [7, 33, 141, 561]
|
||||
end
|
||||
|
||||
include("FreeGroup-tests.jl")
|
||||
include("AutGroup-tests.jl")
|
||||
include("FPGroup-tests.jl")
|
||||
include("free_groups.jl")
|
||||
include("fp_groups.jl")
|
||||
|
||||
@testset "New FPGroups" begin
|
||||
using Groups.New
|
||||
using KnuthBendix
|
||||
include("AutFn.jl")
|
||||
|
||||
using GroupsCore
|
||||
include(joinpath(pathof(GroupsCore), "..", "..", "test", "conformance_test.jl"))
|
||||
|
||||
include("free_groups.jl")
|
||||
include("fp_groups.jl")
|
||||
|
||||
include("AutFn.jl")
|
||||
include("AutSigma_41.jl")
|
||||
|
||||
if !haskey(ENV, "CI")
|
||||
include("benchmarks.jl")
|
||||
end
|
||||
end
|
||||
# if !haskey(ENV, "CI")
|
||||
# include("benchmarks.jl")
|
||||
# end
|
||||
end
|
||||
|
||||
@ -1,31 +0,0 @@
|
||||
import AbstractAlgebra
|
||||
using GroupsCore
|
||||
|
||||
# disambiguation
|
||||
GroupsCore.order(
|
||||
::Type{I},
|
||||
G::AbstractAlgebra.Generic.SymmetricGroup,
|
||||
) where {I<:Integer} = I(factorial(G.n))
|
||||
|
||||
# disambiguation
|
||||
GroupsCore.order(
|
||||
::Type{I},
|
||||
g::AbstractAlgebra.Generic.Perm,
|
||||
) where {I<:Integer} =
|
||||
I(foldl(lcm, length(c) for c in AbstractAlgebra.cycles(g)))
|
||||
|
||||
# correct the AA length:
|
||||
Base.length(G::AbstractAlgebra.Generic.SymmetricGroup) = order(Int, G)
|
||||
|
||||
# genuinely new methods:
|
||||
Base.IteratorSize(::Type{<:AbstractAlgebra.AbstractPermutationGroup}) = Base.HasLength()
|
||||
|
||||
function GroupsCore.gens(G::AbstractAlgebra.Generic.SymmetricGroup{I}) where {I}
|
||||
a, b = one(G), one(G)
|
||||
circshift!(a.d, b.d, -1)
|
||||
b.d[1], b.d[2] = 2, 1
|
||||
return [a, b]
|
||||
end
|
||||
|
||||
Base.deepcopy_internal(g::AbstractAlgebra.Generic.Perm, ::IdDict) =
|
||||
AbstractAlgebra.Generic.Perm(deepcopy(g.d), false)
|
||||
Loading…
Reference in New Issue
Block a user