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154 lines
5.9 KiB
Markdown
154 lines
5.9 KiB
Markdown
# Groups
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[](https://github.com/kalmarek/Groups.jl/actions/workflows/runtests.yml)
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[](https://codecov.io/gh/kalmarek/Groups.jl)
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An implementation of finitely-presented groups together with normalization (using Knuth-Bendix procedure).
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The package implements `AbstractFPGroup` with three concrete types: `FreeGroup`, `FPGroup` and `AutomorphismGroup`. Here's an example usage:
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```julia
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julia> using Groups, GroupsCore
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julia> A = Alphabet([:a, :A, :b, :B, :c, :C], [2, 1, 4, 3, 6, 5])
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Alphabet of Symbol:
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1. :a = (:A)⁻¹
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2. :A = (:a)⁻¹
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3. :b = (:B)⁻¹
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4. :B = (:b)⁻¹
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5. :c = (:C)⁻¹
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6. :C = (:c)⁻¹
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julia> F = FreeGroup(A)
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free group on 3 generators
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julia> a,b,c = gens(F)
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3-element Vector{FPGroupElement{FreeGroup{Symbol}, KnuthBendix.Word{UInt8}}}:
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a
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b
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c
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julia> a*inv(a)
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(empty word)
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julia> (a*b)^2
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a*b*a*b
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julia> commutator(a, b)
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A*B*a*b
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julia> x = a*b; y = inv(b)*a;
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julia> x*y
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a^2
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```
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Let's create a quotient of the free group above:
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```julia
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julia> ε = one(F);
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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 ])
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┌ Warning: Maximum number of rules (100) reached. The rewriting system may not be confluent.
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│ You may retry `knuthbendix` with a larger `maxrules` kwarg.
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└ @ KnuthBendix ~/.julia/packages/KnuthBendix/i93Np/src/kbs.jl:6
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⟨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)⟩
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```
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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.
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```julia
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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)
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⟨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)⟩
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```
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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`.
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```julia
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julia> using Random; Random.seed!(1); w = Groups.Word(rand(1:length(A), 16))
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KnuthBendix.Word{UInt16}: 4·6·1·1·1·6·5·1·5·2·3·6·2·4·2·6
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julia> F(w) # freely reduced w
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B*C*a^4*c*A*b*C*A*B*A*C
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julia> G(w) # w as an element of G
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B*a*b*a*B*a*C^2
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julia> F(w) # freely reduced w
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B*C*a^4*c*A*b*C*A*B*A*C
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julia> word(ans) # the underlying word in A
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KnuthBendix.Word{UInt8}: 4·6·1·1·1·1·5·2·3·6·2·4·2·6
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julia> G(w) # w as an element of G
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B*a*b*a*B*a*C^2
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julia> word(ans) # the underlying word in A
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KnuthBendix.Word{UInt8}: 4·1·3·1·4·1·6·6
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```
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As we can see the underlying words change according to where they are reduced.
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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.
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### Automorphism Groups
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Relatively complete is the support for the automorphisms of free groups, as given by Gersten presentation:
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```julia
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julia> saut = SpecialAutomorphismGroup(F, maxrules=100)
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┌ Warning: Maximum number of rules (100) reached. The rewriting system may not be confluent.
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│ You may retry `knuthbendix` with a larger `maxrules` kwarg.
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└ @ KnuthBendix ~/.julia/packages/KnuthBendix/i93Np/src/kbs.jl:6
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automorphism group of free group on 3 generators
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julia> S = gens(saut)
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12-element Vector{Automorphism{FreeGroup{Symbol},…}}:
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ϱ₁.₂
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ϱ₁.₃
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ϱ₂.₁
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ϱ₂.₃
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ϱ₃.₁
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ϱ₃.₂
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λ₁.₂
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λ₁.₃
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λ₂.₁
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λ₂.₃
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λ₃.₁
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λ₃.₂
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julia> x, y, z = S[1], S[12], S[6];
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julia> f = x*y*inv(z)
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ϱ₁.₂*λ₃.₂*ϱ₃.₂^-1
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julia> g = inv(z)*y*x
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ϱ₃.₂^-1*ϱ₁.₂*λ₃.₂
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julia> word(f), word(g)
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(KnuthBendix.Word{UInt8}: 1·12·18, KnuthBendix.Word{UInt8}: 18·1·12)
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```
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Even though Knuth-Bendix did not finish successfully in automorphism groups we have another ace in our sleeve to solve the word problem: evaluation.
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Lets have a look at the images of generators under those automorphisms:
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```julia
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julia> evaluate(f) # or to be more verbose...
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(a*b, b, b*c*B)
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julia> Groups.domain(g)
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(a, b, c)
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julia> Groups.evaluate!(Groups.domain(g), g)
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(a*b, b, b*c*B)
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```
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Since these automorphism map the standard generating set to the same new generating set, they should be considered as equal! And indeed they are:
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```julia
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julia> f == g
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true
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```
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This is what is happening behind the scenes:
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1. words are reduced using a rewriting system
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2. if resulting words are equal `true` is returned
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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.
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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).
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----
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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.
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