130 lines
3.3 KiB
Julia
130 lines
3.3 KiB
Julia
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function KnuthBendix.Alphabet(S::AbstractVector{<:GSymbol})
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S = unique!([S; inv.(S)])
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inversions = [findfirst(==(inv(s)), S) for s in S]
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return Alphabet(S, inversions)
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end
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struct AutomorphismGroup{G<:Group, T, R, S} <: AbstractFPGroup
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group::G
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gens::Vector{T}
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rws::R
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domain::S
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end
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object(G::AutomorphismGroup) = G.group
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function SpecialAutomorphismGroup(F::FreeGroup;
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ordering=KnuthBendix.LenLex, kwargs...)
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n = length(KnuthBendix.alphabet(F))÷2
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A, rels = gersten_relations(n, commutative=false)
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S = KnuthBendix.letters(A)[1:2(n^2 - n)]
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rws = KnuthBendix.RewritingSystem(rels, ordering(A))
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KnuthBendix.knuthbendix!(rws; kwargs...)
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return AutomorphismGroup(F, S, rws, ntuple(i->gens(F, i), n))
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end
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KnuthBendix.alphabet(G::AutomorphismGroup{<:FreeGroup}) = alphabet(rewriting(G))
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rewriting(G::AutomorphismGroup) = G.rws
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function relations(G::AutomorphismGroup)
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n = length(KnuthBendix.alphabet(object(G)))÷2
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return last(gersten_relations(n, commutative=false))
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end
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_hashing_data(f::FPGroupElement{<:AutomorphismGroup}) = normalform!.(evaluate(f))
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function Base.:(==)(g::A, h::A) where A<:FPGroupElement{<:AutomorphismGroup}
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@assert parent(g) === parent(h)
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if _isvalidhash(g) && _isvalidhash(h)
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hash(g) != hash(h) && return false
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end
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normalform!(g)
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normalform!(h)
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word(g) == word(h) && return true
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@assert isnormalform(g)
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@assert isnormalform(h)
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img_computed, imh_computed = false, false
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if !_isvalidhash(g)
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img = _hashing_data(g)
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_update_savedhash!(g, img)
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img_computed = true
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end
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if !_isvalidhash(h)
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imh = _hashing_data(h)
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_update_savedhash!(h, imh)
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imh_computed = true
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end
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@assert _isvalidhash(g)
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@assert _isvalidhash(h)
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hash(g) != hash(h) && return false
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# words are different, but hashes agree
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if !img_computed
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img = _hashing_data(g)
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end
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if !imh_computed
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imh = _hashing_data(h)
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end
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res = img == imh
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!res && @warn "hash collision in == :" g h
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return res
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end
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# eye-candy
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Base.show(io::IO, ::Type{<:FPGroupElement{<:AutomorphismGroup{T}}}) where T <: FreeGroup = print(io, "Automorphism{$T}")
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## Automorphism Evaluation
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domain(f::FPGroupElement{<:AutomorphismGroup}) = deepcopy(parent(f).domain)
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# tuple(gens(object(parent(f)))...)
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evaluate(f::FPGroupElement{<:AutomorphismGroup{<:FreeGroup}}) =
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evaluate!(domain(f), f)
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function evaluate!(t::NTuple{N, T}, f::FPGroupElement{<:AutomorphismGroup{<:FreeGroup}}) where {T<:FPGroupElement, N}
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A = alphabet(f)
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for idx in word(f)
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t = evaluate!(t, A[idx])::NTuple{N, T}
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end
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return t
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end
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function evaluate!(v::NTuple{N, T}, s::AutSymbol) where {N, T}
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@assert s.pow in (-1, 1)
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return evaluate!(v, s.fn, isone(s.pow))::NTuple{N, T}
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end
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function evaluate!(v, ϱ::RTransvect, flag)
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if flag
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append!(New.word(v[ϱ.i]), New.word(v[ϱ.j] ))
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else
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append!(New.word(v[ϱ.i]), New.word(v[ϱ.j]^-1))
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end
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_setnormalform!(v[ϱ.i], false)
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_setvalidhash!(v[ϱ.i], false)
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return v
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end
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function evaluate!(v, λ::LTransvect, flag)
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if flag
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prepend!(New.word(v[λ.i]), New.word(v[λ.j] ))
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else
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prepend!(New.word(v[λ.i]), New.word(v[λ.j]^-1))
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end
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_setnormalform!(v[λ.i], false)
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_setvalidhash!(v[λ.i], false)
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return v
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end
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