mirror of
https://github.com/kalmarek/PropertyT.jl.git
synced 2024-07-17 10:55:30 +02:00
328 lines
8.5 KiB
Julia
328 lines
8.5 KiB
Julia
module FreeGroups
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export GSymbol, AutSymbol, Word, GWord, FGWord, AutWord, FGAutomorphism
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import Base: length, ==, hash, show, convert
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import Base: *, ^, convert
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import Base: one, inv, reduce, push!, unshift!
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abstract GSymbol
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immutable FGSymbol <: GSymbol
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gen::String
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pow::Int
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end
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immutable AutSymbol <: GSymbol
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gen::String
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pow::Int
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ex::Expr
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end
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IDSymbol(::Type{FGSymbol}) = FGSymbol("(id)", 0)
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IDSymbol(::Type{AutSymbol}) = AutSymbol("(id)", 0, :(IDAutomorphism(N)))
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FGSymbol(x::String) = FGSymbol(x,1)
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function show(io::IO, s::GSymbol)
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if s.pow == 1
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print(io, (s.gen))
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elseif s.pow == 0
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print(io, "(id)")
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else
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print(io, (s.gen)*"^$(s.pow)")
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end
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end
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(==)(s::GSymbol, t::GSymbol) = s.gen == t.gen && s.pow == t.pow
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length(s::GSymbol) = (s.pow == 0 ? 0 : 1)
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one{T<:GSymbol}(::Type{T}) = IDSymbol(T)
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one(s::GSymbol) = one(typeof(s))
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inv(s::FGSymbol) = FGSymbol(s.gen, -s.pow)
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convert(::Type{FGSymbol}, x::String) = FGSymbol(x)
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reduce(s::GSymbol) = (s.pow == 0 ? one(s) : s)
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change_pow(s::FGSymbol, n::Int) = reduce(FGSymbol(s.gen, n))
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change_pow(s::AutSymbol, n::Int) = reduce(AutSymbol(s.gen, n, s.ex))
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(^)(s::GSymbol, n::Integer) = change_pow(s, s.pow*n)
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function inv(f::AutSymbol)
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symbol = f.ex.args[1]
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if symbol == :ɛ
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return FreeGroups.change_pow(f, f.pow % 2)
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elseif symbol == :σ
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perm = invperm(f.ex.args[2])
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gen = string('σ', [Char(8320 + i) for i in perm]...)
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return AutSymbol(gen, f.pow, :(σ($perm)))
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elseif symbol == :(ϱ) || symbol == :λ
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return AutSymbol(f.gen, -f.pow, f.ex)
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elseif symbol == :IDAutomorphism
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return f
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else
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throw(ArgumentError("Don't know how to invert $f (of type $symbol)"))
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end
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end
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function (*){T<:GSymbol}(s::T, t::T)
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return GWord{T}([s])*t
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end
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abstract Word
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#=
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@ScottPJones
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If so, I'd recommend
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1) making GWord a type, not an immutable
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2) add fields
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savedhash::UInt and
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modified::Bool
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3) make any function that modifies the contents of .symbols set the modified flag,
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4) make the hash function
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a) check that flag:
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if false, return the savedhash field,
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otherwise, call reduce!,
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b) clear the modified flag, and
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c) calculate a hash value simply by calling hash(symbols)
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d) save that back into the savedhash field
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5) for ==, I don't think you need to do all that checking for length or length == 0, that will already be handled by comparing the symbols vectors (possibly faster)
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function (==){T}(W::GWord{T}, Z::GWord{T})
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W.modified && reduce!(W) # reduce could actually clear the flag and recalculate the hash
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Z.modified && reduce!(Z)
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W.hash == Z.hash && W.symbols == Z.symbols
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end
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hash{T}(W::GWord{T}) = (W.modified && reduce!(W); W.hash)
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(and last lines of reduce! would have W.modified = false ; W.hash = hash(W.symbols))
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=#
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immutable GWord{T<:GSymbol} <: Word
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symbols::Vector{T}
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end
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typealias FGWord GWord{FGSymbol}
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typealias AutWord GWord{AutSymbol}
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GWord{T<:GSymbol}(s::T) = GWord{T}([s])
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FGWord(s::FGSymbol) = FGWord([s])
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IDWord{T<:GSymbol}(::Type{T}) = GWord(one(T))
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IDWord{T<:GSymbol}(W::GWord{T}) = IDWord(T)
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function length(W::GWord)
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return sum([abs(s.pow) for s in W.symbols])
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end
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one{T}(::Type{GWord{T}}) = IDWord(T)
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one{T}(w::GWord{T}) = one(GWord{T})
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function inv{T}(W::GWord{T})
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if length(W) == 0
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return W
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else
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return prod(reverse([inv(s) for s in W.symbols]))
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end
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end
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function free_group_reduction!(W::GWord)
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reduced = true
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for i in 1:length(W.symbols) - 1
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if W.symbols[i].gen == W.symbols[i+1].gen
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reduced = false
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p1 = W.symbols[i].pow
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p2 = W.symbols[i+1].pow
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W.symbols[i+1] = change_pow(W.symbols[i], p1 + p2)
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W.symbols[i] = one(W.symbols[i])
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end
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end
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return reduced
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end
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function reduce!{T}(W::GWord{T}, reduce_func::Function=free_group_reduction!)
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if length(W) < 2
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deleteat!(W.symbols, find(x -> x.pow == 0, W.symbols))
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return W
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end
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reduced = false
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while !reduced
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reduced = reduce_func(W)
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deleteat!(W.symbols, find(x -> x.pow == 0, W.symbols))
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end
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return W
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end
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reduce(W::GWord) = reduce!(deepcopy(W))
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function (==){T}(W::GWord{T}, Z::GWord{T})
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reduce!(W)
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reduce!(Z)
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if length(W) != length(Z)
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return false
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elseif length(W) == 0
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return true
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else
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return W.symbols == Z.symbols
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end
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end
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function show(io::IO, W::GWord)
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if length(W) == 0
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print(io, "(id)")
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else
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join(io, [string(s) for s in W.symbols], "*")
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end
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end
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push!(W::GWord, x) = push!(W.symbols, x...)
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unshift!(W::GWord, x) = unshift!(W.symbols, x...)
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function r_multiply!(W::GWord, x; reduced::Bool=true)
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if length(x) > 0
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push!(W, x)
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end
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if reduced
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reduce!(W)
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end
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return W
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end
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function l_multiply!(W::GWord, x; reduced::Bool=true)
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if length(x) > 0
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unshift!(W, reverse(x))
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end
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if reduced
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reduce!(W)
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end
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return W
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end
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r_multiply(W::GWord, x; reduced::Bool=true) =
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r_multiply!(deepcopy(W),x, reduced=reduced)
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l_multiply(W::GWord, x; reduced::Bool=true) =
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l_multiply!(deepcopy(W),x, reduced=reduced)
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(*){T}(W::GWord{T}, Z::GWord{T}) = FreeGroups.r_multiply(W, Z.symbols)
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(*)(W::GWord, s::GSymbol) = W*GWord(s)
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(*)(s::GSymbol, W::GWord) = GWord(s)*W
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function power_by_squaring{T}(x::GWord{T}, p::Integer)
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if p < 0
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return power_by_squaring(inv(x), -p)
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elseif p == 0
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return one(x)
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elseif p == 1
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return deepcopy(x)
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elseif p == 2
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return x*x
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end
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t = trailing_zeros(p) + 1
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p >>= t
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while (t -= 1) > 0
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x *= x
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end
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y = x
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while p > 0
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t = trailing_zeros(p) + 1
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p >>= t
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while (t -= 1) >= 0
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x *= x
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end
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y *= x
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end
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return reduce!(y)
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end
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(^)(x::GWord, n::Integer) = power_by_squaring(x,n)
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type FGAutomorphism{T<:GSymbol}
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domain::Vector{T}
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image::Vector{GWord{T}}
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map::Function
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function FGAutomorphism{T}(domain::Vector{T}, image::Vector{GWord{T}}, map::Function)
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length(domain) == length(unique(domain)) ||
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throw(ArgumentError("The elements of $domain are not unique"))
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length(domain) == length(image) ||
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throw(ArgumentError("Dimensions of image and domain must match"))
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# Set(vcat([[s.gen for s in reduce!(x).symbols]
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# for x in image]...)) == Set(s.gen for s in domain) ||
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# throw(ArgumentError("Are You sure that $image defines an automorphism??"))
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new(domain, image, map)
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end
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end
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function show(io::IO, X::FGAutomorphism)
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title = "Endomorphism of Free Group on $(length(X.domain)) generators, sending"
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map = ["$x ⟶ $y" for (x,y) in zip(X.domain, X.image)]
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join(io, vcat(title,map), "\n")
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end
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(==)(f::FGAutomorphism, g::FGAutomorphism) =
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f.domain == g.domain && f.image == g.image
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function aut_func_from_table(table::Vector{Tuple{Int,Int}}, GroupIdentity=one(FGWord))
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if length(table) == 0
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# warn("The map is not an automorphism")
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nothing
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end
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return v->reduce(*,GroupIdentity, v[idx]^power for (idx, power) in table)
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end
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function aut_func_from_word(domain, w::GWord)
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table = Vector{Tuple{Int, Int}}()
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for s in w.symbols
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pair = (findfirst([x.gen for x in domain], s.gen), s.pow)
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push!(table, pair)
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end
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return aut_func_from_table(table)
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end
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function FGMap(domain::Vector{FGSymbol}, image::Vector{GWord})
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function_vector = Vector{Function}()
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for word in image
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push!(function_vector, aut_func_from_word(domain, word))
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end
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return v -> Vector{FGWord}([f(v) for f in function_vector])
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end
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FGAutomorphism(domain::Vector{FGSymbol}, image::Vector{GWord}) =
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FGAutomorphism(domain, image, FGMap(domain, image))
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FGAutomorphism(domain::Vector{FGSymbol}, image::Vector{FGSymbol}) =
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FGAutomorphism(domain, Vector{GWord}(image))
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function FGAutomorphism(domain::Vector, image::Vector)
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FGAutomorphism(Vector{FGSymbol}(domain), Vector{GWord}(image))
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end
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function FGAutomorphism(domain, image)
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FGAutomorphism([domain...], [image...])
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end
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"""Computes the composition g∘f of two morphisms"""
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function compose(f::FGAutomorphism, g::FGAutomorphism)
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if length(f.image) != length(g.domain)
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throw(ArgumentError("Cannot compose $f and $g"))
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else
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h(v) = g.map(f.map(v))
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return FGAutomorphism(f.domain, h(f.domain), h)
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end
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end
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(*)(f::FGAutomorphism, g::FGAutomorphism) = compose(f,g)
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end
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