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Splitted into Groups.jl package
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module FreeGroups
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using Groups
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import Base: inv, convert
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export FGSymbol, IDSymbol
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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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IDSymbol(::Type{FGSymbol}) = FGSymbol("(id)", 0)
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FGSymbol(x::String) = FGSymbol(x,1)
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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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change_pow(s::FGSymbol, n::Int) = reduce(FGSymbol(s.gen, n))
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typealias FGWord GWord{FGSymbol}
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FGWord(s::FGSymbol) = FGWord([s])
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end #end of module FreeGroups
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173
Groups.jl
173
Groups.jl
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module Groups
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import Base: length, ==, hash, show
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import Base: one, inv, reduce, *, ^
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export GSymbol, GWord
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export IdSymbol, change_pow
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abstract GSymbol
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function show(io::IO, s::GSymbol)
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if s.pow == 0
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print(io, "(id)")
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elseif s.pow == 1
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print(io, s.gen)
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else
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print(io, (s.gen)*"^$(s.pow)")
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end
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end
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length(s::GSymbol) = (s.pow == 0 ? 0 : 1)
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IdSymbol(T::Type{GSymbol}) = throw(ArgumentError("Define IdSymbol(::Type{$T}) which is the identity element for Your type!"))
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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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(*){T<:GSymbol}(s::T, t::T) = return GWord{T}([s])*t
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change_pow(s::GSymbol, n::Int) = throw(ArgumentError("Define change_pow function for $(typeof(s))!"))
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abstract Word
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type GWord{T<:GSymbol} <: Word
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symbols::Vector{T}
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savedhash::UInt
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modified::Bool
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function GWord(symbols::Vector{T})
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return new(symbols, hash(symbols), false)
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end
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end
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GWord{T<:GSymbol}(s::T) = GWord{T}([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 freegroup_reduce!(GWord{T}(reverse([inv(s) for s in W.symbols])))
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end
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end
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function join_free_symbols!(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 freegroup_reduce!{T}(W::GWord{T})
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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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else
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reduced = false
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while !reduced
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reduced = join_free_symbols!(W)
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deleteat!(W.symbols, find(x -> x.pow == 0, W.symbols))
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end
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end
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W.modified = false
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W.savedhash = hash(W.symbols,hash(typeof(W)))
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return W
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end
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freegroup_reduce(W::GWord) = freegroup_reduce!(deepcopy(W))
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hash{T}(W::GWord{T}) = (W.modified && freegroup_reduce!(W); W.savedhash)
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function (==){T}(W::GWord{T}, Z::GWord{T})
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W.modified && freegroup_reduce!(W) # reduce could actually clear the flag and recalculate the hash
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Z.modified && freegroup_reduce!(Z)
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return W.savedhash == Z.savedhash && W.symbols == Z.symbols
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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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function r_multiply!(W::GWord, x; reduced::Bool=true)
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if length(x) > 0
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push!(W.symbols, x...)
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end
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if reduced
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freegroup_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.symbols, reverse(x)...)
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end
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if reduced
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freegroup_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}) = 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 freegroup_reduce!(y)
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end
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(^)(x::GWord, n::Integer) = power_by_squaring(x,n)
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end
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