From 9d7596acf1f0cdb69001d93f5a66d8f6ab7da59f Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 5 May 2021 01:10:28 +0200
Subject: [PATCH] first try on the new FPGroups

---
 src/Groups.jl     |   6 ++
 src/new_types.jl  | 211 ++++++++++++++++++++++++++++++++++++++++++++++
 src/normalform.jl |  76 +++++++++++++++++
 3 files changed, 293 insertions(+)
 create mode 100644 src/new_types.jl
 create mode 100644 src/normalform.jl

diff --git a/src/Groups.jl b/src/Groups.jl
index 0789323..9d60ad1 100644
--- a/src/Groups.jl
+++ b/src/Groups.jl
@@ -21,6 +21,12 @@ include("freereduce.jl")
 include("arithmetic.jl")
 include("findreplace.jl")
 
+module New
+include("new_types.jl")
+include("normalform.jl")
+
+end # module New
+
 ###############################################################################
 #
 #   String I/O
diff --git a/src/new_types.jl b/src/new_types.jl
new file mode 100644
index 0000000..ff49ee5
--- /dev/null
+++ b/src/new_types.jl
@@ -0,0 +1,211 @@
+using GroupsCore
+# using Groups
+# import Groups.AbstractFPGroup
+import KnuthBendix
+import KnuthBendix: AbstractWord, Alphabet, Word, RewritingSystem
+import KnuthBendix: alphabet
+using Random
+
+## "Abstract" definitions
+
+"""
+    AbstractFPGroup
+
+An Abstract type representing finitely presented groups. Every instance `` must implement
+ * `KnuthBendix.alphabet(G::MyFPGroup)`
+
+By default `word_type(::Type{MyFPGroup}) = Word{UInt16}` returns the default
+word type used for group elements. You may wish to override this if e.g. to
+> `word_type(::Type{MyFPGroup}) = Word{UInt8}`
+if your group has less than `255` generators.
+"""
+abstract type AbstractFPGroup <: GroupsCore.Group end
+
+word_type(G::AbstractFPGroup) = word_type(typeof(G))
+# the default:
+word_type(::Type{<:AbstractFPGroup}) = Word{UInt16}
+
+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
+
+## Group Interface
+
+Base.one(G::AbstractFPGroup) = FPGroupElement(one(word_type(G)), G)
+
+Base.eltype(::Type{FPG}) where {FPG<:AbstractFPGroup} =
+    FPGroupElement{FPG, word_type(FPG)}
+
+struct FPGroupIter{GEl}
+    elts::Vector{GEl}
+    seen::Set{GEl}
+    u::GEl
+    v::GEl
+end
+
+FPGroupIter(G::AbstractFPGroup) =
+    FPGroupIter([one(G)], Set([one(G)]), one(G), one(G))
+
+Base.iterate(G::AbstractFPGroup) = one(G), (FPGroupIter(G), 1, 1)
+@inline function Base.iterate(G::AbstractFPGroup, state)
+    iter, elt_idx, gen_idx = state
+
+    if gen_idx > length(alphabet(G))
+        elt_idx == length(iter.elts) && return nothing
+        gen_idx = 1
+        elt_idx += 1
+    end
+
+
+    res = let (u, v) = (iter.u, iter.v), elt = iter.elts[elt_idx]
+        copyto!(v, elt) # this invalidates normalform of v
+        @assert !isnormalform(v)
+        push!(word(v), gen_idx)
+        resize!(word(u), 0)
+
+        normalform!(u, v)
+    end
+
+    if res in iter.seen
+        return iterate(G, (iter, elt_idx, gen_idx+1))
+    else
+        w = deepcopy(res)
+        @assert isnormalform(w)
+        push!(iter.elts, w)
+        push!(iter.seen, w)
+        state = (iter, elt_idx, gen_idx+1)
+        return w, state
+    end
+end
+
+# the default:
+# Base.IteratorSize(::Type{<:AbstractFPGroup}) = Base.SizeUnknown()
+
+GroupsCore.ngens(G::AbstractFPGroup) = length(G.gens)
+
+function GroupsCore.gens(G::AbstractFPGroup, i::Integer)
+    @boundscheck 1<=i<=GroupsCore.ngens(G)
+    return FPGroupElement(word_type(G)([i]), 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[]
+    symmgens_len = length(alphabet(F))
+    return FPGroupElement(word_type(G)(rand(1:symmgens_len, l)), G)
+end
+
+## FPGroupElement
+
+mutable struct FPGroupElement{G<:AbstractFPGroup, W<:AbstractWord} <: GroupElement
+    word::W
+    savedhash::UInt
+    parent::G
+
+    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)
+    print(io, KnuthBendix.string_repr(word(f), alphabet(f)))
+end
+
+## Hashing
+
+# We store hash of a word in field `savedhash` to use it as cheap replacement
+# for equality checking. In the last bit of `savehash` we store the information
+# whether the word is already reduced (in normal form) or not. Hence
+isnormalform(g::FPGroupElement) = Bool(g.savedhash & 1)
+
+# To update hash use this internal method, possibly only after computing the
+# normal form of `g`:
+_update_savedhash!(g::FPGroupElement, h = hash(word(g))) =
+    (g.savedhash = h | 1; return g)
+
+# If `g` has been mutated (e.g. `word(g)` has been modified, `_invalidate_hash`
+# must be called.
+_invalidate_hash!(g::FPGroupElement) = g.savedhash = g.savedhash & 0
+
+# Accessor for the saved hash
+@inline _savedhash(f::FPGroupElement) = f.savedhash
+
+function Base.hash(g::FPGroupElement, h::UInt)
+    normalform!(g)
+    # compute the best approximation of the normal form
+    return hash(parent(g), _savedhash(g) ⊻ h)
+end
+
+function Base.copyto!(res::FPGroupElement, g::FPGroupElement)
+    resize!(word(res), length(word(g)))
+    copyto!(word(res), word(g))
+    _invalidate_hash!(res)
+    return res
+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)
+    hash(g) != hash(h) && return false
+    # hash reduces to normal form behind the scenes
+    return word(g) == word(h)
+end
+
+function Base.deepcopy_internal(g::FPGroupElement, stackdict::IdDict)
+    return FPGroupElement(copy(word(g)), _savedhash(g), 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 : throw("Not Implemented")
+
+# 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))
+    return FreeGroup(KnuthBendix.letters(a), a)
+end
+
+Base.show(io::IO, F::FreeGroup) = print(io, "free group on $(ngens(F)) generators")
+
+# mandatory methods:
+KnuthBendix.alphabet(F::FreeGroup) = F.alphabet
diff --git a/src/normalform.jl b/src/normalform.jl
new file mode 100644
index 0000000..945cfc2
--- /dev/null
+++ b/src/normalform.jl
@@ -0,0 +1,76 @@
+"""
+    normalform!(g::FPGroupElement)
+Compute the normal form of `g`, possibly modifying `g` in-place.
+"""
+@inline function normalform!(g::FPGroupElement)
+    isnormalform(g) && return g
+
+    let w = one(word(g))
+        w = normalform!(w, g)
+        resize!(word(g), length(w))
+        copyto!(word(g), w)
+    end
+
+    g = _update_savedhash!(g)
+    @assert isnormalform(g)
+    return g
+end
+
+"""
+    normalform!(res::GEl, g::GEl) where GEl<:FPGroupElement
+Compute the normal fom of `g`, storing it in `res`.
+"""
+function normalform!(res::GEl, g::GEl) where GEl<:FPGroupElement
+    @boundscheck @assert parent(res) === parent(g)
+    if isnormalform(g)
+        copyto!(res, g)
+        _update_savedhash!(res, g.savedhash)
+    else
+        resize!(word(res), 0)
+        normalform!(word(res), g)
+        res = _update_savedhash!(res)
+    end
+    return res
+end
+
+"""
+    normalform!(res::AbstractWord, g::FPGroupElement)
+Append the normal form of `g` to word `res`, modifying `res` in place.
+
+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 free_rewrite!(res, word(g), alphabet(g))
+end
+
+"""
+    free_rewrite!(v::AbstractWord, w::AbstractWord, A::Alphabet)
+Append `w` to `v` applying free reductions as defined by the inverses of `A`.
+"""
+free_rewrite!(v::AbstractWord, w::AbstractWord, A::Alphabet) =
+    KnuthBendix.rewrite_from_left!(v, w, A)
+
+function KnuthBendix.rewrite_from_left!(
+    v::AbstractWord,
+    w::AbstractWord,
+    A::Alphabet
+    )
+    while !isone(w)
+        if isone(v)
+            push!(v, popfirst!(w))
+        else
+            # the first check is for monoids only
+            if KnuthBendix.hasinverse(last(v), A) && inv(A, last(v)) == first(w)
+                pop!(v)
+                popfirst!(w)
+            else
+                push!(v, popfirst!(w))
+            end
+        end
+    end
+    return v
+end