AbstractAlgebra.Generic.groebner_basis(::Vector{FreeAssAlgebraElem}): Support for precomputed groebner basis added. (#2076)

Co-authored-by: Lars Göttgens <lars.goettgens@gmail.com>

Author: Sequenzer

src/generic/FreeAssociativeAlgebraGroebner.jl

@@ -584,16 +584,32 @@ end
 function groebner_basis_buchberger(
     g::Vector{FreeAssociativeAlgebraElem{T}},
     reduction_bound::Int = typemax(Int),
-    remove_redundancies::Bool = false
+    remove_redundancies::Bool = false;
+    obstruction_free_set::Vector{FreeAssociativeAlgebraElem{T}}=FreeAssociativeAlgebraElem{T}[]
 ) where T<:FieldElement
-    g = copy(g)
+
+    if isempty(obstruction_free_set)
+      g = copy(g)
+      obstruction_queue = get_obstructions(g)
+    else
+      temp_g = copy(obstruction_free_set)
+      obstruction_queue =  PriorityQueue{Obstruction{T},FreeAssociativeAlgebraElem{T}}()
+      sizehint!(temp_g, length(g)+length(obstruction_free_set))
+
+      for p in g
+        push!(temp_g,p)
+        add_obstructions!(obstruction_queue,temp_g)
+      end
+      g = temp_g
+    end
+
     #   interreduce!(g) # on some small examples, this increases running time, so it might not be optimal to use this here
     nonzero_reductions = 0
     # compute the aho corasick automaton
     # to make normal form computation more efficient
     aut = AhoCorasickAutomaton([g_i.exps[1] for g_i in g])
     # step 1 from Thm. 5.2.12 Noncommutative Groebner Bases and Applications, Xingqiang Xiu
-    obstruction_queue = get_obstructions(g)
+
     while !isempty(obstruction_queue) # step 2
         obstruction = popfirst!(obstruction_queue)[1]
         # step3
@@ -607,6 +623,7 @@ function groebner_basis_buchberger(
         push!(g, Sp)
         insert_keyword!(aut, Sp.exps[1], length(g))
         if nonzero_reductions >= reduction_bound
+            @warn "The computation terminated because it exceeded the limit of $reduction_bound reduction steps."
             return g
         end
         add_obstructions!(obstruction_queue, g)
@@ -618,7 +635,7 @@ function groebner_basis_buchberger(
 end
 
 @doc """
-    groebner_basis(g::Vector{FreeAssociativeAlgebraElem{T}}, reduction_bound::Int = typemax(Int), remove_redundancies::Bool = false)
+    groebner_basis(g::Vector{FreeAssociativeAlgebraElem{T}}, reduction_bound::Int = typemax(Int), remove_redundancies::Bool = false; obstruction_free_set::Vector{FreeAssociativeAlgebraElem{T}})
 
 Compute a Groebner basis for the ideal generated by `g`. Stop when `reduction_bound` many
 non-zero entries have been added to the Groebner basis. If the computation stops due to the bound being exceeded, 
@@ -627,11 +644,42 @@ respect to this incomplete Groebner basis is `0`, it will also be `0` with respe
 
 If `remove_redundancies` is set to true, some redundant obstructions will be removed during the computation, which might save
 time, however in practice it seems to inflate the running time regularly.
+
+One can provide the keyword argument `obstruction_free_set` with a precomputed partial Groebner basis. 
+It is assumed (but not checked) that all entries of `obstruction_free_set` are indeed contained in the ideal generated by `g`.
+
+# Example
+
+```jldoctest
+julia> R, (a, b) = free_associative_algebra(QQ, ["a", "b"]);
+
+julia> f1 = a^2 -1;
+
+julia> f2 = b^3 -1;
+
+julia> f3 = a*b*a*b^2*a*b*a - b;
+
+julia> g = AbstractAlgebra.groebner_basis([(a*b*a*b^2)^2 - 1]; obstruction_free_set = [f1,f2,f3])
+12-element Vector{AbstractAlgebra.Generic.FreeAssociativeAlgebraElem{Rational{BigInt}}}:
+ a^2 - 1
+ b^3 - 1
+ a*b*a*b^2*a*b*a - b
+ a*b*a*b^2*a*b*a*b^2 - 1
+ -b*a*b^2*a*b*a*b^2 + a
+ -b*a*b^2*a*b*a + a*b
+ -b*a*b^2*a*b + a*b*a
+ -a*b^2*a*b + b^2*a*b*a
+ b^2*a*b*a*b^2 - a*b^2*a
+ -a*b*a*b^2 + b*a*b^2*a
+ b^2*a*b*a*b*a*b - a*b*a*b*a
+ -a*b*a*b*a*b + b*a*b*a*b*a
+```
 """ 
 function groebner_basis(
     g::Vector{FreeAssociativeAlgebraElem{T}},
     reduction_bound::Int = typemax(Int),
-    remove_redundancies::Bool = false
+    remove_redundancies::Bool = false;
+    obstruction_free_set::Vector{FreeAssociativeAlgebraElem{T}}=FreeAssociativeAlgebraElem{T}[]
 ) where T<:FieldElement
-    return groebner_basis_buchberger(g, reduction_bound, remove_redundancies)
+    return groebner_basis_buchberger(g, reduction_bound, remove_redundancies; obstruction_free_set=obstruction_free_set)
 end

test/generic/FreeAssociativeAlgebraGroebner-test.jl

@@ -9,7 +9,7 @@ include("AhoCorasick-test.jl")
 
     # Example 6.1 Kreuzer & Xiu
     R, (a, b) = free_associative_algebra(QQ, ["a", "b"])
- 
+
     g = AbstractAlgebra.groebner_basis([a^2 - 1, b^3 - 1, (a*b*a*b^2)^2 - 1])
     AbstractAlgebra.interreduce!(g)
     @test length(g) == 5
@@ -22,6 +22,15 @@ include("AhoCorasick-test.jl")
     @test all(u ->iszero(normal_form(u, g2)), g) # make sure removing redundant obstructions in the computation does not change the groebner basis
     @test all(u ->iszero(normal_form(u, g)), g2) 
 
+    f1 = a^2 -1;
+    f2 = b^3 -1;
+    f3 = a*b*a*b^2*a*b*a - b;
+    g = AbstractAlgebra.groebner_basis([(a*b*a*b^2)^2 - 1]; obstruction_free_set = [f1,f2,f3])
+    @test length(g) >= 5
+    AbstractAlgebra.interreduce!(g)
+    @test length(g) == 5
+
+
 end
 
 @testset "Generic.free_associative_algebra.groebner.normal_form" begin