Merge pull request #926 from AlgebraicJulia/unique_hom

Require `homomorphism` to be unique

Project.toml

@@ -2,7 +2,7 @@ name = "Catlab"
 uuid = "134e5e36-593f-5add-ad60-77f754baafbe"
 license = "MIT"
 authors = ["Evan Patterson <[email protected]>"]
-version = "0.16.15"
+version = "0.16.16"
 
 [deps]
 ACSets = "227ef7b5-1206-438b-ac65-934d6da304b8"

docs/literate/graphics/graphviz_graphs.jl

@@ -38,7 +38,7 @@ to_graphviz(g, node_attrs=Dict(:color => "cornflowerblue"),
 
 using Catlab.CategoricalAlgebra
 
-f = homomorphism(cycle_graph(Graph, 4), complete_graph(Graph, 2))
+f = homomorphisms(cycle_graph(Graph, 4), complete_graph(Graph, 2)) |> first
 
 # By default, the domain and codomain graph are both drawn, as well the vertex
 # mapping between them.

docs/literate/graphs/graphs.jl

@@ -260,7 +260,7 @@ draw(id(K₃))
 length(homomorphisms(T, esym))
 
 # but we can use 3 colors to color T.
-draw(homomorphism(T, K₃))
+draw(homomorphism(T, K₃; any=true))
 
 # ### Exercise:
 # 1. Find a graph that is not 3-colorable

src/categorical_algebra/CSets.jl

@@ -451,10 +451,14 @@ function coerce_components(S, components, X::ACSet{<:PT}, Y) where PT
   return merge(ocomps, acomps)
 end 
 
+# Enforces that function has a valid domain (but not necessarily codomain)
 function coerce_component(ob::Symbol, f::FinFunction{Int,Int},
                           dom_size::Int, codom_size::Int; kw...)
-  length(dom(f)) == dom_size || error("Domain error in component $ob")
-  # length(codom(f)) == codom_size || error("Codomain error in component $ob") # codom size is now Maxpart not nparts
+  if haskey(kw, :dom_parts)
+    !any(i -> f(i) == 0, kw[:dom_parts]) # check domain of mark as deleted
+  else                         
+    length(dom(f)) == dom_size # check domain of dense parts
+  end || error("Domain error in component $ob")
   return f 
 end
 
@@ -472,8 +476,8 @@ end
 function coerce_attrvar_component(
     ob::Symbol, f::VarFunction,::TypeSet{T},::TypeSet{T},
     dom_size::Int, codom_size::Int; kw...) where {T}
-  # length(dom(f.fun)) == dom_size || error("Domain error in component $ob: $(dom(f.fun))!=$dom_size")
-  length(f.codom) == codom_size || error("Codomain error in component $ob: $(f.fun.codom)!=$codom_size")
+  length(f.codom) == codom_size || error(
+    "Codomain error in component $ob: $(f.fun.codom)!=$codom_size")
   return f
 end
 
@@ -1119,9 +1123,10 @@ end
 const SubCSet{S} = Subobject{<:StructCSet{S}}
 const SubACSet{S} = Subobject{<:StructACSet{S}}
 
-# Componentwise subobjects
+# Componentwise subobjects: coerce VarFunctions to FinFunctions
 components(A::SubACSet{S}) where S = 
-  NamedTuple(k => Subobject(vs) for (k,vs) in pairs(components(hom(A)))
+  NamedTuple(k => Subobject(k ∈ ob(S) ? vs : FinFunction(vs)) for (k,vs) in 
+             pairs(components(hom(A)))
 )
 
 force(A::SubACSet) = Subobject(force(hom(A)))

src/categorical_algebra/CatElements.jl

@@ -68,7 +68,7 @@ function elements(f::ACSetTransformation)
   end
   pts = vcat([collect(f[o]).+off for (o, off) in zip(ob(S), offs)]...)
   # *strict* ACSet transformation uniquely determined by its action on vertices
-  return only(homomorphisms(X, Y; initial=Dict([:El=>pts])))
+  return homomorphism(X, Y; initial=Dict([:El=>pts]))
 end
 
 

src/categorical_algebra/FinSets.jl

@@ -348,7 +348,7 @@ VarFunction(f::AbstractVector{Int},cod::Int) = VarFunction(FinFunction(f,cod))
 VarFunction(f::FinDomFunction) = VarFunction{Union{}}(AttrVar.(collect(f)),codom(f))
 VarFunction{T}(f::FinDomFunction,cod::FinSet) where T = VarFunction{T}(collect(f),cod)
 FinFunction(f::VarFunction{T}) where T = FinFunction(
-  [f.fun(i) isa AttrVar ? f.fun(i).val : error("Cannot cast to FinFunction") 
+  [f(i) isa AttrVar ? f(i).val : error("Cannot cast to FinFunction") 
    for i in dom(f)], f.codom)
 FinDomFunction(f::VarFunction{T}) where T = f.fun
 Base.length(f::AbsVarFunction{T}) where T = length(collect(f.fun))

src/categorical_algebra/HomSearch.jl

@@ -54,7 +54,8 @@ to infinite ``C``-sets when ``C`` is infinite (but possibly finitely presented).
 """
 struct HomomorphismQuery <: ACSetHomomorphismAlgorithm end
 
-""" Find a homomorphism between two attributed ``C``-sets.
+""" Find a unique homomorphism between two attributed ``C``-sets (subject to a
+variety of constraints), if one exists.
 
 Returns `nothing` if no homomorphism exists. For many categories ``C``, the
 ``C``-set homomorphism problem is NP-complete and thus this procedure generally
@@ -94,17 +95,17 @@ In both of these cases, it's possible to compute homomorphisms when there are
 the domain), as each such variable has a finite number of possibilities for it
 to be mapped to.
 
+Setting `any=true` relaxes the constraint that the returned homomorphism is 
+unique.
+
 See also: [`homomorphisms`](@ref), [`isomorphism`](@ref).
 """
 homomorphism(X::ACSet, Y::ACSet; alg=BacktrackingSearch(), kw...) =
   homomorphism(X, Y, alg; kw...)
 
-function homomorphism(X::ACSet, Y::ACSet, alg::BacktrackingSearch; kw...)
-  result = nothing
-  backtracking_search(X, Y; kw...) do α
-    result = α; return true
-  end
-  result
+function homomorphism(X::ACSet, Y::ACSet, alg::BacktrackingSearch; any=false, kw...)
+  res = homomorphisms(X, Y, alg; Dict((any ? :take : :max) => 1)..., kw...)
+  isempty(res) ? nothing : only(res)
 end
 
 """ Find all homomorphisms between two attributed ``C``-sets.
@@ -115,10 +116,30 @@ homomorphisms exist, it is exactly as expensive.
 homomorphisms(X::ACSet, Y::ACSet; alg=BacktrackingSearch(), kw...) =
   homomorphisms(X, Y, alg; kw...)
 
-function homomorphisms(X::ACSet, Y::ACSet, alg::BacktrackingSearch; kw...) 
+""" Find all homomorphisms between two attributed ``C``-sets via BackTracking Search.
+
+take = number of homomorphisms requested (stop the search process early if this 
+       number is reached)
+max = throw an error if we take more than this many morphisms (e.g. set max=1 if 
+      one expects 0 or 1 morphism)
+filter = only consider morphisms which meet some criteria, expressed as a Julia 
+         function of type ACSetTransformation -> Bool
+
+It does not make sense to specify both `take` and `max`.
+"""
+function homomorphisms(X::ACSet, Y::ACSet, alg::BacktrackingSearch; 
+                       take=nothing, max=nothing, filter=nothing, kw...) 
   results = []
-  backtracking_search(X, Y; kw...) do α
-    push!(results, map_components(deepcopy, α)); return false
+  isnothing(take) || isnothing(max) || error(
+    "Cannot set both `take`=$take and `max`=$max for `homomorphisms`")
+  backtracking_search(X, Y; kw...) do αs
+    for α in αs
+      isnothing(filter) || filter(α) || continue
+      length(results) == max && error("Exceeded $max: $([results; α])")
+      push!(results, map_components(deepcopy, α));
+      length(results) == take && return true
+    end 
+    return false
   end
   results
 end
@@ -132,7 +153,7 @@ is_homomorphic(X::ACSet, Y::ACSet; alg=BacktrackingSearch(), kw...) =
   is_homomorphic(X, Y, alg; kw...)
 
 is_homomorphic(X::ACSet, Y::ACSet, alg::BacktrackingSearch; kw...) =
-  !isnothing(homomorphism(X, Y, alg; kw...))
+  !isempty(homomorphisms(X, Y, alg; take=1, kw...))
 
 """ Find an isomorphism between two attributed ``C``-sets, if one exists.
 
@@ -152,8 +173,8 @@ homomorphisms exist, it is exactly as expensive.
 isomorphisms(X::ACSet, Y::ACSet; alg=BacktrackingSearch(), kw...) =
   isomorphisms(X, Y, alg; kw...)
 
-isomorphisms(X::ACSet, Y::ACSet, alg::BacktrackingSearch; initial=(;)) =
-  homomorphisms(X, Y, alg; iso=true, initial=initial)
+isomorphisms(X::ACSet, Y::ACSet, alg::BacktrackingSearch; initial=(;), kw...) =
+  homomorphisms(X, Y, alg; iso=true, initial=initial, kw...)
 
 """ Are the two attributed ``C``-sets isomorphic?
 
@@ -164,7 +185,7 @@ is_isomorphic(X::ACSet, Y::ACSet; alg=BacktrackingSearch(), kw...) =
   is_isomorphic(X, Y, alg; kw...)
 
 is_isomorphic(X::ACSet, Y::ACSet, alg::BacktrackingSearch; kw...) =
-  !isnothing(isomorphism(X, Y, alg; kw...))
+  !isempty(isomorphisms(X, Y, alg; take=1, kw...))
 
 # Backtracking search
 #--------------------
@@ -198,7 +219,6 @@ struct BacktrackingState{
   predicates::Predicates
   image::Image # Negative of image for epic components or if finding an epimorphism
   unassigned::Unassign # "# of unassigned elems in domain of a component 
-
 end
 
 function backtracking_search(f, X::ACSet, Y::ACSet;
@@ -304,44 +324,22 @@ function backtracking_search(f, X::ACSet, Y::ACSet;
   backtracking_search(f, state, 1; random=random)
 end
 
+"""
+Note: a successful search returns an *iterator* of solutions, rather than 
+a single solution. See `postprocess_search_results`.
+"""
 function backtracking_search(f, state::BacktrackingState, depth::Int; 
                               random=false) 
   # Choose the next unassigned element.
   mrv, mrv_elem = find_mrv_elem(state, depth)
   if isnothing(mrv_elem)
     # No unassigned elements remain, so we have a complete assignment.
     if any(!=(identity), state.type_components)
-      return f(LooseACSetTransformation(
-        state.assignment, state.type_components, state.dom, state.codom))
+      return f([LooseACSetTransformation(
+        state.assignment, state.type_components, state.dom, state.codom)])
     else
-      S = acset_schema(state.dom)
-      od = Dict{Symbol,Vector{Int}}(k=>(state.assignment[k]) for k in objects(S))
-
-      # Compute possible assignments for all free variables
-      free_data = map(attrtypes(S)) do k
-        monic = !isnothing(state.inv_assignment[k])
-        assigned = [v.val for (_, v) in state.assignment[k] if v isa AttrVar]
-        valid_targets = setdiff(parts(state.codom, k), monic ? assigned : [])
-        free_vars = findall(==(AttrVar(0)), last.(state.assignment[k]))
-        N = length(free_vars)
-        prod_iter = Iterators.product(fill(valid_targets, N)...)
-        if monic
-          prod_iter = Iterators.filter(x->length(x)==length(unique(x)), prod_iter)
-        end
-        (free_vars, prod_iter) # prod_iter = valid assignments for this attrtype
-      end
-
-      # Homomorphism for each element in the product of the prod_iters
-      for combo in Iterators.product(last.(free_data)...) 
-        ad = Dict(map(zip(attrtypes(S), first.(free_data), combo)) do (k, xs, vs)
-          vec = last.(state.assignment[k])
-          vec[xs] = AttrVar.(collect(vs))
-          k => vec
-        end)
-        comps = merge(NamedTuple(od),NamedTuple(ad))
-        f(ACSetTransformation(comps, state.dom, state.codom))
-      end
-      return false
+      m = Dict(k=>!isnothing(v) for (k,v) in pairs(state.inv_assignment))
+      return f(postprocess_search_results(state.dom, state.codom, state.assignment, m))
     end
   elseif mrv == 0
     # An element has no allowable assignment, so we must backtrack.
@@ -509,6 +507,48 @@ unassign_elem!(state::BacktrackingState{<:DynamicACSet}, depth, c, x) =
   end
 end
 
+""" 
+A hom search result might not have all the data for an ACSetTransformation
+explicitly specified. For example, if there is a cartesian product of possible
+assignments which could not possibly constrain each other, then we should
+iterate through this product at the very end rather than having the backtracking
+search navigate the product space. Currently, this is only done with assignments
+for floating attribute variables, but in principle this could be applied in the
+future to, e.g., free-floating vertices of a graph or other coproducts of 
+representables.
+
+This function takes a result assignment from backtracking search and returns an
+iterator of the implicit set of homomorphisms that it specifies.
+"""
+function postprocess_search_results(dom, codom, assgn, monic)
+  S = acset_schema(dom)
+  od = Dict{Symbol,Vector{Int}}(k=>(assgn[k]) for k in objects(S))
+
+  # Compute possible assignments for all free variables
+  free_data = map(attrtypes(S)) do k
+    assigned = [v.val for (_, v) in assgn[k] if v isa AttrVar]
+    valid_targets = setdiff(parts(codom, k), monic[k] ? assigned : [])
+    free_vars = findall(==(AttrVar(0)), last.(assgn[k]))
+    N = length(free_vars)
+    prod_iter = Iterators.product(fill(valid_targets, N)...)
+    if monic[k]
+      prod_iter = Iterators.filter(x->length(x)==length(unique(x)), prod_iter)
+    end
+    (free_vars, prod_iter) # prod_iter = valid assignments for this attrtype
+  end
+
+  # Homomorphism for each element in the product of the prod_iters
+  return Iterators.map(Iterators.product(last.(free_data)...) ) do combo 
+    ad = Dict(map(zip(attrtypes(S), first.(free_data), combo)) do (k, xs, vs)
+      vec = last.(assgn[k])
+      vec[xs] = AttrVar.(collect(vs))
+      k => vec
+    end)
+    comps = merge(NamedTuple(od),NamedTuple(ad))
+    ACSetTransformation(comps, dom, codom)
+  end
+end
+
 # Macros 
 ########
 

test/categorical_algebra/CSets.jl

@@ -135,9 +135,9 @@ d = naturality_failures(β)
 G = @acset Graph begin V=2; E=1; src=1; tgt=2 end
 H = @acset Graph begin V=2; E=2; src=1; tgt=2 end
 I = @acset Graph begin V=2; E=2; src=[1,2]; tgt=[1,2] end
-f_ = homomorphism(G, H; monic=true)
+f_ = homomorphism(G, H; monic=true, any=true)
 g_ = homomorphism(H, G)
-h_ = homomorphism(G, I)
+h_ = homomorphism(G, I; initial=(V=[1,1],))
 @test is_monic(f_)
 @test !is_epic(f_)
 @test !is_monic(g_)
@@ -689,7 +689,7 @@ rem_part!(X, :E, 2)
 A = @acset WG{Symbol} begin V=1;E=2;Weight=1;src=1;tgt=1;weight=[AttrVar(1),:X] end
 B = @acset WG{Symbol} begin V=1;E=2;Weight=1;src=1;tgt=1;weight=[:X, :Y] end
 C = B ⊕ @acset WG{Symbol} begin V=1 end
-AC = homomorphism(A,C)
+AC = homomorphism(A,C; initial=(E=[1,1],))
 BC = CSetTransformation(B,C; V=[1],E=[1,2], Weight=[:X])
 @test all(is_natural,[AC,BC])
 p1, p2 = product(A,A; cset=true);
@@ -701,8 +701,8 @@ g0, g1, g2 = WG{Symbol}.([2,3,2])
 add_edges!(g0, [1,1,2], [1,2,2]; weight=[:X,:Y,:Z])
 add_edges!(g1, [1,2,3], [2,3,3]; weight=[:Y,:Z,AttrVar(add_part!(g1,:Weight))])
 add_edges!(g2, [1,2,2], [1,2,2]; weight=[AttrVar(add_part!(g2,:Weight)), :Z,:Z])
-ϕ = only(homomorphisms(g1, g0)) |> CSetTransformation
-ψ = only(homomorphisms(g2, g0; initial=(V=[1,2],))) |> CSetTransformation
+ϕ = homomorphism(g1, g0) |> CSetTransformation
+ψ = homomorphism(g2, g0; initial=(V=[1,2],)) |> CSetTransformation
 @test is_natural(ϕ) && is_natural(ψ)
 lim = pullback(ϕ, ψ)
 @test nv(ob(lim)) == 3
@@ -733,29 +733,35 @@ X = @acset VES begin V=6; E=5; Label=5
   src=[1,2,3,4,4]; tgt=[3,3,4,5,6];
   vlabel=[:a,:b,:c,:d,:e,:f]; elabel=AttrVar.(1:5)
 end
-A, B = Subobject(X, V=1:4, E=1:3, Label=1:3), Subobject(X, V=3:6, E=3:5, Label=3:5)
-@test A ∧ B |> force == Subobject(X, V=3:4, E=3:3, Label=3:3) |> force
-expected = @acset VES begin V=2; E=1; Label=1;
-  src=1; tgt=2; vlabel=[:c,:d]; elabel=[AttrVar(1)]
-end
-@test is_isomorphic(dom(hom(A ∧ B )), expected)
-@test A ∨ B |> force == Subobject(X, V=1:6, E=1:5, Label=1:5) |> force
-@test ⊤(X) |> force == A ∨ B |> force
-@test ⊥(X) |> force == Subobject(X, V=1:0, E=1:0, Label=1:0) |> force
-@test force(implies(A, B)) == force(¬(A) ∨ B)
-@test ¬(A ∧ B) == ¬(A) ∨ ¬(B)
-@test ¬(A ∧ B) != ¬(A) ∨ B
-@test (A ∧ implies(A,B)) == B ∧ (A ∧ implies(A,B))
-@test (B ∧ implies(B,A)) == A ∧ (B ∧ implies(B,A))
-@test ¬(A ∨ (¬B)) == ¬(A) ∧ ¬(¬(B))
-@test ¬(A ∨ (¬B)) == ¬(A) ∧ B
-@test A ∧ ¬(¬(A)) == ¬(¬(A))
-@test implies((A∧B), A) == A∨B
-@test dom(hom(subtract(A,B))) == @acset VES begin V=3; E=2; Label=2
-  src=[1,2]; tgt=3; vlabel=[:a,:b,:c]; elabel=AttrVar.(1:2)
-end
 
-@test nv(dom(hom(~A))) == 3
+A′ = Subobject(X, V=1:4, E=1:3, Label=1:3) # component-wise representation
+B′ = Subobject(X, V=3:6, E=3:5, Label=3:5)
+A′′, B′′ = Subobject.(hom.([A′,B′])) # hom representation
+
+for (A,B) in [A′=>B′, A′′ =>B′′]
+  @test A ∧ B |> force == Subobject(X, V=3:4, E=3:3, Label=3:3) |> force
+  expected = @acset VES begin V=2; E=1; Label=1;
+    src=1; tgt=2; vlabel=[:c,:d]; elabel=[AttrVar(1)]
+  end
+  @test is_isomorphic(dom(hom(A ∧ B )), expected)
+  @test A ∨ B |> force == Subobject(X, V=1:6, E=1:5, Label=1:5) |> force
+  @test ⊤(X) |> force == A ∨ B |> force
+  @test ⊥(X) |> force == Subobject(X, V=1:0, E=1:0, Label=1:0) |> force
+  @test force(implies(A, B)) == force(¬(A) ∨ B)
+  @test ¬(A ∧ B) == ¬(A) ∨ ¬(B)
+  @test ¬(A ∧ B) != ¬(A) ∨ B
+  @test (A ∧ implies(A,B)) == B ∧ (A ∧ implies(A,B))
+  @test (B ∧ implies(B,A)) == A ∧ (B ∧ implies(B,A))
+  @test ¬(A ∨ (¬B)) == ¬(A) ∧ ¬(¬(B))
+  @test ¬(A ∨ (¬B)) == ¬(A) ∧ B
+  @test A ∧ ¬(¬(A)) == ¬(¬(A))
+  @test implies((A∧B), A) == A∨B
+  @test dom(hom(subtract(A,B))) == @acset VES begin V=3; E=2; Label=2
+    src=[1,2]; tgt=3; vlabel=[:a,:b,:c]; elabel=AttrVar.(1:2)
+  end
+
+  @test nv(dom(hom(~A))) == 3
+end
 
 # Limits of CSetTransformations between ACSets
 #---------------------------------------------