Interop with SymbolicUtils

Project.toml

@@ -9,12 +9,14 @@ ACSets = "227ef7b5-1206-438b-ac65-934d6da304b8"
 Catlab = "134e5e36-593f-5add-ad60-77f754baafbe"
 DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
 MLStyle = "d8e11817-5142-5d16-987a-aa16d5891078"
+SymbolicUtils = "d1185830-fcd6-423d-90d6-eec64667417b"
 Unicode = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"
 
 [compat]
 ACSets = "0.2"
 Catlab = "0.15, 0.16"
 DataStructures = "0.18.13"
 MLStyle = "0.4.17"
+SymbolicUtils = "3.1.2"
 Unicode = "1.6"
 julia = "1.6"

src/DiagrammaticEquations.jl

@@ -61,6 +61,8 @@ include("colanguage.jl")
 include("openoperators.jl")
 include("deca/Deca.jl")
 include("learn/Learn.jl")
+include("ThDEC.jl")
+include("decasymbolic.jl")
 
 using .Deca
 

src/ThDEC.jl

@@ -0,0 +1,270 @@
+module ThDEC
+using MLStyle
+
+import Base: +, -, *
+
+struct SortError <: Exception
+    message::String
+end
+
+@data Sort begin
+    Scalar()
+    Form(dim::Int, isdual::Bool)
+    VField(isdual::Bool)
+end
+export Sort, Scalar, Form, VField
+
+const SORT_LOOKUP = Dict(
+    :Form0 => Form(0, false),
+    :Form1 => Form(1, false),
+    :Form2 => Form(2, false),
+    :DualForm0 => Form(0, true),
+    :DualForm1 => Form(1, true),
+    :DualForm2 => Form(2, true),
+    :Constant => Scalar()
+)
+
+function Base.nameof(s::Scalar)
+    :Constant
+end
+
+function Base.nameof(f::Form)
+    dual = isdual(f) ? "Dual" : ""
+    Symbol("$(dual)Form$(dim(f))")
+end
+
+const VF = VField
+
+dim(ω::Form) = ω.dim
+isdual(ω::Form) = ω.isdual
+
+isdual(v::VField) = v.isdual
+
+# convenience functions
+PrimalForm(i::Int) = Form(i, false)
+export PrimalForm
+
+DualForm(i::Int) = Form(i, true)
+export DualForm
+
+PrimalVF() = VF(false)
+export PrimalVF
+
+DualVF() = VF(true)
+export DualVF
+
+# show methods
+show_duality(ω::Form) = isdual(ω) ? "dual" : "primal"
+
+function Base.show(io::IO, ω::Form)
+    print(io, isdual(ω) ? "DualForm($(dim(ω)))" : "PrimalForm($(dim(ω)))")
+end
+
+@nospecialize
+function +(s1::Sort, s2::Sort)
+    @match (s1, s2) begin
+        (Scalar(), Scalar()) => Scalar()
+        (Scalar(), Form(i, isdual)) ||
+            (Form(i, isdual), Scalar()) => Form(i, isdual)
+        (Form(i1, isdual1), Form(i2, isdual2)) =>
+            if (i1 == i2) && (isdual1 == isdual2)
+                Form(i1, isdual1)
+            else
+                throw(SortError("Cannot add two forms of different dimensions/dualities: $((i1,isdual1)) and $((i2,isdual2))"))
+            end
+    end
+end
+
+# Type-checking inverse of addition follows addition
+-(s1::Sort, s2::Sort) = +(s1, s2)
+
+# TODO error for Forms
+
+# Negation is always valid
+-(s::Sort) = s
+
+@nospecialize
+function *(s1::Sort, s2::Sort)
+    @match (s1, s2) begin
+        (Scalar(), Scalar()) => Scalar()
+        (Scalar(), Form(i, isdual)) ||
+            (Form(i, isdual), Scalar()) => Form(i, isdual)
+        (Form(_, _), Form(_, _)) => throw(SortError("Cannot scalar multiply a form with a form. Maybe try `∧`??"))
+    end
+end
+
+const SUBSCRIPT_DIGIT_0 = '₀'
+
+function as_sub(n::Int)
+    join(map(d -> SUBSCRIPT_DIGIT_0 + d, digits(n)))
+end
+
+@nospecialize
+function ∧(s1::Sort, s2::Sort)
+    @match (s1, s2) begin
+        (Form(i, isdual), Scalar()) || (Scalar(), Form(i, isdual)) => Form(i, isdual)
+        (Form(i1, isdual), Form(i2, isdual)) =>
+            if i1 + i2 <= 2
+                Form(i1 + i2, isdual)
+            else
+                throw(SortError("Can only take a wedge product when the dimensions of the forms add to less than 2: tried to wedge product $i1 and $i2"))
+            end
+        _ => throw(SortError("Can only take a wedge product of two forms of the same duality"))
+    end
+end
+
+function Base.nameof(::typeof(∧), s1, s2)
+    Symbol("∧$(as_sub(dim(s1)))$(as_sub(dim(s2)))")
+end
+
+@nospecialize
+∂ₜ(s::Sort) = s
+
+@nospecialize
+function d(s::Sort)
+    @match s begin
+        Scalar() => throw(SortError("Cannot take exterior derivative of a scalar"))
+        Form(i, isdual) =>
+            if i <= 1
+                Form(i + 1, isdual)
+            else
+                throw(SortError("Cannot take exterior derivative of a n-form for n >= 1"))
+            end
+    end
+end
+
+function Base.nameof(::typeof(d), s)
+    Symbol("d$(as_sub(dim(s)))")
+end
+
+@nospecialize
+function ★(s::Sort)
+    @match s begin
+        Scalar() => throw(SortError("Cannot take Hodge star of a scalar"))
+        Form(i, isdual) => Form(2 - i, !isdual)
+    end
+end
+
+function Base.nameof(::typeof(★), s)
+    inv = isdual(s) ? "⁻¹" : ""
+    Symbol("★$(as_sub(isdual(s) ? 2 - dim(s) : dim(s)))$(inv)")
+end
+
+@nospecialize
+function ι(s1::Sort, s2::Sort)
+    @match (s1, s2) begin
+        (VF(true), Form(i, true)) => PrimalForm() # wrong
+        (VF(true), Form(i, false)) => DualForm()
+        _ => throw(SortError("Can only define the discrete interior product on:
+                PrimalVF, DualForm(i)
+                DualVF(), PrimalForm(i)
+                ."))
+    end
+end
+
+# in practice, a scalar may be treated as a constant 0-form.
+function ♯(s::Sort)
+    @match s begin
+        Scalar() => PrimalVF()
+        Form(1, isdual) => VF(isdual)
+        _ => throw(SortError("Can only take ♯ to 1-forms"))
+    end
+end
+# musical isos may be defined for any combination of (primal/dual) form -> (primal/dual) vf.
+
+function ♭(s::Sort)
+    @match s begin
+        VF(true) => PrimalForm(1)
+        _ => throw(SortError("Can only apply ♭ to dual vector fields"))
+    end
+end
+
+# OTHER
+
+function ♭♯(s::Sort)
+    @match s begin
+        Form(i, isdual) => Form(i, !isdual)
+        _ => throw(SortError("♭♯ is only defined on forms."))
+    end
+end
+
+# Δ = ★d⋆d, but we check signature here to throw a more helpful error
+function Δ(s::Sort)
+    @match s begin
+        Form(0, isdual) => Form(0, isdual)
+        _ => throw(SortError("Δ is not defined for $s"))
+    end
+end
+
+const OPERATOR_LOOKUP = Dict(
+    :⋆₀ => ★,
+    :⋆₁ => ★,
+    :⋆₂ => ★,
+
+    # Inverse Hodge Stars
+    :⋆₀⁻¹ => ★,
+    :⋆₁⁻¹ => ★,
+    :⋆₂⁻¹ => ★,
+
+    # Differentials
+    :d₀ => d,
+    :d₁ => d,
+
+    # Dual Differentials
+    :dual_d₀ => d,
+    :d̃₀ => d,
+    :dual_d₁ => d,
+    :d̃₁ => d,
+
+    # Wedge Products
+    :∧₀₁ => ∧,
+    :∧₁₀ => ∧,
+    :∧₀₂ => ∧,
+    :∧₂₀ => ∧,
+    :∧₁₁ => ∧,
+
+    # Primal-Dual Wedge Products
+    :∧ᵖᵈ₁₁ => ∧,
+    :∧ᵖᵈ₀₁ => ∧,
+    :∧ᵈᵖ₁₁ => ∧,
+    :∧ᵈᵖ₁₀ => ∧,
+
+    # Dual-Dual Wedge Products
+    :∧ᵈᵈ₁₁ => ∧,
+    :∧ᵈᵈ₁₀ => ∧,
+    :∧ᵈᵈ₀₁ => ∧,
+
+    # Dual-Dual Interior Products
+    :ι₁₁ => ι,
+    :ι₁₂ => ι,
+
+    # Dual-Dual Lie Derivatives
+    # :ℒ₁ => ℒ,
+
+    # Dual Laplacians
+    # :Δᵈ₀ => Δ,
+    # :Δᵈ₁ => Δ,
+
+    # Musical Isomorphisms
+    :♯ => ♯,
+    :♯ᵈ => ♯, :♭ => ♭,
+
+    # Averaging Operator
+    # :avg₀₁ => avg,
+
+    # Negatives
+    :neg => -,
+
+    # Basics
+
+    :- => -,
+    :+ => +,
+    :* => *,
+    :/ => /,
+    :.- => .-,
+    :.+ => .+,
+    :.* => .*,
+    :./ => ./,
+)
+
+end