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{-# OPTIONS --without-K --safe #-}
open import Categories.Category using (Category)
open import Level using (Level)
open import Category.Dagger.Semiadditive using (IdempotentSemiadditiveDagger)
module Data.WiringDiagram.Core
{o ℓ e : Level}
{𝒞 : Category o ℓ e}
(S : IdempotentSemiadditiveDagger 𝒞)
where
open import Categories.Category.Monoidal.Utilities using (module Shorthands)
open import Relation.Binary using (IsEquivalence)
open Category 𝒞 using (Obj; _∘_; _⇒_; id; _≈_; module Equiv)
open IdempotentSemiadditiveDagger S using (_⊕₀_; _⊕₁_; p₂; +-monoidal; △; ▽; _†)
open Shorthands +-monoidal using (α⇒)
-- A "Box" is a pair of objects from the underlying category,
-- representing input and output ports
-- +-----------+
-- | |
-- Aᵢ A Aₒ
-- | |
-- +-----------+
record Box : Set o where
constructor _□_
field
ᵢ : Obj
ₒ : Obj
infix 4 _□_
-- A Wiring Diagram between two boxes
-- is a pair of morphisms from the underlying category
-- (which can be thought of as generalized relations):
-- 1. A relation from the output of the inner box
-- plus the input of the outer box to the input of
-- the inner box
-- 2. A relation from the output of the inner box
-- to the output of the outer box
-- This choice of definition gives a way to represent
-- feedback from output to input.
--
-- outer box (B)
-- +--------------------------------+
-- | |
-- | /-------------------\ |
-- | | | |
-- | | +-------+ | |
-- | \----| | | |
-- | Aₒ ⊕ Bᵢ | inner |------+------|
-- | ⇒ Aᵢ | box | Aₒ ⇒ Bₒ |
-- |----------| (A) | output |
-- | input | | relation |
-- | relation +-------+ |
-- | |
-- +--------------------------------+
record WiringDiagram (A B : Box) : Set ℓ where
constructor _⧈_
private
module A = Box A
module B = Box B
field
input : A.ₒ ⊕₀ B.ᵢ ⇒ A.ᵢ
output : A.ₒ ⇒ B.ₒ
infix 4 _⧈_
-- Two wiring diagrams are equivalent when their
-- input and output relations are equivalent as
-- morphism in the underlying category.
record _≈-⧈_ {A B : Box} (f g : WiringDiagram A B) : Set e where
constructor _⌸_
private
module f = WiringDiagram f
module g = WiringDiagram g
field
≈i : f.input ≈ g.input
≈o : f.output ≈ g.output
infix 4 _≈-⧈_
-- Equivalence of boxes is a legitimate equivalence relation
module _ {A B : Box} where
open Equiv
≈-refl : {x : WiringDiagram A B} → x ≈-⧈ x
≈-refl = refl ⌸ refl
≈-sym : {x y : WiringDiagram A B} → x ≈-⧈ y → y ≈-⧈ x
≈-sym (≈i ⌸ ≈o) = sym ≈i ⌸ sym ≈o
≈-trans : {x y z : WiringDiagram A B} → x ≈-⧈ y → y ≈-⧈ z → x ≈-⧈ z
≈-trans (≈i₁ ⌸ ≈o₁) (≈i₂ ⌸ ≈o₂) = trans ≈i₁ ≈i₂ ⌸ trans ≈o₁ ≈o₂
≈-isEquiv : IsEquivalence (_≈-⧈_ {A} {B})
≈-isEquiv = record
{ refl = ≈-refl
; sym = ≈-sym
; trans = ≈-trans
}
-- The identity wiring diagram
id-⧈ : {A : Box} → WiringDiagram A A
id-⧈ = p₂ ⧈ id
-- Composition of wiring diagrams
_⌻_ : {A B C : Box} → WiringDiagram B C → WiringDiagram A B → WiringDiagram A C
_⌻_ {Aᵢ □ Aₒ} {Bᵢ □ Bₒ} {Cᵢ □ Cₒ} (f′ ⧈ g′) (f ⧈ g) = f″ ⧈ g′ ∘ g
where
f″ : Aₒ ⊕₀ Cᵢ ⇒ Aᵢ
f″ = f ∘ id ⊕₁ (f′ ∘ g ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
infixr 9 _⌻_
-- Special wiring diagrams
loop : {A : Obj} → WiringDiagram (A □ A) (A □ A)
loop = ▽ ⧈ id
pulsh : {A B C D : Obj} → A ⇒ B → C ⇒ D → WiringDiagram (B □ C) (A □ D)
pulsh f g = f ∘ p₂ ⧈ g
push : {A B : Obj} → A ⇒ B → WiringDiagram (A □ A) (B □ B)
push f = pulsh (f †) f
pull : {A B : Obj} → A ⇒ B → WiringDiagram (B □ B) (A □ A)
pull f = pulsh f (f †)
merge : {A B : Obj} → A ⇒ B → WiringDiagram (A □ A) (B □ B)
merge f = f † ∘ ▽ ∘ f ⊕₁ id ⧈ f
split : {A B : Obj} → A ⇒ B → WiringDiagram (B □ B) (A □ A)
split f = ▽ ∘ id ⊕₁ f ⧈ f †
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