Adjoint Folds and Unfolds Or: Scything Through the Thicket of - PowerPoint PPT Presentation

Adjoint Folds and Unfolds — Prologue MPC 2010 Adjoint Folds and Unfolds Or: Scything Through the Thicket of Morphisms Ralf Hinze Computing Laboratory, University of Oxford Wolfson Building, Parks Road, Oxford, OX1 3QD, England ralf.hinze@comlab.ox.ac.uk http://www.comlab.ox.ac.uk/ralf.hinze/ June 2010 University of Oxford — Ralf Hinze 1-48

Adjoint Folds and Unfolds — Prologue MPC 2010 Section 1 Prologue University of Oxford — Ralf Hinze 2-48

Adjoint Folds and Unfolds — Prologue MPC 2010 • Mathematics of Program Construction, 375th Anniversary of the Groningen University, International Conference Groningen, The Netherlands, June 26–30 1989. LNCS 375. • Mike Spivey, A categorical approach to the theory of lists . • Grant Malcolm, Homomorphisms and promotability . University of Oxford — Ralf Hinze 3-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.1 Catamorphism f = � b � ⇐ ⇒ f · in = b · F f University of Oxford — Ralf Hinze 4-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.1 Banana-split law � h � △ � k � = � h · F outl △ k · F outr � University of Oxford — Ralf Hinze 5-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.1 Proof of banana-split law ( � h � △ � k � ) · in = { split-fusion } � h � · in △ � k � · in = { fold-computation } h · F � h � △ k · F � k � = { split-computation } h · F ( outl · ( � h � △ � k � )) △ k · F ( outr · ( � h � △ � k � )) { F functor } = h · F outl · F ( � h � △ � k � ) △ k · F outr · F ( � h � △ � k � ) = { split-fusion } ( h · F outl △ k · F outr ) · F ( � h � △ � k � ) University of Oxford — Ralf Hinze 6-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.2 Example: total data Stack = Empty | Push ( Nat , Stack ) total : Stack → Nat total ( Empty ) = 0 total ( Push ( n , s )) = n + total s University of Oxford — Ralf Hinze 7-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.3 Counterexample: stack stack : ( Stack , Stack ) → Stack stack ( Empty , bs ) = bs stack ( Push ( a , as ), bs ) = Push ( a , stack ( as , bs )) University of Oxford — Ralf Hinze 8-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.3 Counterexample: even and odd even : Stack → Bool even ( Empty ) = True even ( Push ( 0 , x )) = even x even ( Push ( 1 , x )) = odd x odd : Stack → Bool odd ( Empty ) = False odd ( Push ( 0 , x )) = odd x odd ( Push ( 1 , x )) = even x University of Oxford — Ralf Hinze 9-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.3 Counterexamples: fac and fib data Nat = Zero | Succ Nat fac : Nat → Nat fac ( Zero ) = 1 fac ( Succ n ) = Succ n × fac n fib : Nat → Nat fib ( Zero ) = Zero fib ( Succ Zero ) = Succ Zero fib ( Succ ( Succ n )) = fib n + fib ( Succ n ) University of Oxford — Ralf Hinze 10-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.3 Counterexample: sum data List a = Nil | Cons ( a , List a ) sum : List Nat → Nat sum ( Nil ) = 0 sum ( Cons ( a , as )) = a + sum as University of Oxford — Ralf Hinze 11-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.3 Counterexample: append append : ∀ a . ( List a , List a ) → List a append ( Nil , bs ) = bs append ( Cons ( a , as ), bs ) = Cons ( a , append ( as , bs )) University of Oxford — Ralf Hinze 12-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.3 Counterexample: concat concat : ∀ a . List ( List a ) → List a concat ( Nil ) = Nil concat ( Cons ( l , ls )) = append ( l , concat ls ) University of Oxford — Ralf Hinze 13-48

Adjoint Folds and Unfolds — Prologue MPC 2010 1.4 Thicket of morphisms • Catamorphisms, • anamorphism, • paramorphism, • apomorphisms, • mutomorphisms, • zygomorphisms, • histomorphisms, • futomorphisms. University of Oxford — Ralf Hinze 14-48

Adjoint Folds and Unfolds — Mendler-style folds MPC 2010 Section 2 Mendler-style folds University of Oxford — Ralf Hinze 15-48

Adjoint Folds and Unfolds — Mendler-style folds MPC 2010 2.1 Semantics of recursive datatypes data Stack = Empty | Push ( Nat , Stack ) University of Oxford — Ralf Hinze 16-48

Adjoint Folds and Unfolds — Mendler-style folds MPC 2010 2.1 Two-level types Abstracting away from the recursive call. data Stack stack = Empty | Push ( Nat , stack ) instance Functor Stack where fmap f ( Empty ) = Empty fmap f ( Push ( n , s )) = Push ( n , f s ) Tying the recursive knot. newtype µ f = In { in ◦ : f (µ f ) } type Stack = µ Stack University of Oxford — Ralf Hinze 17-48

Adjoint Folds and Unfolds — Mendler-style folds MPC 2010 2.2 Semantics of recursive functions total : µ Stack → Nat total ( In ( Empty )) = 0 total ( In ( Push ( n , s ))) = n + total s University of Oxford — Ralf Hinze 18-48

Adjoint Folds and Unfolds — Mendler-style folds MPC 2010 2.2 Abstracting away from the recursive call total : (µ Stack → Nat ) → (µ Stack → Nat ) total total ( In ( Empty )) = 0 total total ( In ( Push ( n , s ))) = n + total s A function of this type has many fixed points. University of Oxford — Ralf Hinze 19-48

Adjoint Folds and Unfolds — Mendler-style folds MPC 2010 2.2 . . . and removing In Abstracting away from the recursive call and removing In . total : ∀ x . ( x → Nat ) → ( Stack x → Nat ) total total ( Empty ) = 0 total total ( Push ( n , s )) = n + total s A function of this type has a unique fixed point. Tying the recursive knot. total : µ Stack → Nat total ( In l ) = total total l University of Oxford — Ralf Hinze 20-48

Adjoint Folds and Unfolds — Mendler-style folds MPC 2010 2.3 Initial fixed-point equations An initial fixed-point equation in the unknown x : ❈ (µ F , A ) has the syntactic form x · in = Ψ x , where the base function Ψ has type Ψ : ∀ X . ❈ ( X , A ) → ❈ ( F X , A ) . The naturality of Ψ ensures that x is uniquely defined. University of Oxford — Ralf Hinze 21-48

Adjoint Folds and Unfolds — Mendler-style folds MPC 2010 2.3 Mendler-style folds x = � Ψ � Id ⇐ ⇒ x · in = Ψ x University of Oxford — Ralf Hinze 22-48

Adjoint Folds and Unfolds — Mendler-style folds MPC 2010 2.3 Proof of uniqueness φ : ❈ ( F A , A ) ≅ ( ∀ X . ❈ ( X , A ) → ❈ ( F X , A )) x · in = Ψ x ⇐ ⇒ { isomorphism } x · in = φ (φ ◦ Ψ ) x { definition of φ ◦ , that is, φ ◦ Ψ = Ψ id } ⇐ ⇒ x · in = φ ( Ψ id ) x { definition of φ , that is, φ f = λ κ . f · F κ } ⇐ ⇒ x · in = Ψ id · F x ⇐ ⇒ { initial algebras } x = � Ψ id � University of Oxford — Ralf Hinze 23-48

Adjoint Folds and Unfolds — Adjoint folds MPC 2010 Section 3 Adjoint folds University of Oxford — Ralf Hinze 24-48

Adjoint Folds and Unfolds — Adjoint folds MPC 2010 3.1 Recall stack stack : (µ Stack , Stack ) → Stack stack ( In ( Empty ), bs ) = bs stack ( In ( Push ( a , as )), bs ) = In ( Push ( a , stack ( as , bs ))) University of Oxford — Ralf Hinze 25-48

Adjoint Folds and Unfolds — Adjoint folds MPC 2010 3.2 Adjoint fixed-point equations Idea: model the context by a functor. x · L in = Ψ x Requirement: the functor has to be left adjoint: L ⊣ R . University of Oxford — Ralf Hinze 26-48

Adjoint Folds and Unfolds — Adjoint folds MPC 2010 3.2 Adjunction L ❈ ≺ ⊥ ≻ ❉ R φ : ∀ A B . ❈ ( L A , B ) ≅ ❉ ( A , R B ) University of Oxford — Ralf Hinze 27-48

Adjoint Folds and Unfolds — Adjoint folds MPC 2010 3.2 Adjoint initial fixed-point equations An adjoint initial fixed-point equation in the unknown x : ❈ ( L (µ F ), A ) has the syntactic form x · L in = Ψ x , where the base function Ψ has type Ψ : ∀ X : ❉ . ❈ ( L X , A ) → ❈ ( L ( F X ), A ) . The unique solution is called an adjoint fold . Furthermore, φ x is called the transposed fold . University of Oxford — Ralf Hinze 28-48

Adjoint Folds and Unfolds — Adjoint folds MPC 2010 3.2 Adjoint folds x = � Ψ � L ⇐ ⇒ x · L in = Ψ x University of Oxford — Ralf Hinze 29-48

Adjoint Folds and Unfolds — Adjoint folds MPC 2010 3.2 Proof of uniqueness x · L in = Ψ x ⇐ ⇒ { adjunction } φ ( x · L in ) = φ ( Ψ x ) ⇐ ⇒ { naturality of φ , that is, φ f · h = φ ( f · L h ) } φ x · in = φ ( Ψ x ) ⇐ ⇒ { adjunction } φ x · in = (φ · Ψ · φ ◦ ) (φ x ) { universal property of Mendler-style folds } ⇐ ⇒ φ x = � φ · Ψ · φ ◦ � Id ⇐ ⇒ { adjunction } x = φ ◦ � φ · Ψ · φ ◦ � Id University of Oxford — Ralf Hinze 30-48

Adjoint Folds and Unfolds — Adjoint folds MPC 2010 3.3 Banana-split law = � λ x . Φ ( outl · x ) △ Ψ ( outr · x ) � L � Φ � L △ � Ψ � L University of Oxford — Ralf Hinze 31-48

Adjoint Folds and Unfolds — Adjoint folds MPC 2010 3.3 Proof of banana-split law ( � Φ � L △ � Ψ � L ) · L in = { split-fusion } � Φ � L · L in △ � Ψ � L · L in = { fold-computation } Φ � Φ � L △ Ψ � Ψ � L = { split-computation } Φ ( outl · ( � Φ � L △ � Ψ � L )) △ Ψ ( outl · ( � Φ � L △ � Ψ � L )) University of Oxford — Ralf Hinze 32-48

Adjoint Folds and Unfolds — Adjunctions MPC 2010 Section 4 Adjunctions University of Oxford — Ralf Hinze 33-48