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Type Fusion — Prologue AMAST 2010

Type Fusion

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

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Type Fusion — Prologue AMAST 2010

Section 1 Prologue

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Type Fusion — Prologue AMAST 2010

1.1 Memoisation

Say, you want to memoise the function f : Nat → V so that it caches previously computed values.

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Type Fusion — Prologue AMAST 2010

1.1 Table: interface

Given the interface data Table v lookup : ∀ v . Table v → (Nat → v) tabulate : ∀ v . (Nat → v) → Table v, we can memoize f as follows memo-f : Nat → V memo-f = lookup (tabulate f ).

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Type Fusion — Prologue AMAST 2010

1.1 Table: implementation

data Nat = Zero | Succ Nat data Table v = Node {zero : v, succ : Table v} lookup (Node {zero = t }) Zero = t lookup (Node {succ = t }) (Succ n) = lookup t n tabulate f = Node {zero = f Zero, succ = tabulate (λn → f (Succ n))}

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Type Fusion — Prologue AMAST 2010

1.1 Correctness

tabulate : V Nat ≅ Table V : lookup

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Type Fusion — Prologue AMAST 2010

1.2 Type firstification

The first-order datatype data Stack = Empty | Push (Nat, Stack) is an instance of the second-order datatype data List a = Nil | Cons (a, List a).

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1.2 Correctness

Λ-drop : List Nat ≅ Stack : Λ-lift

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1.3 Type specialisation

Lists of optional values, List (Maybe A) with data Maybe a = Nothing | Just a, can be represented more compactly using the tailor-made data Sequ a = Done | Skip (Sequ a) | Yield (a, Sequ a).

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1.3 Correctness

List (Maybe A) ≅ Sequ A

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Section 2 Type fusion

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2.1 Type fusion

❈ ≺ G G ≻ ❈ ≺ L ⊥ R ≻ ❉ ≺ F F ≻ ❉ L (µF) ≅ µG ⇐ = L ◦ F ≅ G ◦ L νF ≅ R (νG) ⇐ = F ◦ R ≅ R ◦ G

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Type Fusion — Type fusion AMAST 2010

2.2 Interlude: adjoint folds

An adjoint initial fixed-point equation in the unknown x : ❈(L (µF), 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.

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Type Fusion — Type fusion AMAST 2010

τ : L (µF) ≅ µG ⇐ = swap : L ◦ F ≅ G ◦ L

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2.3 Definition of τ and τ◦

G (L (µF)) L (F (µF)) ≺ swap swap◦ ≻ G (µG) ≺ G τ G τ◦ ≻ L (µF) L in L in ⋎ ≺ τ◦ τ ≻ µG in in ⋎ τ·L in = in·G τ·swap and τ◦·in = L in·swap◦·G τ◦

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2.3 Proof of τ · τ◦ = idµG

(τ · τ◦) · in = { definition of τ◦ } τ · L in · swap◦ · G τ◦ = { definition of τ } in · G τ · swap · swap◦ · G τ◦ = { inverses } in · G τ · G τ◦ = { G functor } in · G (τ · τ◦) The equation x · in = in · G x has a unique solution. Since id is also a solution, the result follows.

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Type Fusion — Type fusion AMAST 2010

2.3 Proof of τ◦ · τ = idL (µF)

(τ◦ · τ) · L in = { definition of τ } τ◦ · in · G τ · swap = { definition of τ◦ } L in · swap◦ · G τ◦ · G τ · swap = { G functor } L in · swap◦ · G (τ◦ · τ) · swap Again, x · L in = L in · swap◦ · G x · swap has a unique solution. And again, id is also solution, which implies the result.

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Type Fusion — Applications AMAST 2010

Section 3 Applications

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Type Fusion — Applications AMAST 2010

3.1 Recall: type firstification

The first-order datatype data Stack = Empty | Push (Nat, Stack) is an instance of the second-order datatype data List a = Nil | Cons (a, List a). Correctness: Λ-drop : List Nat ≅ Stack : Λ-lift.

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3.1 Speaking categorically

AppNat (µLIST) ≅ µStack ⇐ = AppNat ◦ LIST ≅ Stack ◦ AppNat where AppX : ❈❉ → ❈ AppX F = F X AppX ☛ = ☛ X is ‘type application’.

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3.2 Recall: type specialisation

Lists of optional values, List ◦ Maybe with data Maybe a = Nothing | Just a, can be represented more compactly using the tailor-made data Sequ a = Done | Skip (Sequ a) | Yield (a, Sequ a). Correctness: List ◦ Maybe ≅ Sequ.

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Type Fusion — Applications AMAST 2010

3.2 Speaking categorically

PreMaybe (µLIST) ≅ µSEQU ⇐ = PreMaybe ◦ LIST ≅ SEQU ◦ PreMaybe where PreJ : ❊❉ → ❊❈ PreJ F = F ◦ J PreJ ☛ = ☛ ◦ J is pre-composition, also written ❊J.

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3.3 Recall: memoisation

Functions from the natural numbers data Nat = Zero | Succ Nat can be memoised using streams data Table val = Node {zero : val, succ : Table val}. Correctness: (−)Nat ≅ Table

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3.3 Speaking categorically

νTABLE ≅ Exp (µNat) ⇐ = TABLE ◦ Exp ≅ Exp ◦ Nat where Exp : ❈ → (❈❈)op Exp K = Λ V . V K Exp f = Λ V . V f is curried (!) exponentiation.

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3.3 Laws of exponentials

V 0 ≅ 1 V 1 ≅ V V A+B ≅ V A × V B V A×B ≅ (V B)A Exp 0 ≅ K 1 Exp 1 ≅ Id Exp (A + B) ≅ Exp A ˙ × Exp B Exp (A × B) ≅ Exp A ◦ Exp B

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3.3 Exp is a left adjoint

(❈❈)op ≺ G G ≻ (❈❈)op ≺ Exp ⊥ Sel ≻ ❈ ≺ F F ≻ ❈

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3.3 Deriving the right adjoint

(❈❈)op(Exp A, B) ≅ { definition of −op } ❈❈(B, Exp A) ≅ { natural transformation as an end } ∀ X : ❈ . ❈(B X, Exp A X) ≅ { definition of Exp } ∀ X : ❈ . ❈(B X, X A) ≅ { − × Y ⊣ (−)Y and Y × Z ≅ Z × Y } ∀ X : ❈ . ❈(A, X B X ) ≅ { the functor ❈(A, −) preserves ends } ❈(A, ∀ X : ❈ . X B X ) ≅ { define Sel B = ∀ X : ❈ . X B X } ❈(A, Sel B)

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Type Fusion — Applications AMAST 2010

• Since Exp is a contravariant functor, τ and swap live in

an opposite category. (❈❈)op ≺ G G ≻ (❈❈)op ≺ Exp ⊥ Sel ≻ ❈ ≺ F F ≻ ❈

• Type fusion in terms of arrows in ❈❈:

τ : νG ≅ Exp (µF) ⇐ = swap : G ◦ Exp ≅ Exp ◦ F.

• The isomorphism τ : νG ˙

→ Exp (µF) is a curried look-up function that maps a memo table to an exponential.

• The inverse τ◦ : Exp (µF) ˙

→ νG is a transformation that tabulates a given exponential.

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Type Fusion — Epilogue AMAST 2010

Section 4 Epilogue

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Type Fusion — Epilogue AMAST 2010

4.0 Summary and related work

Summary:

• Type fusion generalises
• type firstification,
• type specialisation,
• memoisation or tabulation.
• Adjunctions play a central role.

Related work:

• Backhouse, Bijsterveld, van Geldrop, van der Woude,

Categorical fixed point calculus. CTCS ’95.

• Hinze, Adjoint folds and unfolds. MPC ’10.

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