[PPT] - Composing and Decomposing Data Types Data Types ` a la Carte with PowerPoint Presentation

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Faculty of Science

Composing and Decomposing Data Types

Data Types ` a la Carte with Closed Type Families Patrick Bahr

University of Copenhagen, Department of Computer Science paba@di.ku.dk

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Introduction

Goal

Improve Haskell implementation of Data Types ` a la Carte:

More flexible
Improved error reporting
New use cases

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Introduction

Goal

Improve Haskell implementation of Data Types ` a la Carte:

More flexible
Improved error reporting
New use cases

How?

Using closed type families

New feature in latest version of GHC
Type-level functions
Pattern matching similar(-ish) to term-level functions

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Data Types ` a la Carte

Idea: Decompose data types into two-level types:

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Data Types ` a la Carte

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp

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Data Types ` a la Carte

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 3

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Data Types ` a la Carte

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith data Fix f = In (f (Fix f ))

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 3

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Data Types ` a la Carte

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 3

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Data Types ` a la Carte

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith Functors can be combined by coproduct construction :+:

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Data Types ` a la Carte

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith Functors can be combined by coproduct construction :+: data Mul a = Mul a a type Exp′ = Fix (Arith :+: Mul)

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Data Types ` a la Carte

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith Functors can be combined by coproduct construction :+: data (f :+: g) a = Inl (f a) | Inr (g a) data Mul a = Mul a a type Exp′ = Fix (Arith :+: Mul)

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Data Types ` a la Carte (cont.)

Subtyping constraint : ≺:

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a)

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Data Types ` a la Carte (cont.)

Subtyping constraint : ≺:

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a)

Example: smart constructors

add :: (Arith : ≺: f ) ⇒ Fix f → Fix f → Fix f add x y = In (inj (Add x y))

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Data Types ` a la Carte (cont.)

Subtyping constraint : ≺:

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a)

Example: smart constructors

add :: (Arith : ≺: f ) ⇒ Fix f → Fix f → Fix f add x y = In (inj (Add x y)) exp :: Fix (Arith :+: Mul) exp = val 1 ‘add‘ (val 2 ‘mul‘ val 3)

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Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance (f : ≺: f1) ⇒ f : ≺: (f1 :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

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Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance (f : ≺: f1) ⇒ f : ≺: (f1 :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

No backtracking!

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Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

No backtracking!

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Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

No backtracking!
Asymmetric treatment of :+:
Left-hand side is not inspected

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Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

No backtracking!
Asymmetric treatment of :+:

A : ≺: A :+: (B :+: C)

Left-hand side is not inspected

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 5

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Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

No backtracking!
Asymmetric treatment of :+:

A : ≺: (A :+: B) :+: C

Left-hand side is not inspected

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 5

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Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

No backtracking!
Asymmetric treatment of :+:

A : ≺: (A :+: B) :+: C

Left-hand side is not inspected

A :+: B : ≺: (A :+: B) :+: C

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 5

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Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

No backtracking!
Asymmetric treatment of :+:

A : ≺: (A :+: B) :+: C

Left-hand side is not inspected

A :+: B : ≺: A :+: (B :+: C)

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 5

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Contributions We re-implemented : ≺: such that:

Subtyping behaves as intuitively expected
Ambiguous subtyping are avoided
We can express isomorphism :≃:

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Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

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Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ “set of signatures in f ” ⊆ “set of signatures in g”

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Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ “set of signatures in f ” ⊆ “set of signatures in g” C :+: A : ≺: A :+: B :+: C

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Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ “set of signatures in f ” ⊆ “set of signatures in g” C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected:

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Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ “set of signatures in f ” ⊆ “set of signatures in g” C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B

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Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ “set of signatures in f ” ⊆ “set of signatures in g” C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B injection not unique!

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Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ “set of signatures in f ” ⊆ “set of signatures in g” C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B injection not unique!

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Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ “set of signatures in f ” ⊆ “set of signatures in g” C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B injection not unique! “injection” not injective!

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Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ “set of signatures in f ” ⊆ “set of signatures in g” C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B injection not unique! “injection” not injective!

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 7

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Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ “set of signatures in f ” = “set of signatures in g”

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Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ “set of signatures in f ” = “set of signatures in g” Easy to implement: f :≃: g = (f : ≺: g, g : ≺: f )

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Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ “set of signatures in f ” = “set of signatures in g” Easy to implement: f :≃: g = (f : ≺: g, g : ≺: f )

Use case: improved projection function

The type of the projection function is unsatisfying: prj :: (f : ≺: g) ⇒ g a → Maybe (f a)

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Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ “set of signatures in f ” = “set of signatures in g” Easy to implement: f :≃: g = (f : ≺: g, g : ≺: f )

Use case: improved projection function

The type of the projection function is unsatisfying: prj :: (f : ≺: g) ⇒ g a → Maybe (f a) With :≃: we can do better: split :: (g :≃: f :+: r) ⇒ g a → Either (f a) (r a)

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Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ “set of signatures in f ” = “set of signatures in g” Easy to implement: f :≃: g = (f : ≺: g, g : ≺: f )

Use case: improved projection function

The type of the projection function is unsatisfying: prj :: (f : ≺: g) ⇒ g a → Maybe (f a) With :≃: we can do better: split :: (g :≃: f :+: r) ⇒ g a → (f a → b) → (r a → b) → b

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Example: Desugaring

data Dbl a = Double a

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Example: Desugaring

data Dbl a = Double a class Desug f g where desugAlg :: f (Fix g) → Fix g

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Example: Desugaring

data Dbl a = Double a class Desug f g where desugAlg :: f (Fix g) → Fix g instance (Desug f1 g, Desug f2 g) ⇒ Desug (f1 :+: f2) g where desugAlg (Inl x) = desugAlg x desugAlg (Inr x) = desugAlg x instance (Arith : ≺: g) ⇒ Desug Dbl g where desugAlg (Double x) = add x x instance (f : ≺: g) ⇒ Desug f g where desugAlg = In . inj

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Example: Desugaring

data Dbl a = Double a class Desug f g where desugAlg :: f (Fix g) → Fix g instance (Desug f1 g, Desug f2 g) ⇒ Desug (f1 :+: f2) g where desugAlg (Inl x) = desugAlg x desugAlg (Inr x) = desugAlg x instance (Arith : ≺: g) ⇒ Desug Dbl g where desugAlg (Double x) = add x x instance (f : ≺: g) ⇒ Desug f g where desugAlg = In . inj desugar :: (Desug f g, Functor f ) ⇒ Fix f → Fix g desugar = fold desugAlg

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Example: Desugaring

data Dbl a = Double a class Desug f g where desugAlg :: f (Fix g) → Fix g instance (Desug f1 g, Desug f2 g) ⇒ Desug (f1 :+: f2) g where desugAlg (Inl x) = desugAlg x desugAlg (Inr x) = desugAlg x instance (Arith : ≺: g) ⇒ Desug Dbl g where desugAlg (Double x) = add x x instance (f : ≺: g) ⇒ Desug f g where desugAlg = In . inj desugar :: Fix (Dbl :+: Arith :+: Mul) → Fix (Arith :+: Mul) desugar = fold desugAlg

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Example: Desugaring (cont.)

desugar :: (f :≃: g :+: Dbl, Arith : ≺: g, Functor f ) ⇒ Fix f → Fix g desugar = fold desugAlg

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Example: Desugaring (cont.)

desugar :: (f :≃: g :+: Dbl, Arith : ≺: g, Functor f ) ⇒ Fix f → Fix g desugar = fold desugAlg desugAlg :: (f :≃: g :+: Dbl, Arith : ≺: g) ⇒ f (Fix g) → Fix g desugAlg e = split e (λx → In x) (λ(Double x) → add x x)

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Example: Desugaring (cont.)

desugar :: (f :≃: g :+: Dbl, Arith : ≺: g, Functor f ) ⇒ Fix f → Fix g desugar = fold desugAlg desugAlg :: (f :≃: g :+: Dbl, Arith : ≺: g) ⇒ f (Fix g) → Fix g desugAlg e = split e (λx → In x) (λ(Double x) → add x x) desugAlg′ :: (f :≃: g :+: Dbl, Arith : ≺: g, Mul : ≺: g) ⇒ f (Fix g) → Fix g desugAlg′ e = split e (λx → In x) (λ(Double x) → mul (val 2) x)

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Implementation of : ≺:

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Idea

Type-level function Embed:

take two signatures f , g as arguments
check whether f :

≺: g

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Idea

Type-level function Embed:

take two signatures f , g as arguments
check whether f :

≺: g Derive implementation of inj and prj: ???

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 12

SLIDE 49

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

take two signatures f , g as arguments
check whether f :

≺: g

if check is successful: produce proof object for f :

≺: g Derive implementation of inj and prj:

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 12

SLIDE 50

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

take two signatures f , g as arguments
check whether f :

≺: g

if check is successful: produce proof object for f :

≺: g Derive implementation of inj and prj:

also use a type class
But: use proof object as oracle in instance declarations

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 12

SLIDE 51

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

take two signatures f , g as arguments
check whether f :

≺: g

if check is successful: produce proof object for f :

≺: g Derive implementation of inj and prj:

also use a type class
But: use proof object as oracle in instance declarations

No singleton types. This all happens at compile time!

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 12

SLIDE 52

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Pos = Here | Left Pos | Right Pos | Sum Pos Pos

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 13

SLIDE 53

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Pos = Here | Left Pos | Right Pos | Sum Pos Pos Here : f : ≺: f

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 13

SLIDE 54

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Pos = Here | Left Pos | Right Pos | Sum Pos Pos Here : f : ≺: f p : f : ≺: g1 Left p : f : ≺: g1 :+: g2 p : f : ≺: g2 Right p : f : ≺: g1 :+: g2

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 13

SLIDE 55

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Pos = Here | Left Pos | Right Pos | Sum Pos Pos Here : f : ≺: f p : f : ≺: g1 Left p : f : ≺: g1 :+: g2 p : f : ≺: g2 Right p : f : ≺: g1 :+: g2 p1 : f1 : ≺: g p2 : f2 : ≺: g Sum p1 p2 : f1 :+: f2 : ≺: g

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 13

SLIDE 56

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Pos = Here | Left Pos | Right Pos | Sum Pos Pos Here : f : ≺: f p : f : ≺: g1 Left p : f : ≺: g1 :+: g2 p : f : ≺: g2 Right p : f : ≺: g1 :+: g2 p1 : f1 : ≺: g p2 : f2 : ≺: g Sum p1 p2 : f1 :+: f2 : ≺: g kind

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 13

SLIDE 57

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Pos = Here | Left Pos | Right Pos | Sum Pos Pos Here : f : ≺: f p : f : ≺: g1 Left p : f : ≺: g1 :+: g2 p : f : ≺: g2 Right p : f : ≺: g1 :+: g2 p1 : f1 : ≺: g p2 : f2 : ≺: g Sum p1 p2 : f1 :+: f2 : ≺: g kind type

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 13

SLIDE 58

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Pos = Here | Left Pos | Right Pos | Sum Pos Pos Here : f : ≺: f p : f : ≺: g1 Left p : f : ≺: g1 :+: g2 p : f : ≺: g2 Right p : f : ≺: g1 :+: g2 p1 : f1 : ≺: g p2 : f2 : ≺: g Sum p1 p2 : f1 :+: f2 : ≺: g kind type type constructor

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 13

SLIDE 59

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Construct Proof Objects

data Emb = Found Pos | NotFound | Ambiguous

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 14

SLIDE 60

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Construct Proof Objects

data Emb = Found Pos | NotFound | Ambiguous type family Embed (f :: ∗ → ∗) (g :: ∗ → ∗) :: Emb where

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 14

SLIDE 61

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Construct Proof Objects

data Emb = Found Pos | NotFound | Ambiguous type family Embed (f :: ∗ → ∗) (g :: ∗ → ∗) :: Emb where Embed f f = Found Here Embed (f1 :+: f2) g = Sum′ (Embed f1 g) (Embed f2 g) Embed f (g1 :+: g2) = Choose (Embed f g1) (Embed f g2) Embed f g = NotFound

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 14

SLIDE 62

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Construct Proof Objects

data Emb = Found Pos | NotFound | Ambiguous type family Embed (f :: ∗ → ∗) (g :: ∗ → ∗) :: Emb where Embed f f = Found Here Embed (f1 :+: f2) g = Sum′ (Embed f1 g) (Embed f2 g) Embed f (g1 :+: g2) = Choose (Embed f g1) (Embed f g2) Embed f g = NotFound type family Choose (e1 :: Emb) (e2 :: Emb) :: Emb where Choose (Found p1) (Found p1) = Ambiguous Choose Ambiguous e2 = Ambiguous Choose e1 Ambiguous = Ambiguous Choose (Found p1) e2 = Found (Left p1) Choose e1 (Found p2) = Found (Right p2) Choose NotFound NotFound = NotFound

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 14

SLIDE 63

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Post-Processing

This is almost what we want.

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 15

SLIDE 64

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Post-Processing

This is almost what we want.

We avoid ambiguity on the right-hand side:

A : ≺: A :+: A :+: C

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 15

SLIDE 65

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Post-Processing

This is almost what we want.

We avoid ambiguity on the right-hand side:

A : ≺: A :+: A :+: C

We still have ambiguity on the left-hand side:

A :+: A : ≺: A :+: B

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 15

SLIDE 66

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Post-Processing

This is almost what we want.

We avoid ambiguity on the right-hand side:

A : ≺: A :+: A :+: C

We still have ambiguity on the left-hand side:

A :+: A : ≺: A :+: B

Solution: check for duplicates in Pos

type family Dupl (p :: Pos) :: Bool where . . .

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 15

SLIDE 67

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Post-Processing

This is almost what we want.

We avoid ambiguity on the right-hand side:

A : ≺: A :+: A :+: C

We still have ambiguity on the left-hand side:

A :+: A : ≺: A :+: B

Solution: check for duplicates in Pos

type family Dupl (p :: Pos) :: Bool where . . . Sum (Left Here) (Left Here)

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 15

SLIDE 68

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Are we there yet?

Check whether f :

≺: g

Construct proof for f :

≺: g

Derive inj and prj

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 16

SLIDE 69

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Are we there yet?

Check whether f :

≺: g

Construct proof for f :

≺: g

Derive inj and prj

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 16

SLIDE 70

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Are we there yet?

Check whether f :

≺: g

Construct proof for f :

≺: g

Derive inj and prj

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 16

SLIDE 71

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Are we there yet?

Check whether f :

≺: g

Construct proof for f :

≺: g

Derive inj and prj

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 16

SLIDE 72

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a) instance f : ≺: f where . . . instance f : ≺: (f :+: g2) where . . . instance f : ≺: g2 ⇒ f : ≺: (g1 :+: g2) where . . .

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 73

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class Sub f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub f f where . . . instance Sub f (f :+: g2) where . . . instance Sub f g2 ⇒ Sub f (g1 :+: g2) where . . .

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 74

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class Sub f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub f f where . . . instance Sub f g1 ⇒ Sub f (g1 :+: g2) where . . . instance Sub f g2 ⇒ Sub f (g1 :+: g2) where . . .

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 75

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class Sub f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub f f where . . . instance Sub f g1 ⇒ Sub f (g1 :+: g2) where . . . instance Sub f g2 ⇒ Sub f (g1 :+: g2) where . . . instance (Sub f1 g, Sub f2 g) ⇒ Sub (f1 :+: f2) g where . . .

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 76

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class Sub (e :: Emb) f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub f f where . . . instance Sub f g1 ⇒ Sub f (g1 :+: g2) where . . . instance Sub f g2 ⇒ Sub f (g1 :+: g2) where . . . instance (Sub f1 g, Sub f2 g) ⇒ Sub (f1 :+: f2) g where . . .

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 77

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class Sub (e :: Emb) f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub (Found Here) f f where . . . instance Sub f g1 ⇒ Sub f (g1 :+: g2) where . . . instance Sub f g2 ⇒ Sub f (g1 :+: g2) where . . . instance (Sub f1 g, Sub f2 g) ⇒ Sub (f1 :+: f2) g where . . . Here : f : ≺: f

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 78

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class Sub (e :: Emb) f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub (Found Here) f f where . . . instance Sub (Found p) f g1 ⇒ Sub (Found (Left p)) f (g1 :+: g2) where . . . instance Sub f g2 ⇒ Sub f (g1 :+: g2) where . . . instance (Sub f1 g, Sub f2 g) ⇒ Sub (f1 :+: f2) g where . . . p : f : ≺: g1 Left p : f : ≺: g1 :+: g2

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 79

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class Sub (e :: Emb) f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub (Found Here) f f where . . . instance Sub (Found p) f g1 ⇒ Sub (Found (Left p)) f (g1 :+: g2) where . . . instance Sub (Found p) f g2 ⇒ Sub (Found (Right p)) f (g1 :+: g2) where . . . instance (Sub f1 g, Sub f2 g) ⇒ Sub (f1 :+: f2) g where . . . p : f : ≺: g2 Right p : f : ≺: g1 :+: g2

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 80

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class Sub (e :: Emb) f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub (Found Here) f f where . . . instance Sub (Found p) f g1 ⇒ Sub (Found (Left p)) f (g1 :+: g2) where . . . instance Sub (Found p) f g2 ⇒ Sub (Found (Right p)) f (g1 :+: g2) where . . . instance (Sub (Found p1) f1 g, Sub (Found p2) f2 g) ⇒ Sub (Found (Sum p1 p2)) (f1 :+: f2) g where . . . p1 : f1 : ≺: g p2 : f2 : ≺: g Sum p1 p2 : f1 :+: f2 : ≺: g

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 81

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Derive inj and prj

class Sub (e :: Emb) f g where inj :: Proxy e → f a → g a prj :: Proxy e → g a → Maybe (f a) instance Sub (Found Here) f f where . . . instance Sub (Found p) f g1 ⇒ Sub (Found (Left p)) f (g1 :+: g2) where . . . instance Sub (Found p) f g2 ⇒ Sub (Found (Right p)) f (g1 :+: g2) where . . . instance (Sub (Found p1) f1 g, Sub (Found p2) f2 g) ⇒ Sub (Found (Sum p1 p2)) (f1 :+: f2) g where . . .

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 17

SLIDE 82

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Are we there yet?

Check whether f :

≺: g

Construct proof for f :

≺: g

Derive inj and prj

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 18

SLIDE 83

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Are we there yet?

Check whether f :

≺: g

Construct proof for f :

≺: g

Derive inj and prj
Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014

Slide 18

SLIDE 84

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Are we there yet?

Check whether f :

≺: g

Construct proof for f :

≺: g

Derive inj and prj

(sort of)

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 18

SLIDE 85

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Final Implementation of : ≺:

class Sub (e :: Emb) f g where inj :: Proxy e → f a → g a prj :: Proxy e → g a → Maybe (f a)

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 19

SLIDE 86

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Final Implementation of : ≺:

class Sub (e :: Emb) f g where inj :: Proxy e → f a → g a prj :: Proxy e → g a → Maybe (f a) type f : ≺: g = Sub (Post (Embed f g)) f g

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 19

SLIDE 87

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Final Implementation of : ≺:

class Sub (e :: Emb) f g where inj′ :: Proxy e → f a → g a prj′ :: Proxy e → g a → Maybe (f a) type f : ≺: g = Sub (Post (Embed f g)) f g

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 19

SLIDE 88

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Final Implementation of : ≺:

class Sub (e :: Emb) f g where inj′ :: Proxy e → f a → g a prj′ :: Proxy e → g a → Maybe (f a) type f : ≺: g = Sub (Post (Embed f g)) f g inj :: (f : ≺: g) ⇒ f a → g a inj = inj′ (P :: Proxy (Post (Embed f g))) prj :: (f : ≺: g) ⇒ g a → Maybe (f a) prj = prj′ (P :: Proxy (Post (Embed f g)))

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 19

SLIDE 89

Type-Level Programming in Haskell

SLIDE 90

Type-Level Programming in Haskell

Now :

≺: has the properties we want / expect

Avoid “ambiguous” subtyping
New isomorphism constraint :≃:

SLIDE 91

Type-Level Programming in Haskell

Now :

≺: has the properties we want / expect

Avoid “ambiguous” subtyping
New isomorphism constraint :≃:
You can try it:

> cabal install compdata

SLIDE 92

Type-Level Programming in Haskell

SLIDE 93

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

If done “wrong”, this implementation can be very slow!

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 22

SLIDE 94

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

If done “wrong”, this implementation can be very slow!
Implementation presented here: O(n2)

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 22

SLIDE 95

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

If done “wrong”, this implementation can be very slow!
Implementation presented here: O(n2)
Slightly different implementation: O(2n)

(but essentially the same)

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 22

SLIDE 96

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

If done “wrong”, this implementation can be very slow!
Implementation presented here: O(n2)
Slightly different implementation: O(2n)

(but essentially the same)

micro benchmark:
derive F :

≺: G

9 summands in F and G

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 22

SLIDE 97

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

If done “wrong”, this implementation can be very slow!
Implementation presented here: O(n2)
Slightly different implementation: O(2n)

(but essentially the same)

micro benchmark:
derive F :

≺: G

9 summands in F and G
Implementation presented here: 0.5s

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 22

SLIDE 98

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

If done “wrong”, this implementation can be very slow!
Implementation presented here: O(n2)
Slightly different implementation: O(2n)

(but essentially the same)

micro benchmark:
derive F :

≺: G

9 summands in F and G
Implementation presented here: 0.5s
Naive implementation: 45s

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 22

SLIDE 99

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

If done “wrong”, this implementation can be very slow!
Implementation presented here: O(n2)
Slightly different implementation: O(2n)

(but essentially the same)

micro benchmark:
derive F :

≺: G

9 summands in F and G
Implementation presented here: 0.5s
Naive implementation: 45s
Type families on kind ∗ are expensive!

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 22

SLIDE 100

Type-Level Programming in Haskell

SLIDE 101

Type-Level Programming in Haskell

SLIDE 102

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Error Messages

A :

≺: B :+: C ?

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 24

SLIDE 103

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Error Messages

A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 24

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Error Messages

A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C)) The original implementation would give: No instance for (A :<: C)

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 24

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Error Messages

A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

A :+: A :

≺: A :+: B ? No instance for (Sub Ambiguous (A :+: A) (A :+: B))

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 24

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Error Messages

A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

A :+: A :

≺: A :+: B ? No instance for (Sub Ambiguous (A :+: A) (A :+: B))

A :

≺: A :+: B ?

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 24

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Error Messages

A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

A :+: A :

≺: A :+: B ? No instance for (Sub Ambiguous (A :+: A) (A :+: B))

a :

≺: a :+: B ?

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 24

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Error Messages

A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

A :+: A :

≺: A :+: B ? No instance for (Sub Ambiguous (A :+: A) (A :+: B))

a :

≺: a :+: B ? No instance for (Sub (Post (Embed a (a :+: B))) a (a :+: B))

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 24

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Conclusion

We can do cool stuff with closed type families.

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 25

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Conclusion

We can do cool stuff with closed type families.
But:
Compile time performance unpredictable.
We need a way to customise error messages.

Patrick Bahr — Composing and Decomposing Data Types — 20th June, 2014 Slide 25