Cody Roux - Pure Type Systems - Boston Haskell Meetup

Pure Type Systems:Dependents When You Need Them

Cody Roux

Draper Laboratories

February 17, 2015

Cody Roux (Draper Labs) PTSes February 17, 2015 1 / 38

Introduction

This talk is not about Haskell!

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Introduction

Or is it?

Wait, which Haskell?

good ol’ Haskell 98

-XTypeFamilies

-XExistentialQuantification

-XRank2Types

-XRankNTypes

-XDataKinds

-XPolyKinds

-XGADTs

-XConstraintKinds

-XImpredicativeTypes

etc.

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Introduction

This talk is about abstraction!

We want to understand -XFooBar in a unified framework

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Abstraction

The simplest form of abstraction

We have an expression 2 + 2

We can abstract it as x + x where x = 2

Have we gained anything?

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Abstraction

We can form the λ-abstraction

λx . x + x

This is already a very powerful idea!

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STLC

The Simply Typed λ-Calculus

Some base types A,B ,C , ...

Higher-order functions λx .λf .f x : A → (A → B) → B

A small miracle: every function is terminating.

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Polymorphism

We want to have polymorphic functions

(λx .x) 3 → 3

(λx .x) true → true

How do we add this feature?

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Polymorphic formulas

There are 2 possible answers!

First

Add type-level variables, X ,Y ,Z , ...

Add polymorphic quantification

∀X .X → X

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Polymorphic formulas

What does ∀X .T quantify over?

1 Only the simple types

2 Any type from the extended language

These lead to dramatically different systems!

In the first case, the extension is conservative (no “new” functions)

In the second case, it is not (system F)

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Dependent types

We can add term-level information to types:

[1, 2, 3] : ListN

[1, 2, 3] : VecN 3

We can add quantification as well:

reverse : ∀n,VecN n → VecN n

When is this kind of dependency conservative?

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Pure Type Systems

Pure type systems:

are a generic framework for logics/programming lang.

only allow universal quantification/dependent function space

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Pure Type Systems

Pure type systems are:

1 Expressive: ∃ a PTS that can express set theory

2 Well studied: invented in the 80s (Barendregt) and studied ever since!

3 Flexible: found at the core of several functional languages, includingHaskell, Agda, Coq.

4 Can be complex! There are several longstanding open questionsincluding

1 Typed Conversion ⇔ Untyped Conversion

2 Weak Normalization ⇔ Strong Normalization

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Pure Type Systems

Can we answer our questions using PTS?

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Pure Type Systems

A Pure Type System is defined as

1 A set of Sorts S

2 A set of Axioms A ⊆ S × S

3 A set of Rules R ⊆ S × S × S

That’s it!

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Pure Type Systems

Informally

Elements ∗,�, ι, ... ∈ S represent a category of objects.

For example

∗ may represent the category of propositions

� may represent the category of types

ι may represent the category of natural numbers

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Pure Type Systems

(s1, s2) ∈ A informally means:

s1 is a member of the category s2

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Pure Type Systems

(s1, s2, s3) ∈ R informally means:

Quantifying over an element of s2 parametrized over an element of s1gives a result in s3

if A : s1 and B(x) : s2 when x : A

then ∀x : A.B(x) : s3

We will write Πx : A.B instead of ∀x : A.B(x) (tradition)

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Pure Type Systems

Given a PTS P we have the following type system:

Type/Sort formation

Γ ⊢axiom (s1, s2) ∈ A

Γ ⊢ s1 : s2

Γ ⊢ A : s1 Γ, x : A ⊢ B : s2prod (s1, s2, s3) ∈ R

Γ ⊢ Πx : A.B : s3

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Pure Type Systems

Term formation

Γ ⊢ A : svar s ∈ S

Γ, x : A ⊢ x : A

Γ, x : A ⊢ t : B Γ ⊢ Πx : A.B : sabs s ∈ S

Γ ⊢ λx : A.t : Πx : A.B

Γ ⊢ t : Πx : A.B Γ ⊢ u : AappΓ ⊢ t u : B [x 7→ u]

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Pure Type Systems

Conversion

Γ ⊢ t : A Γ ⊢ A′ : sconv A ≃β A

′, s ∈ S

Γ ⊢ t : A′

Where ≃β is β-equality

(λx : A.t)u ≃β t[x 7→ u]

We omit the boring rules...

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Pure Type Systems

The rest of this talk

Understanding this definition!

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Simply Typed Lambda Calculus

We can model the STLC using

S = {∗,�}

A = {(∗,�)}

R = {(∗, ∗, ∗)}

We have e.g.A : ∗ ⊢ λx : A.x : A → A

taking A → A = Πx : A. A

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The λ-cube

Some more examples, contained in a family called the λ-cube:

The sorts are ∗,�

∗ : �

The rules are (k1, k2, k2) with ki = ∗ or �

Each dimension of the cube highlights a different feature

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The λ-cube

STLC

F

λΠ

λ2

λω

Fω

λΠω

CC

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λ-cube

STLC = (∗, ∗)

F = (∗, ∗) (�, ∗)

λω = (∗, ∗) (�,�)

λΠ = (∗, ∗) (∗,�)

λ2 = (∗, ∗) (∗,�) (�, ∗)

Fω = (∗, ∗) (�, ∗) (�,�)

λΠω = (∗, ∗) (∗,�) (�,�)

CC = (∗, ∗) (∗,�) (�, ∗) (�,�)

STLC

F

λΠ

λ2

λω

Fω

λΠω

CC

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λ-cube features

Calculus Rule Feature Example

STLC (∗, ∗) Ordinary (higher-order) functions id : N → N

F (�, ∗) Impredicative polymorphism id : ∀X .X → X

λω (�,�) Type constructors rev : List A → List A

λΠ (∗,�) Dependent Types head : VecN (n + 1) → N

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Example

Let’s work out an example in CC :

Induction on lists

∀A P l , P (nil A) → (∀a r , P r → P (cons A y r)) → P l

Π(A : ∗)(P : List A → ∗)(l : List A). P (nil A) →(

Π(a : A)(r : List A). P r → P (cons A y r))

→ P l

X → Y still means Π : A. B

Whiteboard time!

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Example

No whiteboard?

List : ∗ → ∗

nil : ΠA : ∗. List A

cons : ΠA : ∗. A → List A → List A

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Example

⊢ ∗ : �

A : ∗ ⊢ List A : ∗ . . . ⊢ ∗ : �A : ∗ ⊢ List A → ∗ : �

. . . ⊢ P (nil A) : ∗...

. . . ⊢ . . . : ∗

...A : ∗ ⊢ Π(P : List A → ∗)(l : List A) . . . : ∗

⊢ Π(A : ∗)(P : List A → ∗)(l : List A). P (nil A) →(

Π(a : A)(r : List A). P r → P (cons A y r))

→ P l : ∗

Cody Roux (Draper Labs) PTSes February 17, 2015 30 / 38

Other Calculi

Here are a few other examples:

Name Sorts Axioms Rules

STLC(1 base type) ι, ∗ (ι, ∗) (∗, ∗, ∗)

STLC ∗,� (∗,�) (∗, ∗, ∗)

∗ : ∗ ∗ (∗, ∗) (∗, ∗, ∗)

System F ∗,� (∗,�) (∗, ∗, ∗), (�, ∗, ∗)

CC ∗,� (∗,�) (∗, ∗, ∗), (�, ∗, ∗),(∗,�,�), (�,�,�)

U− ∗,�,△ (∗,�), (∗, ∗, ∗), (�, ∗, ∗),(�,△) (�,�,�), (△,�,�)

CCω ∗,�i , (∗,�i ), (∗, ∗, ∗), (�i , ∗, ∗),(core of Coq) i ∈ N (�i ,�j ), i < j (�i ,�j ,�k), k ≥ max(i , j)

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Normalization

A PTS is normalizing ⇔ Γ ⊢ t : T ⇒ t has a β-normal form.

Normalization is a central property:

1 It ensures decidability of type-checking

2 It implies consistency of the system as a logic

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Normalization

Normalization is hard to predict:

Name Axioms Rules Norm.

STLC(1 base type) (ι, ∗) (∗, ∗, ∗) Yes

STLC (∗,�) (∗, ∗, ∗) Yes

∗ : ∗ (∗, ∗) (∗, ∗, ∗) No

System F (∗,�) (∗, ∗, ∗), (�, ∗, ∗) Yes

CC (∗,�) (∗, ∗, ∗), (�, ∗, ∗), Yes(∗,�,�), (�,�,�)

U− (∗,�), (∗, ∗, ∗), (�, ∗, ∗), No(�,△) (�,�,�), (△,�,�)

CCω (∗,�i ), (∗, ∗, ∗), (�i , ∗, ∗), Yes(core of Coq) (�i ,�j), i < j (�i ,�j ,�k), k ≥ max(i , j)

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Other Features

PTSes can capture things like predicative polymorphism:

Only instantiate ∀s with monomorphic types

∀X .X → X 7→ N → N yes

∀X .X → X 7→ (∀Y .Y → Y ) → (∀Y .Y → Y ) no

Sorts: ∗, ∗̂,�

Axioms: ∗ : �, ∗̂ : �

Rules: STLC + {(�, ∗, ∗̂), (�, ∗̂, ∗̂)}

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Other Features

We can seperate type-level data and program-level data

Sorts: ∗t , ∗p ,�t ,�p

Axioms: ∗t : �t , ∗p : �p

Rules:{(∗t , ∗t , ∗t), (∗p , ∗p , ∗p), (∗t ,�p ,�p)}

Nt lives in ∗t , Np lives in ∗p

Similar to GADTs!

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More about U−

Remember U−:

R = {(∗, ∗, ∗), (�, ∗, ∗), (�,�,�), (△,�,�)}

This corresponds to Kind Polymorphism!

But...

It is inconsistent!U− ⊢ t : ∀X . X

This is (maybe) bad news for constraint kinds!

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Conclusion

Pure Type Systems are functional languages with simple syntax

They can explain many aspects of the Haskell Type System.

Pure Type Systems give fine grained ways of extending the typingrules.

The meta-theory can be studied in a single generic framework.

There are still hard theory questions about PTS.

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The End

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