Motivating Example Types and Subtyping Function Types and Operators Conclusion

Gradual Typing with Union and Intersection Types

Giuseppe Castagna, Victor Lanvin

ICFP ’17

September 6, 2017

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Outline

1 Motivating Example

2 Types and Subtyping

3 Function Types and Operators

4 Conclusion

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Motivating Example (1/3)

[Siek and Vachharajani, 2008]

let succ : Int -> Int = ...let not : Bool -> Bool = ...

let f (condition : Bool) (x :

?

) :

?

=if condition then

succ xelse

not x

→ Cannot be typed with simple types, but valid with gradual types.→ What if we apply it to a string?

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Motivating Example (1/3)

[Siek and Vachharajani, 2008]

let succ : Int -> Int = ...let not : Bool -> Bool = ...

let f (condition : Bool) (x : ?) : ? =if condition then

succ xelse

not x

→ Cannot be typed with simple types, but valid with gradual types.

→ What if we apply it to a string?

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Motivating Example (1/3)

[Siek and Vachharajani, 2008]

let succ : Int -> Int = ...let not : Bool -> Bool = ...

let f (condition : Bool) (x : ?) : ? =if condition then

succ xelse

not x

→ Cannot be typed with simple types, but valid with gradual types.→ What if we apply it to a string?

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Motivating Example (2/3)

Set-theoretic version:

let f (condition : Bool) (x : (Int | Bool)): (Int | Bool) =

if condition thenif x ∈ Int then succ x else assert false

elseif x ∈ Bool then not x else assert false

→ Syntactically heavy, but safe

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Motivating Example (2/3)

Set-theoretic version:

let f (condition : Bool) (x : (Int | Bool)): (Int | Bool) =

if condition thenif x ∈ Int then succ x else assert false

elseif x ∈ Bool then not x else assert false

→ Syntactically heavy, but safe

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Motivating Example (3/3)

Mixing the two:

let f (condition : Bool) (x : (Int | Bool) & ?): (Int | Bool) =

if condition thensucc x

elsenot x

→ Cannot be applied to something else than an integer or aboolean, and has a precise return type→ Syntactically straightforward

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Motivating Example (3/3)

Mixing the two:

let f (condition : Bool) (x : (Int | Bool) & ?): (Int | Bool) =

if condition thensucc x

elsenot x

→ Cannot be applied to something else than an integer or aboolean, and has a precise return type→ Syntactically straightforward

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Summary of the Motivations

– Gradualization of single expressions is sometimes too coarse

– Set-theoretic types are powerful but syntactically heavy (noreconstruction)

Mixing the two would:

– Make the transition between dynamic types and static typessmoother

– Reduce the syntactic overhead of set-theoretic types

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Summary of the Motivations

– Gradualization of single expressions is sometimes too coarse

– Set-theoretic types are powerful but syntactically heavy (noreconstruction)

Mixing the two would:

– Make the transition between dynamic types and static typessmoother

– Reduce the syntactic overhead of set-theoretic types

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Outline

1 Motivating Example

2 Types and Subtyping

3 Function Types and Operators

4 Conclusion

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Syntax

t ∈ STypes ::= b | t → t | t ∨ t | t ∧ t | ¬t | Empty | Any

τ ∈ GTypes ::= ? | b | τ → τ | τ ∨ τ | τ ∧ τ | ¬t | Empty | Any

Note: ? 6= Any

Any = unknown type, explicitly deconstructed? = unknown type, implicitly deconstructed

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Syntax

t ∈ STypes ::= b | t → t | t ∨ t | t ∧ t | ¬t | Empty | Any

τ ∈ GTypes ::= ? | b | τ → τ | τ ∨ τ | τ ∧ τ | ¬t | Empty | Any

Note: ? 6= Any

Any = unknown type, explicitly deconstructed? = unknown type, implicitly deconstructed

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Concretization

First idea: apply AGT [Garcia et al., 2016]

We define a concretization function γ : GTypes→ P(STypes).

γ(?) = STypesγ(τ1 ∨ τ2) = {t1 ∨ t2 | ti ∈ γ(τi )}γ(τ1 → τ2) = {t1 → t2 | ti ∈ γ(τi )}

γ(b) = {b}etc...

For example,

γ((?→ Int)∧ ?) = {(t → Int) ∧ t ′ | (t, t ′) ∈ STypes2}

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Concretization

First idea: apply AGT [Garcia et al., 2016]

We define a concretization function γ : GTypes→ P(STypes).

γ(?) = STypesγ(τ1 ∨ τ2) = {t1 ∨ t2 | ti ∈ γ(τi )}γ(τ1 → τ2) = {t1 → t2 | ti ∈ γ(τi )}

γ(b) = {b}etc...

For example,

γ((?→ Int)∧ ?) = {(t → Int) ∧ t ′ | (t, t ′) ∈ STypes2}

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Concretization

First idea: apply AGT [Garcia et al., 2016]

We define a concretization function γ : GTypes→ P(STypes).

γ(?) = STypesγ(τ1 ∨ τ2) = {t1 ∨ t2 | ti ∈ γ(τi )}γ(τ1 → τ2) = {t1 → t2 | ti ∈ γ(τi )}

γ(b) = {b}etc...

For example,

γ((?→ Int)∧ ?) = {(t → Int) ∧ t ′ | (t, t ′) ∈ STypes2}

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Subtyping (1/2)

Consistent subtyping [Garcia et al., 2016]

σ ≤̃ τ ⇐⇒ ∃(s, t) ∈ γ(σ)× γ(τ), s ≤ t

However, we can show the existence of “extremal” concretizations:

∀t ∈ γ(τ), τ⇓ ≤ t ≤ τ⇑

(?→?)⇓ = (Any→ Empty) (?→?)⇑ = (Empty→ Any)

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Subtyping (1/2)

Consistent subtyping [Garcia et al., 2016]

σ ≤̃ τ ⇐⇒ ∃(s, t) ∈ γ(σ)× γ(τ), s ≤ t

However, we can show the existence of “extremal” concretizations:

∀t ∈ γ(τ), τ⇓ ≤ t ≤ τ⇑

(?→?)⇓ = (Any→ Empty) (?→?)⇑ = (Empty→ Any)

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Subtyping (1/2)

Consistent subtyping [Garcia et al., 2016]

σ ≤̃ τ ⇐⇒ ∃(s, t) ∈ γ(σ)× γ(τ), s ≤ t

However, we can show the existence of “extremal” concretizations:

∀t ∈ γ(τ), τ⇓ ≤ t ≤ τ⇑

(?→?)⇓ = (Any→ Empty) (?→?)⇑ = (Empty→ Any)

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Subtyping (2/2)

Consistent subtyping

σ ≤̃ τ ⇐⇒ σ⇓ ≤ τ⇑

=⇒ Consistent subtyping reduces in linear time to semanticsubtyping!

Note: emphasizes the fact that consistent subtyping is nottransitive.

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Subtyping (2/2)

Consistent subtyping

σ ≤̃ τ ⇐⇒ σ⇓ ≤ τ⇑

=⇒ Consistent subtyping reduces in linear time to semanticsubtyping!

Note: emphasizes the fact that consistent subtyping is nottransitive.

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Subtyping (2/2)

Consistent subtyping

σ ≤̃ τ ⇐⇒ σ⇓ ≤ τ⇑

=⇒ Consistent subtyping reduces in linear time to semanticsubtyping!

Note: emphasizes the fact that consistent subtyping is nottransitive.

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty})

=

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})= Int ∨

?

α({Int ∨ Bool, Bool ∨ Int}) =

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty})

=

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})= Int ∨

?

α({Int ∨ Bool, Bool ∨ Int}) =

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty}) =

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})= Int ∨

?

α({Int ∨ Bool, Bool ∨ Int}) =

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty}) =

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})= Int ∨

?

α({Int ∨ Bool, Bool ∨ Int}) =

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty})

=

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})

= Int ∨

?

α({Int ∨ Bool, Bool ∨ Int}) =

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty})

=

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})= Int ∨

?

α({Int ∨ Bool, Bool ∨ Int}) =

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty})

=

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})= Int ∨ ?

α({Int ∨ Bool, Bool ∨ Int}) =

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty})

=

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})= Int ∨ ?

α({Int ∨ Bool, Bool ∨ Int}) = ???

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty})

=

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})= Int ∨ ?

α({Int ∨ Bool, Bool ∨ Int}) = ? ∨ ?

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Type Semantics: Abstraction?

What about the abstraction function?

Problem: ∧ and ∨ are connectives, not constructors.

α({Int∨Bool,Int∨Int, Int∨Empty})

=

= α({Int, Int, Int}) ∨ α({Bool, Int, Empty})= Int ∨ ?

α({Int ∨ Bool, Bool ∨ Int}) = Int ∨ Bool

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Outline

1 Motivating Example

2 Types and Subtyping

3 Function Types and Operators

4 Conclusion

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Typing Applications: Modus Ponens

We start from the usual rule:

Γ ` e1 : σ1 → τ1 Γ ` e2 : σ2 σ2 ≤̃ σ1

Γ ` e1e2 : τ1

Three components:

– Domain– Subtyping check– Result

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Typing Applications: Modus Ponens

We start from the usual rule:

Γ ` e1 : σ1 → τ1 Γ ` e2 : σ2 σ2 ≤̃ σ1

Γ ` e1e2 : τ1

Three components:– Domain

– Subtyping check– Result

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Typing Applications: Modus Ponens

We start from the usual rule:

Γ ` e1 : σ1 → τ1 Γ ` e2 : σ2 σ2 ≤̃ σ1

Γ ` e1e2 : τ1

Three components:– Domain– Subtyping check

– Result

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Typing Applications: Modus Ponens

We start from the usual rule:

Γ ` e1 : σ1 → τ1 Γ ` e2 : σ2 σ2 ≤̃ σ1

Γ ` e1e2 : τ1

Three components:– Domain– Subtyping check– Result

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Typing Applications: Modus Ponens Revisited

Problem: what is the domain of((Int→ Bool) ∧ ¬Int) ∨ (¬(Bool→ Int) ∧ (Int→ Int))?

Int

What is the result of((Int→ Bool) ∧ (Bool→ Int)) ∨ (Nat→ Nat) applied to Nat?

Nat ∨ Bool

Γ ` e1 : σ Γ ` e2 : τ τ ≤̃ d̃om(σ)

Γ ` e1e2 : σ ◦̃ τ

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Typing Applications: Modus Ponens Revisited

Problem: what is the domain of((Int→ Bool) ∧ ¬Int) ∨ (¬(Bool→ Int) ∧ (Int→ Int))?

Int

What is the result of((Int→ Bool) ∧ (Bool→ Int)) ∨ (Nat→ Nat) applied to Nat?

Nat ∨ Bool

Γ ` e1 : σ Γ ` e2 : τ τ ≤̃ d̃om(σ)

Γ ` e1e2 : σ ◦̃ τ

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Typing Applications: Modus Ponens Revisited

Problem: what is the domain of((Int→ Bool) ∧ ¬Int) ∨ (¬(Bool→ Int) ∧ (Int→ Int))?

Int

What is the result of((Int→ Bool) ∧ (Bool→ Int)) ∨ (Nat→ Nat) applied to Nat?

Nat ∨ Bool

Γ ` e1 : σ Γ ` e2 : τ τ ≤̃ d̃om(σ)

Γ ` e1e2 : σ ◦̃ τ

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Typing Applications: Modus Ponens Revisited

Problem: what is the domain of((Int→ Bool) ∧ ¬Int) ∨ (¬(Bool→ Int) ∧ (Int→ Int))?

Int

What is the result of((Int→ Bool) ∧ (Bool→ Int)) ∨ (Nat→ Nat) applied to Nat?

Nat ∨ Bool

Γ ` e1 : σ Γ ` e2 : τ τ ≤̃ d̃om(σ)

Γ ` e1e2 : σ ◦̃ τ

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Function Operators: Examples

d̃om((Int→ Int) ∧ (?→?)) =

Any since it can accept any value

d̃om((Int→ Int) ∨ (?→?)) =

Int

((?→ Int) ∧ (?→ Bool))◦̃Int =

Int ∨ Bool

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Function Operators: Examples

d̃om((Int→ Int) ∧ (?→?)) = Any since it can accept any value

d̃om((Int→ Int) ∨ (?→?)) =

Int

((?→ Int) ∧ (?→ Bool))◦̃Int =

Int ∨ Bool

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Function Operators: Examples

d̃om((Int→ Int) ∧ (?→?)) = Any since it can accept any value

d̃om((Int→ Int) ∨ (?→?)) =

Int

((?→ Int) ∧ (?→ Bool))◦̃Int =

Int ∨ Bool

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Function Operators: Examples

d̃om((Int→ Int) ∧ (?→?)) = Any since it can accept any value

d̃om((Int→ Int) ∨ (?→?)) = Int

((?→ Int) ∧ (?→ Bool))◦̃Int =

Int ∨ Bool

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Function Operators: Examples

d̃om((Int→ Int) ∧ (?→?)) = Any since it can accept any value

d̃om((Int→ Int) ∨ (?→?)) = Int

((?→ Int) ∧ (?→ Bool))◦̃Int =

Int ∨ Bool

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Function Operators: Examples

d̃om((Int→ Int) ∧ (?→?)) = Any since it can accept any value

d̃om((Int→ Int) ∨ (?→?)) = Int

((?→ Int) ∧ (?→ Bool))◦̃Int = Int ∨ Bool

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Outline

1 Motivating Example

2 Types and Subtyping

3 Function Types and Operators

4 Conclusion

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Conclusion and Future Work

Current results:+ Efficient characterization of consistent subtyping+ Sound system+ Supports polymorphism (W.I.P.)– No blame theorem, and no gradual guarantee yet

Future work:– Provide a statically-typed cast-calculus and prove the gradual

guarantee– Study blame

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Motivating Example Types and Subtyping Function Types and Operators Conclusion

Conclusion and Future Work

Current results:+ Efficient characterization of consistent subtyping+ Sound system+ Supports polymorphism (W.I.P.)– No blame theorem, and no gradual guarantee yet

Future work:– Provide a statically-typed cast-calculus and prove the gradual

guarantee– Study blame

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