Type assignment
for intersections and unions
in call-by-value languages
Joshua Dunfield and Frank Pfenning
Triple Project
Carnegie Mellon University
8 April 2003
FOSSACS ’03, Warsaw, Poland
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.1/41
Outline
Motivation
Language
Datasort refinements
Index refinements
Intersections
Indefinite types
Related work
Conclusion & future work
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.2/41
Motivation
Conventional type systems: ML, Java, . . .
Tractable and easy to use
Limited expressive power
Fully dependent type systems: Nuprl, . . .
Undecidable
Extremely expressive
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.3/41
Motivation
Conventional type systems: ML, Java, . . .
Tractable and easy to use
Limited expressive power
Refined type systems:
Tractable and easy to use
Better expressive power
Fully dependent type systems: Nuprl, . . .
Undecidable
Extremely expressive
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.3/41
Push the envelope
The goal of the Triple Project: create refined type
systems:
Two major subareas:
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.4/41
Push the envelope
The goal of the Triple Project: create refined type
systems:
Two major subareas:
Ephemeral refinements
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.4/41
Push the envelope
The goal of the Triple Project: create refined type
systems:
Two major subareas:
Ephemeral refinements
Persistent refinements
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.4/41
Push the envelope
The goal of the Triple Project: create refined type
systems:
Two major subareas:
Ephemeral refinements
Persistent refinements
The present work: Persistent refinements of
algebraic datatypes
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.4/41
Type Assignment
A type assignment system à la Curry
Terms do not contain types
Undecidable
A bidirectional system
Terms contain some type annotations
Two judgments: inference e ↑ A, checking e ↓ A.
This paper: Type assignment only
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.5/41
Fun With Types
Some types build things
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.6/41
Fun With Types
Some types build things
Example: tuples
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.6/41
Fun With Types
Some types build things
Example: tuples
Others talk about things
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.6/41
Fun With Types
Some types build things
Example: tuples
Others talk about things
Example: parametric polymorphism ∀α. α → α
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.6/41
Fun With Types
Some types build things
Example: tuples
Others talk about things
Example: parametric polymorphism ∀α. α → α
∀ describing behavior of the identity fn
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.6/41
Fun With Types
Some types build things
Example: tuples
Others talk about things
Example: parametric polymorphism ∀α. α → α
∀ describing behavior of the identity fn
We call the latter property types
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.6/41
Property Types
In our system:
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.7/41
Property Types
In our system:
δ(i): Datasort and index refinement
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.7/41
Property Types
In our system:
δ(i): Datasort and index refinement
∧: Intersection: v : A ∧ B means v has type A and
type B
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.7/41
Property Types
In our system:
δ(i): Datasort and index refinement
∧: Intersection: v : A ∧ B means v has type A and
type B
>: Greatest type (0-ary ∧)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.7/41
Property Types
In our system:
δ(i): Datasort and index refinement
∧: Intersection: v : A ∧ B means v has type A and
type B
>: Greatest type (0-ary ∧)
Π: Universal quantifier over indices (infinitary ∧)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.7/41
Property Types
In our system:
δ(i): Datasort and index refinement
∧: Intersection: v : A ∧ B means v has type A and
type B
>: Greatest type (0-ary ∧)
Π: Universal quantifier over indices (infinitary ∧)
∨: Union: v : A ∨ B means v has type A or type B
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.7/41
Property Types
In our system:
δ(i): Datasort and index refinement
∧: Intersection: v : A ∧ B means v has type A and
type B
>: Greatest type (0-ary ∧)
Π: Universal quantifier over indices (infinitary ∧)
∨: Union: v : A ∨ B means v has type A or type B
⊥: Empty type (0-ary ∨)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.7/41
Property Types
In our system:
δ(i): Datasort and index refinement
∧: Intersection: v : A ∧ B means v has type A and
type B
>: Greatest type (0-ary ∧)
Π: Universal quantifier over indices (infinitary ∧)
∨: Union: v : A ∨ B means v has type A or type B
⊥: Empty type (0-ary ∨)
Σ: Existential quantifier over indices (infinitary ∨)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.7/41
Initial Language
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.8/41
Language + Initial Typing
A, B, C, D ::= 1 | A → B
e ::= x | f | () | λx. e | e1 (e2 ) | fix f. e
A2 ≤ B 2
B1 ≤ A 1
(→)
A1 → A 2 ≤ B 1 → B 2
Γ (x) = A
(var)
Γ `x:A
1 ≤ 1
Γ (f) = A
(fixvar)
Γ `f:A
Γ ` e1 : A → B Γ ` e 2 : A
(→E)
Γ ` e1 (e2 ) : B
Γ, x:A ` e : B (→I)
Γ ` λx. e : A → B
(1)
Γ `e:A A ≤ B
(sub)
Γ `e:B
Γ, f:A ` e : A (fix)
Γ ` fix f. e : A
Γ ` () : 1
(1I)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.9/41
cbv Semantics
Values
v ::= x | () | λx. e
Evaluation contexts E ::= [] | E(e) | v(E)
e0 →
7 R e 00
E[e 0 ] →
7
E[e 00 ]
(λx. e) v 7→R [v/x] e
fix f. e 7→R [fix f. e / f] e
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.10/41
Key Properties
A practical type system should satisfy
Preservation: If ` e : A and e 7→ e 0 then ` e 0 : A.
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.11/41
Key Properties
A practical type system should satisfy
Preservation: If ` e : A and e 7→ e 0 then ` e 0 : A.
Progress: If ` e : A then e is a value or ∃e 0 . e 7→ e 0 .
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.11/41
Datasort Refinements
a.k.a. refinement types (Freeman, Pfenning, Davies)
Refine an algebraic datatype by a datasort δ
Example: Lists of integers
Nil : 1 → even
Nil : 1 → list
Cons : int ∗ list → list
Cons : (int ∗ odd → even)
∧ (int ∗ even → odd)
∧ (int ∗ list → list)
δ:
list
>
odd
Z
}
Z
Z
even
Intersections essential
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.12/41
Index Refinements
a.k.a. dependent types restricted to a decidable
constraint domain (Xi & Pfenning)
Refine an algebraic datatype by an index
Indices drawn from any decidable constraint domain,
here N
Example: Lists indexed by their length
Nil : 1 → list(0)
Nil : 1 → list
Cons : Πa:N . int∗list(a) → list(a + 1)
Cons : int ∗ list → list
Example:
append : Πa:N . Πb:N . list(a) ∗ list(b) → list(a+b)
Universal quantifier Π essential
Existential quantifier Σ also essential
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.13/41
Intersection Types
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.14/41
Intersection Types
Typing:
Γ ` e : A1 Γ ` e : A2
(∧I)
Γ ` e : A 1 ∧ A2
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.15/41
Intersection Types
Typing:
Γ ` e : A1 Γ ` e : A2
(∧I)
Γ ` e : A 1 ∧ A2
Unsound with mutable references.
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.15/41
Intersection Types
Typing:
Γ ` e : A1 Γ ` e : A2
(∧I)
Γ ` e : A 1 ∧ A2
Unsound with mutable references.
Γ ` : A1 Γ ` : A 2
(∧I)
Γ ` : A 1 ∧ A2
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.15/41
Intersection Types
Typing:
Γ ` e : A1 Γ ` e : A2
(∧I)
Γ ` e : A 1 ∧ A2
Unsound with mutable references.
Γ ` v : A1 Γ ` v : A2
(∧I)
Γ ` v : A 1 ∧ A2
The fix: a value restriction [Davies & Pfenning ’00].
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.15/41
Intersection Types
Subtyping:
A ≤ B1 A ≤ B2
(∧R)
A ≤ B 1 ∧ B2
A1 ≤ B
A2 ≤ B
(∧L1 )
(∧L2 )
A1 ∧ A2 ≤ B
A1 ∧ A2 ≤ B
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.16/41
Intersection Types
Subtyping:
A ≤ B1 A ≤ B2
(∧R)
A ≤ B 1 ∧ B2
A1 ≤ B
A2 ≤ B
(∧L1 )
(∧L2 )
A1 ∧ A2 ≤ B
A1 ∧ A2 ≤ B
Distributivity?
(A → B) ∧ (A → B 0 ) ≤ A → (B ∧ B 0 )
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.16/41
Intersection Types
Subtyping:
A ≤ B1 A ≤ B2
(∧R)
A ≤ B 1 ∧ B2
A1 ≤ B
A2 ≤ B
(∧L1 )
(∧L2 )
A1 ∧ A2 ≤ B
A1 ∧ A2 ≤ B
Distributivity?
(A → B) ∧ (A → B 0 ) ≤ A → (B ∧ B 0 )
Again unsound with mutable references. [Davies &
Pfenning ’00]
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.16/41
Indefinite Types
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.17/41
Motivation
filter : (int→bool) → list → list
filter : Πa:N . (int→bool) → list(a) → list( )
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.18/41
Motivation
filter : (int→bool) → list → list
filter : Πa:N . (int→bool) → list(a) → list( )
filter : Πa:N . (int→bool) → list(a) → (Σb:N . list(b))
Σb:N . B quantifies existentially over b in B
[Xi & Pfenning]
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.18/41
Indefinite Types: Motivation
Definitely need Σ
The binary analogue of Σ is the union type ∨
Need ∨? Possibly just a convenience
Issues similar for both, but easier to talk about ∨
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.19/41
Union Types
Subtyping for ∨ just the dual of ∧:
A1 ≤ B A 2 ≤ B
(∨L)
A1 ∨A2 ≤ B
A ≤ B1
(∨R1 )
A ≤ B1 ∨B2
A ≤ B2
(∨R2 )
A ≤ B1 ∨B2
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.20/41
Union Types
Subtyping for ∨ just the dual of ∧:
A1 ≤ B A 2 ≤ B
(∨L)
A1 ∨A2 ≤ B
A ≤ B1
(∨R1 )
A ≤ B1 ∨B2
A ≤ B2
(∨R2 )
A ≤ B1 ∨B2
Introduction rules straightforward:
Γ ` e : A (∨I )
1
Γ `e:A∨B
Γ ` e : B (∨I )
2
Γ `e:A∨B
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.20/41
Union Types
Subtyping for ∨ just the dual of ∧:
A1 ≤ B A 2 ≤ B
(∨L)
A1 ∨A2 ≤ B
A ≤ B1
(∨R1 )
A ≤ B1 ∨B2
A ≤ B2
(∨R2 )
A ≤ B1 ∨B2
Introduction rules straightforward:
Γ ` e : A (∨I )
1
Γ `e:A∨B
Γ ` e : B (∨I )
2
Γ `e:A∨B
Elimination rule?
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.20/41
Rough Road
Recall: for ∧, introduction rule (∧I) was tricky
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.21/41
Rough Road
Recall: for ∧, introduction rule (∧I) was tricky
Duality suggests (∨E) will be interesting too
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.21/41
∨ Elimination
Want to reason by cases: For x : A ∨ B, show for
x : A and for x : B.
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.22/41
∨ Elimination
Want to reason by cases: For x : A ∨ B, show for
x : A and for x : B.
Substitution approach (e.g. [MacQueen et al.’86]),
reminiscent of natural deduction:
Γ, x:A ` e : C
Γ ` e 0 : A ∨ B Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.22/41
∨ Elimination
Want to reason by cases: For x : A ∨ B, show for
x : A and for x : B.
Substitution approach (e.g. [MacQueen et al.’86]),
reminiscent of natural deduction:
Γ, x:A ` e : C
Γ ` e 0 : A ∨ B Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Suppose Γ ` h : (A → C) ∧ (B → C), Γ ` v : A ∨ B
Γ `v:A∨B
Γ, x:A ` f x : C
Γ `fv:C
Γ, x:B ` f x : C
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.22/41
∨ Elimination
Γ ` e0 : A ∨ B
Γ, x:A ` e : C
Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Consider a nondeterministic choice e1 ⊕ e2 :
e1 ⊕ e2 7→ e1
e1 ⊕ e2 7→ e2
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.23/41
∨ Elimination
Γ ` e0 : A ∨ B
Γ, x:A ` e : C
Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Consider a nondeterministic choice e1 ⊕ e2 :
e1 ⊕ e2 7→ e1
e1 ⊕ e2 7→ e2
Suppose Γ ` f : (A → A → C) ∧ (B → B → C),
Γ ` y : A, Γ ` z : B.
Γ ` y ⊕ z : A ∨ B Γ, x:A ` f x x : C Γ, x:B ` f x x : C
Γ ` f (y ⊕ z) (y ⊕ z)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.23/41
∨ Elimination
Γ ` e0 : A ∨ B
Γ, x:A ` e : C
Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Consider a nondeterministic choice e1 ⊕ e2 :
e1 ⊕ e2 7→ e1
e1 ⊕ e2 7→ e2
Suppose Γ ` f : (A → A → C) ∧ (B → B → C),
Γ ` y : A, Γ ` z : B.
Γ ` y ⊕ z : A ∨ B Γ, x:A ` f x x : C Γ, x:B ` f x x : C
Γ ` f (y ⊕ z) (y ⊕ z)
f (y ⊕ z) (y ⊕ z) 7→ f y (y ⊕ z) 7→ f y z
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.23/41
∨ Elimination
Γ ` e0 : A ∨ B
Γ, x:A ` e : C
Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Consider a nondeterministic choice e1 ⊕ e2 :
e1 ⊕ e2 7→ e1
e1 ⊕ e2 7→ e2
Suppose Γ ` f : (A → A → C) ∧ (B → B → C),
Γ ` y : A, Γ ` z : B.
Γ ` y ⊕ z : A ∨ B Γ, x:A ` f x x : C Γ, x:B ` f x x : C
Γ ` f (y ⊕ z) (y ⊕ z)
f (y ⊕ z) (y ⊕ z) 7→ f y (y ⊕ z) 7→ f y z
Unsound whenever a term can evaluate to more than
one value
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.23/41
∨ Elimination
Γ ` e0 : A ∨ B
Γ, x:A ` e : C
Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Consider a nondeterministic choice e1 ⊕ e2 :
e1 ⊕ e2 7→ e1
e1 ⊕ e2 7→ e2
Suppose Γ ` f : (A → A → C) ∧ (B → B → C),
Γ ` y : A, Γ ` z : B.
Γ ` y ⊕ z : A ∨ B Γ, x:A ` f x x : C Γ, x:B ` f x x : C
Γ ` f (y ⊕ z) (y ⊕ z)
f (y ⊕ z) (y ⊕ z) 7→ f y (y ⊕ z) 7→ f y z
Unsound whenever a term can evaluate to more than
one value
Hence unsound with mutable references
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.23/41
∨ Elimination
Γ ` e0 : A ∨ B
Γ, x:A ` e : C
Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Restrict to exactly one occurrence of e 0 ?
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.24/41
∨ Elimination
Γ ` e0 : A ∨ B
Γ, x:A ` e : C
Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Restrict to exactly one occurrence of e 0 ?
Can fail if e 0 inside a λ (see paper)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.24/41
∨ Elimination
Γ ` e0 : A ∨ B
Γ, x:A ` e : C
Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Restrict to exactly one occurrence of e 0 ?
Can fail if e 0 inside a λ (see paper)
Restrict e 0 to a value? [van Bakel ’99]
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.24/41
∨ Elimination
Γ ` e0 : A ∨ B
Γ, x:A ` e : C
Γ, x:B ` e : C
Γ ` [e 0 /x] e : C
Restrict to exactly one occurrence of e 0 ?
Can fail if e 0 inside a λ (see paper)
Restrict e 0 to a value? [van Bakel ’99]
Sound—but still must guess occurrences of e 0
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.24/41
∨ Elimination: Our Solution
Γ ` e0 : A ∨ B
Γ`
Γ, x:A ` E[x] : C
Γ, x:B ` E[x] : C
E[e 0 ]
:C
(∨E)
Exactly one e 0 , in evaluation position
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.25/41
∨ Elimination: Our Solution
Γ ` e0 : A ∨ B
Γ`
Γ, x:A ` E[x] : C
Γ, x:B ` E[x] : C
E[e 0 ]
:C
(∨E)
Exactly one e 0 , in evaluation position
So e 0 is the next thing to be evaluated
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.25/41
∨ Elimination: Our Solution
Γ ` e0 : A ∨ B
Γ`
Γ, x:A ` E[x] : C
Γ, x:B ` E[x] : C
E[e 0 ]
:C
(∨E)
Exactly one e 0 , in evaluation position
So e 0 is the next thing to be evaluated
Remainder of computation (E) on the result of e 0
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.25/41
The Empty Type ⊥
⊥ is the empty or void type
⊥ ≤ A
(⊥L)
Elimination rule: by analogy with (∨E)
Γ ` e 0 : ⊥ (⊥E)
Γ ` E[e 0 ] : C
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.26/41
A Generalization?
Must we restrict the rules to an evaluation context?
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.27/41
A Generalization?
Must we restrict the rules to an evaluation context?
Or could we also allow cases where e 0 must be
evaluated exactly once but might not be in evaluation
position?
Γ ` e 0 : ⊥ e 0 eval’d exactly once in e
Γ `e:C
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.27/41
A Generalization?
Must we restrict the rules to an evaluation context?
Or could we also allow cases where e 0 must be
evaluated exactly once but might not be in evaluation
position?
Γ ` e 0 : ⊥ e 0 eval’d exactly once in e
Γ `e:C
. . . let ω = (fix f. f). Then ω : ⊥.
Γ `ω:⊥
Γ ` (()())ω : C
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.27/41
A Generalization?
Must we restrict the rules to an evaluation context?
Or could we also allow cases where e 0 must be
evaluated exactly once but might not be in evaluation
position?
Γ ` e 0 : ⊥ e 0 eval’d exactly once in e
Γ `e:C
. . . let ω = (fix f. f). Then ω : ⊥.
Γ `ω:⊥
Γ ` (()())ω : C
But ()() is not a value and does not step
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.27/41
Type Safety
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.28/41
Type Safety
Γ ` e0 : A ∨ B
Γ`
Γ, x:A ` E[x] : C
Γ, x:B ` E[x] : C
E[e 0 ]
:C
(∨E)
X
Preservation and progress hold with
δ, i, ∧, >, Π, ∨, ⊥, Σ
(Fully written out for δ, i, ∧, Π in [Dunfield ’02])
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.29/41
Closely Related Work
Davies ’97, ’00*
δ, ∧; ↑↓
Xi ’98, ’99*, ’00
i, Π, Σ; ↑↓
@
@
@
R
Dunfield ’02
δ, i, ∧, Π; ↑↓
?
This paper*
δ, i, ∧, Π, >, ∨, Σ, ⊥;
:
Ongoing work
δ, i, ∧, Π, >, ∨, Σ, ⊥;
↑↓
?
* & Pfenning
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.30/41
Related Work
[Coppo et al.’81]: ∧ can characterize normal forms
(termination); hence undecidable
[Reynolds ’96]: F ORSYTHE with ∧ (& type
annotations)
[Pierce ’91]: Language with ∧, ∨, syntactic markers
[Barbanera et al.’95]: With unrestricted ∨-elim rule,
types not preserved under β-reduction
[Wells et al.’02], [Palsberg & Pavlopoulou ’01]: ∧, ∨
for control flow analysis
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.31/41
Conclusion
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.32/41
Summary
Goal: express more invariants
Two especially useful refinements already developed:
Datasort refinements
Index refinements
Refinements motivate ∧, ∨, Π, Σ
(∧I): Value restriction
(∨E): Eval. context restriction
Type safety
The present system: Undecidable, hence impractical
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.33/41
Future Work
Bidirectional formulation X
Let-normal form X
Mutable references
Implementation
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.34/41
Acknowledgments
National Science Foundation: CCR-0204248: Type
Refinements; 0121633: ITR/SY+SI: Language
Technology for Trustless Software Dissemination;
Graduate Research Fellowship
Feedback on the paper: Brigitte Pientka and the
anonymous referees
Feedback on the talk: Aleksey Kliger, Stephen Magill,
Jonathan Moody, Frank Pfenning
“ ”: Henrik Christian Grove, available at a CTAN
mirror near you
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.35/41
The End
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.36/41
Formulating Refinements
.
P ::= ⊥ | i = j | . . .
Γ ::= · | Γ, x:A | Γ, a:γ | Γ, P
e ::= . . . | c(e) | case e of ms
v ::= . . . | c(v)
E ::= . . . | c(E) | case E of ms
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.37/41
Formulating Refinements
.
P ::= ⊥ | i = j | . . .
Γ ::= · | Γ, x:A | Γ, a:γ | Γ, P
e ::= . . . | c(e) | case e of ms
v ::= . . . | c(v)
E ::= . . . | c(E) | case E of ms
Atomic subtyping
.
δ1 δ 2 Γ ` i = j
(δ)
Γ ` δ1 (i) ≤ δ2 (j)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.37/41
Formulating Refinements
.
P ::= ⊥ | i = j | . . .
Γ ::= · | Γ, x:A | Γ, a:γ | Γ, P
e ::= . . . | c(e) | case e of ms
v ::= . . . | c(v)
E ::= . . . | c(E) | case E of ms
Atomic subtyping
.
δ1 δ 2 Γ ` i = j
(δ)
Γ ` δ1 (i) ≤ δ2 (j)
Γ ` c : A → δ(i) Γ ` e : A
(δI)
Γ ` c(e) : δ(i)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.37/41
Formulating Refinements
Atomic subtyping
.
δ1 δ 2 Γ ` i = j
(δ)
Γ ` δ1 (i) ≤ δ2 (j)
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.38/41
Evaluation Order
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.39/41
Evaluation Order
Binary and 0-ary indefinite type elim rules:
Γ, x:A ` E[x] : C
0 :⊥
Γ ` e 0 : A ∨ B Γ, x:B ` E[x] : C
Γ
`
e
(⊥E)
(∨E)
0
0
Γ ` E[e ] : C
Γ ` E[e ] : C
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.40/41
Evaluation Order
Binary and 0-ary indefinite type elim rules:
Γ, x:A ` E[x] : C
0 :⊥
Γ ` e 0 : A ∨ B Γ, x:B ` E[x] : C
Γ
`
e
(⊥E)
(∨E)
0
0
Γ ` E[e ] : C
Γ ` E[e ] : C
A corresponding unary rule?
Γ ` e 0 : A Γ, x:A ` E[x] : C
(direct)
0
Γ ` E[e ] : C
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.40/41
Evaluation Order
Binary and 0-ary indefinite type elim rules:
Γ, x:A ` E[x] : C
0 :⊥
Γ ` e 0 : A ∨ B Γ, x:B ` E[x] : C
Γ
`
e
(⊥E)
(∨E)
0
0
Γ ` E[e ] : C
Γ ` E[e ] : C
A corresponding unary rule?
Γ ` e 0 : A Γ, x:A ` E[x] : C
(direct)
0
Γ ` E[e ] : C
Actually a derivable rule: (∨E) with A ∨ A
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.40/41
Evaluation Order
Binary and 0-ary indefinite type elim rules:
Γ, x:A ` E[x] : C
0 :⊥
Γ ` e 0 : A ∨ B Γ, x:B ` E[x] : C
Γ
`
e
(⊥E)
(∨E)
0
0
Γ ` E[e ] : C
Γ ` E[e ] : C
A corresponding unary rule?
Γ ` e 0 : A Γ, x:A ` E[x] : C
(direct)
0
Γ ` E[e ] : C
Actually a derivable rule: (∨E) with A ∨ A
But not derivable in a bidirectional system
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.40/41
Evaluation Order
Γ ` e 0 : A Γ, x:A ` E[x] : C
(direct)
0
Γ ` E[e ] : C
Rule needed so that Π’s can be instantiated with
existential Σ index variables
Problem first recognized by Xi
Formulated a translation to let-normal form
Added “good” programs, but removed them too
Current work: a sound, complete let-normal
translation
Type assignment for intersections and unions in call-by-value languages—FOSSACS ’03 – p.41/41