CS-XXX: Graduate Programming Languages. Lecture 27 Higher-Order Polymorphism. Matthew Fluet 2012
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1 CS-XXX: Graduate Programming Languages Lecture 27 Higher-Order Polymorphism Matthew Fluet 2012
2 Looking back, looking forward Have defined System F. Metatheory (what properties does it have) What (else) is it good for How/why ML is more restrictive and implicit Recursive types (also use type variables, but differently) Existential types (dual to universal types) Next: Type operators and type-level computations Matthew Fluet CS-XXX 2012, Lecture 27 2
3 System F with Recursive and Existential Types e cbv e e ::= c x λx:τ. e e e Λα. e e [τ ] pack α. τ (τ, e) unpack e as (α, x) in e roll µα. τ (e) unroll(e) v ::= c λx:τ. e Λα. e pack α. τ (τ, v) roll µα. τ (v) (λx: τ. e b ) v a cbv e b [v a/x] e f cbv e f e f e a cbv e f ea e a cbv ea v f e a cbv v f e a (Λα. e b ) [τ a] cbv e b [τ a/α] e f cbv e f e f [τ a] cbv e f [τa] e a cbv e a pack α. τ (τ w, e a) cbv pack α. τ (τ w, e a ) e a cbv e a unpack e a as (α, x) in e b cbv unpack e a as (α, x) in e b unpack pack α. τ (τ w, v a) as (α, x) in e b cbv e b [τ w/α][v a/x] e a cbv e a unroll(e a) cbv unroll(e a ) unroll(rollµα. τ (va)) cbv va Matthew Fluet CS-XXX 2012, Lecture 27 3
4 System F with Recursive and Existential Types τ ::= int τ τ α α. τ α. τ µα. τ ::=, α Γ ::= Γ, x:τ ; Γ e : τ ; Γ c : int Γ(x) = τ ; Γ x : τ τ a ; Γ, x : τ a e b : τ r ; Γ e f : τ a τ r ; Γ e a : τ a ; Γ λx:τ a. e b : τ a τ r ; Γ e f e a : τ r, α; Γ e b : τ r ; Γ e f : α. τ r τ a ; Γ Λα. e b : α. τ r ; Γ e f [τ a] : τ r[τ a/α] ; Γ e a : τ [τ w/α] ; Γ pack α. τ (τ w, e a) : α. τ ; Γ e a : τ [(µα. τ )/α] ; Γ roll µα. τ (e a) : µα. τ ; Γ e a : α. τ, α; Γ, x:τ e b : τ r τ r ; Γ unpack e a as (α, x) in e b : τ r ; Γ e a : µα. τ ; Γ unroll(e a) : τ [(µα. τ )/α] Matthew Fluet CS-XXX 2012, Lecture 27 4
5 Goal Understand what this interface means and why it matters: type 'a list val empty : 'a list val cons : 'a -> 'a list -> 'a list val unlist : 'a list -> (' a * 'a list ) option val size : 'a list -> int val map : (' a -> 'b) -> 'a list -> 'b list Story so far: Recursive types to define list data structure Universal types to keep element type abstract in library Existential types to keep list type abstract in client But, cheated when abstracting the list type in client: considered just intlist. Matthew Fluet CS-XXX 2012, Lecture 27 5
6 (Integer) List Library with List library is an existential package: pack(µξ. unit + (int ξ), list library) as L. {empty : L; cons : int L L; unlist : L unit + (int L); map : (int int) L L;...} The witness type is integer lists: µξ. unit + (int ξ). The existential type variable L represents integer lists. List operations are monomorphic in element type (int). The map function only allows mapping integer lists to integer lists. Matthew Fluet CS-XXX 2012, Lecture 27 6
7 (Polymorphic?) List Library with / List library is a type abstraction that yields an existential package: Λα. pack(µξ. unit + (α ξ), list library) as L. {empty : L; cons : α L L; unlist : L unit + (α L); map : (α α) L L;...} The witness type is α lists: µξ. unit + (α ξ). The existential type variable L represents α lists. List operations are monomorphic in element type (α). The map function only allows mapping α lists to α lists. Matthew Fluet CS-XXX 2012, Lecture 27 7
8 Type Abbreviations and Type Operators Reasonable enough to provide list type as a (parametric) type abbreviation: L α = µξ. unit + (α ξ) replace occurrences of L τ in programs with (µξ. unit + (α ξ))[τ /α] Gives an informal notion of functions at the type-level. But, doesn t help with with list library, because this exposes the definition of list type. How modular and safe are libraries built from cpp macros? Matthew Fluet CS-XXX 2012, Lecture 27 8
9 Type Abbreviations and Type Operators Instead, provide list type as a type operator: a function from types to types L = λα. µξ. unit + (α ξ) Gives a formal notion of functions at the type-level. abstraction and application at the type-level equivalence of type-level expressions well-formedness of type-level expressions List library will be an existential package that hides a type operator, (rather than a type). Matthew Fluet CS-XXX 2012, Lecture 27 9
10 Type-level Expressions Abstraction and application at the type level makes it possible to write the same type with different syntax. Id = λα. α int bool int Id bool Id int bool Id int Id bool Id (int bool) Id (Id (int bool))... Matthew Fluet CS-XXX 2012, Lecture 27 10
11 Type-level Expressions Abstraction and application at the type level makes it possible to write the same type with different syntax. Id = λα. α int bool int Id bool Id int bool Id int Id bool Id (int bool) Id (Id (int bool))... Require a precise definition of when two types are the same: τ τ... (λα. τ b ) τ a τ b [α/τ a ]... Matthew Fluet CS-XXX 2012, Lecture 27 10
12 Type-level Expressions Abstraction and application at the type level makes it possible to write the same type with different syntax. Id = λα. α int bool int Id bool Id int bool Id int Id bool Id (int bool) Id (Id (int bool))... Require a typing rule to exploit types that are the same: ; Γ e : τ... ; Γ e : τ τ τ ; Γ e : τ... Matthew Fluet CS-XXX 2012, Lecture 27 10
13 Type-level Expressions Abstraction and application at the type level makes it possible to write the same type with different syntax. Id = λα. α int bool int Id bool Id int bool Id int Id bool Id (int bool) Id (Id (int bool))... Admits wrong/bad/meaningless types:... bool int (Id bool) int bool (Id int)... Matthew Fluet CS-XXX 2012, Lecture 27 10
14 Type-level Expressions Abstraction and application at the type level makes it possible to write the same type with different syntax. Id = λα. α int bool int Id bool Id int bool Id int Id bool Id (int bool) Id (Id (int bool))... Require a type system for types: τ :: κ... τ f :: κ a κ r τ a :: κ a... τ f τ a :: κ r Matthew Fluet CS-XXX 2012, Lecture 27 10
15 Terms, Types, and Kinds, Oh My Matthew Fluet CS-XXX 2012, Lecture 27 11
16 Terms, Types, and Kinds, Oh My Terms: e ::= c x λx:τ. e e e Λα::κ. e e [τ ] v ::= c λx:τ. e Λα::κ. e atomic values (e.g., c) and operations (e.g., e + e) compound values (e.g., (v,v)) and operations (e.g., e.1) value abstraction and application type abstraction and application classified by types (but not all terms have a type) Matthew Fluet CS-XXX 2012, Lecture 27 11
17 Terms, Types, and Kinds, Oh My Terms: e ::= c x λx:τ. e e e Λα::κ. e e [τ ] v ::= c λx:τ. e Λα::κ. e atomic values (e.g., c) and operations (e.g., e + e) compound values (e.g., (v,v)) and operations (e.g., e.1) value abstraction and application type abstraction and application classified by types (but not all terms have a type) Types: τ ::= int τ τ α α::κ. τ λα::κ. τ τ τ atomic types (e.g., int) classify the terms that evaluate to atomic values compound types (e.g., τ τ ) classify the terms that evaluate to compound values function types τ τ classify the terms that evaluate to value abstractions universal types α. τ classify the terms that evaluate to type abstractions type abstraction and application type abstractions do not classify terms, but can be applied to type arguments to form types that do classify terms classified by kinds (but not all types have a kind) Matthew Fluet CS-XXX 2012, Lecture 27 11
18 Terms, Types, and Kinds, Oh My Types: τ ::= int τ τ α α::κ. τ λα::κ. τ τ τ atomic types (e.g., int) classify the terms that evaluate to atomic values compound types (e.g., τ τ ) classify the terms that evaluate to compound values function types τ τ classify the terms that evaluate to value abstractions universal types α. τ classify the terms that evaluate to type abstractions type abstraction and application type abstractions do not classify terms, but can be applied to type arguments to form types that do classify terms classified by kinds (but not all types have a kind) Matthew Fluet CS-XXX 2012, Lecture 27 11
19 Terms, Types, and Kinds, Oh My Types: τ ::= int τ τ α α::κ. τ λα::κ. τ τ τ atomic types (e.g., int) classify the terms that evaluate to atomic values compound types (e.g., τ τ ) classify the terms that evaluate to compound values function types τ τ classify the terms that evaluate to value abstractions universal types α. τ classify the terms that evaluate to type abstractions type abstraction and application type abstractions do not classify terms, but can be applied to type arguments to form types that do classify terms classified by kinds (but not all types have a kind) Kinds κ ::= κ κ kind of proper types classify the types (that are the same as the types) that classify terms arrow kinds κ κ classify the types (that are the same as the types) that are type abstractions Matthew Fluet CS-XXX 2012, Lecture 27 11
20 Kind Examples Matthew Fluet CS-XXX 2012, Lecture 27 12
21 Kind Examples the kind of proper types Bool, Bool Bool,... Matthew Fluet CS-XXX 2012, Lecture 27 12
22 Kind Examples the kind of proper types Bool, Bool Bool,... the kind of (unary) type operators List, Maybe,... Matthew Fluet CS-XXX 2012, Lecture 27 12
23 Kind Examples the kind of proper types Bool, Bool Bool, Maybe Bool, Maybe Bool Maybe Bool,... the kind of (unary) type operators List, Maybe,... Matthew Fluet CS-XXX 2012, Lecture 27 12
24 Kind Examples the kind of proper types Bool, Bool Bool, Maybe Bool, Maybe Bool Maybe Bool,... the kind of (unary) type operators List, Maybe,... the kind of (binary) type operators Either, Map,... Matthew Fluet CS-XXX 2012, Lecture 27 12
25 Kind Examples the kind of proper types Bool, Bool Bool, Maybe Bool, Maybe Bool Maybe Bool,... the kind of (unary) type operators List, Maybe, Map Int, Either (List Bool),... the kind of (binary) type operators Either, Map,... Matthew Fluet CS-XXX 2012, Lecture 27 12
26 Kind Examples the kind of proper types Bool, Bool Bool, Maybe Bool, Maybe Bool Maybe Bool,... the kind of (unary) type operators List, Maybe, Map Int, Either (List Bool),... the kind of (binary) type operators Either, Map,... ( ) the kind of higher-order type operators taking unary type operators to proper types???,... Matthew Fluet CS-XXX 2012, Lecture 27 12
27 Kind Examples the kind of proper types Bool, Bool Bool, Maybe Bool, Maybe Bool Maybe Bool,... the kind of (unary) type operators List, Maybe, Map Int, Either (List Bool),... the kind of (binary) type operators Either, Map,... ( ) the kind of higher-order type operators taking unary type operators to proper types???,... ( ) the kind of higher-order type operators taking unary type operators to unary type operators MaybeT, ListT,... Matthew Fluet CS-XXX 2012, Lecture 27 12
28 Kind Examples the kind of proper types Bool, Bool Bool, Maybe Bool, Maybe Bool Maybe Bool,... the kind of (unary) type operators List, Maybe, Map Int, Either (List Bool), ListT Maybe,... the kind of (binary) type operators Either, Map,... ( ) the kind of higher-order type operators taking unary type operators to proper types???,... ( ) the kind of higher-order type operators taking unary type operators to unary type operators MaybeT, ListT,... Matthew Fluet CS-XXX 2012, Lecture 27 12
29 System F ω : Syntax e ::= c x λx:τ. e e e Λα::κ. e e [τ ] v ::= c λx:τ. e Λα::κ. e Γ ::= Γ, x:τ τ ::= int τ τ α α::κ. τ λα::κ. τ τ τ ::=, α::κ κ ::= κ κ New things: Types: type abstraction and type application Kinds: the types of types : kind of proper types κa κ r : kind of type operators Matthew Fluet CS-XXX 2012, Lecture 27 13
30 System F ω : Operational Semantics Small-step, call-by-value (CBV), left-to-right operational semantics: e cbv e (λx: τ. e b ) v a cbv e b [v a /x] e f cbv e f e f e a cbv e f e a e a cbv e a v f e a cbv v f e a (Λα::κ a. e b ) [τ a ] cbv e b [τ a /α] e f cbv e f e f [τ a ] cbv e f [τ a] Unchanged! All of the new action is at the type-level. Matthew Fluet CS-XXX 2012, Lecture 27 14
31 System F ω : Type System, part 1 In the context the type τ has kind κ: τ :: κ int :: (α) = κ α :: κ τ a :: τ r :: τ a τ r ::, α :: κ a τ r :: α::κ a. τ r ::, α :: κ a τ b :: κ r λα::κ a. τ b :: κ a κ r τ f :: κ a κ r τ a :: κ a τ f τ a :: κ r Should look familiar: Matthew Fluet CS-XXX 2012, Lecture 27 15
32 System F ω : Type System, part 1 In the context the type τ has kind κ: τ :: κ int :: (α) = κ α :: κ τ a :: τ r :: τ a τ r ::, α :: κ a τ r :: α::κ a. τ r ::, α :: κ a τ b :: κ r λα::κ a. τ b :: κ a κ r τ f :: κ a κ r τ a :: κ a τ f τ a :: κ r Should look familiar: the typing rules of the Simply-Typed Lambda Calculus one level up Matthew Fluet CS-XXX 2012, Lecture 27 15
33 System F ω : Type System, part 2 Definitional Equivalence of τ and τ : τ τ τ τ τ 2 τ 1 τ 1 τ 2 τ 1 τ 2 τ 2 τ 3 τ 1 τ 3 τ a1 τ a2 τ r1 τ r2 τ r1 τ r2 τ a1 τ r1 τ a2 τ r2 α::κ a. τ r1 α::κ a. τ r2 τ b1 τ b2 λα::κ a. τ b1 λα::κ a. τ b2 τ f1 τ f2 τ a1 τ a2 τ f1 τ a1 τ f2 τ a2 (λα::κ a. τ b ) τ a τ b [α/τ a ] Should look familiar: Matthew Fluet CS-XXX 2012, Lecture 27 16
34 System F ω : Type System, part 2 Definitional Equivalence of τ and τ : τ τ τ τ τ 2 τ 1 τ 1 τ 2 τ 1 τ 2 τ 2 τ 3 τ 1 τ 3 τ a1 τ a2 τ r1 τ r2 τ r1 τ r2 τ a1 τ r1 τ a2 τ r2 α::κ a. τ r1 α::κ a. τ r2 τ b1 τ b2 λα::κ a. τ b1 λα::κ a. τ b2 τ f1 τ f2 τ a1 τ a2 τ f1 τ a1 τ f2 τ a2 (λα::κ a. τ b ) τ a τ b [α/τ a ] Should look familiar: the full reduction rules of the Lambda Calculus one level up Matthew Fluet CS-XXX 2012, Lecture 27 16
35 System F ω : Type System, part 3 In the contexts and Γ the expression e has type τ : ; Γ e : τ ; Γ c : int Γ(x) = τ ; Γ x : τ τ a :: ; Γ, x : τ a e b : τ r ; Γ e f : τ a τ r ; Γ e a : τ a ; Γ λx:τ a. e b : τ a τ r ; Γ e f e a : τ r, α :: κ a; Γ e b : τ r ; Γ e f : α::κ a. τ r τ a :: κ a ; Γ Λα. e b : α::κ a. τ r ; Γ e f [τ a] : τ r[τ a/α] ; Γ e : τ τ τ τ :: ; Γ e : τ Matthew Fluet CS-XXX 2012, Lecture 27 17
36 System F ω : Type System, part 3 In the contexts and Γ the expression e has type τ : ; Γ e : τ ; Γ c : int Γ(x) = τ ; Γ x : τ τ a :: ; Γ, x : τ a e b : τ r ; Γ e f : τ a τ r ; Γ e a : τ a ; Γ λx:τ a. e b : τ a τ r ; Γ e f e a : τ r, α :: κ a; Γ e b : τ r ; Γ e f : α::κ a. τ r τ a :: κ a ; Γ Λα. e b : α::κ a. τ r ; Γ e f [τ a] : τ r[τ a/α] ; Γ e : τ τ τ τ :: ; Γ e : τ Syntax and type system easily extended with recursive and existential types. Matthew Fluet CS-XXX 2012, Lecture 27 17
37 Polymorphic List Library with higher-order List library is an existential package: pack(λα::. µξ::. unit + (α ξ), list library) as L::. {empty : α::. L α; cons : α::. α L α L α; unlist : α::. L α unit + (α L α); map : α::. β::. (α β) L α L β;...} The witness type operator is poly.lists: λα::. µξ::. unit + (α ξ). The existential type operator variable L represents poly. lists. List operations are polymorphic in element type. The map function only allows mapping α lists to β lists. Matthew Fluet CS-XXX 2012, Lecture 27 18
38 Other Kinds of Kinds Kinding systems for checking and tracking properties of type expressions: Record kinds records at the type-level; define systems of mutually recursive types Polymorphic kinds kind abstraction and application in types; System F one level up Dependent kinds dependent types one level up Row kinds describe pieces of record types for record polymorphism Power kinds alternative presentation of subtyping Singleton kinds formalize module systems with type sharing Matthew Fluet CS-XXX 2012, Lecture 27 19
39 Metatheory System F ω is type safe. Matthew Fluet CS-XXX 2012, Lecture 27 20
40 Metatheory System F ω is type safe. Preservation: Induction on typing derivation, using substitution lemmas: Term Substitution: if 1, 2 ; Γ 1, x : τ x, Γ 2 e 1 : τ and 1 ; Γ 1 e 2 : τ x, then 1, 2 ; Γ 1, Γ 2 e 1 [e 2 /x] : τ. Type Substitution: if 1, α::κ α, 2 τ 1 :: κ and 1 τ 2 :: κ α, then 1, 2 τ 1 [τ 2 /α] :: κ. Type Substitution: if τ 1 τ 2, then τ 1 [τ /α] τ 2 [τ /α]. Type Substitution: if 1, α::κ α, 2 ; Γ 1, Γ 2 e 1 : τ and 1 τ 2 :: κ α, then 1, 2 ; Γ 1, Γ 2 [τ 2 /α] e 1 [τ 2 /α] : τ. All straightforward inductions, using various weakening and exchange lemmas. Matthew Fluet CS-XXX 2012, Lecture 27 20
41 Metatheory System F ω is type safe. Progress: Induction on typing derivation, using canonical form lemmas: If ; v : int, then v = c. If ; v : τa τ r, then v = λx:τ a. e b. If ; v : α::κa. τ r, then v = Λα::κ a. e b. Complicated by typing derivations that end with: ; Γ e : τ τ τ τ :: ; Γ e : τ (just like with subtyping and subsumption). Matthew Fluet CS-XXX 2012, Lecture 27 20
42 Definitional Equivalence and Parallel Reduction Parallel Reduction of τ to τ : τ τ τ τ τ a1 τ a2 τ r1 τ r2 τ r1 τ r2 τ a1 τ r1 τ a2 τ r2 α::κ a. τ r1 α::κ a. τ r2 τ b1 τ b2 λα::κ a. τ b1 λα::κ a. τ b2 τ f1 τ f2 τ a1 τ a2 τ f1 τ a1 τ f2 τ a2 τ b τ b τ a τ a (λα::κ a. τ b) τ a τ b[α/τ a] A more computational relation. Matthew Fluet CS-XXX 2012, Lecture 27 21
43 Definitional Equivalence and Parallel Reduction Key properties: Matthew Fluet CS-XXX 2012, Lecture 27 22
44 Definitional Equivalence and Parallel Reduction Key properties: Transitive and symmetric closure of parallel reduction and type equivalence coincide: τ τ iff τ τ Matthew Fluet CS-XXX 2012, Lecture 27 22
45 Definitional Equivalence and Parallel Reduction Key properties: Transitive and symmetric closure of parallel reduction and type equivalence coincide: τ τ iff τ τ Parallel reduction has the Church-Rosser property: If τ τ 1 and τ τ 2, then there exists τ such that τ 1 τ and τ 2 τ Matthew Fluet CS-XXX 2012, Lecture 27 22
46 Definitional Equivalence and Parallel Reduction Key properties: Transitive and symmetric closure of parallel reduction and type equivalence coincide: τ τ iff τ τ Parallel reduction has the Church-Rosser property: If τ τ 1 and τ τ 2, then there exists τ such that τ 1 τ and τ 2 τ Equivalent types share a common reduct: If τ 1 τ 2, then there exists τ such that τ 1 τ and τ 2 τ Matthew Fluet CS-XXX 2012, Lecture 27 22
47 Definitional Equivalence and Parallel Reduction Key properties: Transitive and symmetric closure of parallel reduction and type equivalence coincide: τ τ iff τ τ Parallel reduction has the Church-Rosser property: If τ τ 1 and τ τ 2, then there exists τ such that τ 1 τ and τ 2 τ Equivalent types share a common reduct: If τ 1 τ 2, then there exists τ such that τ 1 τ and τ 2 τ Reduction preserves shapes: If int τ, then τ = int If τa τ r τ, then τ = τ a τ r and τ a τ a and τ r τ r If α::κ a. τ r τ, then τ = α::κ a. τ r and τ r τ r Matthew Fluet CS-XXX 2012, Lecture 27 22
48 Canonical Forms If ; v : τ a τ r, then v = λx:τ a. e b. Proof: By cases on the form of v: Matthew Fluet CS-XXX 2012, Lecture 27 23
49 Canonical Forms If ; v : τ a τ r, then v = λx:τ a. e b. Proof: By cases on the form of v: v = λx:τ a. e b. We have that v = λx:τ a. e b. Matthew Fluet CS-XXX 2012, Lecture 27 23
50 Canonical Forms If ; v : τ a τ r, then v = λx:τ a. e b. Proof: By cases on the form of v: v = c. Derivation of ; v : τ a τ r must be of the form:. ; c : int int τ 1 ; c : τ 1. ; c : τ n 1 τ n 1 τ n ; c : τ n τ n τ a τ r ; c : τ a τ r Therefore, we can construct the derivation int τ a τ r. We can find a common reduct: int τ and τ a τ r τ. Reduction preserves shape: int τ implies τ = int. Reduction preserves shape: τ a τ r τ implies τ = τ a τ r. But, τ = int and τ = τ a τ r is a contradiction. Matthew Fluet CS-XXX 2012, Lecture 27 23
51 Canonical Forms If ; v : τ a τ r, then v = λx:τ a. e b. Proof: By cases on the form of v: v = Λα::κ a. e b. Derivation of ; v : τ a τ r must be of the form:. ; Λα::κ a. e b : α::κ a. τ z α::κ a. τ z τ 1 ; Λα::κ a. e b : τ 1. ; Λα::κ a. e b : τ n 1 τ n 1 τ n ; Λα::κ a. e b : τ n τ n τ a τ r ; Λα::κ a. e b : τ a τ r Therefore, we can construct the derivation α::κ a. τ z τ a τ r. We can find a common reduct: α::κ a. τ z τ and τ a τ r τ. Reduction preserves shape: α::κ a. τ z τ implies τ = α::κ a. τ z. Reduction preserves shape: τ a τ r τ implies τ = τ a τ r. But, τ = α::κ a. τ z and τ = τ a τ r is a contradiction. Matthew Fluet CS-XXX 2012, Lecture 27 23
52 Metatheory System F ω is type safe. Where was the τ :: κ judgement used in the proof? Matthew Fluet CS-XXX 2012, Lecture 27 24
53 Metatheory System F ω is type safe. Where was the τ :: κ judgement used in the proof? In Type Substitution lemmas, but only in an inessential way. Matthew Fluet CS-XXX 2012, Lecture 27 24
54 Metatheory System F ω is type safe. Where was the τ :: κ judgement used in the proof? In Type Substitution lemmas, but only in an inessential way. After weeks of thinking about type systems, kinding seems natural; but kinding is not required for type safety! Matthew Fluet CS-XXX 2012, Lecture 27 24
55 System F ω without Kinds / System F with Type-Level Abstraction and Application e ::= c x λx:τ. e e e Λα. e e [τ ] v ::= c λx:τ. e Λα. e τ ::= int τ τ α α. τ λα. τ τ τ Γ ::= Γ, x:τ ::=, α Matthew Fluet CS-XXX 2012, Lecture 27 25
56 System F ω without Kinds / System F with Type-Level Abstraction and Application e ::= c x λx:τ. e e e Λα. e e [τ ] v ::= c λx:τ. e Λα. e τ ::= int τ τ α α. τ λα. τ τ τ Γ ::= Γ, x:τ ::=, α e cbv e (λx: τ. e b) v a cbv e b[v a/x] e f cbv e f e a cbv e a e f e a cbv e f e a v f e a cbv v f e a (Λα. e b) [τ a] cbv e b[τ a/α] e f cbv e f e f [τ a] cbv e f [τ a] Matthew Fluet CS-XXX 2012, Lecture 27 25
57 System F ω without Kinds / System F with Type-Level Abstraction and Application e ::= c x λx:τ. e e e Λα. e e [τ ] v ::= c λx:τ. e Λα. e τ ::= int τ τ α α. τ λα. τ τ τ Γ ::= Γ, x:τ ::=, α τ :: int :: α α ::, α τ b :: λα. τ b :: τ a :: τ r :: τ a τ r ::, α τ r :: α. τ r :: τ f :: τ a :: τ f τ a :: Check that free type variables of τ are in, but nothing else. Matthew Fluet CS-XXX 2012, Lecture 27 25
58 System F ω without Kinds / System F with Type-Level Abstraction and Application e ::= c x λx:τ. e e e Λα. e e [τ ] v ::= c λx:τ. e Λα. e τ ::= int τ τ α α. τ λα. τ τ τ Γ ::= Γ, x:τ ::=, α τ τ τ τ τ 2 τ 1 τ 1 τ 2 τ 1 τ 2 τ 2 τ 3 τ 1 τ 3 τ a1 τ a2 τ r1 τ r2 τ r1 τ r2 τ a1 τ r1 τ a2 τ r2 α. τ r1 α. τ r2 τ b1 τ b2 λα. τ b1 λα. τ b2 τ f1 τ f2 τ a1 τ a2 τ f1 τ a1 τ f2 τ a2 (λα. τ b) τ a τ b[α/τ a] Matthew Fluet CS-XXX 2012, Lecture 27 25
59 System F ω without Kinds / System F with Type-Level Abstraction and Application e ::= c x λx:τ. e e e Λα. e e [τ ] v ::= c λx:τ. e Λα. e τ ::= int τ τ α α. τ λα. τ τ τ Γ ::= Γ, x:τ ::=, α ; Γ e : τ ; Γ c : int Γ(x) = τ ; Γ x : τ τ a :: ; Γ, x : τ a e b : τ r ; Γ e f : τ a τ r ; Γ e a : τ a ; Γ λx:τ a. e b : τ a τ r ; Γ e f e a : τ r, α; Γ e b : τ r ; Γ e f : α. τ r τ a :: ; Γ Λα. e b : α. τ r ; Γ e f [τ a] : τ r[τ a/α] ; Γ e : τ τ τ ; Γ e : τ Matthew Fluet CS-XXX 2012, Lecture 27 25
60 System F ω without Kinds / System F with Type-Level Abstraction and Application This language is type safe. Matthew Fluet CS-XXX 2012, Lecture 27 26
61 System F ω without Kinds / System F with Type-Level Abstraction and Application This language is type safe. Preservation: Induction on typing derivation, using substitution lemmas: Term Substitution: if 1, 2 ; Γ 1, x : τ x, Γ 2 e 1 : τ and 1 ; Γ 1 e 2 : τ x, then 1, 2 ; Γ 1, Γ 2 e 1 [e 2 /x] : τ. Type Substitution: if 1, α, 2 τ 1 :: and 1 τ 2 ::, then 1, 2 τ 1 [τ 2 /α] ::. Type Substitution: if τ 1 τ 2, then τ 1 [τ /α] τ 2 [τ /α]. Type Substitution: if 1, α, 2 ; Γ 1, Γ 2 e 1 : τ and 1 τ 2 ::, then 1, 2 ; Γ 1, Γ 2 [τ 2 /α] e 1 [τ 2 /α] : τ. All straightforward inductions, using various weakening and exchange lemmas. Matthew Fluet CS-XXX 2012, Lecture 27 26
62 System F ω without Kinds / System F with Type-Level Abstraction and Application This language is type safe. Progress: Induction on typing derivation, using canonical form lemmas: If ; v : int, then v = c. If ; v : τa τ r, then v = λx:τ a. e b. If ; v : α. τr, then v = Λα. e b. Using parallel reduction relation. Matthew Fluet CS-XXX 2012, Lecture 27 26
63 Why Kinds? Why aren t kinds required for type safety? Matthew Fluet CS-XXX 2012, Lecture 27 27
64 Why Kinds? Why aren t kinds required for type safety? Recall statement of type safety: If ; e : τ, then e does not get stuck. Matthew Fluet CS-XXX 2012, Lecture 27 27
65 Why Kinds? Why aren t kinds required for type safety? Recall statement of type safety: If ; e : τ, then e does not get stuck. The typing derivation ; e : τ includes definitional-equivalence sub-derivations τ τ, which are explicit evidence that τ and τ are the same. E.g., to show that the natural type of the function expression in an application is equivalent to an arrow type:. ; Γ e f : τ f. τ f τ a τ r ; Γ e f : τ a τ r ; Γ e f e a : τ r. ; Γ e a : τ a Matthew Fluet CS-XXX 2012, Lecture 27 27
66 Why Kinds? Why aren t kinds required for type safety? Recall statement of type safety: If ; e : τ, then e does not get stuck. The typing derivation ; e : τ includes definitional-equivalence sub-derivations τ τ, which are explicit evidence that τ and τ are the same. Definitional equivalence (τ τ ) and parallel reduction (τ τ ) do not require well-kinded types (although they preserve the kinds of well-kinded types). E.g., (λα. α α) (int int) (int int) (int int) Matthew Fluet CS-XXX 2012, Lecture 27 27
67 Why Kinds? Why aren t kinds required for type safety? Recall statement of type safety: If ; e : τ, then e does not get stuck. The typing derivation ; e : τ includes definitional-equivalence sub-derivations τ τ, which are explicit evidence that τ and τ are the same. Definitional equivalence (τ τ ) and parallel reduction (τ τ ) do not require well-kinded types (although they preserve the kinds of well-kinded types). Type (and kind) erasure means that wrong/bad/meaningless types do not affect run-time behavior. Ill-kinded types can t make well-typed terms get stuck. Matthew Fluet CS-XXX 2012, Lecture 27 27
68 Why Kinds? Kinds aren t for type safety: Because a typing derivation (even with ill-kinded types), carries enough evidence to guarantee that expressions don t get stuck. Matthew Fluet CS-XXX 2012, Lecture 27 28
69 Why Kinds? Kinds aren t for type safety: Because a typing derivation (even with ill-kinded types), carries enough evidence to guarantee that expressions don t get stuck. Kinds are for type checking: Because programmers write programs, not typing derivations. Because type checkers are algorithms. Matthew Fluet CS-XXX 2012, Lecture 27 28
70 Why Kinds? Kinds are for type checking: Because programmers write programs, not typing derivations. Because type checkers are algorithms. Matthew Fluet CS-XXX 2012, Lecture 27 28
71 Why Kinds? Kinds are for type checking: Because programmers write programs, not typing derivations. Because type checkers are algorithms. Recall the statement of type checking: Given, Γ, and e, does there exist τ such that ; Γ e : τ. Matthew Fluet CS-XXX 2012, Lecture 27 28
72 Why Kinds? Kinds are for type checking: Because programmers write programs, not typing derivations. Because type checkers are algorithms. Recall the statement of type checking: Given, Γ, and e, does there exist τ such that ; Γ e : τ. Two issues: ; Γ e : τ τ τ τ :: ; Γ e : τ is a non-syntax-directed rule τ τ is a non-syntax-directed relation One non-issue: τ :: κ is a syntax-directed relation (STLC one level up ) Matthew Fluet CS-XXX 2012, Lecture 27 28
73 Type Checking for System F ω Remove non-syntax-directed rules and relations: ; Γ e : τ ; Γ c : int Γ(x) = τ ; Γ x : τ τ a :: ; Γ, x : τ a e b : τ r, α::κ a; Γ e b : τ r ; Γ λx:τ a. e b : τ a τ r ; Γ Λα. e b : α::κ a. τ r ; Γ e f : τ f τ f τ f τ f = τ fa τ fr ; Γ e a : τ a τ a τ a τ fa = τ a ; Γ e f e a : τ fr ; Γ e f : τ f τ f τ f τ f = α::κ fa. τ fr τ a :: κ a κ fa = κ a ; Γ e f [τ a] : τ fr[τ a/α] Matthew Fluet CS-XXX 2012, Lecture 27 29
74 Type Checking for System F ω Kinds are for type checking. Given, Γ, and e, does there exist τ such that ; Γ e : τ. Metatheory for kind system: Matthew Fluet CS-XXX 2012, Lecture 27 30
75 Type Checking for System F ω Kinds are for type checking. Given, Γ, and e, does there exist τ such that ; Γ e : τ. Metatheory for kind system: Well-kinded types don t get stuck. If τ :: κ and τ τ, then either τ is in (weak-head) normal form (i.e., a type-level value ) or τ τ. Proofs by Progress and Preservation on kinding and parallel reduction derivations. Matthew Fluet CS-XXX 2012, Lecture 27 30
76 Type Checking for System F ω Kinds are for type checking. Given, Γ, and e, does there exist τ such that ; Γ e : τ. Metatheory for kind system: Well-kinded types don t get stuck. If τ :: κ and τ τ, then either τ is in (weak-head) normal form (i.e., a type-level value ) or τ τ. Proofs by Progress and Preservation on kinding and parallel reduction derivations. But, irrelevant for type checking of expressions. If τ f τ f gets stuck at a type τ f that is not an arrow type, then the application typing rule does not apply and a typing derivation does not exist. Matthew Fluet CS-XXX 2012, Lecture 27 30
77 Type Checking for System F ω Kinds are for type checking. Given, Γ, and e, does there exist τ such that ; Γ e : τ. Metatheory for kind system: Well-kinded types don t get stuck. If τ :: κ and τ τ, then either τ is in (weak-head) normal form (i.e., a type-level value ) or τ τ. But, irrelevant for type checking of expressions. Matthew Fluet CS-XXX 2012, Lecture 27 30
78 Type Checking for System F ω Kinds are for type checking. Given, Γ, and e, does there exist τ such that ; Γ e : τ. Metatheory for kind system: Well-kinded types don t get stuck. If τ :: κ and τ τ, then either τ is in (weak-head) normal form (i.e., a type-level value ) or τ τ. But, irrelevant for type checking of expressions. Well-kinded types terminate. If τ :: κ, then there exists τ such that τ τ. Proof is similar to that of termination of STLC. Matthew Fluet CS-XXX 2012, Lecture 27 30
79 Type Checking for System F ω Kinds are for type checking. Given, Γ, and e, does there exist τ such that ; Γ e : τ. Metatheory for kind system: Well-kinded types don t get stuck. If τ :: κ and τ τ, then either τ is in (weak-head) normal form (i.e., a type-level value ) or τ τ. But, irrelevant for type checking of expressions. Well-kinded types terminate. If τ :: κ, then there exists τ such that τ τ. Proof is similar to that of termination of STLC. Type checking for System F ω is decidable. Matthew Fluet CS-XXX 2012, Lecture 27 30
80 Going Further This is just the tip of an iceberg. Pure type systems Why stop at three levels of expressions (terms, types, and kinds)? Allow abstraction and application at the level of kinds, and introduce sorts to classify kinds. Why stop at four levels of expressions?... For programming languages, however, three levels have proved sufficient. Matthew Fluet CS-XXX 2012, Lecture 27 31
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