The Great Type Hope Philip Wadler, Avaya Labs wadler@avaya.com - - PowerPoint PPT Presentation

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The Great Type Hope Philip Wadler, Avaya Labs wadler@avaya.com - - PowerPoint PPT Presentation

Dec 16, 2022 617 likes •1.15k views

The Great Type Hope Philip Wadler, Avaya Labs wadler@avaya.com Part I A logical coincidence Coincidences Coincidences Curry-Howard Hindley-Milner Girard-Reynolds Simply typed lambda calculus Id x 1 : A 1 , . . . , x n : A n x i : A i

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SLIDE 1

The Great Type Hope

Philip Wadler, Avaya Labs wadler@avaya.com

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SLIDE 2

Part I

A logical coincidence

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SLIDE 3

Coincidences

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Coincidences Curry-Howard Hindley-Milner Girard-Reynolds

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Simply typed lambda calculus

Id x1 : A1, . . . , xn : An ⊢ xi : Ai Γ, x : A ⊢ u : B →-I Γ ⊢ λxA. u : A → B Γ ⊢ s : A → B Γ ⊢ t : A →-E Γ ⊢ s t : B

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Simply typed lambda calculus

Id x1 : A1, . . . , xn : An ⊢ xi : Ai Γ, x : A ⊢ u : B →-I Γ ⊢ λxA. u : A → B Γ ⊢ s : A → B Γ ⊢ t : A →-E Γ ⊢ s t : B

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Polymorphic lambda calculus

Id x1 : A1, . . . , xn : An ⊢ xi : Ai Γ, x : A ⊢ u : B →-I Γ ⊢ λxA. u : A → B Γ ⊢ s : A → B Γ ⊢ t : A →-E Γ ⊢ s t : B Γ ⊢ u : B ∀2-I (X not free in Γ) Γ ⊢ ΛX. u : ∀X. B Γ ⊢ s : ∀X. B ∀2-E Γ ⊢ s A : B[X := A]

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Polymorphic lambda calculus

Id x1 : A1, . . . , xn : An ⊢ xi : Ai Γ, x : A ⊢ u : B →-I Γ ⊢ λxA. u : A → B Γ ⊢ s : A → B Γ ⊢ t : A →-E Γ ⊢ s t : B Γ ⊢ u : B ∀2-I (X not free in Γ) Γ ⊢ ΛX. u : ∀X. B Γ ⊢ s : ∀X. B ∀2-E Γ ⊢ s A : B[X := A]

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The Church numeral one

Γ ⊢ s : X → X Γ ⊢ z : X →-E s : X → X, z : X ⊢ s z : X →-I s : X → X ⊢ λzX. s z : X → X →-I ⊢ λsX→X. λzX. s z : (X → X) → X → X ∀2-I ⊢ ΛX. λsX→X. λzX. s z : ∀X. (X → X) → X → X Γ ≡ s : X → X, z : X

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The Church numeral one

Γ ⊢ s : X → X Γ ⊢ z : X →-E s : X → X, z : X ⊢ s z : X →-I s : X → X ⊢ λzX. s z : X → X →-I ⊢ λsX→X. λzX. s z : (X → X) → X → X ∀2-I ⊢ ΛX. λsX→X. λzX. s z : ∀X. (X → X) → X → X Γ ≡ s : X → X, z : X

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Products

Γ ⊢ t : A Γ ⊢ u : B ×-I Γ ⊢ (t, u) : A × B Γ ⊢ s : A × B ×-E Γ ⊢ fst s : A Γ ⊢ s : A × B Γ ⊢ snd s : B

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Products

Γ ⊢ t : A Γ ⊢ u : B ×-I Γ ⊢ (t, u) : A × B Γ ⊢ s : A × B ×-E Γ ⊢ fst s : A Γ ⊢ s : A × B Γ ⊢ snd s : B

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Products

Γ ⊢ t : A Γ ⊢ u : B ×-I Γ ⊢ (t, u) : A × B Γ ⊢ s : A × B ×-E Γ ⊢ fst s : A Γ ⊢ s : A × B Γ ⊢ snd s : B A × B ≡ ∀X. (A → B → X) → X (t, u) ≡ ΛX. λkA→B→X. k t u fst s ≡ s A (λxA. λyB. x) snd s ≡ s A (λxA. λyB. x)

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Sums

Γ ⊢ t : A +-I Γ ⊢ inl t : A + B Γ ⊢ u : B Γ ⊢ inr u : A + B Γ ⊢ s : A + B Γ, x : A ⊢ t : C Γ ⊢ y : B ⊢ u : C +-E Γ ⊢ case t of inl x → u; inr y → v : C

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Sums

Γ ⊢ t : A +-I Γ ⊢ inl t : A + B Γ ⊢ u : B Γ ⊢ inr u : A + B Γ ⊢ s : A + B Γ, x : A ⊢ t : C Γ ⊢ y : B ⊢ u : C +-E Γ ⊢ case t of inl x → u; inr y → v : C

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SLIDE 16

Sums

Γ ⊢ t : A +-I Γ ⊢ inl t : A + B Γ ⊢ u : B Γ ⊢ inr u : A + B Γ ⊢ s : A + B Γ, x : A ⊢ t : C Γ ⊢ y : B ⊢ u : C +-E Γ ⊢ case t of inl x → u; inr y → v : C A + B ≡ ∀X. (A → X) → (B → X) → X inl t ≡ ΛX. λjA→X. λkB→X. j t inr u ≡ ΛX. λjA→X. λkB→X. k u case s of inl x → t; inr y → u ≡ s C (λxA. t) (λyB. u)

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The Triumph of Type ML Haskell Java XML/XQuery Erlang?

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The Curry-Howard homeomorphism

LC'90

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Part II

Typed Erlang

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Typed Erlang

  • deftype tree(A,B) =

T when T = empty | {branch,A,B,T,T}.

  • type new() -> tree(0,0).

new() -> empty.

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Inferred type

new() -> A when empty <= A

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Simplified type

new() -> empty

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Typed Erlang

  • type insert(A,B,tree(A,B)) -> tree(A,B).

insert(K0,V0,empty) -> {branch,K0,V0,empty,empty}; insert(K0,V0,{branch,K,V,L,R}) -> if K0 < K -> {branch,K,V,insert(K0,V0,L),R}; K0 == K -> {branch,K0,V0,L,R}; true

  • >

{branch,K,V,L,insert(K0,V0,R)} end.

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Inferred type

insert(B, C, D) -> A when branchE,F,G,A <= A; branchB,C,G,H <= A; branchE,F,A,H <= A; branchB,C,empty,empty <= A; D <= empty | branchE,F,G,H; G <= empty | branchE,F,G,H; H <= empty | branchE,F,G,H; H <= D; G <= D.

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Simplified type

insert(D, E, F) -> A when empty | branchD,E,A,A <= A; F <= empty | branchD,E,F,F.

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Typed Erlang

  • type lookup(A,tree(A,B)) -> B | error

when B \ error. lookup(K0,empty) -> error; lookup(K0,{branch,K,V,L,R}) -> if K0 < K -> lookup(K0,L); K0 == K -> V; true

  • > lookup(K0,R)

end.

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SLIDE 27

Inferred type

lookup(B, C) -> A when error <= A C <= empty | branchD,E,F,G; F <= empty | branchD,E,F,G; G <= empty | branchD,E,F,G; E <= A; F <= C; G <= C.

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Simplified type

lookup(1, B) -> error | A when B <= empty | branch1, error | A, B, B; A error.

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Part III

Details

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Syntax

f, g function names c, d constructors X, Y, Z variables E ::= X expression | f(E) | c{E} | case E0 of c1{X1} → E1; · · · ; cn{Xn} → En; X → En+1 prog ::= f1(X1) → E1; · · · ; fn(Xn) → En program

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Types

c, d constructors α, β type variables U, V ::= P | U union type | R P, Q ::= c{U} prime type R ::= αcs remainder | 1cs |

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Typing rules

F; A, X : U; C ⊢ X : U (Var) F; A; C ⊢ E : U C U ⊆ V F; A; C ⊢ E : V (Sub) F; A; C ⊢ E1 : U1 . . . F; A; C ⊢ En : Un F; A; C ⊢ E : U (Multi)

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Typing rules

F, f : ∀α.(U) → V when D; A; C, D[V /α] ⊢ f : ((U) → V )[V /α] (Fun) F; A; C ⊢ f : (U) → V F; A; C ⊢ E : U F; A; C ⊢ f(E) : V (Call) F, f : ((U) → V when C); X : U; C ⊢ E : V FTV((U) → V when C) = α F; ∅; C ⊢ f(X) → E : (∀α.(U) → V when C) (Def)

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Typing rules

F; A; C ⊢ E : U F; A; C ⊢ c{E} : c{U} (Con) F; A; C ⊢ E0 : c1{U1} | . . . | cn{Un} | U F; A, X1 : U1; C ⊢ E1 : V . . . F; A, Xn : Un; C ⊢ En : V F; A, X : U; C ⊢ En+1 : V F; A; C ⊢ (case E0 of c1{X1} → E1; . . . cn{Xn} → En; X → En+1 end) : V (Case)

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Constraint reduction

P | U ⊆ V ⇒ P ⊆ V, U ⊆ V 0 ⊆ U ⇒ none 1cs ⊆ 0 ⇒ fail 1cs ⊆ c{U} | U ⇒ 1 ⊆ U, 1cs ⊆ U if c / ∈ cs 1cs ⊆ U

  • therwise

1cs ⊆ 1ds ⇒ none if ds ⊆ cs fail

  • therwise

1cs ⊆ αds ⇒ 1cs ⊆ αds if ds ⊆ cs fail

  • therwise
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Constraint reduction

c{U} ⊆ 0 ⇒ fail c{U} ⊆ c′{U

′} | U ⇒ U ⊆ U ′

if c = c′ c{U} ⊆ U

  • therwise

c{U} ⊆ 1cs ⇒ none if c / ∈ cs fail

  • therwise

c{U} ⊆ αcs ⇒ c{U} ⊆ αcs if c / ∈ cs fail

  • therwise

U ⊆ αcs, αcs ⊆ V ⇒ U ⊆ V, U ⊆ αcs, αcs ⊆ V

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Part IV

A fly in the ointment

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And

  • datatype bool() = true | false.
  • type and(bool(),bool()) -> bool().

and(true,true) -> true; and(false,X)

  • > false;

and(X,false)

  • > false.
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Uh oh

  • type and(1,false) -> false | true.

and(X,Y) -> let Z = (case Y of false -> false end) in case X of true

  • >

case Y of true -> true; X -> Z end; false -> false; X -> Z end.

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Part V

A simpler approach?

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Typed Erlang, simplified

  • deftype tree(A,B) =

empty | {branch,A,B,T,T}.

  • type new() -> tree(A,B).

new() -> empty.

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Typed Erlang

  • type insert(A,B,tree(A,B)) -> tree(A,B).

insert(K0,V0,empty) -> {branch,K0,V0,empty,empty}; insert(K0,V0,{branch,K,V,L,R}) -> if K0 < K -> {branch,K,V,insert(K0,V0,L),R}; K0 == K -> {branch,K0,V0,L,R}; true

  • >

{branch,K,V,L,insert(K0,V0,R)} end.

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SLIDE 43

Typed Erlang

  • deftype sum(A,B) =

inl(A) | inr(B).

  • deftype error =

error

  • type lookup(A,tree(A,B)) -> inl(B) | inr(error)

lookup(K0,empty) -> inr(error); lookup(K0,{branch,K,V,L,R}) -> if K0 < K -> lookup(K0,L); K0 == K -> inl(V); true

  • > lookup(K0,R)

end.

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Part VI

A simpler but more powerful approach?

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Types and logic

s ∈ A → B ≡ ∀x. x ∈ A → s x ∈ B

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Retrofitting types

  • type lookup(A,tree(A,B)) -> B | error

when B \ error. lookup(K0,empty) -> error; lookup(K0,{branch,K,V,L,R}) -> if K0 < K -> lookup(K0,L); K0 == K -> V; true

  • > lookup(K0,R)

end.

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Retrofitting types

  • assert

K in A & T in tree(A,B) & V = lookup(K,T) & not (error in B)

  • > V in B \/ V in error.

lookup(K0,empty) -> error; lookup(K0,{branch,K,V,L,R}) -> if K0 < K -> lookup(K0,L); K0 == K -> V; true

  • > lookup(K0,R)

end.

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Part VII

Conclusions

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Conclusions Types are good Erlang is good Typed Erlang could be better

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SLIDE 50

Conclusions Types are good Erlang is good Typed Erlang could be better

Long live λ calculus!

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Further reading

Simon Marlow and Philip Wadler, A practical subtyping system for Erlang, 2’nd International Conference on Functional Programming, Amsterdam, June 1997. Philip Wadler, New Languages, Old Logic, Dr Dobbs Journal, special supplement on Software in the 21st century, December 2000. (See also, 19th century logic and 21st century computing, on my web page.) Philip Wadler, The Girard-Reynolds isomorphism, Theoretical Aspects of Computer Software Sendai, Japan, October 2001. Journal version to appear in Information and Computation.