A Polymorphic Type and System for Multi-Staged Exceptions

Transcription

1 A Polymorphic Type System for Multi-Staged Exceptions Seoul National University 08/04/2006 This is a joint work with In-Sook Kim and Kwangkeun Yi

2 Outlie 1. Introduction and Examples 2. Operational Semantics 3. Monomorphic Type System (APLAS 2006) 4. Polymorphic Type System 5. Conclusion

3 Introduction and Examples

4 Multi-Staged Languages Macros, partial evaluation, code generation, etc. Normal computation (at stage 0) λ-calculus Code composition (at stage > 0) quasi-quote in Lisp backquote ( ): create code template comma (,): code substitution eval (eval): execute code template let x = 1 let y = (,x+2) in eval y

5 Exceptions Control diverter Raised exceptions can escape control structures Error handler When error occurs, we raise an exception. Then, handlers can catch the raised exception to handle it. Possible safety hole Uncaught exceptions cause abnormal termination of programs fun find [] x = raise NotFound find h::t x = if h=x then raise Found else find t x find [1,2,3] 2 handle Found => true NotFound => false

6 Exceptions in Multi-Staged Languages Restriction exceptions must be raised and handled only at stage 0 Most interesting feature exceptions raised during code composition can be raised and handled at stage 0 can cross stages upwards by comma(,) and downwards by backquote( )

7 Staged Exception Examples (1/4) fun g [] = 1 g x::r = (,x *,(g r)) fun f ls = (a *,(g ls)) input: 2:: 0:: 3::[] output: (a * 2 * 0 * 3 * 1)

8 Staged Exception Examples (2/4) Raise exception Zero when input has 0 fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) fun f ls = (a *,(g ls)) input: 2:: 0:: 3::[] output: uncaught exception Zero

9 Staged Exception Examples (3/4) We can handle Zero at stage 0 in the code composition fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) fun f ls = (a *,((g ls) handle Zero => 0)) input: 2:: 0:: 3::[] output: (a * 0)

10 Staged Exception Examples (4/4) Or, we can handle Zero at stage 0 outside the code composition fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) fun f ls = (a *,(g ls)) handle Zero => 0 input: 2:: 0:: 3::[] output: 0

11 Goal A static type system that supports Lisp/Scheme s quasi-quote operators and exception facilities

12 Idea effect E: set of possible uncaught exceptions expression e has effect E and c E means that the expression may raise uncaught exception c. code s type: annotated with latent effect evaluation of code (raise c) : (Γ A, {c}), eval (raise c) : A, {c}

13 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 handler at stage 0 f : ( int, ) list ({a : int} int, ), fun f ls = (a *,(g ls) handle Zero => 0) handler at stage 1 f : ( int, ) list {Zero} ({a : int} int, ),

14 Operational Semantics

15 Language e Exp ::= i c x λx.e e 1e 2 box e code template e unbox k e code substitution, e eval e code execution eval e raise e exception raise handle e 1 c e 2 exception handle Evaluation where e n r n: a stage number r: a value v or raised exception c

16 Operational Semantics Exceptions must be raised and handled only at stage 0 Normal computations (at stage 0) and Propagation of code compositions (at stage n > 0) raised exceptions (ERAISE) (EHANDLE) e 0 c raise e 0 c n e v e raise e n (n > 0) raise v e 1 0 v handle e 1 c e 2 0 v e 1 0 c e 2 0 v handle e 1 c e 2 0 v e 1 0 c handle e 1 c e 2 0 c n c raise e n (n 0) c n e 1 c n (n > 0) handle e 1 c e 2 c n e 1 c handle e 1 c n (n > 0) e 2 c n e 2 c n (n > 0) handle e 1 c e 2 c

17 Operational Semantics Exceptions can cross stages upwards or downwards Normal computations (at stage 0) and Propagation of code compositions (at stage n > 0) raised exceptions (EBOX) (EUNBOX) (EEVAL) e n+1 v e box e n (n 0) box v e 0 box v unbox k e n (n = k > 0) v v e unbox k e n (n > k > 0) unbox k v e n k e 0 box v 1 v 1 0 v 0 eval e 0 v 0 n e v e eval e n (n > 0) eval v n+1 c box e n (n 0) c n k c unbox k e n (n k > 0) c n c eval e n (n 0) c

18 Monomorphic Type System

19 Type and effect A, B Type ::= int exn(e) A E B (Γ A, E) E Effects = 2 Exn c Exn = set of exception names Typing judgment Γ 0 Γ n e : A, E n E 0

20 Typing rules c E Γ 0 Γ n c : exn(e), E n 1 E 0 (TEXN) Γ 0 Γ n e : exn(e), E n E n 1 E 0 Γ 0 Γ n raise e : A, (E E n )E n 1 E 0 (TRAISE) Γ 0 Γ n e 1 : A, E n E n 1 E 0 Γ 0 Γ n e 2 : A, E n E n 1 (THANDLE) E 0 Γ 0 Γ n handle e 1 c e 2 : A, ((E n \ {c}) E n )(E n 1 E n 1 ) (E 0 E 0 )

21 Typing rules Γ 0 Γ n Γ e : A, EE n E 0 Γ 0 Γ n box e : (Γ A, E), E n E 0 (TBOX) Γ 0 Γ n e : (Γ n+k A, E), E n E 0 Γ 0 Γ n Γ n+k unbox k e : A, E k 1 E n E 0 (TUNBOX) Γ 0 Γ n e : ( A, E), E n E n 1 E 0 Γ 0 Γ n eval e : A, (E E n )E n 1 E 0 (TEVAL)

22 Typing rules Γ 0 Γ n + x : A e : B, E n E n 1 E 0 Γ 0 Γ n λx.e : A En B, E n 1 E 0 (TABS) Γ 0 Γ n e 1 : A E B, E n E n 1 E 0 Γ 0 Γ n e 2 : A, E ne n 1 E 0 Γ 0 Γ n e 1 e 2 : B, (E E n E n )(E n 1 E n 1 ) (E 0 E 0 ) (TAPP) Γ 0 Γ n e : A, E n E 0 E n E n E 0 E 0 Γ 0 Γ n e : A, E n E 0 (TSUB)

23 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 ( int, ), {Zero}

24 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 ( int, ), {Zero}

25 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 int, {Zero}

26 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 ({a : int} int, ), {Zero}

27 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 ({a : int} int, ),

28 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 f : ( int, ) list ({a : int} int, ),

29 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 handler at stage 0 int, {Zero} fun f ls = (a *,(g ls) handle Zero => 0) handler at stage 1 int, {Zero}

30 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 handler at stage 0 ({a : int} int, ), {Zero} fun f ls = (a *,(g ls) handle Zero => 0) handler at stage 1 int, {Zero}

31 Example fun g [] = 1 g x::r = if x = 0 then raise Zero else (,x *,(g r)) g : ( int, ) list {Zero} ( int, ), fun f ls = (a *,(g ls)) handle Zero => 0 handler at stage 0 ({a : int} int, ), fun f ls = (a *,(g ls) handle Zero => 0) handler at stage 1 ({a : int} int, ), {Zero}

32 Soundness Lemma (Demotion and Promotion) Suppose Γ 1 Γ n v : A, E n E If Γ 1 = then Γ 1 Γ n v : A, E n E For all Γ 1 Γ m, E m E 1, Γ 1 Γ mγ 1 Γ n v : A, E n E 1E m E 1E 0. Lemma (Empty Effect of v 0 ) If Γ 0 v : A, E then Γ 0 v : A,. Theorem (Soundness) Suppose Γ 1 Γ n e : A, E n E If e n v then Γ 1 Γ n v : A, E n E If e n c then E n {c}.

33 Polymorphic Type System

34 What s the type of the following function? λx.box (unbox 1 (raise x)) How can we generalize it? We need variables. exn({c}) {c} ( int, ) exn({c }) {c } ( exn({c}), {c }) exn({c, c }) {c,c } ( int {c } int, ) α, ρ, ϕ, ϕ.exn(ϕ) ϕ (ρ α, ϕ ) Problem: set operations (, \), and subset order ( ) for variables. Two approaches Bounded polymorphism Row polymorphism our approache

35 Row Polymorphism Record type f = λx.(x.a) : {a : int} int f {a : 1} : int - O.K. f {a : 1, b : true} : - Error!! Row polymorphism f = λx.(x.a) : {a : int; ρ} int f {a : 1} : int - O.K. f {a : 1, b : true} : int - O.K. ρ represents all of the extra fields Set operations (, \) and subset order ( ): Unification

36 Type and effect A, B Type ::= α int exn(e) A E B (Γ A, E) α, β TyVar Γ TyEnv ::= ρ x : F ; Γ ρ TyEnvVar F Field ::= θ A θ FieldVar E Effect ::= ϕ c : π; E ϕ EffectVar π Presence ::= Pre δ δ PresenceVar

37 Type and effect τ TyScheme ::= ξ.τ A ξ Var ::= α ρ θ ϕ δ µ FieldScheme ::= θ τ TySchemeEnv ::= ρ x : µ; Typing judgment 0 n e : A, E n E 0

38 Typing Rules n (x) A 0 n x : A, E n E 0 (TVAR) 0 n + x : A e : B, EE n 1 E 0 0 n λx.e : A E B, E n E 0 (TABS) 0 n e 1 : A E B, EE n 1 E 0 0 n e 2 : A, EE n 1 E 0 (TAPP) 0 n e 1 e 2 : B, EE n 1 E 0 0 n e 1 : A, E n E 0 0 n + x : GEN A ( 0 n, E n E 0 ) e 2 : B, E n E 0 0 n let (x e 1 ) e 2 : B, E n E 0 (TLET) GEN A ( 0 n, E n E 0 ) = ξ 1 ξ n.a such that {ξ 1 ξ n } = F V (A) \ (F V ( 0 n ) F V (E n E 0 ))

39 Typing Rules 0 n e : A, EE n E 0 0 n box e : (Γ A, E), E n E 0 (TBOX) 0 n e : (Γ n+k A, E n+k ), E n E 0 n+k Γ n+k 0 n n+k unbox k e : A, E n+k E n E 0 (TUNBOX) 0 n e : ( A, E), EE n 1 E 0 0 n eval e : A, EE n 1 E 0 (TEVAL) Rank-1 polymorphism: Arguments and results of a function can not be polymorphic.

40 Typing Rules 0 n c : exn(c : Pre; E), E n E 0 (TEXN) 0 n e : exn(e), EE n 1 E 0 0 n raise e : A, EE n 1 E 0 (TRAISE) 0 n e 1 : A, (c : Pre; E)E n 1 E 0 0 n e 2 : A, (c : π; E)E n 1 E 0 0 n handle e 1 c e 2 : A, (c : π; E)E n 1 E 0 (THANDLE)

41 Example ρ λx.box (unbox 1 (raise x)) : α, ϕ ϕ 1 {α = α 1 α2 } ρ 1 box (unbox 1 (raise x)) : α 2, ϕ 1 {ρ 1 = (x : α 1 ; ρ 2 ), ρ 2 = ρ} ρ 1 ρ 3 unbox 1 (raise x) : α 3, ϕ 2 ϕ 1 {α 2 = (ρ 3 α 3, ϕ 2 )} ρ 1 raise x : (ρ 3 α 4, ϕ 2 ), ϕ 1 ρ 1 x : exn(ϕ 1 ), ϕ 1 {(x : α 1 ; ρ 2 ) = (x : exn(ϕ 1 ); ρ 4 )} {α 1 = exn(ϕ 1 )} {α = exn(ϕ 1 ) ϕ1 (ρ 3 α 3, ϕ 2 )} ρ λx.box (unbox 1 (raise x)) : exn(ϕ 1 ) ϕ1 (ρ 3 α 3, ϕ 2 ), ϕ

42 Example ρ let (f λx.box (unbox 1 (raise x))) f c : α, ϕ ρ λx.box (unbox 1 (raise x)) : α 1, ϕ {α 1 = exn(ϕ 1) ϕ 1 (ρ 1 α 2, ϕ 2)} ρ 2 f c : α, ϕ {ρ 2 = f : α 2ρ 1ϕ 1ϕ 2.exn(ϕ 1) ϕ 1 (ρ 1 α 2, ϕ 2); ρ 3, ρ 3 = ρ} ϕ ρ 2 f : α 3 α, ϕ {ϕ = ϕ 1, α 3 = exn(ϕ 1), α = (ρ 1 α 2, ϕ 2)} ρ 2 c : α 3, ϕ {α 3 = exn(c : Pre; ϕ 3), ϕ 1 = (c : Pre; ϕ 3)} {α = (ρ 1 α 2, ϕ 2)} {ϕ = (c : Pre; ϕ 3)} May-uncaught exception c ρ let (f λx.box (unbox 1 (raise x))) f c : (ρ 1 α 2, ϕ 2), (c : Pre; ϕ 3)

43 Example ρ let (f λx.box (unbox 1 (raise x))) handle (f c) c (box 1) : α, ϕ ρ λx.box (unbox 1 (raise x)) : α 1, ϕ {α 1 = exn(ϕ 1) ϕ 1 (ρ 1 α 2, ϕ 2)} ρ 2 handle (f c) c (box 1) : α, ϕ {ρ 2 = f : α 2ρ 1ϕ 1ϕ 2.exn(ϕ 1) ϕ 1 (ρ 1 α 2, ϕ 2); ρ 3, ρ 3 = ρ} ρ 2 f c : α, ϕ {α = (ρ 1 α 2, ϕ 2), (c : Pre; ϕ 4) = (c : Pre; ϕ 3)} ρ 2 box 1 : α, (c : δ; ϕ 4) {ϕ = (c : δ; ϕ 4), α = (ρ 3 α 4, ϕ 5)} ρ 2ρ 3 1 : α 4, ϕ 5(c : δ; ϕ 4) {α 4 = int} {α = (ρ 3 int, ϕ 5)} {ϕ = (c : δ; ϕ 3)} No uncaught exception!!

44 Conclusion

45 Conclusion A type system for λ-calculus + Lisp s quasi-quote + exception exception-raise and -handle can appear at any stage exceptions (raised during code composition) can escape stages our effect type system safely supports such features empty effect implies no uncaught exceptions