Type assignment for intersections and unions in call-by-value languages

Transcription

1 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.1/42

2 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.2/42

3 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.3/42

4 Push the envelope The goal of the Triple Project: create refined type systems. The present work: Persistent refinements of algebraic datatypes Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.4/42

5 Type Assignment A type assignment system a 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.5/42

6 Property Types Some types build values Example: tuples Others talk about values 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.6/42

7 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.7/42

8 Language Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.8/42

9 Language + Initial Typing A, B, C, D ::= 1 A B e ::= x f () λx. e e 1 (e 2 ) fix f. e B 1 A 1 A 2 B 2 A 1 A 2 B 1 B 2 ( ) 1 1 (1) Γ(x) = A Γ x : A (var) Γ(f) = A Γ f : A (fixvar) Γ e : A A B Γ e : B (sub) Γ e 1 : A B Γ e 2 : A Γ e 1 (e 2 ) : B ( E) Γ, x:a e : B Γ λx. e : A B ( I) Γ, f:a e : A Γ fix f. e : A (fix) Γ () : 1 (1I) Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.9/42

10 cbv Semantics Values v ::= x () λx. e Evaluation contexts E ::= [] E(e) v(e) e R e E[e ] E[e ] (λx. e) v R fix f. e R [v/x] e [fix f. e / f] e Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.10/42

11 Key Properties A practical type system should satisfy Preservation: If e : A and e e then e : A. Progress: If e : A then e is a value or e. e e. Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.11/42

12 Datasort Refinements a.k.a. refinement types (Freeman, Pfenning, Davies) Refine an algebraic datatype by a datasort δ Example: Lists of integers Nil : 1 list Cons : int list list Intersections essential Nil : 1 even Cons : (int odd even) δ : (int even odd) (int list list) list odd 6 even Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.12/42

13 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 Cons : int list list Nil : 1 list(0) Cons : Πa:N. int list(a) list(a + 1) 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.13/42

14 Intersection Types Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.14/42

15 Typing Intersections Typing: Γ e : A 1 A 2 Γ e : A 1 ( E 1 ) Γ e : A 1 A 2 Γ e : A 2 ( E 2 ) Γ e : A 1 Γ e : A 2 Γ e : A 1 A 2 ( I) ( I) unsound with mutable references. Γ v : A 1 Γ v : A 2 Γ v : A 1 A 2 ( I) The fix: a value restriction [Davies & Pfenning 00]. Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.15/42

16 Indefinite Types Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.16/42

17 Indefinite Types: 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.17/42

18 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.18/42

19 Union Types Introduction rules straightforward: Γ e : A Γ e : A B ( I 1) Elimination rule? Γ e : B Γ e : A B ( I 2) Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.19/42

20 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.20/42

21 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 : A B Γ, x:b e : C Γ [e /x] e : C Suppose Γ h : (A C) (B C), Γ v : A B Γ v : A B Γ, x:a h x : C Γ, x:b h x : C Γ h v : C Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.21/42

22 Elimination Γ e : A B Γ [e /x] e : C Γ, x:a e : C Γ, x:b e : C Consider a nondeterministic choice e 1 e 2 : e 1 e 2 e 1 e 1 e 2 e 2 Suppose Γ g : (A A C) (B B C), Γ y : A, Γ z : B. Γ y z : A B Γ, x:a g x x : C Γ, x:b g x x : C Γ g (y z) (y z) : C g (y z) (y z) g y (y z) g y z Unsound whenever a term evaluates to more than one value Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.22/42

23 Elimination Γ e : A B Γ [e /x] e : C Γ, x:a e : C Γ, x:b e : C Restrict to exactly one occurrence of e? Can fail if e inside a λ (see paper) Restrict e to a value? [van Bakel 99] Sound but still must guess occurrences of e Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.23/42

24 Elimination: Our Solution Γ e : A B Γ E[e ] : C Γ, x:a E[x] : C Γ, x:b E[x] : C ( E) Exactly one e, in evaluation position So e is the next thing to be evaluated Remainder of computation (E) on the result of e Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.24/42

25 The Empty Type is the empty or void type A ( L) Elimination rule: by analogy with ( E) Γ e : Γ E[e ] : C ( E) Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.25/42

26 A Generalization? Must we restrict the rules to an evaluation context? Or could we also allow cases where e must be evaluated exactly once but might not be in evaluation position? Γ e : e 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.26/42

27 Type Safety Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.27/42

28 Type Safety Γ e : A B Γ E[e ] : C Γ, x:a E[x] : C Γ, x:b E[x] : C ( E) Preservation and progress hold with δ, i,,, Π,,, Σ Lemma: Values always have definite types. v : If v : A B then v : A or v : B If v : Σa:N. A then v : [i/a] A for some i Above: a special case of the actual lemma (Fully written out for δ, i,, Π in [Dunfield 02]) Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.28/42

29 Closely Related Work Davies 97, 00* δ, ; Dunfield 02 δ, i,, Π; Xi 98, 99*, 00 i, Π, Σ; * & Pfenning This paper* δ, i,, Π,,, Σ, ; : Ongoing work δ, i,, Π,,, Σ, ; Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.29/42

30 Related Work [Coppo et al. 81]: can characterize normal forms (termination); hence undecidable [Reynolds 96]: FORSYTHE 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.30/42

31 Conclusion Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.31/42

32 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.32/42

33 Future Work Bidirectional formulation Let-normal form Mutable references Implementation Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.33/42

34 Acknowledgments National Science Foundation: CCR : Type Refinements; : 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.34/42

35 The End Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.35/42

36 Intersections: Subtyping Subtyping: A B 1 A B 2 A B 1 B 2 A 1 B A 1 A 2 B ( L 1) Distributivity? ( R) A 2 B A 1 A 2 B ( L 2) (A B) (A B ) A (B B ) Again unsound with mutable references. [Davies & Pfenning 00] Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.36/42

37 Unions: Subtyping Subtyping for just the dual of : A 1 B A 2 B A 1 A 2 B ( L) A B 1 A B 1 B 2 ( R 1 ) A B 2 A B 1 B 2 ( R 2 ) Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.37/42

38 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 δ 1 δ 2 Γ i =. j Γ δ 1 (i) δ 2 (j) (δ) Γ c : A δ(i) Γ e : A Γ c(e) : δ(i) (δi) Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.38/42

39 Formulating Refinements Atomic subtyping δ 1 δ 2 Γ i =. j Γ δ 1 (i) δ 2 (j) (δ) Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.39/42

40 Evaluation Order Type assignment for intersections and unions in call-by-value languages FOSSACS 03 p.40/42

41 Evaluation Order Binary and 0-ary indefinite type elim rules: Γ e : A B Γ E[e ] : C Γ, x:a E[x] : C Γ, x:b E[x] : C A corresponding unary rule? ( E) Γ e : A Γ, x:a E[x] : C Γ E[e ] : C Γ e : Γ E[e ] : C ( E) (direct) 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.41/42

42 Evaluation Order Γ e : A Γ, x:a E[x] : C Γ E[e ] : C (direct) 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.42/42