Logical Types for Untyped Languages

From ICFP 2010.

Transcript

1. Logical Types for Untyped Languages Sam Tobin-Hochstadt & Matthias Felleisen PLT @ Northeastern University ICFP, September 27, 2010
2. Types for Untyped Languages: • Ruby [Furr et al 09] • Perl [Tang 06] • Thorn [Wrigstad et al 09] • ActionScript [Adobe 06] • Typed Racket
3. Types for Untyped Languages: • Reynolds 68 • Cartwright 75 • Wright & Cartwright 94 • Henglein & Rehof 95
4. "Some account should be taken of the premises in conditional expressions." Reynolds, 1968
5. "Type testing predicates aggravate the loss of static type information." Henglein & Rehof, 1995
6. Types and Predicates
7. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))]))
8. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))])) n : Peano n n : Peano
9. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))])) n : Peano n n : Peano
10. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))])) n : 'Z 0 n : 'Z
11. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))])) n : (List 'S Peano) n n : List 'S Peano
12. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))]))
13. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))])) s : (U String Symbol) s* : (U String Symbol) s s : U String Symbol s* s* : U String Symbol
14. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))])) s : String s* : String s s : String s* s* : String
15. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))])) s : String s* : Symbol s s : String s* s* : Symbol
16. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))])) s : Symbol s* : String s s : Symbol s* s* : String
17. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))])) s : Symbol s* : Symbol s s : Symbol s* s* : Symbol
18. Types and Propositions
19. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))]))
20. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))])) n : (List 'S Peano) add1 convert rest n n : List 'S Peano
21. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))])) ⊢ (List 'S Peano) @ n add1 convert rest n ⊢ List 'S Peano @ n
22. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))])) ⊢ (List 'S Peano) @ n symbol? n ⊢ Symbol @ n
23. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))])) ⊢ Peano @ n ⊢ Symbol @ n ⊢ (List 'S Peano) @ n symbol? n ⊢ Symbol @ n
24. (define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))])) ⊢ (U 'Z (List 'S Peano)) @ n ⊢ Symbol @ n ⊢ (List 'S Peano) @ n symbol? n ⊢ Symbol @ n
25. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))]))
26. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))])) ⊢ String @ s ⊢ String @ s* string-append s s* ⊢ String @ s ⊢ String @ s*
27. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))])) ⊢ String @ s ∧ String @ s* ⊢ String @ s ⊢ String @ s* string? s String @ s string? s* String @ s*
28. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))])) ⊢ Symbol @ s* string-append s symbol->string s* Symbol @ s*
29. (: combine : (U String Symbol) (U String Symbol) -> String) (define (combine s s*) (cond [(and (string? s) (string? s*)) (string-append s s*)] [(string? s) (string-append s (symbol->string s*))] [(string? s*) (string-append (symbol->string s) s*)] [else (string-append (symbol->string s) (symbol->string s*))])) ⊢ String @ s ⊢ String @ s ⊃ String @ s* ⊢ Symbol @ s* and string? s string? s* String @ s ⊃ String @ s* string? s String @ s
30. (string? s )
31. (string? s ) String @ s
32. (string? s ) String @ s | String @ s
33. (x:Any -> Bool : String@x | String@x) s (string? s ) String @ s | String @ s x:Any -> Bool : String@x | String@x string? s s
34. Latent Propositions (x:Any -> Bool : String@x | String@x) Objects s (string? s ) String @ s | String @ s Propositions x:Any -> Bool : String@x | String@x string? s s
35. Latent Propositions (x:Any -> Bool : String@x | String@x) Objects car(s) (string? (car s)) String @ car(s) | String @ car(s) Propositions x:Any -> Bool : String@x | String@x string? car s car(s)
36. (λ ([s : (Pair Any Any)]) (string? (car s))) String @ car(s) | String @ car(s)
37. (λ ([s : (Pair Any Any)]) (string? (car s))) (s:(Pair Any Any) -> Bool : String @ car(s) | String @ car(s))
38. Propositional Logic
39. Judgments Γ ⊢ e : T ; φ1 | φ2 ; o
40. Judgments Γ ⊢ e : T ; φ1 | φ2 ; o e ::= n | c | (λ x : T . e) | (e e) | (if e e e)
41. Judgments Γ ⊢ e : T ; φ1 | φ2 ; o T ::= Number | (U T ...) | #t | #f | (x:T -> T : φ|φ)
42. Judgments Γ ⊢ e : T ; φ1 | φ2 ; o φ ::= T@π(x) | T@π(x) | φ1 ∨ φ2 | φ1 ∧ φ2 | φ1 ⊃ φ2
43. Judgments Γ ⊢ e : T ; φ1 | φ2 ; o Γ ::= x:T ...
44. Judgments Γ ⊢ e : T ; φ1 | φ2 ; o Γ ::= x:T T@π(x) ...
45. Judgments Γ ⊢ e : T ; φ1 | φ2 ; o Γ ::= x:T T@π(x) φ ...
46. Judgments Γ ⊢ φ
47. Judgments Γ ⊢ φ Number @ x ∨ String @ y , Number @ x ⊢ String @ y
48. Typing (if e1 e2 e3 )
49. Typing Γ,φ+ ⊢ e2 : T ; φ1+ |φ1- ; o Γ,φ- ⊢ e3 : T ; φ2+ |φ2- ; o Γ ⊢ e1 : T' ; φ+ |φ- ; o' (if e1 e2 e3 )
50. Typing Γ,φ+ ⊢ e2 : T ; φ1+ |φ1- ; o Γ,φ- ⊢ e3 : T ; φ2+ |φ2- ; o Γ ⊢ e1 : T' ; φ+ |φ- ; o' Γ ⊢ (if e1 e2 e3 ) : T ; φ1+ ∨φ2+ | φ1- ∨φ2- ; o
51. Typing Γ ⊢ x : T
52. Typing Γ ⊢ Tx Γ ⊢ x : T
53. Evaluation
54. Scaling Multiple Arguments Multiple Values User-defined Datatypes Local Binding Adapting Mutable Structures Mutable Variables
55. Local Binding Γ ⊢ e0 : S ; φ*+ | φ*- ; o Γ, S@x, #f @ x ⊃ φ*+ ⊢ e1 : T ; φ+ | φ- ; o* Γ ⊢ (let [x e0 ] e1 ) : T[o/x] ; φ+ | φ- [o/x] ; o*[o/x]
56. Empirical Evaluation Estimated usage of two idioms in Racket code base (600k lines) • Local binding with or: ~470 uses • Predicates with Selectors: ~440 uses
57. Prior Work None of the Examples Shivers 91, Henglein & Rehof 95, Crary et al 98, ... Just convert Aiken et al 94, Wright 94, Flanagan 97, Komondoor et al 2005 Everything but abstraction Bierman et al 2010
58. Conclusions Propositions can relate types and terms
59. Conclusions Propositions can relate types and terms Existing programs are a source of type system ideas
60. Thank You Code and Documentation http://www.racket-lang.org samth@ccs.neu.edu
61. Thank You Code and Documentation http://www.racket-lang.org samth@ccs.neu.edu