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Logical Types for Untyped Languages Sam Tobin-Hochstadt & Matthias Felleisen PLT @ Northeastern University ICFP, September 27, 2010

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Types for Untyped Languages: • Ruby [Furr et al 09] • Perl [Tang 06] • Thorn [Wrigstad et al 09] • ActionScript [Adobe 06] • Typed Racket

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Types for Untyped Languages: • Reynolds 68 • Cartwright 75 • Wright & Cartwright 94 • Henglein & Rehof 95

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"Some account should be taken of the premises in conditional expressions." Reynolds, 1968

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"Type testing predicates aggravate the loss of static type information." Henglein & Rehof, 1995

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Types and Predicates

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(define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))]))

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(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

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(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

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(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

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(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

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(: 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*))]))

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(: 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

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(: 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

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(: 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

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(: 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

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(: 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

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Types and Propositions

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(define-type Peano (U 'Z (List 'S Peano))) (: convert : Peano -> Number) (define (convert n) (cond [(symbol? n) 0] [else (add1 (convert (rest n)))]))

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(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

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(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

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(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

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(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

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(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

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(: 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*))]))

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(: 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*

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(: 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*

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(: 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*

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(: 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

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(string? s )

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(string? s ) String @ s

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(string? s ) String @ s | String @ s

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(x:Any -> Bool : String@x | String@x) s (string? s ) String @ s | String @ s x:Any -> Bool : String@x | String@x string? s s

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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

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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)

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(λ ([s : (Pair Any Any)]) (string? (car s))) String @ car(s) | String @ car(s)

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(λ ([s : (Pair Any Any)]) (string? (car s))) (s:(Pair Any Any) -> Bool : String @ car(s) | String @ car(s))

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Propositional Logic

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Judgments Γ ⊢ e : T ; φ1 | φ2 ; o

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Judgments Γ ⊢ e : T ; φ1 | φ2 ; o e ::= n | c | (λ x : T . e) | (e e) | (if e e e)

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Judgments Γ ⊢ e : T ; φ1 | φ2 ; o T ::= Number | (U T ...) | #t | #f | (x:T -> T : φ|φ)

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Judgments Γ ⊢ e : T ; φ1 | φ2 ; o φ ::= T@π(x) | T@π(x) | φ1 ∨ φ2 | φ1 ∧ φ2 | φ1 ⊃ φ2

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Judgments Γ ⊢ e : T ; φ1 | φ2 ; o Γ ::= x:T ...

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Judgments Γ ⊢ e : T ; φ1 | φ2 ; o Γ ::= x:T T@π(x) ...

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Judgments Γ ⊢ e : T ; φ1 | φ2 ; o Γ ::= x:T T@π(x) φ ...

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Judgments Γ ⊢ φ

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Judgments Γ ⊢ φ Number @ x ∨ String @ y , Number @ x ⊢ String @ y

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Typing (if e1 e2 e3 )

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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 )

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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

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Typing Γ ⊢ x : T

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Typing Γ ⊢ Tx Γ ⊢ x : T

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Evaluation

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Scaling Multiple Arguments Multiple Values User-defined Datatypes Local Binding Adapting Mutable Structures Mutable Variables

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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]

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Empirical Evaluation Estimated usage of two idioms in Racket code base (600k lines) • Local binding with or: ~470 uses • Predicates with Selectors: ~440 uses

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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

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Conclusions Propositions can relate types and terms

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Conclusions Propositions can relate types and terms Existing programs are a source of type system ideas

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Thank You Code and Documentation http://www.racket-lang.org [email protected]

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Thank You Code and Documentation http://www.racket-lang.org [email protected]