Compiling to -calculus and church encodings e ::= (lambda (x) e) - - PowerPoint PPT Presentation

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Compiling to -calculus and church encodings e ::= (lambda (x) e) - - PowerPoint PPT Presentation

Aug 20, 2023 365 likes •920 views

Compiling to -calculus and church encodings e ::= (lambda (x) e) | (e e) | x Alonzo Church = ( U U ) = (( (u) (u u)) ( (u) (u u))) (u u)[u U ] = (( (u) (u u)) ( (u) (u u))) Church encoding e ::= (letrec ([x

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Compiling to

λ-calculus and church encodings

λ

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e ::= (lambda (x) e) | (e e) | x

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Alonzo Church

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Ω = (U U) = ((λ (u) (u u)) Ω (λ (u) (u u))) (u u)[u ← U] = ((λ (u) (u u)) (λ (u) (u u)))

β

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Church encoding

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e ::= (letrec ([x (lambda (x …) e)]))
 | (let ([x e] …) e) | (lambda (x …) e) | (e e …) | x | (if e e e) | (+ e e) | (* e e) | (cons e e) | (car e) | (cdr e) | d d ::= ℕ | #t | #f | ‘() x ::= <vars>

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e ::= (letrec ([x (lambda (x …) e)]))
 | (let ([x e] …) e) | (lambda (x …) e) | (e e …) | x | (if e e e) | (+ e e) | (* e e) | (cons e e) | (car e) | (cdr e) | d d ::= ℕ | #t | #f | ‘() x ::= <vars> e ::= ) ) e)

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Desugaring Let

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(let ([x e] …) ebody)

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((λ (x …) ebody) e …) (let ([x e] …) ebody)

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Currying

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Gottlob Frege Moses Schönfinkel

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(λ (x y z) e) (λ (x) e) (λ () e) (λ (x) e) (λ (_) e) (λ (x) (λ (y) (λ (z) e)))

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(f a b c d) ((((f a) b) c) d) (f a) (f a) (f) (f f)

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e ::= (letrec ([x (lambda (x) e)])) | (lambda (x) e) | (e e) | x | (if e e e) | ((+ e) e) | ((* e) e) | ((cons e) e) | (car e) | (cdr e) | d d ::= ℕ | #t | #f | ‘() x ::= <vars>

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Conditionals & Booleans

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(if #t eT eF) eT (if #f eT eF) eF

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(if #t eT eF) (if #f eT eF) ((λ (t f) t) ((λ (t f) f) ((λ (t f) t) vT vF) ((λ (t f) f) vT vF) vT vF

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((λ (t f) t) eT Ω)

.........

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((λ () eT)) eT ((λ (t f) (t)) (λ () eT) (λ () Ω)) vT

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Turn all values into verbs!

(Focus on the behaviors that are implicit in values.)

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e ::= (letrec ([x (lambda (x) e)])) | (lambda (x) e) | (e e) | x | ((+ e) e) | ((* e) e) | ((cons e) e) | (car e) | (cdr e) | d d ::= ℕ | ‘() x ::= <vars>

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Natural Numbers

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(λ (f) (λ (x) (fN x))) 0: (λ (f) (λ (x) x)) 1: (λ (f) (λ (x) (f x))) 2: (λ (f) (λ (x) (f (f x)))) 3: (λ (f) (λ (x) (f (f (f x)))))

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church+ = (λ (n) (λ (m) (λ (f) (λ (x) …))))

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church+ = (λ (n) (λ (m) (λ (f) (λ (x) ((n f) ((m f) x))))))

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church* = (λ (n) (λ (m) (λ (f) (λ (x) …))))

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church* = (λ (n) (λ (m) (λ (f) (λ (x) ((n (m f)) x)))))

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fN = fN*M

M

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e ::= (letrec ([x (lambda (x) e)])) | (lambda (x) e) | (e e) | x | ((cons e) e) | (car e) | (cdr e) | d d ::= ‘() x ::= <vars>

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Lists

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We need to be able to case-split.

The fundamental problem:

We take two callbacks as with #t, #f!

The solution:

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‘() = (λ (when-cons) (λ (when-null) (when-null))) (cons a b) = (λ (when-cons) (λ (when-null) (when-cons a b)))

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e ::= (letrec ([x (lambda (x) e)])) | (lambda (x) e) | (e e) | x x ::= <vars>

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Y combinator

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(letrec ([fact (λ (n) (if (= n 0) 1 (* n (fact (- n 1)))))]) (fact 5))

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(letrec ([fact (λ (fact) (λ (n) (if (= n 0) 1 (* n (fact (- n 1)))))]) (fact 5))

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(letrec ([mk (λ (fact) (λ (n) (if (= n 0) 1 (* n (fact (- n 1)))))]) ((mk mk) 5))

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(letrec ([mk (λ (mk) (λ (n) (if (= n 0) 1 (* n ((mk mk) (- n 1)))))]) ((mk mk) 5))

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Y

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(Y f) = f (Y f)

(It’s a fixed-point combinator!)

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Y = (U (λ (y) (λ (f) (f (λ (x) (((y y) f) x)))))

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(letrec ([fact (Y (λ (fact) (λ (n) (if (= n 0) 1 (* n (fact (- n 1)))))]) (fact 5))

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e ::= (lambda (x) e) | (e e) | x

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(define (church->nat cv) ) (define (church->list cv) ) (define (church->bool cv) )

De-churching

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(define (church->nat cv) ((cv add1) 0)) (define (church->list cv) ) (define (church->bool cv) )

De-churching

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(define (church->nat cv) ((cv add1) 0)) (define (church->list cv) ((cv (λ (car) (λ (cdr) (cons car (church->list cdr))))) (λ (na) ‘()))) (define (church->bool cv) )

De-churching

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(define (church->nat cv) ((cv add1) 0)) (define (church->list cv) ((cv (λ (car) (λ (cdr) (cons car (church->list cdr))))) (λ (na) ‘()))) (define (church->bool cv) ((cv (λ () #t)) (λ () #f)))

De-churching

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(letrec ([map (λ (f lst) (if (null? lst) '() (cons (f (car lst)) (map f (cdr lst)))))]) (map (λ (x) (+ 1 x)) '(0 5 3)))

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(define lst (((((((((λ (Y-comb) (λ (church:null?) (λ (church:cons) (λ (church:car) (λ (church:cdr) (λ (church:+) (λ (church:*) (λ (church:not) ((λ (map) ((map (λ (x) ((church:+ (λ (f) (λ (x) (f x)))) x))) ((church:cons (λ (f) (λ (x) x))) ((church:cons (λ (f) (λ (x) (f (f (f (f (f x)))))))) ((church:cons (λ (f) (λ (x) (f (f (f x)))))) (λ (when-cons) (λ (when-null) (when-null (λ (x) x))))))))) (Y-comb (λ (map)

> (map church->nat (church->list lst)) '(1 6 4)

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Now we can write a church encoder!