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Exploring Function Definition and Evaluation in Lambda Calculus

This document provides an analysis of a piece of code utilizing lambda calculus to define and evaluate mathematical functions. It highlights the definition of a function (f), which determines behavior based on the input value, along with another function (foo) that utilizes a helper function (bar). We observe the evaluations for specific inputs, gaining insight into how the functions interact. The process emphasizes the significance of lambda expressions in functional programming and the evaluation process, offering clarity on the logic and structure behind recursive functions.

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Exploring Function Definition and Evaluation in Lambda Calculus

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  1. GE f: P1 para:x body:(if … ) (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y))))

  2. GE f: foo: P1 P2 para:x body:(if … ) para:bar,x,y body:(let … ) (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y))))

  3. GE f: foo: P1 P2 bar:P1 x:1 y:-2 para:x body:(if … ) para:bar,x,y body:(let … ) (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  4. GE f: foo: P1 P2 bar:P1 x:1 y:-2 para:x body:(if … ) para:bar,x,y body:(let … ) Evaluating: (bar y) => (P1 -2) (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  5. GE f: foo: P1 P2 bar:P1 x:1 y:-2 x:-2 para:x body:(if … ) para:bar,x,y body:(let … ) Evaluating: (bar y) => (P1 -2) (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  6. GE f: foo: P1 P2 bar:P1 x:1 y:-2 x:-2 para:x body:(if … ) para:bar,x,y body:(let … ) Evaluating: (if (< x 0) … ) (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  7. GE f: foo: P1 P2 bar:P1 x:1 y:-2 x:-2 para:x body:(if … ) para:bar,x,y body:(let … ) P3 para:y body:(- y x) (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  8. GE f: foo: P1 P2 bar:P1 x:1 y:-2 x:-2 para:x body:(if … ) para:bar,x,y body:(let … ) P3 para:y body:(- y x) g: (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  9. GE f: foo: P1 P2 bar:P1 x:1 y:-2 x:-2 para:x body:(if … ) para:bar,x,y body:(let … ) P3 para:y body:(- y x) g: Evaluating: (g x) => (P3 1) (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  10. GE f: foo: P1 P2 bar:P1 x:1 y:-2 x:-2 y:1 para:x body:(if … ) para:bar,x,y body:(let … ) P3 Evaluating: (- y x) => 3 para:y body:(- y x) g: (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  11. GE f: foo: P1 P2 bar:P1 x:1 y:-2 x:-2 y:1 para:x body:(if … ) para:bar,x,y body:(let … ) P3 para:y body:(- y x) g: Evaluating: (g y) => (P3 -2) (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  12. GE f: foo: P1 P2 bar:P1 x:1 y:-2 x:-2 y:1 para:x body:(if … ) para:bar,x,y body:(let … ) P3 y:-2 Evaluating: (- y x) => 0 para:y body:(- y x) g: (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

  13. GE f: foo: P1 P2 bar:P1 x:1 y:-2 x:-2 y:1 para:x body:(if … ) para:bar,x,y body:(let … ) P3 y:-2 para:y body:(- y x) g: Evaluating: (+ 3 0) => 3 (define (f x) (if (< x 0) (lambda (y) (- y x)) (lambda (y) (- x y)))) (define (foo bar x y) (let ((g (bar y))) (+ (g x) (g y)))) (foo f 1 -2)

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