CS 3015 Lecture 5 Higher Order Procedures

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CS 3015 Lecture 5Higher Order Procedures
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Number of Nodes in a Binary Tree

• Basis nodes(ltgt) 0
• Induction nodes(tree(L,a,R))
• 1 nodes(L) nodes(R)

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PreOrder Traversal

• pre BinTreesA -gt ListsA
• Basis pre(ltgt) ltgt
• Induction pre(tree(L,a,R))
• (list a (pre L) (pre R))
• pre(tree(ltgt,A,tree(tree(ltgt,B,ltgt),C,ltgt)))
• (list A () (list C (list B () ()) ()))

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Higher-Order Procedures

• Procedures give us a method of creating
  abstractions
• Could get along without procedure cube by always
  writing expressions like ( 5 5 5)
• But this would place us at a serious disadvantage
• Powerful programming languages must always
  provide methods of abstraction
• Procedures are one way to provide this

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Not Enough

• However, consider functions to do the following
• Compute the sum of the integers a through b
• Compute the sum of the cubes of the integers in a
  certain range
• Compute the sum of a sequence of terms in the the
  series

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Look at the Scheme Functions

• (define (sum-integers a b)  (if (gt a b)      0 
       ( a (sum-integers ( a 1) b))))
• (define (sum-cubes a b)  (if (gt a b)      0    
    ( (cube a) (sum-cubes 
• ( a 1) b))))

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• (define (pi-sum a b)  (if (gt a b)      0      (
   (/ 1.0 ( a ( a 2)))
•  (pi-sum ( a 4) b))))
• Converges on 1/8 Pi.

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Notice the Pattern

• (define (ltnamegt a b)  (if (gt a b)      0      (
   (lttermgt a)         (ltnamegt (ltnextgt a) b))))
• Wed like to be able to write a program that
  expresses the idea of summing a series without
  committing to computing particular sums

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Procedures as Arguments

• (define (sum term a next b)  (if (gt a b)      0
        ( (term a)         (sum term (next a) next
   b))))
• (define (inc n) ( n 1))
• (define (sum-cubes a b)  (sum cube a inc b))

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Defining Sum of Integers

• (define (identity x) x)
• (define (sum-integers a b)  (sum identity a inc b
  ))

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Defining Pi-Sum

• (define (pi-sum a b)  (define (pi-term x)    (/ 
  1.0 ( x ( x 2))))  (define (pi-next x)    ( x
   4))  (sum pi-term a pi-next b))
• ( 8 (pi-sum 1 1000))3.139592655589783

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Defining Further Abstractions

• Once we have sum we can define further
  abstractions
• For example, the definite integral of a function
  f between limits a and b

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Scheme Function to Compute Definite Integral

• (define (integral f a b dx)  (define (add-dx x) (
   x dx))  ( (sum f ( a (/ dx 2.0)) add-dx b)  
     dx))
• (integral cube 0 1 0.01)

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Exercise

• Write a procedure min-fx-gx that takes two
  numerical procedures f and g and a number x as
  inputs, and returns the minimum of applying f to
  x and g to x.

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Exercise

• Write a procedure min-fx-gx that takes two
  numerical procedures f and g and a number x as
  inputs, and returns the minimum of applying f to
  x and g to x.
• (define (min-fx-gx f g x)
• (min (f x) (g x))
• (min-fx-gx square cube -1)
• -1
• (min-fx-gx square cube 2)
• 4

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Exercise contd

• Now generalize your procedure to make the
  function applied to f(x) and g(x) be an input
  parameter also.
• (define (combine-fx-gx h f g x)
• (h (f x) (g x))
• (combine-fx-gx min square cube -1)
• -1

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Constructing Procedures using lambda

• Sometimes its awkward to have to define trivial
  procedures just to pass them as arguments
• pi-next, pi-term
• We introduce a new special form called lambda
• (lambda (x) ( x 4))
• (lambda (x) (/ 1.0 ( x ( x 2))))

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New Definition of Pi-Sum

• (define (pi-sum a b)  (sum (lambda (x) 
• (/ 1.0 ( x ( x 2))))       a       (lambda
   (x) ( x 4))       b))

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New Definition of Integral

• (define (integral f a b dx)  ( (sum f          
  ( a (/ dx 2.0))          (lambda (x) ( x dx)) 
           b)     dx))

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Lambda in General

• General form
• (lambda (ltformal-paramsgt) ltbodygt)
• Read
• (lambda  (x)  (    x     4)
•  the procedure of an argument x  that adds  x and 
  4
• (define (plus4 x) ( x 4))
• is really just shorthand for
• (define plus4 (lambda (x) ( x 4)))

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So Now

• ((lambda (x y z) ( x y (square z))) 
• 1 2 3)

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More Illustration of lambda

• We can use integrals to get another method of
  approximating p, since p/4 arctan 1
• (define (approx-pi dx)
• ( 4 (integral
• (lambda (x)
• (/ 1 ( 1 ( x x))))
• 0 1 dx)))