Transcript (sum-of-squares (+ 5 1) (* 5 2))

Primitive Data Types
 Symbol: any sequence of characters except
()[]{};,”`’#\
 Number
– signed integers
– signed real numbers
– bignums (large integers)
– rationals
– exponential notation for reals
– complex numbers
 Boolean constants: #t #f
Naming
(define pi 3.14159)
(define radius 10)
(* pi (* radius radius))
314.159
(define circumference
(* 2 pi radius))
circumference
62.8318
(define (circumference r)
(* 2 pi r))
(circumference 20)
;; special form
;; redefinition
Example of Evaluation
(* (+ 2 (* 4 6))
390
(+ 3 5 7))
15
26
*
+
2
*
+
24
4
5
3
6
7
Rules of Evaluation
 The value of a numeral is itself.
 The value of a primitive operation is a pointer to the
internal machine instructions to accomplish it.
 The value of a name is the value associated with that
name in an environment.
 The value of a combination is obtained by:
– Evaluating the sub-expressions in any order.
– Applying the value of the operator sub-expression
to the values of the other sub-expressions, where
applying a compound procedure means evaluating
the body of the procedure with each formal
parameter replaced by is corresponding value.
Procedure Definition
(define
(square
(square
(square
(square x) (* x x))
21)
(+ 2 5))
(square 3))
(define (sum-of-squares x y)
(+ (square x) (square y)))
(sum-of-squares 3 4)
(define (f a)
(sum-of-squares (+ a 1) (* a 2)))
(f 5)
Substitution Model
(f 5)
;; (sum-of-squares (+ a 1) (* a 2))
;: (sum-of-squares (+ 5 1) (* 5 2))
;: (+ (square 6) (square 10))
;: (+ (* 6 6) (* 10 10))
;: (+ 36 100)
136
 NOT how the interpreter really works!
 A way to start thinking about the process
 Applicative order of evaluation: eval-and-apply
Normal Order Evaluation
(f 5)
(sum-of-squares (+ 5 1) (* 5 2))
(+ (square (+ 5 1))(square (* 5 2)))
(+ (* (+ 5 1) (+ 5 1))
(* (* 5 2) (* 5 2)))
(+ (* 6 6)
(* 10 10))
(+ 36
100)
136
Evaluation: An Example
(define (p) (p))
(define (test x y)
(if (= x 0)
0
y))
(test 0 (p))
 applicative order: eval-and-apply
infinite loop due to (p)
 normal order: fully-expand-and-reduce
infinite term, safe execution
Conditional Expressions
(define (abs x)
(cond ((> x 0) x)
((= x 0) 0)
((< x 0) (- x))))
(define (abs x)
(cond ((< x 0) (- x))
(else x)))
(define (abs x)
(if (< x 0)
(- x)
x))
Compound Predicates
(and (> x 5) (< x 10))
(define (>= x y)
(or (> x y) (= x y)))
(define (>= x y)
(not (< x y)))