# Evaluating Scheme Expressions

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1 Evaluating Scheme Expressions How Scheme evaluates the expressions? (procedure arg 1... arg n ) Find the value of procedure Find the value of arg 1 Find the value of arg n Apply the value of procedure to the values of arg 1... arg n LISP 1

2 Evaluating Scheme Expressions Example The simple procedure application (+ 3 4) The value of + is the addition procedure The value of 3 is the number 3 The value of 4 is the number 4 Applying the procedure + to 3 and 4 yields 7 The value is the object 7 LISP 2

3 Evaluating Scheme Expressions Example Find the value of the nested expression (* (+ 3 4) 2) The value of * is the multiplication procedure The value of (+ 3 4) is the number 7 The value of 2 is the number 2 Multiplying 7 by 2 we get 14, so answer is 14 LISP 3

4 Evaluating Scheme Expressions This rule works for procedure applications but not for quote expressions Subexpressions of a procedure are evaluated The evaluation of a quote expression is more similar to the evaluation of constant objects The value of a quote expression of the form (quote object) is simply object LISP 4

5 Evaluating Scheme Expressions A Scheme evaluator is free to evaluate the expressions in any order left to right right to left or any other sequential order Subexpressions may be evaluated in different orders for different applications even in the same implementation LISP 5

6 Evaluating Scheme Expressions Example. Determine the value of ((car (list + - * /)) 2 3) Here, procedure is (car (list + - * /)) The value of (car (list + - * /)) is the addition procedure Just as if procedure were simply the variable + LISP 6

7 Variables and Let Expressions Suppose: expr is a Scheme expression that contains a variable var var has the value val when we evaluate expr let a syntactic form let includes a list of variable-expression pairs, along with a sequence of expressions referred to as the body of the let LISP 7

8 Variables and Let Expressions General form of a let expression is (let ((var val)...)exp 1 exp 2...) Variables are bound to the values by the let We refer to variables bound by let as let-bound variables A let expression is often used to simplify an expression that would contain two identical subexpressions LISP 8

9 Variables and Let Expressions A value of the common subexpression is computed only once (+ (* 4 4) (* 4 4)) 32 (let ((a (* 4 4))) (+ a a)) 32 LISP 9

10 Variables and Let Expressions Example of using Scheme's let syntactic form (let ((x 2)) (+ x 3)) 5 (let ((y 3)) (+ 2 y)) 5 (let ((x 2) (y 3)) (+ x y)) 5 LISP 10

11 Variables and Let Expressions Example. What is the results? (let ((list1 '(a b c)) (list2 '(d e f))) (cons (cons (car list1) (car list2)) (cons (car (cdr list1)) (car (cdr list2)))) ((a. d) b. e) LISP 11

12 Variables and Let Expressions Example of using Scheme's let syntactic form (let ((f +)) (f 2 3)) 5 (let ((f +) (x 2)) (f x 3)) 5 (let ((f +) (x 2) (y 3)) (f x y)) 5 LISP 12

13 Variables and Let Expressions The variables bound by let are visible only within the body of the let For example: (let ((+ *)) (+ 2 3)) (+ 2 3) 6 5 LISP 13

14 Variables and Let Expressions Nested let expressions (let ((a 4) (b -3)) (let ((a-squared (* a a)) (b-squared (* b b))) (+ a-squared b-squared))) 25 When nested let expressions bind the same variable, only the binding created by the inner let is visible within its body LISP 14

15 Variables and Let Expressions Example. What is the value of (+ x 1)? What about (+ x x)? (let ((x 1)) (let ((x (+ x 1))) (+ x x))) 4 LISP 15

16 Variables and Let Expressions The inner binding for x is said to shadow the outer binding A let-bound variable is visible everywhere within the body of its let expression except where it is shadowed The region where a variable binding is visible is called its scope LISP 16

17 Variables and Let Expressions The scope of the first x in the example above is the body of the outer let expression minus the body of the inner let expression, where it is shadowed by the second x This form of scoping is referred to as lexical scoping The scope of each binding can be determined by a straightforward textual analysis of the program LISP 17

18 Variables and Let Expressions Shadowing may be avoided by choosing different names for variables The expression above could be rewritten so that variable bound by the inner let is new-x (let ((x 1)) (let ((new-x (+ x 1))) (+ new-x new-x))) 4 LISP 18

19 Lambda Expressions The syntactic form lambda to create a new procedure The general form of a lambda expression is (lambda (var...) exp 1 exp 2...) The variables var... are the formal parameters of the procedure The sequence of expressions exp 1... is its body LISP 19

20 Lambda Expressions Examplary: the procedure that has x as a parameter and has the same body as the let expression (lambda (x) (+ x x)) #<procedure> The most common operation to perform on a procedure is to apply it to one or more values ((lambda (x) (+ x x)) (* 3 4)) 24 LISP 20

21 Lambda Expressions We can establish a procedure as the value of a variable and use the procedure more than once (let ((double (lambda (x) (+ x x)))) (list (double (* 3 4)) (double (/ 99 11)) (double (- 2 7)))) ( ) LISP 21

22 Lambda Expressions The procedure expects its actual parameter to be a number, since it passes the actual parameter on to + The actual parameter may be any sort of object Examplary of a similar procedure that uses cons instead of + (let((double-cons (lambda (x)(cons x x)))) (double-cons 'a)) (a. a) LISP 22

23 Lambda Expressions Noting the similarity between double and double-cons Collapsed into a single procedure by adding an additional argument (let ((double-any (lambda (f x) (f x x)))) (list (double-any + 13) (double-any cons 'a))) (26 (a. a)) LISP 23

24 Lambda Expressions Lambda expressions can be are nested within other lambda or let expressions (let ((x 'a)) (let ((f (lambda (y) (list x y)))) (f 'b))) (a b) LISP 24

25 Lambda Expressions The occurrence of x within the lambda expression refers to the x outside the lambda that is bound by the outer let expression The variable x is said to occur free in the lambda expression or to be a free variable of the lambda expression LISP 25

26 Lambda Expressions The variable y does not occur free in the lambda expression since it is bound by the lambda expression A variable that occurs free in a lambda expression should be bound by an enclosing lambda or let expression LISP 26

27 Lambda Expressions What happens when the procedure is applied somewhere outside the scope of the bindings for variables that occur free within the procedure? (let ((f (let ((x 'sam)) (lambda (y z) (list x y z))))) (f 'i 'am)) (sam i am) The same bindings that were in effect when the procedure was created are in effect again when the procedure is applied LISP 27

28 Lambda Expressions This is true even if another binding for x is visible where the procedure is applied (let ((f (let ((x 'sam)) (lambda (y z) (list x y z))))) (let ((x 'not-sam)) (f 'i 'am))) (sam i am) In both cases, the value of x within the procedure named f is sam LISP 28

29 Lambda Expressions A let expression is nothing more than the direct application of a lambda expression to a set of argument expressions For example, the two expressions below are equivalent (let ((x 'a)) (cons x x)) ((lambda (x) (cons x x)) 'a) LISP 29

30 Lambda Expressions A let expression is a syntactic extension defined in terms of lambda and procedure application, which are both core syntactic forms Any expression of the form (let ((var val)...) exp 1 exp 2...) is equivalent to the following ((lambda (var...) exp 1 exp 2...) val...) LISP 30

31 Lambda Expressions The formal parameter specification of lambda expression: a proper list of variables (var 1... var n ) a single variable var r an improper list of variables (var 1... var n. var r ) LISP 31

32 Lambda Expressions Consider a few examples: (let ((f (lambda x x))) (f )) ( ) (let ((f (lambda x x))) (f)) () (let ((g (lambda (x. y) (list x y)))) (g )) (1 (2 3 4)) (let ((h (lambda (x y. z)(list x y z)))) (h 'a 'b 'c 'd)) (a b (c d)) LISP 32

33 Top-Level Definitions The variables bound by let and lambda expressions are not visible outside the bodies Suppose you have created an object or procedure, that must be accessible anywhere What you need is a top-level definition, which may be established with define Top-level definitions are visible in every expression you enter, except where shadowed by another binding LISP 33

34 Top-Level Definitions The variables bound by let and lambda expressions are not visible outside the bodies Suppose you have created an object or procedure, that must be accessible anywhere What you need is a top-level definition, which may be established with define Top-level definitions are visible in every expression you enter, except where shadowed by another binding LISP 34

35 Top-Level Definitions Any definition (define var exp) where exp is a lambda expression can be written in a shorter form that suppresses the lambda The exact syntax depends upon the format of the lambda expression's formal parameter i.e.: a proper list of variables a single variable an improper list of variables LISP 35

36 Top-Level Definitions A definition of the form (define var 0 (lambda (var 1... var n ) e 1 e 2...)) may be abbreviated (define (var 0 var 1... var n ) e 1 e 2...) LISP 36

37 Top-Level Definitions A definition of the form (define var 0 (lambda var r e 1 e 2...)) may be abbreviated (define (var 0. var r ) e 1 e 2...) LISP 37

38 Top-Level Definitions A definition of the form (define var 0 (lambda (var 1... var n. var r ) e 1 e 2...)) may be abbreviated (define (var 0 var 1... var n. var r ) e 1 e 2...) LISP 38

39 Top-Level Definitions For example, the definitions of cadr and list may be written as follows: (define (cadr x) (car (cdr x))) (define (list. x) x) LISP 39

40 Top-Level Definitions Let's establish a top-level definition of the double-any procedure (define double-any (lambda (f x) (f x x))) The variable double-any now has the same status as cons or the name of any other primitive procedure LISP 40

41 Top-Level Definitions We can use double-any as if it were a primitive procedure (double-any + 10) 20 (double-any cons 'a) (a. a) A top-level definition may be established for any object, not just for procedures (define sandwich "peanut-butter-and-jelly") sandwich "peanut-butter-and-jelly" LISP 41

42 Top-Level Definitions Top-level definitions may be shadowed by let or lambda bindings (define xyz '(x y z)) (let ((xyz '(z y x))) xyz) (z y x) Variables with top-level definitions act almost as if they were bound by a let expression enclosing all of the expressions you type LISP 42

43 Top-Level Definitions The abbreviations cadr and cddr for the compositions of car with cdr cdr with cdr For example: (cadr list) (car (cdr list)) (cddr list) (cdr (cdr list)) LISP 43

44 Top-Level Definitions They are easily defined as follows: (define cadr (lambda (x) (car (cdr x)))) (define cddr (lambda (x) (cdr (cdr x)))) (cadr '(a b c)) b (cddr '(a b c)) (c) LISP 44

45 Top-Level Definitions Top-level definitions make it easier for us to experiment with a procedure interactively because we need not retype the procedure each time it is used Let's try defining a somewhat more complicated variation of double-any, one that turns an "ordinary" two-argument procedure into a "doubling" one-argument procedure LISP 45

46 Top-Level Definitions (define doubler (lambda (f) (lambda (x) (f x x)))) doubler accepts one argument, f, which must be a procedure that accepts two arguments The procedure returned by doubler accepts one argument, which it uses for both arguments in an application of f LISP 46

47 Top-Level Definitions We can define, with doubler, the simple double and double-cons procedures (define double (doubler +)) (double 13/2) 13 (define double-cons(doubler cons)) (double-cons 'a) (a. a) LISP 47

48 Top-Level Definitions We can also define double-any with doubler (define double-any (lambda (f x) ((doubler f) x))) Within double and double-cons f has the appropriate value, i.e., + or cons, even though the procedures are clearly applied outside the scope of f LISP 48

49 Top-Level Definitions What happens if you attempt to use a variable that is not bound by a let or lambda expression and that does not have a top-level definition? Try using the variable i-am-not-defined to see what happens. (i-am-not-defined 3) Scheme print an error message to inform you that the variable is unbound or undefined LISP 49

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