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Intermezzo 2 List Abbreviations

Using cons to create lists is cumbersome if a list contains many items. Fortunately, Scheme provides the list operation, which consumes an arbitrary number of values and creates a list. Here is Scheme's extended syntax:

   = list

The extended collection of values is

   = (list    ...   )

A simpler way to understand list expressions is to think of them as abbreviations. Specifically, every expression of the shape

(list exp-1 ... exp-n)

stands for a series of n cons expressions:

(cons exp-1 (cons ... (cons exp-n empty)))

Recall that empty is not an item of the list here, but the rest of the list. Here are three examples:

(list 1 2)
= (cons 1 (cons 2 empty))

(list 'Houston 'Dallas 'SanAntonio) 
= (cons 'Houston (cons 'Dallas (cons 'SanAntonio empty)))

(list false true false false) 
= (cons false (cons true (cons false (cons false empty))))

They introduce lists with two, three, and four items, respectively.

Of course, we can apply list not only to values but also to expressions:

  (list (+ 0 1) (+ 1 1))
= (list 1 2)

Before the list is constructed, the expressions must be evaluated. If during the evaluation of an expression an error occurs, the list is never formed:

  (list (/ 1 0) (+ 1 1))
= /: divide by zero

In short, list behaves just like any other primitive operation.

The use of list greatly simplifies the notation for lists with many items and lists that contains lists or structures. Here is an example:

  (list 0 1 2 3 4 5 6 7 8 9)

This list contains 10 items and its formation with cons and empty would require 10 uses of cons and one instance of empty. Similarly, the list

  (list (list 'bob 0 'a) 
        (list 'carl 1 'a) 
        (list 'dana 2 'b) 
        (list 'erik 3 'c) 
        (list 'frank 4 'a) 
        (list 'grant 5 'b) 
        (list 'hank 6 'c) 
        (list 'ian 8 'a) 
        (list 'john 7 'd) 
        (list 'karel 9 'e))

requires 11 uses of list in contrast to 40 of cons and 11 of empty.

Exercises

Exercise 13.0.1

Use cons and empty to construct the equivalent of the following lists:

(list 0 1 2 3 4 5)
(list (list 'adam 0) (list 'eve 1) (list 'louisXIV 2))
(list 1 (list 1 2) (list 1 2 3)).

Solution

Exercise 13.0.2

Use list to construct the equivalent of the following lists:

(cons 'a (cons 'b (cons 'c (cons 'd (cons 'e empty)))))
(cons (cons 1 (cons 2 empty)) empty)
(cons 'a (cons (cons 1 empty) (cons false empty))).
(cons (cons 1 (cons 2 empty)) (cons (cons 2 (cons 3 empty)) empty))

Start by determining how many items each list and each nested list contains. Solution

Exercise 13.0.3

On rare occasions, we encounter lists formed with cons and list. Reformulate the following lists using cons and empty exclusively:

(cons 'a (list 0 false))
(list (cons 1 (cons 13 empty)))
(list empty empty (cons 1 empty))
(cons 'a (cons (list 1) (list false empty))).

Then formulate the lists using list. Solution

Exercise 13.0.4

Determine the values of the following expressions:

(list (symbol=? 'a 'b) (symbol=? 'c 'c) false)
(list (+ 10 20) (* 10 20) (/ 10 20))
(list 'dana 'jane 'mary 'laura)

Solution

Exercise 13.0.5

Determine the values of

(first (list 1 2 3))

(rest (list 1 2 3))

Solution

The use of list makes it significantly easier to evaluate expressions involving lists. Here are the recursive steps from an example from section :

  (sum (list (make-ir 'robot 22.05) (make-ir 'doll 17.95)))
= (+ (ir-price (first (list (make-ir 'robot 22.05) (make-ir 'doll 17.95)))) 
     (sum (rest (list (make-ir 'robot 22.05) (make-ir 'doll 17.95))))) 
= (+ (ir-price (make-ir 'robot 22.05)) 
     (sum (list (make-ir 'doll 17.95)))) 
;At this place, we use one of the equations governing  
;the new primitive operations for the first time:  
= (+ 22.05 
     (sum (list (make-ir 'doll 17.95)))) 
= (+ 22.05 
     (+ (ir-price (first (list (make-ir 'doll 17.95)))) 
        (sum (rest (list (make-ir 'doll 17.95)))))) 
= (+ 22.05 
     (+ (ir-price (make-ir 'doll 17.95)) 
        (sum empty))) 
= (+ 22.05 (+ 17.95 (sum empty))) 
= (+ 22.05 (+ 17.95 0))

Since the laws of first and rest carry over to list values in a natural manner, an evaluation using list does not need to expand list into uses of cons and empty.

Following an old programming language convention, we may abbreviate lists and symbols even further. If a list is formulated with list, we can simply agree to drop list and that each opening parenthesis stands for itself and the word list. For example,

'(1 2 3)
;abbreviates  
(list 1 2 3) 
;or 
(cons 1 (cons 2 (cons 3 empty))) .

Similarly,

'((1 2) (3 4) (5 6))
;stands for 
(list (list 1 2) (list 3 4) (list 5 6)),

which can be further expanded into cons and empty expressions.

If we drop quotes in front of symbols, writing lists of symbols is a breeze:

'(a b c)
;This short-hand is an abbreviation for 
(list 'a 'b 'c)

And, more impressively,

'(<html> 
         (<title> My First Web Page) 
         (<body> Oh!)) 
;stands for 
(list '<html> 
  (list '<title> 'My 'First 'Web 'Page) 
  (list '<body> 'Oh!)) .

Exercises

Exercise 13.0.6

Restore list and quotes where necessary:

```
'(1 a 2 b 3 c)
```

'((alan 1000)
  (barb 2000) 
  (carl 1500) 
  (dawn 2300))

'((My First Paper)
  (Sean Fisler) 
  (Section 1 
    (Subsection 1 Life is difficult) 
    (Subsection 2 But learning things makes it interesting)) 
  (Section 2 
    Conclusion? What conclusion?))

Solution

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