1 1. Define a function `zero?` that expects a single number as an argument, and returns `'true` if that number is `0`, else returns `'false`. Your solution should have a form something like this:
4 zero? match lambda x. FILL_IN_THIS_PART
7 You can use the `if...then...else` construction if you like, but it will make it easier to generalize to later problems if you use the `case EXPRESSION of PATTERN1 then RESULT1; PATTERN2 then RESULT2; ... end` construction instead.
9 2. Define a function `empty?` that expects a sequence of values as an argument (doesn't matter what type of values), and returns `'true` if that sequence is the empty sequence `[]`, else returns `'false`. Here your solution should have a form something like this:
12 empty? match lambda xs. case xs of
17 3. Define a function `tail` that expects a sequence of values as an argument (doesn't matter what type of values), and returns that sequence with the first element (if any) stripped away. (Applying `tail` to the empty sequence `[]` can just give us back the empty sequence.)
19 4. Define a function `drop` that expects two arguments, in the form (*number*, *sequence*), and works like this:
21 drop (0, [10, 20, 30]) # evaluates to [10, 20, 30]
22 drop (1, [10, 20, 30]) # evaluates to [20, 30]
23 drop (2, [10, 20, 30]) # evaluates to [30]
24 drop (3, [10, 20, 30]) # evaluates to []
25 drop (4, [10, 20, 30]) # evaluates to []
27 Your solution should have a form something like this:
30 drop match lambda (n, xs). FILL_IN_THIS_PART
33 What is the relation between `tail` and `drop`?
35 5. Define a function `take` that expects two arguments, in the same form as `drop`, but works like this instead:
37 take (0, [10, 20, 30]) # evaluates to []
38 take (1, [10, 20, 30]) # evaluates to [10]
39 take (2, [10, 20, 30]) # evaluates to [10, 20]
40 take (3, [10, 20, 30]) # evaluates to [10, 20, 30]
41 take (4, [10, 20, 30]) # evaluates to [10, 20, 30]
43 6. Define a function `split` that expects two arguments, in the same form as `drop` and `take`, but this time evaluates to a pair of results. It works like this:
45 split (0, [10, 20, 30]) # evaluates to ([], [10, 20, 30])
46 split (1, [10, 20, 30]) # evaluates to ([10], [20, 30])
47 split (2, [10, 20, 30]) # evaluates to ([10, 20], [30])
48 split (3, [10, 20, 30]) # evaluates to ([10, 20, 30], [])
49 split (4, [10, 20, 30]) # evaluates to ([10, 20, 30], [])
51 Here's a way to answer this problem making use of your answers to previous questions:
54 drop match ... ; # as in problem 4
55 take match ... ; # as in problem 5
56 split match lambda (n, xs). let
62 However, we want you to instead write this function from scratch.
64 7. Write a function `filter` that expects two arguments. The second argument will be a sequence `xs` with elements of some type *t*, for example numbers. The first argument will be a function `p` that itself expects arguments of type *t* and returns `'true` or `'false`. What `filter` should return is a sequence that contains exactly those members of `xs` for which `p` returned `'true`. For example, helping ourself to a function `odd?` that works as you'd expect:
66 filter (odd?, [11, 12, 13, 14]) # evaluates to [11, 13]
67 filter (odd?, [11]) # evaluates to [11]
68 filter (odd?, [12, 14]) # evaluates to []
70 8. Write a function `partition` that expects two arguments, in the same form as `filter`, but this time evaluates to a pair of results. It works like this:
72 partition (odd?, [11, 12, 13, 14]) # evaluates to ([11, 13], [12, 14])
73 partition (odd?, [11]) # evaluates to ([11], [])
74 partition (odd?, [12, 14]) # evaluates to ([], [12, 14])
76 9. Write a function `double` that expects one argument which is a sequence of numbers, and returns a sequence of the same length with the corresponding elements each being twice the value of the original element. For example:
78 double [10, 20, 30] # evaluates to [20, 40, 60]
79 double [] # evaluates to []
81 10. Write a function `map` that generalizes `double`. This function expects a pair of arguments, the second being a sequence `xs` with elements of some type *t*, for example numbers. The first argument will be a function `f` that itself expects arguments of type *t* and returns some type *t'* of result. What `map` should return is a sequence of the results, in the same order as the corresponding original elements. The result should be that we could say:
84 map match lambda (f, xs). FILL_IN_THIS_PART;
85 double match lambda xs. map ((lambda x. 2*x), xs)
88 11. Write a function `map2` that generalizes `map`. This function expects a triple of arguments: the first being a function `f` as for `map`, and the second and third being two sequences. In this case `f` is a function that expects *two* arguments, one from the first of the sequences and the other from the corresponding position in the other sequence. The result should behave like this:
90 map2 ((lambda (x,y). 10*x + y), [1, 2, 3], [4, 5, 6]) # evaluates to [14, 25, 36]
97 <!-- take_while, drop_while, split_while -->
99 <!-- unmap2 (g, xs) where g x \mapsto (y,z), and unmap2 (g, [x1, x2, x3]) \mapsto ([y1, y2, y3], [z1, z2, z3]) -->