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Exercises

4.9. Exercises#

Solutions to most exercises are available. Fall 2022 is the first public release of these solutions. Though they have been available to Cornell students for a few years, it is inevitable that wider circulation will reveal improvements that could be made. We are happy to add or correct solutions. Please make contributions through GitHub.

Exercise: twice, no arguments [★]

Consider the following definitions:

let double x = 2 * x
let square x = x * x
let twice f x = f (f x)
let quad = twice double
let fourth = twice square

Use the toplevel to determine what the types of quad and fourth are. Explain how it can be that quad is not syntactically written as a function that takes an argument, and yet its type shows that it is in fact a function.

Exercise: mystery operator 1 [★★]

What does the following operator do?

let ( $ ) f x = f x

Hint: investigate square $ 2 + 2 vs. square 2 + 2.

Exercise: mystery operator 2 [★★]

What does the following operator do?

let ( @@ ) f g x = x |> g |> f

Hint: investigate String.length @@ string_of_int applied to 1, 10, 100, etc.

Exercise: repeat [★★]

Generalize twice to a function repeat, such that repeat f n x applies f to x a total of n times. That is,

  • repeat f 0 x yields x

  • repeat f 1 x yields f x

  • repeat f 2 x yields f (f x) (which is the same as twice f x)

  • repeat f 3 x yields f (f (f x))

  • …

Exercise: product [★]

Use fold_left to write a function product_left that computes the product of a list of floats. The product of the empty list is 1.0. Hint: recall how we implemented sum in just one line of code in lecture.

Use fold_right to write a function product_right that computes the product of a list of floats. Same hint applies.

Exercise: terse product [★★]

How terse can you make your solutions to the product exercise? Hints: you need only one line of code for each, and you do not need the fun keyword. For fold_left, your function definition does not even need to explicitly take a list argument. If you use ListLabels, the same is true for fold_right.

Exercise: sum_cube_odd [★★]

Write a function sum_cube_odd n that computes the sum of the cubes of all the odd numbers between 0 and n inclusive. Do not write any new recursive functions. Instead, use the functionals map, fold, and filter, and the ( -- ) operator (defined in the discussion of pipelining).

Exercise: sum_cube_odd pipeline [★★]

Rewrite the function sum_cube_odd to use the pipeline operator |>.

Exercise: exists [★★]

Consider writing a function exists: ('a -> bool) -> 'a list -> bool, such that exists p [a1; ...; an] returns whether at least one element of the list satisfies the predicate p. That is, it evaluates the same as (p a1) || (p a2) || ... || (p an). When applied to an empty list, it evaluates to false.

Write three solutions to this problem, as we did above:

  • exists_rec, which must be a recursive function that does not use the List module,

  • exists_fold, which uses either List.fold_left or List.fold_right, but not any other List module functions nor the rec keyword, and

  • exists_lib, which uses any combination of List module functions other than fold_left or fold_right, and does not use the rec keyword.

Exercise: account balance [★★★]

Write a function which, given a list of numbers representing debits, deducts them from an account balance, and finally returns the remaining amount in the balance. Write three versions: fold_left, fold_right, and a direct recursive implementation.

Exercise: library uncurried [★★]

Here is an uncurried version of List.nth:

let uncurried_nth (lst, n) = List.nth lst n

In a similar way, write uncurried versions of these library functions:

  • List.append

  • Char.compare

  • Stdlib.max

Exercise: map composition [★★★]

Show how to replace any expression of the form List.map f (List.map g lst) with an equivalent expression that calls List.map only once.

Exercise: more list fun [★★★]

Write functions that perform the following computations. Each function that you write should use one of List.fold, List.map or List.filter. To choose which of those to use, think about what the computation is doing: combining, transforming, or filtering elements.

  • Find those elements of a list of strings whose length is strictly greater than 3.

  • Add 1.0 to every element of a list of floats.

  • Given a list of strings strs and another string sep, produce the string that contains every element of strs separated by sep. For example, given inputs ["hi";"bye"] and ",", produce "hi,bye", being sure not to produce an extra comma either at the beginning or end of the result string.

Exercise: association list keys [★★★]

Recall that an association list is an implementation of a dictionary in terms of a list of pairs, in which we treat the first component of each pair as a key and the second component as a value.

Write a function keys: ('a * 'b) list -> 'a list that returns a list of the unique keys in an association list. Since they must be unique, no value should appear more than once in the output list. The order of values output does not matter. How compact and efficient can you make your solution? Can you do it in one line and linearithmic space and time? Hint: List.sort_uniq.

Exercise: valid matrix [★★★]

A mathematical matrix can be represented with lists. In row-major representation, this matrix

\[\begin{split} \begin{bmatrix} 1 & 1 & 1 \\ 9 & 8 & 7 \end{bmatrix} \end{split}\]

would be represented as the list [[1; 1; 1]; [9; 8; 7]]. Let’s represent a row vector as an int list. For example, [9; 8; 7] is a row vector.

A valid matrix is an int list list that has at least one row, at least one column, and in which every column has the same number of rows. There are many values of type int list list that are invalid, for example,

  • []

  • [[1; 2]; [3]]

Implement a function is_valid_matrix: int list list -> bool that returns whether the input matrix is valid. Unit test the function.

Exercise: row vector add [★★★]

Implement a function add_row_vectors: int list -> int list -> int list for the element-wise addition of two row vectors. For example, the addition of [1; 1; 1] and [9; 8; 7] is [10; 9; 8]. If the two vectors do not have the same number of entries, the behavior of your function is unspecified—that is, it may do whatever you like. Hint: there is an elegant one-line solution using List.map2. Unit test the function.

Exercise: matrix add [★★★]

Implement a function add_matrices: int list list -> int list list -> int list list for matrix addition. If the two input matrices are not the same size, the behavior is unspecified. Hint: there is an elegant one-line solution using List.map2 and add_row_vectors. Unit test the function.

Exercise: matrix multiply [★★★★]

Implement a function multiply_matrices: int list list -> int list list -> int list list for matrix multiplication. If the two input matrices are not of sizes that can be multiplied together, the behavior is unspecified. Unit test the function. Hint: define functions for matrix transposition and row vector dot product.