List Comprehensions

List Comprehensions

List comprehensions provide a convenient syntactical mechanism to create lists. The syntax is similar to the mathematical set notation.

In Math: { 2 × n | n ∈ N }

In English: for all n in the naturals, two times n

In Haskell:

Test>
[2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, ……] :: [Integer]

The members of the input list can be selectively tested with Bool-typed expressions, separated by commas.

In Math: { n2 | n ∈ N, n even }

In Haskell:

Test>
[4, 16, 36, 64, 100, 144, 196, 256, 324, 400, 484, 576, 676, 784, 900, 1024, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 6084, 6400, 6724, 7056, 7396, 7744, 8100, 8464, 8836, 9216, 9604, 10000, 10404, 10816, 11236, 11664, 12100, 12544, 12996, 13456, 13924, 14400, 14884, 15376, 15876, 16384, 16900, 17424, 17956, 18496, 19044, 19600, 20164, 20736, 21316, 21904, 22500, 23104, 23716, 24336, 24964, 25600, 26244, 26896, 27556, 28224, 28900, 29584, 30276, 30976, 31684, 32400, 33124, 33856, 34596, 35344, 36100, 36864, …, ……] :: [Integer]

That is, the following expression is invalid.

Test> 
<interactive>:1:14: error: Variable not in scope: n :: Integer

Exercise: Powers of 2

List powers of 2 from 20 to 210 in ascending order.

Solution>

Exercise: Double of the First 10 Square Numbers

List the double of the first 10 square numbers.

Solution>

Exercise: False and True Repeated

Generate a list of length 10 whose elements are False, True, False, True, … (alternating)

Solution>

Exercise: Searching Power of 2 Big Enough

Which is the least power of 2 which is bigger than 1020?

Solution>

Hint: Use the head function:

head :: [a] -> a

Exercise: 1024 > 1000, …

Which is the least natural number n for which 1024n > 2 * 1000n?

Solution>

Exercise: Divisors of 60

List the divisors of 60.

Solution>

Exercise: Number of Divisors

How many divisors does 60 have?

Solution>

Exercise: Prime Test

Is 123457 a prime number?

Solution>

Exercise: Hours and Minutes

Generate a list that contains all (hour, minute) pairs in a day.

Solution>

Exercise: 1,1,2,1,2,3,1,2,3,4,…

Generate the following list: [1,1,2,1,2,3,1,2,3,4,…].

Solution>

Exercise: 0,1,0,-1,0,1,2,1,0,-1,-2,-1,0,1,2,3,2,1,0,-1,…

Generate the following list: 0,1,0,-1,0,1,2,1,0,-1,-2,-1,0,1,2,3,2,1,0,-1,…

Solution>

Hint: Use concat.

concat :: Foldable t => t [a] -> [a]

Exercise: Month and Days

Generate a list that contains all (month, day) pairs in a 365-day year.

Solution>

Hint: It is easier with the elem function.

Exercise: Table

Test>

Exercise: Pattern from Small Circles

Test>

Exercise: Chess Table

Test>

Tuples

Tuples may have multiple elements with different types. Remember that lists can contain only one type.

Test>
(True, 'a') :: (Bool, Char)

In the above example (True, ‘a’) is the value of the tuple, and (Bool, Char) is the type.

Unlike a list, a tuple can only be compared to a tuple of the same length and type.

Test>
False :: Bool

Zipping

The function zip combines two lists. Each element from the first list is put in a tuple with the corresponding element from the second list.

zip :: [a] -> [b] -> [(a, b)]
Test>
[('a', 1), ('b', 2)] :: [(Char, Integer)]

Zipping finishes if one of the lists ends.

Exercise: Numbering

Generate the list [(1,'a'),(2,'b'),…(…,'z')].

Solution>

Exercise: Dominoes

Generate a list of all “dominoes”: [(0,0),(0,1),...,(0,9),(1,1),...,(9,9)].

Domino (1,0) shall not be in the list because it is already in it as (0,1).

Note: The dominoes in this exercise should contain at least zero, at most nine dots (i.e. they are numbered from (0,0) to (9,9)).

Solution>

Exercise: Precedence of AND and OR

Find all Boolean values a, b, and c such that

(a || (b && c)) /= ((a || b) && c)
Solution>

The type of the result should be [(Bool,Bool,Bool)]

Exercises with Infinite Lists

You can use the take function to observe the beginning of infinite lists.

take :: Int -> [a] -> [a]
Test>
[1, 3, 5, 7, 9, 11] :: [Integer]

What is the difference between take and !!?

Test>
Test>

Explain this result:

Test>

Exercise: All Pairs of Natural Numbers

Generate the list [(0,10),(1,9),…,(10,0)].

Enumerate all pairs of natural numbers, e.g. [(0,0),(0,1),(1,0),(0,2),(1,1),(2,0), …].

Solution>

Exercise: 1, 2, 2, 3, 3, 3, …

Generate the following list: [1,2,2,3,3,3,4,4,4,4, …]. The ith number appears i times in the list.

Solution>

Exercise: 1, 2, 1, 2, 3, 2, 1, …

Test the concat function. concat will combine two lists.

Test>
concat :: Foldable t => t [a] -> [a]

Generate the following sequence: 1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,2,3,4,5,6,5,4,3,2,1,…

Solution>

Exercise: Stars

Test the unwords function.

Test>
unwords :: [String] -> String

Generate the following infinite string: "* ** *** **** *****…

Solution>

Exercise: Omitting Square Numbers

Generate a list that is derived from the set of natural numbers by omitting the square numbers.

Solution>

Exercise: Finding an Arithmetic Sequence

Consider the following equations: 3+4+5+6=18; 15+16+…+34+35=525. They consist of sequences of ascending numbers whose sum equals the product of the elements on the left and right sides.

Generate all such sequences ordered by the last element.

For testing, only the first 6 sequences must be generated.

Solution>

Hint: Use the init and tails functions from the List module.