Solutions to Haskell from First Principles

January 23, 02022

This is just one of many textbooks I'm working through on my quest to learn Haskell.

Chapter 1. Anything from Almost Nothing

1. $${(\lambda a b c . c b a)zz(\lambda w v.w)}$$ (Currying for clarity) $${(\lambda a . \lambda b . \lambda c . cba)(z)z(\lambda w . \lambda v.w)}$$ (Eliminate the head and apply the variiable) $${(\lambda b. \lambda c. cbz)(z)(\lambda w. \lambda v.w)}$$ (Eliminate from left to right, etc) $${(\lambda c. czz)(\lambda w. \lambda v.w)}$$ $${(\lambda w. \lambda v.w)(z)z}$$ $${(\lambda v . z)(z)}$$ (Becoming the singular:) $${z}$$ 2. $${(\lambda x. \lambda y.xyy)(\lambda a.a)b}$$ $${\lambda y.(\lambda a.a)yy)b}$$ $${\lambda a.a)(b)b}$$ $${b.b}$$

Chapter 2. Basic expressions and functions

Most arithmetic functions in Haskell (and math) are left-associative. So 2*3*4 is evaluated as (2*3)*4. Exponentiation is right-associative, so 2 ^ 3 ^ 4 is evaluated as 2 ^ ( 3 ^ 4). Unlike multiplication, which is commutative and not affected by order of operations, which expression we evaluate with exponentiation is more important. i.e.:

 2 ^ 3 ^ 4

   2417851639229258349412352

Whereas

(2 ^ 3) ^ 4

4096

Exercises: Parentheses and Association:

Here's some revision testing our intuitions around order-of-operations. In Haskell, we can get a numerical representation of the priority of an infix operator by running :info. i.e. :info (*)
7 *

1.

a) 8+7 *9

  b) (8+7)* 9

Yes, parentheses change the evaluation of this equation.

2.

  a) perimeter x y = (x *2) + (y+2)

  b) perimeter x y = x * 2 + y * 2

No, as multiplication is already prioritised over addition, the evaluation result should be the same.

3.

    a) f x = x / 2 + 9

    b) f x = x / (2 + 9)

Yes, a will evaluate x / 2 first, while b will evaluate (2 + 9) first.