Base cases and identities

Sometimes it's hard to work out what the base case of a function should be. Sometimes you can work it out by examining the identities of your operations.

Examples

As a simple example, consider the function sum, which takes a list of numbers and adds them:

sum [] = ???
sum (x:xs) = x + sum xs

where `???` is yet to be determined. It's not obvious what the `sum` of an empty list should be, so let's try to work it out indirectly.

The sum function is about adding things. For non-degenerate cases at least, we want `sum` to obey these rules:

sum [x] == x
sum xs + sum ys == sum (xs ++ ys)

Substituting xs = [] and ys =  gives us:

sum [] + sum  == sum ([] ++ )
=> sum [] + 0 == 0
=> sum [] == 0

...and there's our base case.

Similarly, for the `product` function:

product [x] == x
product xs * product ys == product (xs ++ ys)
=> product [] * product  == product ([] ++ )    -- (using xs = [], ys = )
=> product [] == 1

In both of these cases, the base case is the identity of the underlying operation. This is no accident, and the reason should be obvious:

product [] * product [x] == product ([] ++ [x])
=> product [] * x == x

It follows that `product []` should be the identity for multiplication.

Sometimes there is no identity. Consider this function, for example, which returns the minimum value from a list:

minimum [x] == x
minimum xs `min` minimum ys == minimum (xs ++ ys)
=> minimum [] `min` minimum [x] == minimum ([] ++ [x])
=> minimum [] `min` x == x

The only sensible value for minimum [] is the maximum possible value for whatever type x has. Since there is no such value in general (consider x :: Integer, for example), minimum [] has no sensible value. Better to use a foldr1 or foldl1 pattern instead:

minimum [x] = x
minimum (x:xs) = x `min` minimum xs

Exercises

What are sensible base cases for these functions?

• concat, which appends a list of lists (e.g. concat [,[],[2,3]] == [1,2,3]).
• and, which takes a list of Bool values and logically "ands" (&&) them together.
• or, which takes a list of Bool values and logically "ors" (||) them together.
• xor, which takes a list of bool values and logically "exclusive ors" them together.
• greatest_common_divisor, which returns the GCD of a list of integers. (The GCD of two integers is the largest number which divides evenly into them both.)
• least_common_multiple, which returns the LCM of a list of integers. (The LCM of two integers is the smallest number which they both evenly divide into.)
• compose, which composes a list of "endo"-functions e.g.:
compose [recip,(** 2),sin,(* 2 * pi)] = recip . (** 2) . sin . (* 2 * pi) = \x -> recip (sin (x * 2 * pi) ** 2)
("endo"-function meaning that the function returns something of the same type as it as it takes as input, (from endomorphism in category theory))