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.
1 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 = [0] gives us:
sum [] + sum [0] == sum ([] ++ [0]) => 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 [1] == product ([] ++ [1]) -- (using xs = [], ys = [1]) => 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
2 Exercises
What are sensible base cases for these functions?
-
concat, which appends a list of lists (e.g.concat [[1],[],[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.:
- ("endo"-function meaning that the function returns something of the same type as it as it takes as input, (from EndoMorphism in category theory))