Difference between revisions of "Base cases and identities"
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Revision as of 15:23, 25 February 2008
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 nondegenerate 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
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.:
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))