# Base cases and identities

(Difference between revisions)

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
```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.:
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))