# Write Small Functions Using Examples

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We would like to write code that is correct, and have evidence on hand that it is correct. It can help with both issues is to think about the "size" of a functions. Not in the sense of the amount of code that implements a function, although that is interesting, but rather the size of the mathematical function that our code manifests.

For example, in the game of Go there is a condition called *atari* in which a player's stones may be captured by their opponent: a stone with two or more free spaces adjacent to it (called *liberties*) is not in atari. It can be tricky to count how many liberties a stone has, but knowing that determining atari is easy. We might begin by writing a function like this:

boolean atari (int libertyCount) if libertyCount < 2 true false

This is larger than it looks. A mathematical function can be understood as a set, some subset of the Cartesian product of the sets that are its domain (here, `int`

) and range (here, `boolean`

). If those sets of values were the same size as in Java then there would be `2L*(Integer.MAX_VALUE+(-1L*Integer.MIN_VALUE)+1L)`

or 8,589,934,592 members in the set `int`

×`boolean`

. Half of these are members of the subset that is our function, so to produce compelling evidence that our function is correct we would need to check around 4.3×10^{9} examples.

This is the essence of the claim that test cannot prove the absence of bugs. Tests can demonstrate the presence of features, though. But still we have this issue of size.

The problem domain helps us out. The nature of Go means that number of liberties of a stone is not any `int`

, but exactly one of {1,2,3,4}. So we could alternatively write:

LibertyCount = {1,2,3,4} boolean atari(LibertyCount libertyCount) if libertyCount == 1 true false

This is much more tractable, the function computed is now a set with at most 8 members. In fact, 4 checked examples would constitute evidence of complete certainty that the function is correct. This is one reason why its a good idea to use types closely related to the problem domain to write programs, rather than native types. Using domain–inspired types can often make our functions much smaller.