Write Small Functions Using Examples

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The [http://www.digital60.org/birth/program/firstprog.html earliest recorded program] for an electronic computer is covered in crossings-out, amendments, corrections and expressions of uncertainty. From the very beginning programers have struggled to write code that is correct and that they know is correct. One approach that can help with both issues is to think about the "size" of the functions we write. 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.
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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.
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For example, in the game of Go there is a condition called ''atari'' in which one or more of a player's stones may be captured by their opponent. The rule is that a stone with two or more free spaces adjacent to it on the board (these are called ''liberties'') is not in atari. While it can be surprisingly tricky to work out how many liberties a stone has, once we know that working out if the stone is in atari is easy. We might begin by writing a function (in pseudo–code) like this:
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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. The rule is that a stone with two or more free spaces adjacent to it (these are called ''liberties'') is not in atari. While it can be surprisingly tricky to count how many liberties a stone has, once we know that working out if the stone is in atari is easy. We might begin by writing a function (in pseudo–code) like this:
boolean atari (int libertyCount)
boolean atari (int libertyCount)
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false
false
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This function is simply ''enormous''. A mathematical function can be understood as a set, specifically the product of the sets that are its domain and range. Here the domain is the set of all <code>int</code> values and the range is the set of all <code>boolean</code> values. If these were the same size as the corresponding ones in Java then there would be <code>2L * (Integer.MAX_VALUE + (-1L * Integer.MIN_VALUE) + 1L)</code> or 8,589,934,592 members of the set <code>int * boolean</code>. The function <code>atari</code> as written is also very trivial, but because the function it models is so huge it would be very hard to produce evidence that it is correct. It would be ludicrous to set about creating eight and a half thousand million test cases.
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It might not look it, but this function is pretty huge. A mathematical function can be understood as a set, specifically the Cartesian product of all the sets that are its domain and range. Here the domain is the set of all <code>int</code> values and the range is the set of all <code>boolean</code> values. If these were the same size as in Java then there would be <code>2L * (Integer.MAX_VALUE + (-1L * Integer.MIN_VALUE) + 1L)</code> or 8,589,934,592 members of the set <code>int * boolean</code>. The function <code>atari</code> as written is trivial, but because the function it models is so big it would be very hard to produce strong evidence that it is correct. It would be ludicrous to set about creating 8.6&times;10<sup>9</sup>; test cases. This is the essence of the claim that test cannot prove the absence of bugs.
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Tests can demonstrate the presence of features, though. But still we have the problem of size.
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The problem domain helps us out. The number of liberties is not any integer, or even any <code>int</code>, but exactly one of {1,2,3,4}.

Revision as of 09:55, 2 August 2009

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. The rule is that a stone with two or more free spaces adjacent to it (these are called liberties) is not in atari. While it can be surprisingly tricky to count how many liberties a stone has, once we know that working out if the stone is in atari is easy. We might begin by writing a function (in pseudo–code) like this:

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

It might not look it, but this function is pretty huge. A mathematical function can be understood as a set, specifically the Cartesian product of all the sets that are its domain and range. Here the domain is the set of all int values and the range is the set of all boolean values. If these 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 of the set int * boolean. The function atari as written is trivial, but because the function it models is so big it would be very hard to produce strong evidence that it is correct. It would be ludicrous to set about creating 8.6×109; test cases. 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 the problem of size.

The problem domain helps us out. The number of liberties is not any integer, or even any int, but exactly one of {1,2,3,4}.

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