"The Complexity of Songs" was an article published by Donald Knuth, an example of an in-joke in computer science, namely, in computational complexity theory. The article capitalizes on the tendency of popular songs to evolve from long and content-rich ballads to highly repetitive texts with little or no meaningful content.[1]
With a grain of truth, Knuth writes that "...our ancient ancestors invented the concept of refrain" to reduce the space complexity of songs, which becomes crucial when a large number of songs is to be committed to one's memory. Knuth's Lemma 1 states that if N is the length of a song, then the refrain decreases the song complexity to cN, where c < 1. [2]
Knuth further demonstrates a way of producing songs with O(
) complexity, an approach "...further improved by a Scottish farmer named O. MacDonald" (priority disputed).[2]
More ingenious approaches yield songs of complexity O(logN), a class known as "m
bottles of beer on the wall". Finally, progress during the twentieth
century—stimulated by the fact that "the advent of modern drugs has led
to demands for still less memory"—leads to the ultimate improvement:
Arbitrarily long songs with space complexity O(1), e.g. for a song to
be defined by the recurrence relation[2]
- Vk = 'That's the way,' U 'I like it,' U, for all k > 0
- U = 'uh huh, uh huh'