The Complexity of Songs

"The Complexity of Songs" was a journal article published by computer scientist Donald Knuth in 1977, as an in-joke about computational complexity theory. The article capitalizes on the tendency of popular songs to devolve from long and content-rich ballads to highly repetitive texts with little or no meaningful content. The article notes how some songs can reach a complexity level, for a song of length N words, as formula: O(log N). The gist of the article is repeated below, maintaining the wit of the key concepts.

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 the factor c < 1.

Knuth further demonstrates a way of producing songs with O(${\displaystyle {\sqrt {N}}}$) complexity, an approach "further improved by a Scottish farmer named O. MacDonald".

More ingenious approaches yield songs of complexity O(${\displaystyle \log N}$), a class known as "m bottles of beer on the wall".

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