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The Complexity Of Songs




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.

Knuth further demonstrates a way of producing songs with O (\sqrt N) complexity, an approach "...further improved by a Scottish farmer named O. McDonald " (priority disputed).

More ingenious approaches yield songs of complexity (1), e.g. for a song defined by the Recurrence Relation

:S_0=e, S_k = V_kS_{k-1},\, k> 1,
:V_k = 'That's the way,' U 'I like it,' U, for all k > 0
:U= 'uh huh,' 'uh huh'


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