  Languages across the world exhibit Zipf's law of abbreviation, namely more
frequent words tend to be shorter. The generalized version of the law - an
inverse relationship between the frequency of a unit and its magnitude - holds
also for the behaviours of other species and the genetic code. The apparent
universality of this pattern in human language and its ubiquity in other
domains calls for a theoretical understanding of its origins. To this end, we
generalize the information theoretic concept of mean code length as a mean
energetic cost function over the probability and the magnitude of the types of
the repertoire. We show that the minimization of that cost function and a
negative correlation between probability and the magnitude of types are
intimately related.
