Universal scaling laws for hierarchical complexity in languages, organisms, behaviors and other combinatorial systems.

There are many complex systems in nature where components, or "words", are combined together to make expressions, or "sentences". Such combinatorial systems include: (1) human language, where sentences are composed of words; (2) bird vocalization, where songs are built from syllables; (3) organisms, where organism-expressions (e.g. the tonsil) are made out of cells; (4) behavioral repertoire, where mammalian behavior consists of a temporal arrangement of muscle contractions; (5) universities, where student academic degrees are comprised of departmental concentrations; and (6) electronic devices, where the device's actions are implemented via strings of button-presses. My central aim here is to discover how combinatorial systems accommodate greater numbers of expressions; that is, what changes do combinatorial systems undergo when they "say more things?" Are there general laws characterizing the properties of combinatorial systems as the number of expressions increases? If so, what are they? My main result is that, in all the kinds of combinatorial system mentioned above, there appear to be general laws describing how combinatorial systems change as they become more expressive. In particular, in each of these cases, increase in expression complexity (i.e. number of expressions the combinatorial system allows) is achieved, at least in part, by increasing the number of component types. Each kind of system follows one of two kinds of scaling law. In the first kind of scaling law, expression complexity increase is carried out exclusively by increasing the number of component types; the number of components per expression (i.e. the expression length) remains invariant. This applies to human language over history, bird vocalization, organisms in phylogeny and ontogeny, and universities. In the second kind of scaling law, expression complexity is accomplished by increasing in a law-like manner both the number of component types and the expression length. This applies to two cases of the ontogeny of language-the development of words and sentences, and the development of phonemes and morphemes-and to mammalian behavior. By treating these diverse systems as combinatorial systems we, in addition to elucidating general principles underlying such systems, gain insight into each kind of system mentioned.