  The complex organization of syntax in hierarchical structures is one of the
core design features of human language. Duality of patterning refers for
instance to the organization of the meaningful elements in a language at two
distinct levels: a combinatorial level where meaningless forms are combined
into meaningful forms and a compositional level where meaningful forms are
composed into larger lexical units. The question remains wide open regarding
how such a structure could have emerged. Furthermore a clear mathematical
framework to quantify this phenomenon is still lacking. The aim of this paper
is that of addressing these two aspects in a self-consistent way. First, we
introduce suitable measures to quantify the level of combinatoriality and
compositionality in a language, and present a framework to estimate these
observables in human natural languages. Second, we show that the theoretical
predictions of a multi-agents modeling scheme, namely the Blending Game, are in
surprisingly good agreement with empirical data. In the Blending Game a
population of individuals plays language games aiming at success in
communication. It is remarkable that the two sides of duality of patterning
emerge simultaneously as a consequence of a pure cultural dynamics in a
simulated environment that contains meaningful relations, provided a simple
constraint on message transmission fidelity is also considered.
