Introduction
In a seminal paper, Charles Hockett BIBREF0 identified duality of patterning as one of the core design features of human language. A language exhibits duality of patterning when it is organized at two distinct levels. At a first level, meaningless forms (typically referred to as phonemes) are combined into meaningful units (henceforth this property will be referred to as combinatoriality). For example, the English forms /k/, /a/, and /t/ are combined in different ways to obtain the three words /kat/, /akt/, and /tak/ (respectively written 'cat', 'act' and 'tack'). Because the individual forms in them are meaningless, these words have no relation in meaning in spite of being made of the same forms. This is a very important property, thanks to which all of the many words of the English lexicon can be obtained by relatively simple combinations of about forty phonemes. If phonemes had individual meaning, this degree of compactness would not be possible. At a second level, meaningful units (typically referred to as morphemes) are composed into larger units, the meaning of which is related to the individual meaning of the composing units (henceforth this property will be referred to as compositionality). For example, the meaning of the word 'boyfriend' is related to the meaning of the words 'boy' and 'friend' which composed it. The compositional level includes syntax as well. For example, the meaning of the sentence 'cats eat fishes' is related to the meaning of the words 'cats', 'eat', and 'fishes'. In this paper, for the sake of simplicity, we focus exclusively on the lexicon level. This has to be considered as a first step towards the comprehension of the emergence of complex structures in languages.
Quantifying duality of patterning
In this section we quantify the notion of duality of patterning as observed in real languages in order to provide suitable measures for the combinatoriality and compositionality.
Modelling the emergence of duality of patterning
We now focus on the mechanisms that could lead to the establishment of duality of patterning in a lexicon. There have been a number of previous works devoted to explain the emergence of combinatoriality and compositionality. A thorough review of the attempts presented in literature is far from the scope of the present paper. Here we shall only focus on a few aspects which are relevant for our purposes.
It should be remarked that the two facets of duality of patterning have often been studied independently from each other BIBREF3 , BIBREF4 , BIBREF5 , BIBREF6 . It should also be remarked that often studies in this ares have been focused on evolutionary times scales (e.g., BIBREF7 , BIBREF8 , BIBREF9 ), disregarding in this way the peer-to-peer negotiation taking place on cultural time-scales in large populations. In contrast, there is evidence suggesting that humans are capable of evolving languages with duality of patterning in the course of only one or two generations (consider for instance Nicaraguan Sign Language BIBREF10 or the emergence of Pidgins and Creole languages BIBREF11 ).
Here we aim at explaining in an unitary framework the co-emergence of combinatoriality and compositionality. In addition, unlike previous approaches that looked for the emergence of meaning-symbols compositional mappings out of a small bounded set of predefined symbols available to the population, our approach adopts an open-ended set of forms and it does not rely on any predefined relations between objects/meanings and symbols. For instance we require combinatoriality to emerge out of a virtually infinite set of forms which are freely provided to a blank slate of individuals. Such set can only be limited by means of self-organization through repeated language games, the only purpose being that of communication. In addition, with our simple representation of the conceptual space, modeled as a graph, we do not hypothesize any predefined linguistic category or predefined meaning. This choice also allows to model the effect of differently shaped conceptual spaces and of conceptual spaces that may differ from individual to individual.
Discussion and perspectives
In this paper we have investigated duality of patterning at the lexicon level. We have quantified in particular the notions of combinatoriality and compositionality as observed in real languages as well as in a large-scale dataset produced in the framework of a web-based word association experiment BIBREF1 . We have paralleled this empirical analysis with a modeling scheme, the Blending Game, whose aim is that of identifying the main determinants for the emergence of duality of patterning in language. We analyzed the main properties of the lexicon emerged from the Blending Game as a function of the two parameters of the model, the graph connectivity $p_{link}$ and the memory scale $\tau $ . We found that properties of the emerging lexicon related to the combinatoriality, namely the words length distribution, the frequency of use of the different forms and a measure for the combinatoriality itself, reflect both qualitatively and quantitatively the corresponding properties as measured in human languages, provided that the memory parameter $\tau $ is sufficiently high, that is that a sufficiently high effort is required in order to understand and learn brand new forms. Conversely, the compositional properties of the lexicon are related to the parameter $p_{link}$ , that is a measure of the level of structure of the conceptual graph. For intermediate and low values of $p_{link}$ , semantic relations between objects are more differentiated with respect to the situation of a more dense graph, in which every object is related to anyone else, and compositionality is enhanced. In summary, while the graph connectivity strongly affects the compositionality of the lexicon, noise in communication strongly affects the combinatoriality of the lexicon.
These results are important because they demonstrate for the first time that the two sides of duality of patterning can emerge simultaneously as a consequence of a purely cultural dynamics in a simulated environment which contains meaningful relations. Many directions are open for future investigations. First of all to elucidate the emergence of duality of patterning at the syntax level beyond that of the lexicon. In addition many different manipulations of our modelling scheme are possible. One very interesting consists in relaxing the assumptions that the conceptual space of all the individuals are identical and modelled as a static graph, imaging instead that the conceptual space of each individual gets continuously reshaped by the interactions among the users. In this way one would realize a truly co-evolution of the conceptual spaces of the individuals and of their inventories of associations between objects and words. Finally it is worth mentioning how recent advances in information and communication technologies allow nowadays the realization of focused experiments also in the framework of the emergence of linguistic structures and a general trend is emerging for the adoption of web-games (see for instance the recently introduced Experimental Tribe platform: www.xtribe.eu) as a very interesting laboratory to run experiments in the social-sciences and whenever the contribution of human beings is crucially required for research purposes. This is opening tremendous opportunities to monitor the emergence of specific linguistic features and their co-evolution with the structure of out conceptual spaces.
Acknowledgements
The authors acknowledge support from the KREYON project funded by the Templeton Foundation under contract n. 51663. It is pleasure to warmly thank Bruno Galantucci with whom part of this work has been carried out.

Question: What other models are compared to the Blending Game?
Answer: Unanswerable

Question: What empirical data are the Blending Game predictions compared to?
Answer: words length distribution, the frequency of use of the different forms and a measure for the combinatoriality