.... the origins of language has been dominated by verbal theories, both in scienti c publications (see e.g. 4] and in popular, best selling books (e.g. 1] Recently also mathematical models of the evolution of language, especially those of Martin Nowak et al. have received much attention (e.g. [6]) These models are sometimes seen as a validation of the earlier verbal theories. Steven Pinker, e.g. writes in the accompanying news story of [7] that the paper shows the evolvability of [one of ] the most striking features of language , i.e. its compositionality. Although we appreciate the ....

.... cases, and too seldomly systematically compared with each other and with mathematical models (the review papers [8, 3] are exceptions, although they unfortunately do not discuss 2 mathematical models) In this paper we explore the similarities between a recently published mathematical model [6], our own A life simulations [9] and the model of Kirby [5] We believe that such an approach can eventually both avoid the problematic simpli cations of mathematical models, and the ad hoc ness of many A life models. In the conference presentation we will also discuss some shortcomings of ....

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Martin A. Nowak, Natalia Komarova, and Partha Niyogi. Evolution of universal grammar. Science, 291:114-118, 2001.

.... origins of language has been dominated by verbal theories, both in scienti c publications (see e.g. 9] and in popular, best selling books (e.g. 19, 7] Recently also mathematical models of the evolution of language, especially those of Martin Nowak et al. have received much attention (e.g. [16, 17, 15]) These models are sometimes seen as a validation of the earlier verbal theories. Steven Pinker, e.g. writes in the accompanying news story of [17] that the paper shows the evolvability of [one of ] the most striking features of language , i.e. its compositionality. Although we appreciate the ....

....shed light on both the intricacies of the dynamics of language evolution and the explanatory role of selforganization. In this paper we will rst discuss some of the exposed shortcomings of verbal and mathematical theories. Next we will explore how the combination of a mathematical model (from [15]) and A life simulations can help to adapt and extend the existing language evolution scenario s. We believe that such an approach can eventually both avoid the problematic simpli cations of mathematical models, and the ad hoc ness of many A life models. In some sense, this paper thus aims to ....

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Martin A. Nowak, Natalia Komarova, and Partha Niyogi. Evolution of universal grammar. Science, january 2001.

.... (1994, 1999) All humans, but no animals have a language instinct (Bickerton, 1990; Deacon, 1997; Hauser, 1996; Brandon Hornstein, 1986; Pinker Bloom, 1990) The purpose of this paper is to develop a mathematical theory for the evolutionary and population dynamics of grammar acquisition (Nowak et al. 2001). In accordance with mainstream linguistic theory, we assume that children have a search space that consists of n candidate grammars, G # , 2 , G # . Then they hear sample sentences and decide which of the possible grammars to use. Note that the number of candidate grammars can also be ....

....will be maintained in the population. However, in order to demonstrate that universal grammar has come about by means of Darwinian evolution, it is important to look at competition of di erent types of universal grammars. Equations of type (2) can be used. The rst step has been made in Komarova Nowak (2001), where a one parametric family of universal grammars was considered. All the universal grammars were identical (had the same search space and the same learning mechanism) except for the number, b, of sampling events available to children during the grammar learning phase. As a result, an ....

NOWAK, M. A., KOMAROVA,N.L.&NIYOGI, P. (2001). Evolution of universal grammar. Science, to appear.

....n ) denote the fraction of people who speak grammars G 1 through G n of U 2 . The total population size is scaled to unity: n i1 (x i # y i ) 1. The language acquisition devices are inherited genetically. The system of 2n equations describing the coexistence of U 1 and U 2 is as follows (Nowak et al. 2001): x r(b 1 ) n j 1 x j f (1) j Q ji (b 1 ) x i , 1) y i r(b 2 ) n j 1 y j f (2) j Q ji (b 2 ) y i , i 1, n: 2) The left hand side of these equations contains the timederivative of the frequency of each grammar. We shall now explain the terms in the right hand side ....

....6 j, Q ij (b 2 ) q(b 2 ) i j, 1 q(b 2 ) n 1) i 6 j. A8) The concrete form of the function q(b) is given by the learning mechanism. If the entire population uses U 1 , there are n identical one grammar solutions corresponding to each of the grammars G k (Komarova et al. 2001; Nowak et al. 2001). These solutions exist only if q(b 1 )5q c , where the threshold value of q is given by q c 2 a p 1 # a p # O(1 n) A9) The one grammar solutions have the form x k X , x i (1 X) n 1) for i 6 k; y j 0 8j, A10) where X is given in terms of a, n and q. Also we can show that ....

Nowak, M. A., Komarova, N. L. & Niyogi, P. 2001 Evolution of universal grammar. Science 291, 114^118.

....instance, it often makes sense to talk about a lexical matrix as a formal description of human mental vocabularies. It is introduced to describe the arbitrary relations between discrete words and discrete concepts of human languages ( Hurford 1989] Miller 1996] Regier et al. 2001] Komarova Nowak 2001]) Each column of the lexical matrix corresponds to a particular word meaning (or concept) each row corresponds to a particular word form (or word image) In the Saussurean terminology of arbitrary sign, the lexical matrix provides the link between signifie and signifiant ( Saussure 1983] An ....

.... Evans 1993, Cheney Seyfarth 1990] A classic example of this is alarm calls in primates. There are a finite number of referents that are coded using acoustic signals and decoded appropriately by recipients. Infinite association matrices can be used as a description of human languages ([Nowak et al. 2001], Komarova et al. 2001] Human grammars mediate a complex mapping between form and meaning. There, the space of possible signals is the set of all strings (sentences) over a finite syntactic alphabet and the set of possible meanings is the set of all strings over some semantic alphabet. Most ....

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Nowak, M.A., Komarova, N.L. & Niyogi, P. (2001) Evolution of universal grammar, Science, 291, pp. 114--118. 37

....us to delicate question of the rate of convergence to stable laws and tail estimates for stable laws. Introduction The original motivation for the work in this paper was provided by the first named author s research in learning theory, specifically in various models of language acquisition (see [KNN2001, NKN2001, KN2001]) and more specifically yet by the analysis of the speed of convergence of the memoryless learner algorithm. The setup is described in some detail in section 3.1, but here we will just recall the essentials: there is a collection of concepts R 1 , R n and words which refer to this ....

Nowak, M. A., Komarova, N. L., Niyogi, P. (2001) Evolution of universal grammar, Science 291, 114-118. Appendix A. Tails of the limiting density g Let us study the behavior of the density (22) away from zero. We

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