## Modeling language evolution (2004)

Venue: | Foundations of Computational Mathematics |

Citations: | 15 - 3 self |

### BibTeX

@ARTICLE{Cucker04modelinglanguage,

author = {Felipe Cucker and Steve Smale and Ding-xuan Zhou},

title = {Modeling language evolution},

journal = {Foundations of Computational Mathematics},

year = {2004}

}

### OpenURL

### Abstract

A purpose of this paper is to understand the evolution of the languages used by the agents of a society. We focus on language features in which convexity plays a central role. In our model a language is a function from a set X of objects or meanings to a

### Citations

8973 | Statistical Learning Theory
- Vapnik
- 1998
(Show Context)
Citation Context ...). (4) The empirical target function is computable. It approximates the target function well when the number m of samples is large enough (see e.g. [Cucker and Smale 2002; Haussler 1992; Niyogi 1998; =-=Vapnik 1998-=-]). For our purposes we next state Theorem C* of [Cucker and Smale 2002]. Proposition 2 ([Cucker and Smale 2002]) Let F be a compact and convex subset of C (X; Y ) and ρ a probability measure on Z = X... |

251 |
Acoustic Phonetics
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(Show Context)
Citation Context ... X → Y , which is of course continuous. Example 3 We now modify Example 2 to incorporate a convex space of signals Y , to be interpreted as a space of vowel sounds following [De Boer 2001; Fant 1970; =-=Stevens 1998-=-]. The sound spectrum of a vowel sound is represented by a function from frequencies in Hertz (i.e., vibrations per second) to amplitudes. The frequencies corresponding to the peaks (i.e., local maxim... |

224 | On the mathematical foundations of learning
- Cucker, Smale
(Show Context)
Citation Context ...g theory The sampling and subsequent learning algorithm described in Section 4 are an instance of the general situation in learning theory. We next briefly describe this general situation (following [=-=Cucker and Smale 2002-=-]). Let X and Y be as in Section 2 and Z = X ×IR l . Consider a probability measure ρ in Z. For a function f : X → IR l we define its error (with respect to ρ) by ∫ E(f) := ‖f(x) − y‖ Z 2 Y dρ. (2) It... |

209 |
Acoustic theory of speech production
- Fant
- 1960
(Show Context)
Citation Context ... is any f : X → Y , which is of course continuous. Example 3 We now modify Example 2 to incorporate a convex space of signals Y , to be interpreted as a space of vowel sounds following [De Boer 2001; =-=Fant 1970-=-; Stevens 1998]. The sound spectrum of a vowel sound is represented by a function from frequencies in Hertz (i.e., vibrations per second) to amplitudes. The frequencies corresponding to the peaks (i.e... |

136 | Self-organization in vowel systems - Boer - 2000 |

80 | The emergence of linguistic structure: An overview of the iterated learning model
- Kirby, Hurford
- 2001
(Show Context)
Citation Context ... years the subject “Evolution of Language” has seen much activity. We make no attempt to summarize this except to highlight the following five references [De Boer 2001; Ke, Minett, Au, and Wang 2002; =-=Kirby and Hurford 2001-=-; Niyogi 2003a; Nowak, Komarova, and Niyogi 2001] Conversations with Partha Niyogi have been very useful and the references he has given us have been important for us. Also appreciation to Bill Wang i... |

69 | Evolution of universal grammar - Nowak, Komarova, et al. - 2001 |

60 |
Non-Negative Matrices
- Seneta
- 1973
(Show Context)
Citation Context ...square matrices of dimension ni and nj respectively. Note that in this situation, one can not expect that all the agents will eventually use the same language. The matrix Γ is said to be “reducible” [=-=Seneta 1973-=-]. Remark 2 Throughout this paper we assume that ∑ k j=1 γij > 0 for i = 1, . . . , k. This excludes the possibility of a complete immature agent (one with inertia zero) which is not influenced by the... |

21 |
The Informational Complexity of Learning
- Niyogi
- 1998
(Show Context)
Citation Context ... min f∈F Ez(f). (4) The empirical target function is computable. It approximates the target function well when the number m of samples is large enough (see e.g. [Cucker and Smale 2002; Haussler 1992; =-=Niyogi 1998-=-; Vapnik 1998]). For our purposes we next state Theorem C* of [Cucker and Smale 2002]. Proposition 2 ([Cucker and Smale 2002]) Let F be a compact and convex subset of C (X; Y ) and ρ a probability mea... |

12 |
Stable manifolds for differential equations and diffeomorphisms
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(Show Context)
Citation Context ...k and a number 0 < α∗ < 1 such that, for all v ∈ ∆k and w ∈ W , dΛ(w, ∆k) = ‖Λw‖Λ ≤ α∗‖w‖Λ, and dΛ(v + w, ∆k) = dΛ(v, ∆k). Proof. Part (i) is well-known. To prove part (ii) we note that the proof of [=-=Smale 1963-=-, Theorem 3.1], for instance, yields a constant α∗ < 1 and an inner product 〈 , 〉Λ in W satisfying ‖Λw‖Λ ≤ α∗‖w‖Λ for all w ∈ W . For w ∈ W , (V ′ ) T Λw = (Λ T V ′ ) T w = (V ′ ) T w. Since V ′ ̸= 0 ... |

11 |
Global analysis and economics
- Smale
- 1981
(Show Context)
Citation Context ...ere are great limitations to this picture because it takes into account neither the resources nor the preferences of the agents which could play a role in their offers to buy and sell. One might see [=-=Smale 1981-=-] for some background —economic equilibrium theory. 105 Statement of the Main Result For fi ∈ F, i = 1, . . . , k, write Denoting by Λ the matrix Fi = k∑ j=1 λijfj. Λ := ( ) k λij i,j=1 and writing F... |

10 |
Language and Linguistics: An Introduction
- Lyons
- 1981
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Citation Context ..., which of itself is purely abstract, in that it has no physical existence, but which is actualized on particular occasions in the language-behaviour of individual members of the language community. [=-=Lyons 1981-=-, page 10]. 2 “In the last resort, we should have to admit that everyone has his own individual dialect: that he has his own idiolect, as linguists put it. Every idiolect will differ from every other,... |

4 | Self-organization and natural selection in the emergence of vocabulary - Ke, Minett, et al. - 2002 |

3 | Course in General Linguistics. Duckworth. Translated by Roy Harris - Saussure - 1976 |

2 | The Collected Papers of Stephen Smale - Gucker, Wong - 2000 |

2 | Decision Theoretic Generalizations of the PAC Model for Neural Net and Other Leaming Applications - unknown authors - 1995 |

2 |
Phase transitions in language evolution
- Niyogi
- 2004
(Show Context)
Citation Context ...ution of Language” has seen much activity. We make no attempt to summarize this except to highlight the following five references [De Boer 2001; Ke, Minett, Au, and Wang 2002; Kirby and Hurford 2001; =-=Niyogi 2003-=-a; Nowak, Komarova, and Niyogi 2001] Conversations with Partha Niyogi have been very useful and the references he has given us have been important for us. Also appreciation to Bill Wang is acknowledge... |

1 | The computational nature of language learning and evolution. http://www.cs.uchicago.edu/niyogi - Niyogi |