The Computational Nature of Language Learning and Evolution Chapter 12 The Origin of Communicative Systems: Linguistic Coherence and Communicative Fitness The Computational Nature of Language Learning and Evolution Partha Niyogi

Contents 12.1 General Model 12.2 Dynamics of a Fully Symmetric System 12.3 Fidelity of Learning Algorithms 12.4 Asymmetric A Matrices 12.5 Conclusions

The Class of Languages Mutual Intelligibility Aij is the Probability that a speaker using language μi is understood by a hearer using μj Symmetric case where Aii = 1 and Aij = a if i ≠ j

Fitness, Reproduction, and Learning Population of constant size Fraction of people who speak the language μj is denoted by xj Fitness Mutual intelligibility between μi and μj Average communicative efficiency of a speaker of μj F0 is the background fitness that does not depend on the person’s language

Fitness, Reproduction, and Learning Differential Reproduction Individuals reproduce in proportion to their fitness Learning Transition Probability Qij to learn from a person with language μi and end up speaking language μj

Population Dynamics Proportion of μj users in the next generation First candidate for the differential equations that characterize the dynamics Aditional Constraints Total population size is constatnt Population sizes are positive Modified differential equation

Dynamics of a Fully Symmetric System Fitness in the fully symmetric case Learning fidelity Differential equation with these assumption

Fixed Points We will examine the one-language solutions Only one language has different frequency Factoring the cubic

Fixed Points The solutions Existence of One-Language Solution Real-valued solutions exist only if D ≥ 0 Existence condition q ≥ q1

Fixed Points Properties of Coherence Threshold

Fixed Points Summary remarks Small values of q(<q1), only the uniform solution exists.

Stability of the Fixed Points The Uniform Solution If q > q2, uniform solution loses stability The Asymmetric Solutions The asymmetric solution is stable every where in the domain of its existence

The Bifurcation Scenario Average fitness experience a jump

The Bifurcation Scenario Stability diagram in terms of the error rate One-grammar solution Uniform solution

Fidelity of Learning Algorithms Memoryless Learning Initial hypothesis Updating hypothesis Transition matrix, T(k) for Markov chain which depednts on the teacher’s language, μk The (i, j) element of Q

Fidelity of Learning Algorithms Memoryless learning Learning accuracy and error rate How many (b) examples are needed? One-language solution Uniform solution loses stability if

Fidelity of Learning Algorithms Batch Learning Total comprehensibility score Set of empirically optimal languages to be Probability that U has a unique member Ej is the event that at least one sentences in S is incomprehensible accordignt to uj

Fidelity of Learning Algorithms

Asymmetric A matrices Breaking the Symmetry of the A Matrix Slightest perturbation:

Asymmetric A matrices Random Off-Diagonal Elements

Conclusions Two major assumptions Natural selection Local learning Primary question is when coherence would emerge in the population