1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

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

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

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

10. Fixed Points
Properties of Coherence Threshold

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

12. 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

13. The Bifurcation Scenario
Average fitness experience a jump

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

15. 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

16. 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

17. 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

18. Fidelity of Learning Algorithms

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

20. Asymmetric A matrices
Random Off-Diagonal Elements

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