1. Summarized by In-Hee Lee 2009. 07. 31.
The Computational Nature of Language Learning and Evolution 6. Language Changes – Multiple Languages 6.2 Example 1: A Three Parameter System
Summarized by In-Hee Lee

2. Overview for learning system
Population at t … L1 L4 L2 L7 P1 P4 P2 P7
Child Population …
Sampled sentences are presented
TLA L4
TLA L2
TLA L2
TLA L5
Population at t + 1 …

3. A Learning System Triple
G : three-parameter system -> 8 possible languages.
A : memoryless algorithm -> TLA
±single value constraint
± greediness constraint
{Pi}: sentence distribution of language i -> uniform on degree-0 sentences.
Finite sample sentences (128) to each learner.

4. 6.2.1 Homogeneous Initial Populations
4 variations on learning algorithms
TLA(+Single Value, +Greedy)
TLA(– Single Value, +Greedy)
TLA(±Single Value, – Greedy)

5. 6.2.1 Homogeneous Initial Populations Variation 1: TLA(+Single Value, +Greedy)
All +V2 languages are relatively stable.
Populations speaking –V2 language all drift to +V2 languages.
The rates at which the linguistic composition change vary from language to language.
A homogeneous population can split into different linguistic groups.
The observed instability and drifts are from the learning algorithm (to some extent).

6. Compare the rate of change (drift)
L1 → L2
L5 → L2

7. 6.2.1 Homogeneous Initial Populations Variation 2: TLA(–Single Value, +Greedy)
L4, L6 L8
All populations eventually drift to a similar population mixture. (Why? is left as open question.)
All populations drift to a population mixture of only +V2 languages (L2, L4, L6, L8).

8. 6.2.1 Homogeneous Initial Populations Variation 3,4: TLA(±Single Value, –Greedy)
Dropping greediness constraint
3. single step constraint without greediness: choose any new state that is at most one parameter away from the current one.
4. without both single step and greediness: choose any new state at each step.
Both yield dynamical systems that arrive at the same population mixture after 30 generations.
Final mixture contains all languages in significant proportion. No languages has disappeared.

9. With single step
Without single step
Both yield dynamical systems that arrive at the same population mixture after 30 generations. But the path is slightly different.

10. 6.2.2 Modeling Diachronic Trajectories
Diachronic population trajectories open take “S-shape.”
From the point of view of this book, the trajectories might or might not be S-shaped, and might have varying rate of change.
Other factors that affect evolutionary trajectories
Maturation time : the number of sentences available to learner before it internalize its adult grammar.
Probability distribution : the distribution according to which sentences are presented to the learner.

11. 6.2.2 Modeling Diachronic Trajectories
The effect of maturation time or sample size
TLA(–Single Value, +Greedy)
Sample size from 8, 16, 32, 64, 128, 256
Initial rate of change is highest when the maturation time is smallest.
The stable population composition depends on maturation time.
The trajectories do not have an S-shape curve.
Sample size increasing
L2 proportion

12. 6.2.2 Modeling Diachronic Trajectories
The effect of sentence distributions
TLA(+Single Value, +Greedy)
An example focusing on the interaction between L1 and L2 L1 L2
1-p p
We can drive the population towards certain direction by changing sentence distributions.
We can examine the conditions needed to generate such change.

13. 6.2.3 Nonhomogeneous populations: Phase-space plots
Phase-space plot based on the relationship between the state of the population in one generation and the next.
(t): 8 dimensional vector corresponding to the proportion of each language users. L2
Phase-space plot for TLA(– Single Value, + Greedy)
: Projection to (L1, L2) plane L1

14. 6.2.3 Nonhomogeneous populations: Phase-space plots
Stability issues
Many initial conditions yield trajectories that seem to converge to a single point in state space.
What are the conditions for stability?
How many fixed points are there?
How can we solve for them?
The equations to characterize the stable population mix.

15. 6.2.3 Nonhomogeneous populations: Phase-space plots
The equations to characterize the stable population mix.
(from Section 6.1) 1 2 3

16. 6.2.3 Nonhomogeneous populations: Phase-space plots
By letting (t + 1) = (t), we get