1. Introduction -- 1.1 The Components of a Learning Paradigm -- 1.2 Parametric Hypothesis Spaces -- 1.3 Technical Contents and Major Contributions -- 2. Generalization Error For Neural Nets -- 2.1 Introduction -- 2.2 Definitions and Statement of the Problem -- 2.3 Stating the Problem for Radial Basis Functions -- 2.4 Main Result -- 2.5 Remarks -- 2.6 Implications of the Theorem in Practice: Putting In the Numbers -- 2.7 Conclusion -- 2-A Notations -- 2-B A Useful Decomposition of the Expected Risk -- 2-C A Useful Inequality -- 2-D Proof of the Main Theorem -- 3. Active Learning -- 3.1 A General Framework For Active Approximation -- 3.2 Example 1: A Class of Monotonically Increasing Bounded Functions -- 3.3 Example 2: A Class of Functions with Bounded First Derivative -- 3.4 Conclusions, Extensions, and Open Problems -- 3.5 A Simple Example -- 3.6 Generalizations -- 4. Language Learning -- 4.1 Language Learning and The Poverty of Stimulus -- 4.2 Constrained Grammars-Principles and Parameters -- 4.3 Learning in the Principles and Parameters Framework -- 4.4 Formal Analysis of the Triggering Learning Algorithm -- 4.5 Characterizing Convergence Times for the Markov Chain Model -- 4.6 Exploring Other Points -- 4.7 Batch Learning Upper and Lower Bounds: An Aside -- 4.8 Conclusions, Open Questions, and Future Directions -- 4-A Unembedded Sentences For Parametric Grammars -- 4-B Memoryless Algorithms and Markov Chains -- 4-C Proof of Learnability Theorem -- 4-D Formal Proof -- 5. Language Change -- 5.1 Introduction -- 5.2 Language Change in Parametric Systems -- 5.3 Example 1: A Three Parameter System -- 5.4 Example 2: The Case of Modern French: -- 5.5 Conclusions -- 6. Conclusions -- 6.1 Emergent Themes -- 6.2 Extensions -- 6.3 A Concluding Note -- References.