Knowledge Spaces

0. Overview and Mathematical Glossary.- Main Concepts.- Possible Limitations.- A Practical Application: The Aleks System.- Potential Applications to Other Fields.- On the Content and Organization of this Book.- Glossary of Mathematical Concepts.- Original Sources and Related Works.- 1. Knowledge Structures and Spaces.- Fundamental Concepts.- Knowledge Spaces.- Closure Spaces.- Substructures.- Bases and Atoms.- An Algorithm for Constructing the Base.- An Algorithm for Generating a Space from its Base.- Bases and Atoms: the Infinite Case*.- The Surmise Relation.- Quasi Ordinal Spaces.- Original Sources and Related Works.- Problems.- 2. Well-Graded Knowledge Structures.- Essentially Finite Structures.- A Well-Graded Family of Relations: the Biorders*.- The Infinite Case*.- Finite Learnability.- Original Sources and Related Works.- Problems.- 3. Surmise Systems.- Basic Concepts.- Knowledge Spaces and Surmise Systems.- AND/OR Graphs.- Surmise Functions and Wellgradedness.- Hasse Systems.- Resolubility and Acyclicity.- Original Sources and Related Works.- Problems.- 4. Skill Maps, Labels and Filters.- Skills.- Skill Maps: The Disjunctive Model.- Minimal Skill Maps.- Skill Maps: The Conjunctive Model.- Skill Multimaps: The Competency Model.- Labels and Filters.- Original Sources and Related Works.- Problems.- 5. Entailments and the Maximal Mesh.- Entailments.- Entail Relations.- Meshability of Knowledge Structures.- The Maximal Mesh.- Original Sources and Related Works.- Problems.- 6. Galois Connections*.- Three Exemplary Correspondences.- Closure Operators and Galois Connections.- Lattices and Galois Connections.- Knowledge Structures and Binary Relations.- Granular Knowledge Structures and Granular Attributions.- Knowledge Structures and Associations.- Original Sources and Related Works.- Problems.- 7. Probabilistic Knowledge Structures.- Basic Concepts and Examples.- An Empirical Application.- The Likelihood Ratio Procedure.- Learning Models.- A Combinatorial Result.- Markov Chain Models.- Probabilistic Substructures.- Nomenclatures and Classifications.- Independent Substructures.- Original Sources and Related Works.- Problems.- 8. Stochastic Learning Paths.- A Knowledge Structure in Euclidean Geometry.- Basic Concepts.- Basic Results.- Assumptions on Distributions.- The Learning Latencies.- Empirical Predictions.- Limitations of this Theory.- Simplifying Assumptions.- Remarks on Application and Use of the Theory.- An Application of the Theory to the Case n =2.- Original Sources and Related Works.- Problems.- 9. Descriptive and Assessment Languages.- Languages and Decision Trees.- Terminology.- Recovering Ordinal Knowledge Structures.- Recovering Knowledge Structures.- Original Sources and Related Works.- Problems.- 10. Uncovering the State of an Individual: A Continuous Markov Procedure.- A Deterministic Algorithm.- Outline of a Markovian Stochastic Process.- Notation and Basic Concepts.- Special Cases.- General Results.- Uncovering the Latent State.- Refining the Assessment.- Proofs*.- Original Sources and Related Works.- Problems.- 11. Uncovering the State of an Individual: A Markov Chain Procedure.- Outline of a Markov Chain Procedure.- The Stochastic Assessment Process.- Combinatorial Assumptions on the Structure.- Markov Chains Terminology.- Results for the Fair Case.- Uncovering a Stochastic State: Examples.- Intractable Case.- Original Sources and Related Works.- Problems.- 12. Building the Knowledge Structure in Practice.- Koppen’s Algorithm for Constructing the Knowledge Space.- Kambouri’s Experiment.- Results.- Discussion of Kambouri’s Results.- Cosyn and Thiéry’s Work.- Refining a Knowledge Structure.- Simulation of Various Refinements.- Original Sources and Related Works.- Problems.- References.- Index of Names.- Index of Terms.