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.-

Jean-Paul Doignon is a professor of the mathematics department of the Université Libre de Bruxelles, Belgium. His research covers various aspects of discrete mathematics (graphs, ordered sets, convex polytopes, etc.) and applications to behavioral sciences (preference modelling, choice models, knowledge assessment, etc.). Jean-Claude Falmagne is emeritus professor of cognitive sciences at the Universtiy of California, Irvine. His research spans topics in psychophysics, probabilistic choice theory, combinatorics, measurement theory, educational technology, and philosophy of science.