Full Description
This is the first book which gives a unified theory for countable and uncountable computable structures.
The work treats computable linear orderings, graphs, groups and Boolean algebras unified with computable metric and Banach spaces, profinite groups, and the like. Further, it provides the first account of these that exploits effective versions of dualities, such as Stone and Pontryagin dualities. The themes are effective classification and enumeration.
Topics and features:
· Delivers a self-contained, gentle introduction to priority arguments, directly applying them in algebraic contexts
· Includes extensive exercises that both cement and amplify the materials
· Provides complete introduction to the basics of computable analysis, particularly in the context of computable structures
· Offers the first monograph treatment of computable Polish groups, effective profinite groups via Stone duality, and effective abelian groups via Pontryagin duality
· Presents the first book treatment of Friedberg enumerations of structures
This unique volume is aimed at graduate students and researchers in computability theory, as well as mathematicians seeking to understand the algorithmic content of structure theory. Being self-contained, it provides ample opportunity for self-study.
Contents
Part I Foundation of Computability.- 1. Introduction.- 2. Basics of Computability Theory.- 3. Computable Algebraic Structures.- 4. Computable Separable Spaces.- Part II Computable Duality.- 5. Computable Boolean Algebras.- 6. Computable Stone Spaces.- 7. Computable Abelian Groups.- 8. Computable Connected Compact Spaces.- Part III Computability and Classification Problems. 9. The Analytical Hierarchy and Σ11-completeness.- 10. Computable Categoricity.- 11. Computable Banach Spaces with Applications.- 12. Resource Bounded Computation.- Part IV Non-computability and Randomness. 13. Randomness.- 14. Degree Spectra.- 15.- Computable Transfinite Analysis.
