Algebraic computability and enumeration models: recursion theory and descriptive complexity

Description

"This book, Computability, Algebraic Trees, Enumeration Degree Models, and Applications, presents new techniques with functorial models to address important areas on pure mathematics and computability theory from the algebraic view point. The reader is first introduced to categories and functorial models, with Kleene algebra examples for languages. Functorial models for Peano arithmetic are described toward important computational complexity areas on a Hilbert program, leading to computability with initial models. Infinite language categories are introduced also to explain descriptive complexity with recursive computability with admissible sets and urelements. Algebraic and categorical realizability is staged on several levels, addressing new computability questions with omitting types realizably. Further applications to computing with ultrafilters on sets and Turing degree computability are examined. Functorial models computability are presented with algebraic trees realizing intuitionistic types of models. New homotopy techniques developed in the author's volume on the functorial model theory are applicable to Martin Lof types of computations with model categories. Functorial computability, induction, and recursion are examined in view of the above, presenting new computability techniques with monad transformations and projective sets. This informative volume will give readers a complete new feel for models, computability, recursion sets, complexity, and realizability. This book pulls together functorial thoughts, models, computability, sets, recursion, arithmetic hierarchy, filters, with real tree computing areas, presented in a very intuitive manner for university teaching, with exercises for every chapter. The book will also prove valuable for faculty in computer science and mathematics."-- Read more... Abstract: "This book, Computability, Algebraic Trees, Enumeration Degree Models, and Applications, presents new techniques with functorial models to address important areas on pure mathematics and computability theory from the algebraic view point. The reader is first introduced to categories and functorial models, with Kleene algebra examples for languages. Functorial models for Peano arithmetic are described toward important computational complexity areas on a Hilbert program, leading to computability with initial models. Infinite language categories are introduced also to explain descriptive complexity with recursive computability with admissible sets and urelements. Algebraic and categorical realizability is staged on several levels, addressing new computability questions with omitting types realizably. Further applications to computing with ultrafilters on sets and Turing degree computability are examined. Functorial models computability are presented with algebraic trees realizing intuitionistic types of models. New homotopy techniques developed in the author's volume on the functorial model theory are applicable to Martin Lof types of computations with model categories. Functorial computability, induction, and recursion are examined in view of the above, presenting new computability techniques with monad transformations and projective sets. This informative volume will give readers a complete new feel for models, computability, recursion sets, complexity, and realizability. This book pulls together functorial thoughts, models, computability, sets, recursion, arithmetic hierarchy, filters, with real tree computing areas, presented in a very intuitive manner for university teaching, with exercises for every chapter. The book will also prove valuable for faculty in computer science and mathematics." Content: Introduction -- Computing categories, language fragments, and models -- Functorial admissible models -- Initial tree computing and languages -- Descriptive sets and infinitary languages -- Complexity and computing -- Arithmetic hierarchy and enumeration degrees -- Peano arithmetic models and computability -- Realizability and computability.