Algebraic Computability and Enumeration Models: Recursion Theory and Descriptive Complexity, 1st Edition (Hardback) book cover

Algebraic Computability and Enumeration Models

Recursion Theory and Descriptive Complexity, 1st Edition

By Cyrus F. Nourani

Apple Academic Press

310 pages

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Description

This book, Algebraic Computability and Enumeration Models: Recursion Theory and Descriptive Complexity, presents new techniques with functorial models to address important areas on pure mathematics and computability theory from the algebraic viewpoint. 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 also introduced 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 is presented with algebraic trees realizing intuitionistic types of models. New homotopy techniques are applied to Marin 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.

Table of Contents

Preface

Introduction

Computing Categories, Language Fragments, and Models

Introduction

Limits and Infinitary Languages

Generic Functors and Language String Models

Positive Generic Models

Fragment Consistent Algebras

Generic Products

Positive Morphisms and Models

Positive Consistency and Omitting Types

Positive Fragment Consistency Models

Horn Models

Positive Categories and Horn Fragments

Fragment Consistent Kleene Models

More on Kleene Structures

Process Algebras

Functorial Admissible Models

Infinitary Languages Basics

Admissible Languages

Admissible Models

Infinite Language Categories

A Descriptive Computing

Computing Model Diagrams

Situations and Compatibility

Boolean Computing Diagrams

Description Logic

Functorial Model Theory and HIFI Computing

Generic Functor Initial Models

Initial Tree Algebras and Amplification

Tree Amplifiers and The Sonic Booms

The Recursion Theorem

Tree Amplifiers and Recursion

Admissible Gain Synthesizer

Initial Tree Computing and Languages

Initial Models and Their Algebraic Formulation

The Basics

Canonical Models

Generic diagrams of Initial Models

Initial Algebras and Computable Trees

Tree Rewriting, Algebras, and Infinitary Models

Are There Models for Nothing

Free Proof Trees and Computing Models

Generating Models by Positive Forcing

Algebraically Closed Groups

Word Problems and the SRS Roller Coaster

The Roller Coaster

Private Languages and Wittgenstein’s Paradox

Concluding Comments

Descriptive Sets and Infinitary Languages

Introduction

Admissible Sets and Structures

Basic Descriptive Characterizations

Boolean Valued Models

Admissible Sets and Ordinals Error! Bookmark not defined.

Set Reducibility

Admissible Tree Recursion

Admissible Set Reducibility

Complexity and Computing

Introduction

Forcing, Complexity, and Diaphontine Definability

Technical Preliminaries

Initial Models

Generic Diagrams for Initial Models

Models and Fragment Inductive Closure

Positive Forcing and Infinitary Models

Generating Models by Positive Forcing

Forcing and Computability

Complexity Classes, Models, and Urlements

Functorial Implicit Complexity Error! Bookmark not defined.

Abstract Descriptive Complexity

A Descriptive Computing Example Revisit

Rudiments, KPU, and Recursion

Admissible Hulls

Concrete Descriptive Complexity

Concrete Implicit Complexity

Overview to Arithmetic Hierarchy

Arithmetic Hierarchy and Enumeration Degrees

Introduction

Turing Degrees and Isomorphism Types

Arithmetic Hierarchy and Infinitary Languages

Computability and Hierarchy with Infinitary Languages

Computability on Infinitary Languages

Enumeration Degrees

Enumeration Definability and Turing Jumps

Automorphisms and Lifts on K-Pairs

Enumeration Computability Models

Rudiments, KPU, and Recursion

Computable Categorical Trees

Enumerations Model Theory

Peano Arithmetic Models and Computability

Introduction

Recursion on Arithmetic Fragments

Godel’s Incompleteness and Ordinal Arithmetic

Descriptive Sets and Automata

Finite Models

Fields and Fragments of Peano Arithmetic

Arithmetic Hierarchy and Borel Sets

Infinitary Theories and c=Countable N Models

KPU Ordinal Models

Generic Computability and Filters

Realizability and Computability

Introduction

Categorical Models and Realizability

Categorical Intuitionistic Models

Infinitary Language Product Models

Positive Generic Models

Omitting Types Realizability

Positive Realizability Morphisms and Models

Fragment Product Algebra Realizability

Positive Realizability on Horn Filters

Computability and Positive Realizability

Morphic Realization Functors

Positive Categories and Consistency Models

Horn Computability and Realizability

Intuitionistic Types and Realizability

Realizability on Ultrafilters

Computing Morphisms on Topos

Relative Realizability on Topos

Realizability Triposes

More on Topos Realizability

On PreSheaves Topos Realizability

Index

About the Author

Dr. Cyrus F. Nourani has a national and international reputation in computer science, artificial intelligence, mathematics, virtual haptic computation, information technology, and management. He has many years of experience in the design and implementation of computing systems. Dr. Nourani's academic experience includes faculty positions at the University of Michigan-Ann Arbor, the University of Pennsylvania, the University of Southern California, UCLA, MIT, and the University of California, Santa Barbara. He was also research professor at Simon Frasier University in Burnaby, British Columbia, Canada. He was a visiting professor at Edith Cowan University, Perth, Australia, and a lecturer of management science and IT at the University of Auckland, New Zealand.

Dr. Nourani commenced his university degrees at MIT where he became interested in algebraic semantics. That was pursued with a category theorist at the University of California. Dr. Nourani’s dissertation on computing models and categories proved to have intuitionist forcing developments that were published from his postdoctoral times on at ASL. He has taught AI to the Los Angeles aerospace industry and has authored many R&D and commercial ventures. He has written and co-authored several books. He has over 350 publications in mathematics and computer science and has written on additional topics, such as pure mathematics, AI, EC, and IT management science, decision trees, predictive economics game modeling. In 1987, he founded Ventures for computing R&D. He began independent consulting with clients such as System Development Corporation (SDC), the US Air Force Space Division, and GE Aerospace. Dr. Nourani has designed and developed AI robot planning and reasoning systems at Northrop Research and Technology Center, Palos Verdes, California. He also has comparable AI, software, and computing foundations and R&D experience at GTE Research Labs.

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT012000
MATHEMATICS / Geometry / General
MAT028000
MATHEMATICS / Set Theory
SCI000000
SCIENCE / General