Computability

This survey of computability theory offers the techniques and tools that computer scientists (as well as mathematicians and philosophers studying the mathematical foundations of computing) need to mathematically analyze computational processes and investigate the theoretical limitations of computing. Beginning with an introduction to the mathematisation of "mechanical process" using URM programs, this textbook explains basic theory such as primitive recursive functions and predicates and sequence-coding, partial recursive functions and predicates, and loop programs.

Advanced chapters cover the Ackerman function, Tarski's theorem on the non-representability of truth, Goedel's incompleteness and Rosser's incompleteness theorems, two short proofs of the incompleteness theorem that are based on Lob's deliverability conditions, Church's thesis, the second recursion theorem and applications, a provably recursive universal function for the primitive recursive functions, Oracle computations and various classes of computable functionals, the Arithmetical hierarchy, Turing reducibility and Turing degrees and the priority method, a thorough exposition of various versions of the first recursive theorem, Blum's complexity, Hierarchies of primitive recursive functions, and a machine-independent characterisation of Cobham's feasibly computable functions.

這本關於可計算性理論的調查提供了計算機科學家（以及研究計算數學基礎的數學家和哲學家）所需的技巧和工具，以數學方式分析計算過程並探討計算的理論限制。從使用URM程序對“機械過程”進行數學化的介紹開始，這本教科書解釋了基本理論，如原始遞歸函數和謂詞以及序列編碼，部分遞歸函數和謂詞，以及循環程序。

高級章節涵蓋了阿克曼函數，塔斯基關於真理不可表示性的定理，哥德爾的不完全性定理和羅瑟的不完全性定理，基於洛布可達性條件的不完全性定理的兩個簡短證明，教堂的論點，第二個遞歸定理及其應用，原始遞歸函數的可證明遞歸通用函數，Oracle計算和各種可計算函數類，算術層次結構，圖靈可化約性和圖靈度以及優先方法，對第一個遞歸定理的各種版本進行了詳細的闡述，布魯姆的復雜性，原始遞歸函數的層次結構，以及對Cobham的可行計算函數的機器獨立特徵描述。

George Tourlakis, PHD, is University Professor of Computer Science and Engineering at York University in Toronto, Canada. He has published extensively in his areas of research interest, which include calculational logic, modal logic, computability, and complexity theory. Dr. Tourlakis is the author of Theory of Computation and Mathematical Logic, both published by Wiley, and Lectures in Logic and Set Theory; Volumes 1 and 2 (Cambridge University Press).

George Tourlakis博士是加拿大多倫多約克大學的計算機科學和工程學的大學教授。他在計算邏輯、模態邏輯、可計算性和複雜性理論等研究領域發表了大量的論文。Tourlakis博士是Wiley出版社的《計算理論》和《數學邏輯》的作者，也是劍橋大學出版社的《邏輯和集合論講義》第一卷和第二卷的作者。