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Computability

Autor George Tourlakis
en Limba Engleză Paperback – 3 aug 2023
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, Oraclecomputations 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.
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Specificații

ISBN-13: 9783030832049
ISBN-10: 303083204X
Pagini: 637
Ilustrații: XXVII, 637 p. 12 illus., 10 illus. in color.
Dimensiuni: 155 x 235 x 40 mm
Greutate: 0.92 kg
Ediția:1st ed. 2022
Editura: Springer International Publishing
Colecția Springer
Locul publicării:Cham, Switzerland

Cuprins

Mathematical Background; a Review.- A Theory of Computability.- Primitive Recursive Functions.- Loop Programs.-The Ackermann Function.- (Un)Computability via Church's Thesis.- Semi-Recursiveness.- Yet another number-theoretic characterisation of P.- Godel's Incompleteness Theorem via the Halting Problem.- The Recursion Theorem.- A Universal (non-PR) Function for PR.- Enumerations of Recursive and Semi-Recursive Sets.- Creative and Productive Sets Completeness.- Relativised Computability.- POSSIBILITY: Complexity of P Functions.- Complexity of PR Functions.- Turing Machines and NP-Completeness.

Recenzii

“This textbook is suited for self-study … . As a second reading however a reader interested in rigorous proofs and/or different approaches to known concepts will benefit from this wealth of material.” (Dieter Riebesehl, zbMATH 1507.03002, 2023)

Notă biografică

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).

Textul de pe ultima copertă

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.
Features:
  • Extensive and mathematically complete coverage of the limitations of logic, including Gödel’s incompleteness theorems (first and second), Rosser’s version of the first incompleteness theorem, and Tarski’s non expressibility of “truth”
  • Inability of computability to detect formal theorems effectively, using Church’s proof of the unsolvability of Hilbert’s Entscheidungsproblem
  • Arithmetisation-free proof of the pillars of computability: Kleene’s s-m-n, universal function and normal form theorems — using “Church’s thesis” and a simulation of the URM (“register machine”) by a simultaneous recursion. These three pivotal results lead to the deeper results of the theory
  • Extensive coverage of the advanced topic of  computation with “oracles" including an exposition of the search computability theory of Moschovakis, the first recursion theorem, Turing reducibility and Turing degrees and an application of the Sacks priority method of “preserving agreements”, and the arithmetical hierarchy including Post’s theorem
  • Cobham’s mathematical characterisation of the concept deterministic polynomial time computable function is fully proved
  • A complete proof of Blum’s speed-up theorem

Caracteristici

Features extensive and mathematically complete coverage of limitations of logic Explains the inability of computability to effectively detect formal theorems Provides arithmetisation-free proofs of the pillars of computability