TUHH Open Research
Help
  • Log In
    New user? Click here to register.Have you forgotten your password?
  • English
  • Deutsch
  • Communities & Collections
  • Publications
  • Research Data
  • People
  • Institutions
  • Projects
  • Statistics
  1. Home
  2. TUHH
  3. Publications
  4. Computability theory
 
Options

Computability theory

Citation Link: https://doi.org/10.15480/882.1064
Publikationstyp
Book
Date Issued
2012
Sprache
English
Author(s)
Zimmermann, Karl-Heinz  
Institut
Rechnertechnologie E-13  
TORE-DOI
10.15480/882.1064
TORE-URI
http://tubdok.tub.tuhh.de/handle/11420/1066
Why do we need a formalization of the notion of algorithm or effective computation? In order to show that a specific problem is algorithmically solvable, it is sufficient to provide an algorithm that solves it in a sufficiently precise manner. However, in order to prove that a problem is in principle not solvable by an algorithm, a rigorous formalism is necessary that allows mathematical proofs. The need for such a formalism became apparent in the studies of David Hilbert (1900) on the foundations of mathematics and Kurt Gödel (1931) on the incompleteness of elementary arithmetic.

The first investigations in the field were conducted by the logicians Alonzo Church, Stephen Kleene, Emil Post, and Alan Turing in the early 1930s. They have provided the foundation of computability theory as a branch of theoretical computer science. The fundamental results established Turing computability as the correct formalization of the informal idea of effective calculation. The results have led to Church’s thesis stating that ”everything computable is computable by a Turing machine”. The theory of computability has grown rapidly from its beginning. Its questions and methods are penetrating many other mathematical disciplines. Today, computability theory provides an important theoretical background for logicians, pure mathematicians, and computer scientists. Many mathematical problems are known to be undecidable such as the word problem for groups, the halting problem, and Hilbert’s tenth problem.

This book is a development of class notes for a two-hour lecture including a one-hour lab held for second-year Bachelor students of Computer Science at the Hamburg University of Technology during the last two years. The course aims to present the basic results of computability theory, including mathematical models of computability, primitive recursive and partial recursive functions, Ackermann’s function, Gödel numbering, universal functions, smn theorem, Kleene’s normal form, undecidable sets, theorems of Rice, and word problems. The manuscript has partly grown out of notes taken by the author during his studies at the University of Erlangen-Nuremberg. I would like to thank again my teachers Martin Becker† and Volker Strehl for giving inspiring lectures in this field.

The second edition contains minor changes. In particular, the section on Gödel numbering has been rewritten and a glossary of terms has been added.
Subjects
computability
recursion theory
theoretical computer science
Lizenz
http://doku.b.tu-harburg.de/doku/lic_ohne_pod.php
Loading...
Thumbnail Image
Name

ComputabilityBookAll.pdf

Size

7.42 MB

Format

Adobe PDF

TUHH
Weiterführende Links
  • Contact
  • Send Feedback
  • Cookie settings
  • Privacy policy
  • Impress
DSpace Software

Built with DSpace-CRIS software - Extension maintained and optimized by 4Science
Design by effective webwork GmbH

  • Deutsche NationalbibliothekDeutsche Nationalbibliothek
  • ORCiD Member OrganizationORCiD Member Organization
  • DataCiteDataCite
  • Re3DataRe3Data
  • OpenDOAROpenDOAR
  • OpenAireOpenAire
  • BASE Bielefeld Academic Search EngineBASE Bielefeld Academic Search Engine
Feedback