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The Foundations of Computability Theory (Hardcover, 2nd ed. 2020)
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The Foundations of Computability Theory (Hardcover, 2nd ed. 2020)
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This book offers an original and informative view of the
development of fundamental concepts of computability theory. The
treatment is put into historical context, emphasizing the
motivation for ideas as well as their logical and formal
development. In Part I the author introduces computability theory,
with chapters on the foundational crisis of mathematics in the
early twentieth century, and formalism. In Part II he explains
classical computability theory, with chapters on the quest for
formalization, the Turing Machine, and early successes such as
defining incomputable problems, c.e. (computably enumerable) sets,
and developing methods for proving incomputability. In Part III he
explains relative computability, with chapters on computation with
external help, degrees of unsolvability, the Turing hierarchy of
unsolvability, the class of degrees of unsolvability, c.e. degrees
and the priority method, and the arithmetical hierarchy. Finally,
in the new Part IV the author revisits the computability
(Church-Turing) thesis in greater detail. He offers a systematic
and detailed account of its origins, evolution, and meaning, he
describes more powerful, modern versions of the thesis, and he
discusses recent speculative proposals for new computing paradigms
such as hypercomputing. This is a gentle introduction from
the origins of computability theory up to current research, and it
will be of value as a textbook and guide for advanced undergraduate
and graduate students and researchers in the domains of
computability theory and theoretical computer science. This new
edition is completely revised, with almost one hundred pages of new
material. In particular the author applied more up-to-date, more
consistent terminology, and he addressed some notational
redundancies and minor errors. He developed a glossary relating to
computability theory, expanded the bibliographic references with
new entries, and added the new part described above and other new
sections.
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