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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, 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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Computability Theory
Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s.This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications.Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences.Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results.The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context.This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy.The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field.Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way.
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Computability and Complexity
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Martin Davis on Computability, Computational Logic, and Mathematical Foundations
This book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability.The individual contributions touch on most of the core aspects of Davis’ work and set it in a contemporary context.They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert’s tenth problem.The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions.In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis’ work.As a whole, this book shows how Davis’ scientific work lies at the intersection of computability,theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic.
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