Computability

Beschreibung

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

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.

Autor*in

George Tourlakis

Themen in »Computability«

recursive functions formal definability recursion theorem diagonalisation strong reducibility loop programs Kleene predicate Turing reducibility Register machine Church’s thesis Goedel incompleteness Rosser incompleteness Tarski’s theorem Ackermann function (Unbounded) Register Machine (URM)

Stimmen zu »Computability«

“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)
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