Computability

Bning 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 Godel'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 deteistic polynomial computable function is fully proved
A complete proof of Blum's speed-up theorem



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