ebook9783030832025d4325

Description

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 ‚Äúmechanical process‚Äù 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‚Äôs theorem on the non-representability of truth, Goedel‚Äôs incompleteness and Rosser‚Äôs incompleteness theorems, two short proofs of the incompleteness theorem that are based on Lob’s deliverability conditions, Church‚Äôs 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‚Äôs complexity, Hierarchies of primitive recursive functions, and a machine-independent characterisation of Cobham’s feasibly computable functions.