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020 _a9783030832025
_9978-3-030-83202-5
024 7 _a10.1007/978-3-030-83202-5
_2doi
050 4 _aQA75.5-76.95
072 7 _aUYA
_2bicssc
072 7 _aCOM014000
_2bisacsh
072 7 _aUYA
_2thema
082 0 4 _a004.0151
_223
100 1 _aTourlakis, George.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
245 1 0 _aComputability
_h[electronic resource] /
_cby George Tourlakis.
250 _a1st ed. 2022.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2022.
300 _aXXVII, 637 p. 12 illus., 10 illus. in color.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
505 0 _aMathematical Background; a Review -- A Theory of Computability -- Primitive Recursive Functions -- Loop Programs.-The Ackermann Function -- (Un)Computability via Church's Thesis -- Semi-Recursiveness -- Yet another number-theoretic characterisation of P -- Godel's Incompleteness Theorem via the Halting Problem -- The Recursion Theorem -- A Universal (non-PR) Function for PR -- Enumerations of Recursive and Semi-Recursive Sets -- Creative and Productive Sets Completeness -- Relativised Computability -- POSSIBILITY: Complexity of P Functions -- Complexity of PR Functions -- Turing Machines and NP-Completeness.
520 _aThis 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.
650 0 _aComputer science.
650 0 _aComputable functions.
650 0 _aRecursion theory.
650 0 _aComputational complexity.
650 0 _aTechnology
_xPhilosophy.
650 1 4 _aTheory of Computation.
650 2 4 _aComputability and Recursion Theory.
650 2 4 _aComputational Complexity.
650 2 4 _aModels of Computation.
650 2 4 _aPhilosophy of Technology.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783030832018
776 0 8 _iPrinted edition:
_z9783030832032
776 0 8 _iPrinted edition:
_z9783030832049
856 4 0 _uhttps://doi.org/10.1007/978-3-030-83202-5
912 _aZDB-2-SCS
912 _aZDB-2-SXCS
942 _cSPRINGER
999 _c178318
_d178318