LEADER 05065nam 22007452 450 001 9910790463503321 005 20151005020621.0 010 $a1-107-22428-4 010 $a1-280-48485-3 010 $a1-139-22192-2 010 $a9786613579836 010 $a1-139-21710-0 010 $a1-139-21403-9 010 $a1-139-22363-1 010 $a1-139-22020-9 010 $a1-139-03190-2 035 $a(CKB)2670000000140314 035 $a(EBL)833389 035 $a(OCoLC)775869745 035 $a(SSID)ssj0000640139 035 $a(PQKBManifestationID)11364158 035 $a(PQKBTitleCode)TC0000640139 035 $a(PQKBWorkID)10611513 035 $a(PQKB)11603715 035 $a(UkCbUP)CR9781139031905 035 $a(Au-PeEL)EBL833389 035 $a(CaPaEBR)ebr10533191 035 $a(CaONFJC)MIL357983 035 $a(MiAaPQ)EBC833389 035 $a(PPN)261343718 035 $a(EXLCZ)992670000000140314 100 $a20110223d2012|||| uy| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aProofs and computations /$fHelmut Schwichtenberg, Stanley S. Wainer$b[electronic resource] 210 1$aCambridge :$cCambridge University Press,$d2012. 215 $a1 online resource (xiii, 465 pages) $cdigital, PDF file(s) 225 1 $aPerspectives in logic 300 $aTitle from publisher's bibliographic system (viewed on 05 Oct 2015). 311 $a0-521-51769-9 320 $aIncludes bibliographical references and index. 327 $a1.4. Soundness and completeness of the classical fragment1.4.1. Models.; 1.4.2. Soundness of classical logic.; 1.4.3. Completeness of classical logic.; 1.4.4. Compactness and Lo?wenheim-Skolem theorems.; 1.5. Tait calculus; 1.6. Notes; Chapter 2: RECURSION THEORY; 2.1. Register machines; 2.1.1. Programs.; 2.1.2. Program constructs.; 2.1.3. Register machine computable functions.; 2.2. Elementary functions; 2.2.1. Definition and simple properties.; 2.2.2. Elementary relations.; 2.2.3. The class ?.; 2.2.4. Closure properties of ?.; 2.2.5. Coding finite lists.; 2.3. Kleene's normal form theorem 327 $a2.3.1. Program numbers.2.3.2. Normal form.; 2.3.3. ?o1-definable relations and ?-recursive functions.; 2.3.4. Computable functions.; 2.3.5. Undecidability of the halting problem.; 2.4. Recursive definitions; 2.4.1. Least fixed points of recursive definitions.; 2.4.2. The principles of finite support and monotonicity, and the effective index property.; 2.4.3. Recursion theorem.; 2.4.4. Recursive programs and partial recursive functions.; 2.4.5. Relativized recursion.; 2.5. Primitive recursion and for-loops; 2.5.1. Primitive recursive functions.; 2.5.2. Loop-programs. 327 $a2.5.3. Reduction to primitive recursion.2.5.4. A complexity hierarchy for Prim.; 2.6. The arithmetical hierarchy; 2.6.1. Kleene's second recursion theorem.; 2.6.2. Characterization of ?01-definable and recursive relations.; 2.6.3. Arithmetical relations.; 2.6.4. Closure properties.; 2.6.6. ?0r-complete relations.; 2.7. The analytical hierarchy; 2.7.1. Analytical relations.; 2.7.2. Closure properties.; 2.7.3. Universal ?1r+1-definable relations.; 2.7.4. ?1r-complete relations.; 2.8. Recursive type-2 functionals and well-foundedness; 2.8.1. Computation trees.; 2.8.2. Ordinal assignments 327 $arecursive ordinals. 330 $aDriven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Go?del's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to ?11-CA0. Ordinal analysis and the (Schwichtenberg-Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and ?11-CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic. 410 0$aPerspectives in logic. 517 3 $aProofs & Computations 606 $aComputable functions 606 $aProof theory 615 0$aComputable functions. 615 0$aProof theory. 676 $a511.352 700 $aSchwichtenberg$b Helmut$f1942-$01488562 702 $aWainer$b S. S. 712 02$aAssociation for Symbolic Logic, 801 0$bUkCbUP 801 1$bUkCbUP 906 $aBOOK 912 $a9910790463503321 996 $aProofs and computations$93708817 997 $aUNINA