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Abramsky S., Barwise J., Fine K. — Recursive Functionals. Studies in Logic and the Foundations of Mathematics Volume 131
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Название: Recursive Functionals. Studies in Logic and the Foundations of Mathematics Volume 131Авторы: Abramsky S., Barwise J., Fine K.Аннотация: This work is a self-contained elementary exposition of the theory of recursive functionals, that also includes a number of advanced results. Although aiming basically at a theory of higher order computability, attention is restricted to second order functionals, where the arguments are numerical functions and the values, when defined, are natural numbers. This theory is somewhat special, for to some extent it can be reduced to first order theory, but when properly extended and relativized it requires the full machinery of higher order computations. In the theory of recursive monotonic functionals the author formulates a reasonable notion of computation which provides the right frame for what appears to be a convincing form of the extended Church's thesis. At the same time, the theory provides sufficient room to formulate the classical results that are usually derived in terms of singular functionals. Presented are complete proofs of Gandy's selector theorem, Kleene's theorem on hyperarithmetical predicates, and Grilliot's theorem on effectively discontinuous functionals.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1992Количество страниц: 279Добавлена в каталог: 13.08.2015Операции: Положить на полку |
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Abstraction 161 Algorithm 86 181 Algorithm, deterministic 74 Algorithm, explicit 50 86 Algorithm, recursive 110 Analytic 243 Anti-symmetric 5 Application 1 Application, functional 37 ASSIGNMENT 129 BASIC 37 Basic functional 37 Basic operation 39 Basic, primitive recursion 52 Basic, recursive term 69 Basic, term 47 Boolean 20 Boolean, extension 23 Boolean, monotonic 23 Boolean, values 20 Bottom 6 Bounded 56 Bounded minimalization 58 Bounded product 57 Bounded quantification 60 Bounded sum 56 CALL 111 Call, direct 113 Call, indirect 114 Cartesian product 19 cases 38 Cases, definition 38 Cases, general 40 Characteristic functional 25 35 Characteristic functional, dual 25 35 Characteristic functional, partial 25 35 Church's thesis 67 88 Closed 128 Coenumerable 193 Cohyperenumerable 236 complete 17 229 Computable 36 Computable, recursively 117 Computation 86 110 Computation rules 86 110 Condition 4 Condition, inductive 251 Condition, single-valued 4 Condition, total 4 Configuration 86 110 Conjunction 21 24 Consistent 7 172 227 Consistent, set 7 Constant functional 37 continuous 206 Cover 142 Definable 136 Definable, formally 136 defined 1 Defined mapping 1 Defined well 4 Definition 3 Definition, mapping 3 Definition, predicate 26 Deflector 210 Denotation 129 Discontinuous 209 Discontinuous, effectively 213 Discrete 5 Discrete poset 5 Disjunction 21 distribution 38 Distribution, predicate 42 Domain 8 Dual 26 35 Effectively discontinuous 213 Elemental 243 Enumerable 193 197 Existential quantification 19 Existential quantification, barred 234 Existential quantification, bounded 50 Existential quantification, functional 233 Existential quantification, unbounded 190 Explicit 3 Explicit algorithm 50 Explicit specification 3 50 Extension 4 Extension, Boolean 23 Extension, mapping 4 Factorization 209 Fenstad, J.E. 14 170 Fixed point 9 Formalization 125 Formalization, closed 128 Formalization, language 126 Formalization, linear 128 Formalization, regular 128 Formalization, term 126 Formalization, variable 126 Function 30 Function, number decoding 59 Function, number encoding 59 Functional 16 30 Functional application 37 Functional characteristic 26 Functional complete 17 Functional constant 37 Functional discontinuous 209 Functional effectively discontinuous 213 Functional preorder 165 Functional primitive recursive 54 Functional projection 37 Functional singular 18 Functional successor 51 Functional, continuous 206 Functional, quasi-total 17 55 Functional, recursive 110 Functional, similar 18 Functional, weakly normal 213 Gandy, R.O. 124 232 Graph predicate 26 35 Greatest lower bound 8 Grilliot, T.J. 205 206 211 212 217 Hindley, J.R. 139 156 Hinman, P.G. 14 45 170 187 217 232 258 Hyperarithmetical 239 Hyperenumerable 235 239 Inconsistent 7 INDEX 159 174 Induction 251 Inductive 251 Interpreter 158 Intersection 34 Jump 184 Kechris, A.S. 108 124 Kleene, S.C. 45 95 187 200 217 242 258 260 Kreisel, G. 217 Least upper bound 7 Linear 128 Lower bound 6 Lower bound, greatest 8 machine 111 Mapping 1 Mapping, assignment 129 Mapping, defined 1 Mapping, definition 3 Mapping, monotonic 15 Mapping, partial 2 Mapping, specification 2 Mapping, total 2 Mapping, undefined 1 Maximal 6 Minimal fixed point 10 Minimalization 58 68 Minimalization, bounded 58 Minimalization, unbounded 68 Moldestad, J. 124 Monotonic 15 Monotonic boolean 23 Monotonic extension 23 Moschovakis, Y.N. 108 124 258 Negation 21 24 normal 231 Normal, weakly 213 Norman, D. 217 numerical 30 Numerical function 30 Numerical predicate 34 Numerical substitution 37 40 Operations 39 Operations, 68 Operations, basic 39 Operations, primitive recursive 52 Operations, recursive 110 Oracle 112 Partial characteristic functional 25 35 Partial ordered set 5 Partial ordering 5 Platek, R.A. 124 Point 9 Point, minimal fixed 9 Polish normal form 75 POSET 5 Poset, discrete 5 Predicate 22 33 Predicate, coenumerable 193 Predicate, distribution 42 Predicate, enumerable 193 Predicate, numerical 34 Predicate, partially recursive 192 Predicate, primitive recursive 192 Predicate, recursive 192 Prefix 127 Prefix, 127 Prefix, 127 Preorder functional 165 Preordered 168 Prime number enumeration 58 Primitive recursion 51 Primitive recursion, basic 52 Projection functional 37 Quantifier 21 Quantifier, barred 234 Quantifier, bounded 60 Quantifier, unbounded 190 Quasi-similar 153 Quasi-total 17 RANGE 27 Recursion 51 Recursion, basic 71 Recursion, functional 102 Recursion, operational 110 Recursion, primitive 51 Recursion, simultaneous 106 RECURSIVE 69 Recursive algorithm 110 recursive call 77 Recursive Functional 110 Recursive machine 112 Recursive operation 110 Reflexive structure 158 Regular 6 Regular element 6 Regular poset 6 Regular set 149 Regular, term 101 Representation 172 Restriction assumption 145 Rogers, H. 242 Sanchis, L.E. 13 29 47 58 67 69 71 84 89 95 162 167 170 187 189 232 258 Seldin, J.P. 139 156 Selector property 168 Semantics 127 Semantics, reductional 127 Semantics, structural 127 131 Similar 18 Similar extension 191 240 Single-valued 4 singular 17 Specification 2 Specification, explicit 3 Specification, mapping 2 Specification, relational 3 STACK 74 Stack, algorithm 74 Structure 158 Substitution 37 Substitution, formal 134 Substitution, free 134 Substitution, functional 62 Substitution, numerical 37 Substitution, predicate 43 Support 113 system 110 Term 47 Term, analytic 243 Term, basic 47 Term, elemental 243 Term, functional 98 Term, inductive 251 Term, regular 251 Top 6 total 2 Total condition 4 Total mapping 2 Tourlakis, G.J. 45 124 187 Transformation 9 70 Tugue, T. 187 Tuples 31 Type-(k,m) 126 Type-0 29 Type-1 126 Type-1, functional 126 Type-1, numerical 126 Type-k 126 union 34 Universal quantification 19 Universal quantification, barred 234 Universal quantification, bounded 49 Universal quantification, functional 233 Universal quantification, unbounded 190 Upper bound 6 Variable 126 Variable, bound 128 Variable, formal 126 Variable, free 128 Variable, renamed 145 Variable, renaming 145 Variant 146 Weakly normal 213
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