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SchUtte, Kurt: Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977
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Shepherdson,J.C.; Sturgis, H.E.: Computability of recursive functions, Journal of the ACM 10, 217-255 (1963)
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Shoenfield, Joseph R.: Degrees of unsolvability, North-Holland, Amsterdam, 1971
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505
Index of Notations
(Entries are listed in order of occurrence in text)
iff 1 "R. :=R.+l" 25 1- 1-
w.l .g. 1 "R. :=R. -I" 25 1- 1-
i'J 1 "R.=O" 25 1-
dom 1 IC(k) 25 range 1 6 32 graph 1 Z 32 f:X----> Y 1 pr{k) 32 div 1 IF (k) 34 r1 2 IF (00) 34 p
-1 2 IP (k) 34 f:X->Y 2 IPH 34 fn 2 Sub
m 34 W (X) 2 Prk 34 n W(X) 2 -
II 34 £ 2 11 37 wn 3 1I(k) 37
R 3 <xJl' ... ,xk> 37 w
CON 9 11 ~ ) 38 1-
SF 9 f<x 1,· .. ,xk> 39 CT 9 lln [ ] 39 TF 9 Gr 42
fF 9 PRK 42
tF 9 Var 43 Sim 16 sg 44
Sim 16 sg 44 M 19 A 45
n IC 19 TimeM 49 DC 19 T 53
fM 19 Tk 53
tM 19 L 60 IA 21 R 60 OA 21 [v,a,w] 60
Index of Notations 507
"R.:= R.a" ~ ~
71 [\) 1"" '\)n l 153
"R. :=pop R." 71 \)1 U \)2 153 ~ ~
"Ri top a" 71
\) 79 \)1 11 \)2 153
p(k) 87 WD( \) 0) 153 R(k) 87 FS(\)o) 154 p(oo) 87 K 159 R(oo) 87 KO 159
r.e. 93 ::; 160
<P 103 - 160
u 107 W 163 <P W. 163 ::; 113 ~
113 Z 163 -K 117 Z.
~ 163
<P KO 117 RE 163
<P REC 163 PL 123 PCP 124 Sup 177
SPCP 124 Inf 177
T 130 l.u .b. 178
-4 130 < 178 T
-1
- 178 <P 143 ,. 178 <P 143 < 178 \) 144
-m
N(M) 144 cfA 179 c(\) ) 182
TN(M) 144 sup 227 " N 144 (IiJ (a.) 230
TN 144 OIl 231 n A 145 S 231 \) 145 T (0') 231 a ex \) E 145 lou.b. 235 \) * 145 \) 236
\) G 145 VarA 249 e 145 TmA 249
\)z 145 LA 249
\)2 145 n 250 V 250 n
508 Index of Notations
[p:x/t] 251 IBo 327 val 252 w( i) 327
Lt 258 !; 327
A
Lf 258 TBo 327
A 0 327 n 270 w
[IB -lB ] 327 ° ° ni 271 rz 327
.A 272 U:O 328 (jJi
<I>~ 272 Teo 328
~ Qw 328 W~ 273 ~
[CCo -U:o] 328
KA 274 rz 329
< 279 Yl 329 -T
-T 279 Y2 333
A' 285 Y3 336
A(n) 285 IB 338
cr 286 CC 338 n p[n]
7r 286 338 n
1: 286 [w] 338 n
II 286 TB 339 n
< 300 [Iwll 339 -tt
\I 307 TC 339 co
3 307 [IB -IB] 346 co
C( t) 311 [IB -IN] 346
cOt t) 311 II 346
[0 322 rr(k) 346
lBo 322 It'" ) 346
P 322 II' 346 w
T 322 <Ql'''' ,Qk> 346
T 323 <Pi>i 346 E
0 323 <n,p> 346 E
[P _P] 323 181 T 346 w w iEI i rz 323 TN 346
<l>e n 325 ;j, 349
<l>e < 352
325 --t
< 352 ::; 325
-c e -t 352
Index of Notations 509
- 352 Du 389 e lji 352 d (x ,y) 395
X 352 (M,d) 395
Go 352 B(x;e:) 395 w(p) 352 cls(X) 395 t;(p) 352 0 395
a w'(p) 354 :: 404 t;' (p) 354 sup 404 < 363 U 404 -t
$ 363 cpo 404 e
- t 363 D 404 - 363 1) 404 e K 364 1 404
X KO 364 T 405 X
Bx 409 0 367 °b 412 UB 368 (D1+D2) 416 0:N 368 [1)1+1)2) 416 :M 368 [lJ1 xlJ2) 419 Me 368 (D1-D2) 422 :M ef 368
[IJ1-lJ2) 422 lP 368 Fix(f) 426 [ol·· .. ok) 374
Us 427 [o)i 374
Os 427 [v.o) 374 433 Vo 01 U O2 374
Os _ 444 01 t1 02 374 re 1 (D) 453 [01-0 2 ] 374 Mi n(M) 455 SUPt 376 cpl 455 Inft 376 DOM 455 SUPe 376 DOMo 455 Inf 376 D 456 e
384 _5 TO D 456 tlA 386 5
sum 464 sup( '1 • '1) 386 prd 465 i nf ( l' 2) 386 fct 466 V(x) 388 ~ 467 C[O;l) 389 l 467
510 Index of Notations
sum 474 '< 480 n sum' 474 , 480 n > prd 474 0 481
n m prd' 474 v 484 n p fct 474 v< 484 n 1R 479 v 484 > p 479 C[O,1] 487
p< 479 a 487
p> 479 is 488
M< 479 Z . ml.n 492 M 479 Z 492 > max ,
p 480
Subject Index
Aberth, O. 499 Ackermann, W. 45, 52 Adjustment,
input 21 output 21
Admissible, c- 393 number; ng 114 t- 392
Algebra 220, 232 Algebraic
basis 408 cpo 408
Alphabet 2 input/output 59, 71
Apply function 425 Arithmetic, first order 248 Arithmetical
functi on 253 hierarchy 279, 286, 290 set 253
Arithmetically representable 248 Auxiliary symbols 62 Axiom 248, 258 Axiomatic theory 259 Axioms, Blum's 139, 140, 143
b-complete 414 Baire's
space 339 topology 339
Ball 395 Barendregt, H. 344 Base of neighbourhoods 388 Base, topological 322 Basis 408 Bijective function 1 Bird, R. 23 Bishop, E. 499 Blank 59 Blum, M. 137, 138, 319
complexity measure 306 Blum's axioms 139, 140, 143 Bohling, K.H. 319 Boo los, G. 4 Bottom element 405 Bounded minimization 51 Bounded set 455 Brainerd, W. 4, 52, 137, 319 BraunmUhl v., B. 319 Bridges, 0.5. 499 Brouwer, L.E.J. 499
c-admissible 393 c-clopen 361 c-complete 364 c-creative 363 c-equivalence 352, 363 c-open 360 c-precomplete 380 c-productive 364 c-reducible 352, 363 Calculus 258 Canonical numbering 223, 235 Cantor, G. 37, 145 Cantor's
discontinuum 487 pairing function 37 space 339 topo logy 339
Cauchy-representation 396, 397 Ceitin, G.S. 438, 452, 499 Center 395 Chain 404 Change of encodings 20 Church, A. 92 Church's thesis 79, 87, 91 Clopen 361
c- 361 t- 361 6- 372
Closure 395 Compact element 408 Complete,
b- 414 c- 364 fb- 414, 455 formally 260 m- 205, 364 numbering 191 partial order 338, 404 t- 364 1- 205, 364
Completely productive 210 Completion,
constructive 398 d- 410
Complexity class 311 measure 139 measure, Blum 306 Type 2 496
Compression theorem 312 Computability 26
oracle- 268, 340
512
relative 147, 369 Type 2 319, 320, 348, 494, 496
Computable, A- 272 cpo 444 left- 484 number function 26, 84, 87 0- 268 ordinal 237 real number 484 register 24, 26 right- 484 stack 71, 73, 84 strongly 147, 148, 369, 370 WHILE 55 Word function 66, 88 0- 367 (0,0' )(o,v)(v,v' )XElBo
369 370 147, 148 328
x E(I; 329 A E po 325
(,)
rE [lB- B] 346 r E[lB - lBo] 340 rE[lB o - lB o] 328 r E [ JP - IN] 346 rE[(I;-(I;o] 340 r E: [(I;o - (I;o] 329 r E [P w :--+- P w 1 325
Computation time function 9 Configuration 9
final 9 Consistent 327
forma lly 260 Constable, R.L. 452 Constructive
completion 398 cpo 427 d-completion 455 equivalence 454 function space 464 product 429, 464 sum 464 Type 2 320, 496
Continuity, relative 369, 370 Continuous,
(°1,°2 )- 406 (0,0')- 369 (o,v) 370 (.,.')- 323 strongly 369, 370
Correspondence 124, 147 problem 123
cpo 404 algebraic 408 computable 444 constructive 427 flat 405 super 456
cpo's, isomorphic 407 Creative,
c- 363 set 207, 364 t- 364
Cutland, N. 4 Cylinder 177, 182, 183, 377
d-completion 410 constructive 455
Dataset 7 Davis, M. 4, 69, 137 Decidable
numbering 187 partially 93, 99 set 93, 99, 123, 361 0- 372
Decision problem 123 Deduction rule 248 Degree, Turing 279, 280 Deil, T. 452 Dekker, J. 215 Denotational semantics 404 Dense 395
Subject Index
Diagonalization 46, 89, 90, 160 Diophantine equation 134 Directed set 404, 455 Discontinuum, Cantor's 487 Discrete topology 347 Distinguished element 191 Domain 453
of a function 1 recursive 453
Effective numbering 114, 216 Effectively inseparable 211, 365 Egli, H. 452 Empty word 2 Encoding,
change of 20 input 19 output 19
Enumerable, recursively 93, 97, 99, 163, 165. 168
Enumeration operator 268, 325 reducibility 325
Subject Index
Equivalence, c- 352, 363 constructive 454 m- 113, 150, 178, 179 t- 352, 363 1- 178, 179
Equivalent numberi ngs 113 Turing 279, 280
Ershov, J.L. 137, 158, 204, 215 Evaluation
func ti on 425 mapping 220, 233
Expression 249 Extension theorem 356, 357, 411
fb-complete 414, 455 Fermat's conjecture 263 Fibonacci function 51 Final
configuration 9 state 7 topology 330, 384
First order arithmetic 248 Fixed point,
Computable 467 equations 467 theorem 191, 194, 426
Flat cpo 405 Flowchart 5, 6
semantics of 9 isomorphic 10, 11 similar 17 substituti on 13
Formal language 259 theory 259
Formally complete 260 consistent 260
Formula 249 Free variable 251 Friedberg, R.M. 305, 442, 452 Friedman, H. 499 Function 1
apply 425 arithmetical 253 bijective 1 computation time 9 evaluation 425 injective 1 partial 1 partial recursive 87 productive 207 recursive 88 single step 9 successor 231 surjective 1 tota 1 1
total recursive 87 total step 9 universal 107, 349, 357, 358, 425 ll-recursive 47
Function space, constructive 464 standard 422, 463
Functions, similar 17
Gap theorem 313 General recursive function 88 Generalized
register macbine 28,29 stack machine 73
Generating system 408 Godel, K. 259, 263, 264
numbering 114, 145 Godelization 146 Godel's set 253 Graph
of a function 1 theorem 169
Grzegorczyk, A. 499
Halting problem 58, 117, 118, 364 Hartmanis, J. 319 Head, tape 59 Heidler. K. 4 Hennie, F. 4 Hermes, H. 4, 52, 69
513
Hierarchy, arithmetical 279, 286, 290 Hilbert, D. 134 Homeomorphic 338 Homeomorphism 338 Hopcroft, J. 4, 69, 136, 319
Induced topology 322 Inference rules 259 Initial state 7 Injective function 1 Input
adjustment 21 encoding 19 set 19
Input/Output alphabet 59, 71 Inseparable, effectively 211, 365 Intermediate value theorem 493 Interpretation 252 Isomorphic
cpo's 407 flowcharts 10, 11
Isomorphism of numberings 177, 178 recursive 474 theorem 198, 202
Jones, N.D. 4 Jump 285
514
Klaua, D. 499 Kleene, S.C. 91, 92, 122, 204, 235, 264
hierarchy 279, 286, 290 sets 286
Kleene's normal form theorem 142 Ko, K.-I. 499 Kreisel, G. 438, 452 Kreitz, C. 383, 403, 499 Kushner, B.A. 499
Label 6 Lacombe, D. 438, 452 Landweber, L.H. 4, 52, 137, 319 Language, formal 259 Least upper bound 178, 404, 455 left-computable 484 length of a word 2 Limit number 231 Loeckx, J. 4, 69, 432 Longo, G. 343 Lorenzen, P. 499 Luckham, D.C. 23
m-complete set 205, 364 m-equivalence 113, 150, 178, 179 Machine 5, 19
oracle 265, 266 queue 68 semantics of 19 stack 70, 71 tape 59 Turing 59, 65, 67 Type 2 341
Machtey, M. 4, 52, 137, 264 Mahn, F.-K. 4 Malcev, A.!, 4,52, 137, 158 Manna, Z. 23 Mapping 1
evaluation 220, 233 Markov, A.A. 92 Matijasevic, J.V. 136 Metric space 389, 395 Milne, R. 432 Minimization 34
bounded 51 Minimum 455 Modulus of uniform continuity 488 Moschovakis, Y.N. 438, 452 Muchnik, A.A. 305 Myhill, J. 447,452 Myhill's theorem 177, 179, 190 lJ.-Notation 39 lJ.-recursion 34, 35 lJ.-recursive function 47
Naming theorem 313 Natural number 1 Negative numbering 187 Neighbourhood 322
Subject Index
Neighbourhoods, base of 388 Normal form theorem 142 Notation 155 Number, limit 231 Number function, computable 26, 84, 87 Numbering 113, 144
admissible 114 canonical 223, 235 complete 191 decidable 187 effective 114, 216 Godel 114, 145 negative 187 positive 187 precomplete 191, 203 standard 79, 86, 106, 145, 163,
222, 223, 234, 235, 286, 433 Numberings,
equivalent 113 isomorphic 113
o-computable 268 I-complete 205, 364 I-equivalence 178, 179 Open
ba 11 395 c- 360 set 322 t- 360 6- 372
Operator, enumeration 268, 325 Operators, substitution 34, 35 Oracle
computability 268, 340 machines 265, 266 register machine 266 stack machine 268 tape machine 269
Order function 79 Ordinal
computable 237 numbers 227, 230 recursive 240
Output adjus tment 21 encoding 19 set 19
p-expression 42 Padding function 183 Pairing function 37 Pair list 123 Park, D. 23 Partial function 1
order 177, 404 order, complete 338, 404 recursive function 87
Partially decidable 93, 99 Paterson, M.S. 23
Subject Index
Paul, W. J. 319 Peter, R. 52 Polish school 487 Positive numbering 187 Post, E.L. 92, 123, 136, 176, 215, 305
correspondence problem 123, 129, 130 special 124
Post's problem 284 Precomplete
c- 380 numbering 191, 203 t- 380
Prefix 3 Prefixfree 333 pre-order 177 Primitive
recursion 34, 35 recursive function 42
Priority argument 281 Problem
correspondence 123 decision 123 halting 58, 117, 118, 364 self applicability 117, 274, 364 word 131, 134
Product 419, 463, 464 constructive 429, 464 standard 419, 463 topology 347
Producti on 130 Productive
c- 364 completely 210 functi on 207 set 207, 364 t- 364
Program, WHILE- 53 Progressive 228 Projection theorem 95, 362 Proof system 248, 258 Provable 259
set 93, 99, 361 6- 372
Pseudometric space 396 Pushdown store 70
Queue machine 68
Radius 396 Range of a function 1 r.e. set 93, 97. 99 Real number, computable 484 Recursion,
primitive 34. 35 s i mu ltaneous 51 theorem 191. 194. 195. 381 ~- 34. 35
Recursive A- 272 analysis 487 domain 453 function 88
general 88 primiti ve 42 tota 1 87
isomorphism 474 ordinal 240 set 93. 97. 98, 99. 163, 168
\1- 151 Recursively
515
enumerable 93. 97. 99. 163, 165, 168 A- 272 \1- 151
isomorphic 177, 178 Reducibility
c- 352. 363 enumeration 325 m- 113. 150. 178. 179 numberings 113. 150, 178 1- 177. 178, 179 one-one 177, 178, 179 sets 160. 179 t- 352, 363 truth-table 279. 300 tt- 279. 300 Turi ng- 279
Reduction, proof by 118, 160 Refinement 5. 10, 11, 14 Regi ster 24
computable 24. 26 machine 24. 25
generalized 28. 29 oracle 266
Reiser, A. 226 Relation 1 Relative
computability 147, 369, 370 continuity 369, 370
Relativized complexity 270 Renaming of states 10 Representable. arithmetically 248 Representation 352. 367
c-admissible 393 Cauchy 396. 397 m-adic 481 standard- 329. 333. 336. 349. 353,
368, 374. 389. 401. 479 t-admissible 392
Representations (of) lR 479 Reverse of a word 3 Rice, H.G. 449. 452 Rice's theorem 172. 176, 196, 213, 382 Right-computable 484 Robinson. J. 52
516
Rogers, H. 91, 122, 137, 176, 198, 204, 264, 305, 344, 366, 452
Rosser, J.B. 264 Russian school 487
Saturated set 410, 455 Schnorr, C.P. 4, 137, 319 Scott, D. 23, 344, 432, 476 Scott topology 412 Second number class 230 Segment 227 Self applicability problem 117, 274, 364 Semantics
denotational 404 flowchart 9 machine 19
Semi-Thue system 130, 131 Separable 395 Set
arithmetical 253 bounded 455 creative 207, 364 G6- 352 input 19 productive 207, 364 provable 93, 99, 361 recursive 93, 97, 98, 99, 163, 168 simple 162, 185, 202, 305, 364 Type 2 320
Shapiro, N. 449 Shepherdson, J.C. 41, 92, 447, 452 Shoenfield, J.R. 264, 305, 438 Sieber, K. 432 Signature 219, 231 Similar
fl owcharts 17 functions 16 tests 16
Similarity 16, 17 Simple set 162, 185, 202, 305, 364 Simulate 17 Simulation 5 simultaneous recursion 51 Single
step function 9 -valuedness theorem 170
smn-theorem lID, 143, 270, 435 effecti ve 112 for X 357 for tJ! 359
for $ 349 i njecti ve 196
Soare, R. 305 Solvable 123
Space Baire's 329 metric 389, 395 pseudometric 396 topological 322 To 388
Subject Index
Special Post correspondence problem 124 Speedup theorem 315 Spreen, D. 452 Stack 70
computable 71, 73, 84 machine 70, 71
genera 1 i zed 73 oracle 268
Standard function space 422, 463 k-tupling function 37, 38 numbering 79, 86, 106, 145, 163, 222,
223, 234, 235, 286, 433 product 419, 463 representation 329, 333, 336, 349,
353, 368, 374, 389, 401, 479
sum 415, 463 topology
on IB 0 327
on II 0 328 on IN 347 on Pw 323
tupling function 37, 38, 346 on lB 346
State 6, 7 final 7 initial 7
Stoy, J.E. 432 Strachey, C. 432 Strongly
computable 147, 148, 369, 370 continuous 369, 370
Sturgis, H.E. 41, 92 Subname property 241 Substitution 251
of flowcharts 13, 14 operators 34, 35
Subword 3 Successor 231
functi on 231 Suffix 3 Sum 416, 463, 464
constructive 464 standard 416, 463
Super cpo 456 Surjective function 1
Subject Index
t-admissible 392 t-clopen 361 t-complete 364 t-creative 364 t-equivalence 352, 363 t-open 360 t-reducible 352, 363 t-precomplete 380 T-predicate 142 t-productive 364 tt-reducibility 279, 300 To-space 388 Tape 59
cell 59 head 59 machine 59
oracle 268 Term 249 Test 19 Tests, similar 17 Theory,
axiomatic 259 formal 259
Thue, A. 136 Thue system 134 Topological
base 322 space 322
Topology 322 Baire's 339 Cantor's 339 discrete 347 final 330, 384 induced 322 standard 323, 327, 328, 347
Total function 1 recursive function 87 step function 9
Transfinite induction 228 Translation lemma 110, 113 Tree 219, 231
Truth-table reducibility 279, 300 Troelstra, A.S. 499
517
tupling function, standard 37, 38, 346 Turing, A.M. 59, 69, 92, 122, 499 Turing
degree 279, 280 equivalent 279 machine 59, 65, 67 reducibility 279
Type 2 complexity 496 computable 320, 348, 496 constructive 320, 496 machine 341 sets 320
Ullman, J.D. 4, 69, 136, 319 Uniform continuity 488 Universal function 107, 349, 357,
358, 425 utm-theorem 107, 140, 143, 270
for X 357 for 1jI 359 for W 349
Variable, free 251 Vector recursion 51
weakly v-r.e. 314 Weight of a tree 219, 231 Weihrauch, K. 226, 366, 383, 403,
We ll-order 227 WHILE-computable 55
-programs 53 Word 2
empty 2
452, 499
function, computable 66, 88 problem 131. 134
Young, P. 4, 52, 137, 264, 452