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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