Knowledge

Effective method

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The definition of an effective method involves more than the method itself. In order for a method to be called effective, it must be considered with respect to a class of problems. Because of this, one method may be effective with respect to one class of problems and
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Optionally, it may also be required that the method never returns a result as if it were an answer when the method is applied to a problem from
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is a procedure for solving a problem by any intuitively 'effective' means from a specific class. An effective method is sometimes also called a
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Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions (
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its class. Adding this requirement reduces the set of classes for which there is an effective method.
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A method is formally called effective for a class of problems when it satisfies these criteria:
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In principle, it can be done by a human without any aids except writing materials.
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Metalogic: An Introduction to the Metatheory of Standard First-Order Logic
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An effective method for calculating the values of a function is an
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Problem-solving procedures with certain characteristics
243:"Church's Thesis and the Principles for Mechanisms" 352: 535: 270:; Copeland, Jack; Proudfoot, Diane (June 2000). 78:When it is applied to a problem from its class: 64:be effective with respect to a different class. 518: 338: 278:. Turing Archive for the History of Computing 260: 100:to succeed. In other words, it requires no 525: 511: 345: 331: 159:states that the two notions coincide: any 96:Its instructions need only to be followed 296:The Cambridge Dictionary of Philosophy, 218: 131: 536: 231:, University of California Press, 1971 326: 319:, pp. 233 ff., esp. p. 231. 240: 75:number of exact, finite instructions. 477: 150:recursive or effective computability 88:It always produces a correct answer. 13: 196:Effective results in number theory 163:that is effectively calculable is 14: 570: 85:) after a finite number of steps. 481: 413:Gödel's incompleteness theorems 290: 234: 1: 211: 115: 54: 497:. You can help Knowledge by 408:Gödel's completeness theorem 7: 174: 138:general recursive functions 10: 575: 476: 396:Foundations of mathematics 311:. Reprinted, Dover, 2002, 272:"The Turing-Church Thesis" 364: 161:number-theoretic function 438:Löwenheim–Skolem theorem 463:Use–mention distinction 493:-related article is a 458:Type–token distinction 165:recursively computable 126:effectively calculable 554:Theory of computation 307:S. C. Kleene (1967), 241:Gandy, Robin (1980). 51:method or procedure. 549:Computability theory 381:Church–Turing thesis 375:Entscheidungsproblem 247:The Kleene Symposium 181:Decidability (logic) 157:Church–Turing thesis 132:Computable functions 81:It always finishes ( 37:computability theory 298:effective procedure 206:Undecidable problem 45:effective procedure 309:Mathematical logic 169:mathematical proof 506: 505: 471: 470: 71:It consists of a 566: 527: 520: 513: 485: 478: 391:Effective method 369:Cantor's theorem 347: 340: 333: 324: 323: 300: 294: 288: 287: 285: 283: 264: 258: 257: 255: 253: 238: 232: 225:Hunter, Geoffrey 222: 191:Function problem 186:Decision problem 41:effective method 29:computer science 574: 573: 569: 568: 567: 565: 564: 563: 534: 533: 532: 531: 474: 472: 467: 360: 358:metamathematics 351: 304: 303: 295: 291: 281: 279: 265: 261: 251: 249: 239: 235: 223: 219: 214: 177: 142:Turing machines 134: 118: 57: 17: 12: 11: 5: 572: 562: 561: 556: 551: 546: 530: 529: 522: 515: 507: 504: 503: 486: 469: 468: 466: 465: 460: 455: 450: 448:Satisfiability 445: 440: 435: 433:Interpretation 430: 425: 420: 415: 410: 405: 404: 403: 393: 388: 383: 378: 371: 365: 362: 361: 350: 349: 342: 335: 327: 321: 320: 302: 301: 289: 276:AlanTuring.net 268:Copeland, B.J. 259: 233: 216: 215: 213: 210: 209: 208: 203: 198: 193: 188: 183: 176: 173: 133: 130: 117: 114: 106: 105: 94: 91: 90: 89: 86: 76: 56: 53: 15: 9: 6: 4: 3: 2: 571: 560: 557: 555: 552: 550: 547: 545: 542: 541: 539: 528: 523: 521: 516: 514: 509: 508: 502: 500: 496: 492: 487: 484: 480: 479: 475: 464: 461: 459: 456: 454: 451: 449: 446: 444: 441: 439: 436: 434: 431: 429: 426: 424: 421: 419: 416: 414: 411: 409: 406: 402: 399: 398: 397: 394: 392: 389: 387: 384: 382: 379: 377: 376: 372: 370: 367: 366: 363: 359: 355: 348: 343: 341: 336: 334: 329: 328: 325: 318: 317:0-486-42533-9 314: 310: 306: 305: 299: 293: 277: 273: 269: 263: 248: 244: 237: 230: 226: 221: 217: 207: 204: 202: 201:Recursive set 199: 197: 194: 192: 189: 187: 184: 182: 179: 178: 172: 170: 166: 162: 158: 153: 151: 147: 143: 139: 129: 127: 123: 113: 111: 103: 99: 95: 92: 87: 84: 80: 79: 77: 74: 70: 69: 68: 65: 63: 52: 50: 46: 42: 38: 34: 31:, especially 30: 26: 22: 499:expanding it 488: 473: 453:Independence 428:Decidability 423:Completeness 390: 373: 308: 297: 292: 280:. Retrieved 275: 262: 250:. Retrieved 246: 236: 228: 220: 154: 135: 119: 109: 107: 82: 66: 61: 58: 48: 44: 40: 18: 559:Logic stubs 443:Metatheorem 401:of geometry 386:Consistency 104:to succeed. 25:mathematics 538:Categories 212:References 146:λ-calculus 116:Algorithms 98:rigorously 83:terminates 55:Definition 49:mechanical 544:Metalogic 418:Soundness 354:Metalogic 122:algorithm 102:ingenuity 33:metalogic 282:23 March 252:19 April 175:See also 110:outside 315:  73:finite 491:logic 489:This 39:, an 21:logic 495:stub 356:and 313:ISBN 284:2013 254:2024 155:The 35:and 27:and 62:not 43:or 19:In 540:: 274:. 245:. 227:, 171:. 152:. 144:, 140:, 128:. 23:, 526:e 519:t 512:v 501:. 346:e 339:t 332:v 286:. 256:.

Index

logic
mathematics
computer science
metalogic
computability theory
finite
rigorously
ingenuity
algorithm
effectively calculable
general recursive functions
Turing machines
λ-calculus
recursive or effective computability
Church–Turing thesis
number-theoretic function
recursively computable
mathematical proof
Decidability (logic)
Decision problem
Function problem
Effective results in number theory
Recursive set
Undecidable problem
Hunter, Geoffrey
"Church's Thesis and the Principles for Mechanisms"
Copeland, B.J.
"The Turing-Church Thesis"
ISBN
0-486-42533-9

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