Knowledge (XXG)

Kleene equality

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where the down arrow means that the term on the left side of it is defined. Then it becomes possible to define the strong equality in the following way:
55:, that states that on a given argument either both functions are undefined, or both are defined and their values on that arguments are equal. 435: 595: 655: 684: 679: 425:{\displaystyle (y_{1}\sim y_{2}):\Leftrightarrow ((y_{1}\downarrow \lor y_{2}\downarrow )\longrightarrow y_{1}=y_{2}),} 674: 648: 641: 208: 101: 591: 52: 625: 34: 585: 283: 254: 179: 150: 127: 81: 61: 668: 621: 533: 20: 613: 511:{\displaystyle (f\simeq g):\Leftrightarrow (\forall x.(f(x)\sim g(x))).} 564: 313:
Some authors are using "quasi-equality", which is defined like this:
587:
Computability, an introduction to recursive function theory
569:
Studia Logica: An International Journal for Symbolic Logic
629: 438: 319: 286: 257: 211: 182: 153: 130: 104: 84: 64: 37: 510: 424: 301: 272: 241: 197: 168: 136: 116: 90: 70: 43: 565:"A Set Theory with Support for Partial Functions" 563:Farmer, William M.; Guttman, Joshua D. (2000). 649: 8: 590:. Cambridge University Press. p. 251. 656: 642: 58:For example, if we have partial functions 437: 410: 397: 378: 362: 340: 327: 318: 285: 256: 210: 181: 152: 129: 103: 83: 63: 36: 550: 525: 16:Equality operator on partial functions 7: 610: 608: 628:. You can help Knowledge (XXG) by 460: 14: 612: 502: 499: 496: 490: 481: 475: 469: 457: 451: 439: 416: 390: 387: 384: 368: 355: 352: 346: 320: 296: 290: 267: 261: 236: 230: 221: 215: 192: 186: 163: 157: 1: 51:) is an equality operator on 701: 607: 685:Equivalence (mathematics) 534:"Kleene equality in nLab" 242:{\displaystyle f(x)=g(x)} 117:{\displaystyle f\simeq g} 584:Cutland, Nigel (1980). 44:{\displaystyle \simeq } 624:-related article is a 512: 426: 303: 274: 243: 199: 170: 138: 118: 92: 72: 45: 513: 427: 304: 275: 244: 205:are both defined and 200: 171: 139: 124:means that for every 119: 93: 73: 46: 680:Computability theory 436: 317: 302:{\displaystyle g(x)} 284: 273:{\displaystyle f(x)} 255: 209: 198:{\displaystyle g(x)} 180: 169:{\displaystyle f(x)} 151: 128: 102: 82: 62: 35: 309:are both undefined. 508: 422: 299: 270: 239: 195: 166: 134: 114: 88: 68: 41: 675:Mathematics stubs 637: 636: 597:978-0-521-29465-2 137:{\displaystyle x} 91:{\displaystyle g} 71:{\displaystyle f} 53:partial functions 692: 658: 651: 644: 616: 609: 601: 577: 576: 560: 554: 548: 542: 541: 530: 517: 515: 514: 509: 431: 429: 428: 423: 415: 414: 402: 401: 383: 382: 367: 366: 345: 344: 332: 331: 308: 306: 305: 300: 279: 277: 276: 271: 248: 246: 245: 240: 204: 202: 201: 196: 175: 173: 172: 167: 143: 141: 140: 135: 123: 121: 120: 115: 97: 95: 94: 89: 77: 75: 74: 69: 50: 48: 47: 42: 700: 699: 695: 694: 693: 691: 690: 689: 665: 664: 663: 662: 605: 598: 583: 580: 562: 561: 557: 549: 545: 532: 531: 527: 523: 434: 433: 406: 393: 374: 358: 336: 323: 315: 314: 282: 281: 253: 252: 207: 206: 178: 177: 149: 148: 126: 125: 100: 99: 80: 79: 60: 59: 33: 32: 29:strong equality 25:Kleene equality 17: 12: 11: 5: 698: 696: 688: 687: 682: 677: 667: 666: 661: 660: 653: 646: 638: 635: 634: 617: 603: 602: 596: 579: 578: 555: 543: 524: 522: 519: 507: 504: 501: 498: 495: 492: 489: 486: 483: 480: 477: 474: 471: 468: 465: 462: 459: 456: 453: 450: 447: 444: 441: 421: 418: 413: 409: 405: 400: 396: 392: 389: 386: 381: 377: 373: 370: 365: 361: 357: 354: 351: 348: 343: 339: 335: 330: 326: 322: 311: 310: 298: 295: 292: 289: 269: 266: 263: 260: 249: 238: 235: 232: 229: 226: 223: 220: 217: 214: 194: 191: 188: 185: 165: 162: 159: 156: 133: 113: 110: 107: 87: 67: 40: 15: 13: 10: 9: 6: 4: 3: 2: 697: 686: 683: 681: 678: 676: 673: 672: 670: 659: 654: 652: 647: 645: 640: 639: 633: 631: 627: 623: 618: 615: 611: 606: 599: 593: 589: 588: 582: 581: 574: 570: 566: 559: 556: 552: 547: 544: 539: 535: 529: 526: 520: 518: 505: 493: 487: 484: 478: 472: 466: 463: 454: 448: 445: 442: 419: 411: 407: 403: 398: 394: 379: 375: 371: 363: 359: 349: 341: 337: 333: 328: 324: 293: 287: 264: 258: 250: 233: 227: 224: 218: 212: 189: 183: 160: 154: 147: 146: 145: 131: 111: 108: 105: 85: 65: 56: 54: 38: 30: 26: 22: 630:expanding it 619: 604: 586: 572: 568: 558: 553:, p. 3. 551:Cutland 1980 546: 537: 528: 312: 57: 28: 24: 18: 622:mathematics 575:(1): 59–78. 538:ncatlab.org 21:mathematics 669:Categories 521:References 455::⇔ 350::⇔ 485:∼ 461:∀ 446:≃ 391:⟶ 385:↓ 372:∨ 369:↓ 334:∼ 109:≃ 39:≃ 594:  620:This 27:, or 626:stub 592:ISBN 280:and 176:and 78:and 251:or 31:, ( 19:In 671:: 573:66 571:. 567:. 536:. 144:: 98:, 23:, 657:e 650:t 643:v 632:. 600:. 540:. 506:. 503:) 500:) 497:) 494:x 491:( 488:g 482:) 479:x 476:( 473:f 470:( 467:. 464:x 458:( 452:) 449:g 443:f 440:( 420:, 417:) 412:2 408:y 404:= 399:1 395:y 388:) 380:2 376:y 364:1 360:y 356:( 353:( 347:) 342:2 338:y 329:1 325:y 321:( 297:) 294:x 291:( 288:g 268:) 265:x 262:( 259:f 237:) 234:x 231:( 228:g 225:= 222:) 219:x 216:( 213:f 193:) 190:x 187:( 184:g 164:) 161:x 158:( 155:f 132:x 112:g 106:f 86:g 66:f

Index

mathematics
partial functions
"Kleene equality in nLab"
Cutland 1980
"A Set Theory with Support for Partial Functions"
Computability, an introduction to recursive function theory
ISBN
978-0-521-29465-2
Stub icon
mathematics
stub
expanding it
v
t
e
Categories
Mathematics stubs
Computability theory
Equivalence (mathematics)

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