1844:
1184:'s constant Ω are random or incompressible in the sense that we cannot compute them by a halting algorithm with fewer than n-O(1) bits. However, consider the short but never halting algorithm which systematically lists and runs all possible programs; whenever one of them halts its probability gets added to the output (initialized by zero). After finite time the first n bits of the output will never change any more (it does not matter that this time itself is not computable by a halting program). So there is a short non-halting algorithm whose output converges (after finite time) onto the first n bits of Ω. In other words, the
354:(instantaneous code) on the set of finite binary strings. A simple way to enforce prefix-free-ness is to use machines whose means of input is a binary stream from which bits can be read one at a time. There is no end-of-stream marker; the end of input is determined by when the universal machine decides to stop reading more bits, and the remaining bits are not considered part of the accepted string. Here, the difference between the two notions of program mentioned in the last paragraph becomes clear; one is easily recognized by some grammar, while the other requires arbitrary computation to recognize.
34:
594:, are equivalent to solving the halting problem for special programs (which would basically search for counter-examples and halt if one is found), knowing enough bits of Chaitin's constant would also imply knowing the answer to these problems. But as the halting problem is not generally solvable, and therefore calculating any but the first few bits of Chaitin's constant is not possible in a very concise language, this just reduces hard problems to impossible ones, much like trying to build an
1048:, can be used to characterize the halting probabilities among the left-c.e. reals. One can show that a real number in is a Chaitin constant (i.e. the halting probability of some prefix-free universal computable function) if and only if it is left-c.e. and algorithmically random. Ω is among the few
125:
Although there are infinitely many halting probabilities, one for each (universal, see below) method of encoding programs, it is common to use the letter Ω to refer to them as if there were only one. Because Ω depends on the program encoding used, it is sometimes called
1200:(2000) constructed a limit-computable "Super Ω" which in a sense is much more random than the original limit-computable Ω, as one cannot significantly compress the Super Ω by any enumerating non-halting algorithm.
473:
877:
723:
826:
1126:. Calude, Hertling, Khoussainov, and Wang showed that a recursively enumerable real number is an algorithmically random sequence if and only if it is a Chaitin's Ω number.
1380:
1030:
505:
165:
Such a function, intuitively, represents a programming language with the property that no valid program can be obtained as a proper extension of another valid program.
1253:
1106:
until enough elements of the domain have been found so that the probability they represent is within 2 of Ω. After this point, no additional program of length
1455:(1): 3488–3511 (Theme Issue 'The foundations of computation, physics and mentality: the Turing legacy' compiled and edited by Barry Cooper and Samson Abramsky).
889:
represents the probability that a randomly selected infinite sequence of 0s and 1s begins with a bit string (of some finite length) that is in the domain of
1622:
1218:) is prefixed by a random binary string – can be seen as the non-halting probability of a machine with oracle the third iteration of the
1052:
algorithmically random numbers and is the best-known algorithmically random number, but it is not at all typical of all algorithmically random numbers.
1886:
1419:
1168:
is effectively represented, and thus does not directly reflect the complexity of the axiomatic system. This incompleteness result is similar to
1405:
1268:
1896:
1169:
55:
1135:
571:
fashion, all programs of all lengths are run, until enough have halted to jointly contribute enough probability to match these first
1504:
77:
1843:
1583:
1110:
can be in the domain, because each of these would add 2 to the measure, which is impossible. Thus the set of strings of length
539:
may be denoted simply Ω, although different prefix-free universal computable functions lead to different values of Ω.
1615:
1580:
which appeared in the August 2004 edition of
Mathematics Today, on the occasion of the 50th anniversary of Alan Turing's death.
1123:
150:
982:> Ω is not computably enumerable. (Reason: every left-c.e. real with this property is computable, which Ω isn't.)
1891:
568:
95:
402:
315:
can generally be seen both as the concatenation of a program part and a data part, and as a single program for the function
290:
represents an "interpreter" that parses the script as a prefix of its input and then executes it on the remainder of input.
1749:
1539:(2002). "Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit".
621:
The probability measure on Cantor space, sometimes called the fair-coin measure, is defined so that for any binary string
1068:
digits of the number. This is equivalent to the existence of a program that enumerates the digits of the real number.
837:
176:
that takes one argument, a finite binary string, and possibly returns a single binary string as output. The function
48:
42:
1608:
218:
1818:
960:
358:
579:
hasn't halted yet, then it never will, since its contribution to the halting probability would affect the first
59:
1211:
1204:
670:
591:
1214:(UTM) – namely, the probability that it remains universal even when every input of it (as a
775:
1071:
No halting probability is computable. The proof of this fact relies on an algorithm which, given the first
1679:
1033:
912:
610:
is the collection of all infinite sequences of 0s and 1s. A halting probability can be interpreted as the
20:
1704:
1860:
1273:
1049:
915:(also known as Martin-Löf random or 1-random). This means that the shortest program to output the first
138:
1595:
1536:
1197:
524:
1372:
1865:
1855:
1456:
1385:
1041:
1003:
512:
295:
482:
1699:
1689:
1672:
1084:
986:
938:, which means that its digits are equidistributed as if they were generated by tossing a fair coin.
615:
182:
118:
that a randomly constructed program will halt. These numbers are formed from a construction due to
1573:
1411:
1774:
1739:
1717:
1631:
1500:
1474:
1401:
1292:
942:
142:
1225:
618:
on Cantor space. It is from this interpretation that halting probabilities take their name.
1811:
1806:
1784:
1769:
1734:
1652:
1548:
1464:
1393:
1208:
1189:
1141:
1040:
Not every set that is Turing equivalent to the halting problem is a halting probability. A
968:
351:
173:
91:
945:; there is no computable function that enumerates its binary expansion, as discussed below.
1577:
1219:
1181:
1145:
1076:
997:
949:
552:
370:
119:
927:
bits enable us to find out exactly which programs halt among all those of length at most
1460:
1389:
1667:
1657:
611:
595:
362:
187:
1295:
1880:
1722:
1684:
1371:
Calude, Cristian S.; Hertling, Peter H.; Khoussainov, Bakhadyr; Wang, Yongge (1998),
1215:
1165:
993:
935:
366:
263:
134:
1529:
1789:
1759:
1415:
1149:
607:
1172:
in that it shows that no consistent formal theory for arithmetic can be complete.
1122:
A real number is random if the binary sequence representing the real number is an
1256:
528:
115:
111:
1591:
153:, meaning there is not even any algorithm which can reliably guess its digits.
1552:
1185:
278:
can be used to simulate any computable function of one variable. Informally,
146:
1478:
1469:
1444:
1588:, Gregory Chaitin, originally appeared in Scientific American, March 2006.
1570:
Survey article discussing recent advances in the study of
Chaitin's Omega.
1188:
first n bits of Ω are highly compressible in the sense that they are
1060:
A real number is called computable if there is an algorithm which, given
1557:
Preprint: Algorithmic
Theories of Everything (arXiv: quant-ph/ 0011122)
1397:
1600:
1193:
1114:
in the domain is exactly the set of such strings already enumerated.
161:
The definition of a halting probability relies on the existence of a
1567:
1342:
1340:
190:
that computes it, in the sense that for any finite binary strings
523:. The requirement that the domain be prefix-free, together with
1384:, vol. 1373, Springer Berlin Heidelberg, pp. 596–606,
222:
if the following property holds: for every computable function
1604:
1160:
th can be proven to be 1 or 0 within that system. The constant
27:
1525:
An
Introduction to Kolmogorov Complexity and Its Applications
388:
be the domain of a prefix-free universal computable function
590:
Because many outstanding problems in number theory, such as
1828:
923:− O(1). This is because, as in the Goldbach example, those
907:
Each
Chaitin constant Ω has the following properties:
656:
be a prefix-free universal computable function. The domain
1373:"Recursively Enumerable Reals and Chaitin Ω numbers"
629:
has measure 2. This implies that for each natural number
1541:
International
Journal of Foundations of Computer Science
756:
contains all sequences in cantor space that begin with
1497:
Information and
Randomness: An Algorithmic Perspective
645:) = 1 has measure 1/2, and the set of sequences whose
468:{\displaystyle \Omega _{F}=\sum _{p\in P_{F}}2^{-|p|}}
1228:
1140:
For each specific consistent effectively represented
1006:
840:
778:
673:
485:
405:
357:
The domain of any universal computable function is a
1090:
The algorithm proceeds as follows. Given the first
614:
of a certain subset of Cantor space under the usual
583:
bits. Thus, the halting problem would be solved for
1594:and generalizations of algorithmic information, by
1592:
Limit-computable Super Omega more random than Omega
1445:"Universality probability of a prefix-free machine"
1196:with respect to the set of enumerating algorithms.
149:to compute its digits. Each halting probability is
19:"Omega number" redirects here. For other uses, see
1247:
1024:
871:
820:
717:
499:
467:
282:represents a "script" for the computable function
1449:Philosophical Transactions of the Royal Society A
1358:
1346:
1331:
1319:
963:; a real number with such a property is called a
872:{\displaystyle \bigcup _{i\in \mathbb {N} }S_{i}}
563:for which the halting problem is to be solved be
16:Halting probability of a random computer program
1130:Incompleteness theorem for halting probabilities
664:consists of an infinite set of binary strings
1616:
201:if and only if the Turing machine halts with
130:when not referring to any specific encoding.
8:
712:
680:
1623:
1609:
1601:
346:. This can be rephrased as: the domain of
163:prefix-free universal computable function.
114:that, informally speaking, represents the
1468:
1233:
1227:
1102:, the algorithm enumerates the domain of
1016:
1011:
1005:
863:
853:
852:
845:
839:
811:
803:
799:
783:
777:
718:{\displaystyle P=\{p_{1},p_{2},\ldots \}}
700:
687:
672:
484:
458:
450:
446:
434:
423:
410:
404:
78:Learn how and when to remove this message
21:Omega (disambiguation) § Mathematics
1192:by a very short algorithm; they are not
1180:As mentioned above, the first n bits of
919:bits of Ω must be of size at least
551:bits of Ω, one could calculate the
41:This article includes a list of general
1284:
1203:For an alternative "Super Ω", the
527:, ensures that this sum converges to a
226:of a single variable there is a string
821:{\displaystyle \sum _{p\in P}2^{-|p|}}
649:th element is 0 also has measure 1/2.
596:oracle machine for the halting problem
1516:Algorithmic Randomness and Complexity
1443:Barmpalias, G. and Dowe D.L. (2012).
1156:such that no bit of Ω after the
625:the set of sequences that begin with
7:
1514:Downey, R.; Hirschfeldt, D. (2010).
1425:from the original on 19 January 2004
893:. It is for this reason that Ω
831:represents the measure of the set
543:Relationship to the halting problem
1075:digits of Ω, solves Turing's
1008:
765:. These sets are disjoint because
407:
47:it lacks sufficient corresponding
14:
899:is called a halting probability.
555:for all programs of a size up to
535:is clear from context then Ω
205:on its tape when given the input
1842:
1576:article based on one written by
1523:Li, Ming; Vitányi, Paul (1997).
1136:Chaitin's incompleteness theorem
515:which has one summand for every
32:
1574:Omega and why maths has no TOEs
1269:Gödel's incompleteness theorems
1124:algorithmically random sequence
1083:. Since the halting problem is
1025:{\displaystyle \Delta _{2}^{0}}
769:is a prefix-free set. The sum
602:Interpretation as a probability
507:denotes the length of a string
145:, which means that there is no
1887:Algorithmic information theory
1530:Introduction chapter full-text
1240:
1234:
1170:Gödel's incompleteness theorem
812:
804:
500:{\displaystyle \left|p\right|}
459:
451:
133:Each halting probability is a
96:algorithmic information theory
1:
1499:(second ed.). Springer.
1359:Downey & Hirschfeldt 2010
1347:Downey & Hirschfeldt 2010
1332:Downey & Hirschfeldt 2010
1320:Downey & Hirschfeldt 2010
1087:, Ω cannot be computed.
1079:for programs of length up to
330:if there are no two elements
1495:Calude, Cristian S. (2002).
974:The set of rational numbers
307:on which it is defined. For
1897:Real transcendental numbers
311:that are universal, such a
1913:
1568:Aspects of Chaitin's Omega
1152:, there exists a constant
1133:
934:As a consequence, it is a
637:in Cantor space such that
18:
1851:
1840:
1638:
1553:10.1142/S0129054102001291
747:of Cantor space; the set
359:computably enumerable set
342:is a proper extension of
303:is the set of all inputs
1212:Universal Turing Machine
1205:universality probability
365:. The domain is always
338:in its domain such that
141:real number that is not
1248:{\displaystyle O^{(3)}}
1094:digits of Ω and a
633:, the set of sequences
62:more precise citations.
1470:10.1098/rsta.2011.0319
1249:
1118:Algorithmic randomness
1044:equivalence relation,
1034:arithmetical hierarchy
1026:
913:algorithmically random
873:
822:
729:Each of these strings
719:
501:
469:
392:. The constant Ω
128:Chaitin's construction
1892:Theory of computation
1300:mathworld.wolfram.com
1274:Kolmogorov complexity
1250:
1027:
965:left-c.e. real number
961:computably enumerable
874:
823:
720:
592:Goldbach's conjecture
575:bits. If the program
531:between 0 and 1. If
502:
470:
1585:The Limits of Reason
1296:"Chaitin's Constant"
1257:Turing Jump notation
1226:
1064:, returns the first
1004:
838:
776:
738:determines a subset
671:
483:
403:
104:Chaitin omega number
1537:Schmidhuber, Jürgen
1461:2012RSPTA.370.3488B
1390:1998LNCS.1373..596C
1361:, pp. 228–229.
1164:depends on how the
1046:Solovay equivalence
1021:
987:arithmetical number
883:In this way, Ω
616:probability measure
396:is then defined as
274:. This means that
266:of the two strings
108:halting probability
1632:Irrational numbers
1596:Jürgen Schmidhuber
1518:. Springer-Verlag.
1398:10.1007/bfb0028594
1293:Weisstein, Eric W.
1245:
1198:Jürgen Schmidhuber
1022:
1007:
1000:and thus at level
869:
858:
818:
794:
715:
559:. Let the program
547:Knowing the first
525:Kraft's inequality
497:
465:
441:
230:such that for all
1874:
1873:
1775:Supersilver ratio
1740:Supergolden ratio
1700:Twelfth root of 2
1407:978-3-540-64230-5
1334:, Theorem 5.1.11.
994:Turing equivalent
943:computable number
841:
779:
519:in the domain of
419:
367:Turing equivalent
151:Martin-Löf random
88:
87:
80:
1904:
1846:
1834:
1824:
1812:Square root of 7
1807:Square root of 6
1802:
1785:Square root of 5
1780:
1770:Square root of 3
1765:
1755:
1745:
1735:Square root of 2
1728:
1713:
1695:
1663:
1648:
1625:
1618:
1611:
1602:
1556:
1528:
1519:
1510:
1483:
1482:
1472:
1440:
1434:
1433:
1432:
1430:
1424:
1377:
1368:
1362:
1356:
1350:
1344:
1335:
1329:
1323:
1322:, Theorem 6.1.3.
1317:
1311:
1310:
1308:
1306:
1289:
1254:
1252:
1251:
1246:
1244:
1243:
1190:limit-computable
1150:Peano arithmetic
1142:axiomatic system
1031:
1029:
1028:
1023:
1020:
1015:
969:recursion theory
950:rational numbers
878:
876:
875:
870:
868:
867:
857:
856:
827:
825:
824:
819:
817:
816:
815:
807:
793:
724:
722:
721:
716:
705:
704:
692:
691:
506:
504:
503:
498:
496:
474:
472:
471:
466:
464:
463:
462:
454:
440:
439:
438:
415:
414:
352:prefix-free code
174:partial function
100:Chaitin constant
92:computer science
83:
76:
72:
69:
63:
58:this article by
49:inline citations
36:
35:
28:
1912:
1911:
1907:
1906:
1905:
1903:
1902:
1901:
1877:
1876:
1875:
1870:
1847:
1838:
1832:
1822:
1801:
1793:
1778:
1763:
1753:
1743:
1726:
1708:
1693:
1661:
1646:
1634:
1629:
1578:Gregory Chaitin
1564:
1535:
1522:
1513:
1507:
1494:
1491:
1486:
1442:
1441:
1437:
1428:
1426:
1422:
1408:
1375:
1370:
1369:
1365:
1357:
1353:
1345:
1338:
1330:
1326:
1318:
1314:
1304:
1302:
1291:
1290:
1286:
1282:
1265:
1229:
1224:
1223:
1220:halting problem
1182:Gregory Chaitin
1178:
1146:natural numbers
1138:
1132:
1120:
1077:halting problem
1058:
1056:Uncomputability
1002:
1001:
998:halting problem
959:< Ω is
905:
898:
888:
859:
836:
835:
795:
774:
773:
764:
755:
746:
737:
696:
683:
669:
668:
604:
553:halting problem
545:
538:
486:
481:
480:
442:
430:
406:
401:
400:
395:
387:
379:
371:halting problem
262:represents the
159:
120:Gregory Chaitin
84:
73:
67:
64:
54:Please help to
53:
37:
33:
24:
17:
12:
11:
5:
1910:
1908:
1900:
1899:
1894:
1889:
1879:
1878:
1872:
1871:
1869:
1868:
1863:
1861:Transcendental
1858:
1852:
1849:
1848:
1841:
1839:
1837:
1836:
1826:
1815:
1814:
1809:
1804:
1797:
1787:
1782:
1772:
1767:
1757:
1747:
1737:
1731:
1730:
1720:
1718:Cube root of 2
1715:
1702:
1697:
1687:
1682:
1680:Logarithm of 2
1676:
1675:
1670:
1665:
1655:
1650:
1639:
1636:
1635:
1630:
1628:
1627:
1620:
1613:
1605:
1599:
1598:
1589:
1581:
1571:
1563:
1562:External links
1560:
1559:
1558:
1547:(4): 587–612.
1533:
1520:
1511:
1505:
1490:
1487:
1485:
1484:
1435:
1406:
1363:
1351:
1349:, p. 405.
1336:
1324:
1312:
1283:
1281:
1278:
1277:
1276:
1271:
1264:
1261:
1242:
1239:
1236:
1232:
1177:
1174:
1134:Main article:
1131:
1128:
1119:
1116:
1057:
1054:
1038:
1037:
1019:
1014:
1010:
990:
983:
972:
946:
939:
932:
904:
901:
894:
884:
881:
880:
866:
862:
855:
851:
848:
844:
829:
828:
814:
810:
806:
802:
798:
792:
789:
786:
782:
760:
751:
742:
733:
727:
726:
714:
711:
708:
703:
699:
695:
690:
686:
682:
679:
676:
603:
600:
567:bits long. In
544:
541:
536:
495:
492:
489:
477:
476:
461:
457:
453:
449:
445:
437:
433:
429:
426:
422:
418:
413:
409:
393:
385:
378:
375:
363:computable set
188:Turing machine
186:if there is a
158:
155:
139:transcendental
86:
85:
40:
38:
31:
15:
13:
10:
9:
6:
4:
3:
2:
1909:
1898:
1895:
1893:
1890:
1888:
1885:
1884:
1882:
1867:
1866:Trigonometric
1864:
1862:
1859:
1857:
1856:Schizophrenic
1854:
1853:
1850:
1845:
1830:
1827:
1820:
1817:
1816:
1813:
1810:
1808:
1805:
1800:
1796:
1791:
1788:
1786:
1783:
1776:
1773:
1771:
1768:
1761:
1758:
1751:
1750:Erdős–Borwein
1748:
1741:
1738:
1736:
1733:
1732:
1724:
1723:Plastic ratio
1721:
1719:
1716:
1711:
1706:
1703:
1701:
1698:
1691:
1688:
1686:
1683:
1681:
1678:
1677:
1674:
1671:
1669:
1666:
1659:
1656:
1654:
1651:
1644:
1641:
1640:
1637:
1633:
1626:
1621:
1619:
1614:
1612:
1607:
1606:
1603:
1597:
1593:
1590:
1587:
1586:
1582:
1579:
1575:
1572:
1569:
1566:
1565:
1561:
1554:
1550:
1546:
1542:
1538:
1534:
1531:
1526:
1521:
1517:
1512:
1508:
1506:3-540-43466-6
1502:
1498:
1493:
1492:
1488:
1480:
1476:
1471:
1466:
1462:
1458:
1454:
1450:
1446:
1439:
1436:
1421:
1417:
1413:
1409:
1403:
1399:
1395:
1391:
1387:
1383:
1382:
1374:
1367:
1364:
1360:
1355:
1352:
1348:
1343:
1341:
1337:
1333:
1328:
1325:
1321:
1316:
1313:
1301:
1297:
1294:
1288:
1285:
1279:
1275:
1272:
1270:
1267:
1266:
1262:
1260:
1258:
1237:
1230:
1221:
1217:
1216:binary string
1213:
1210:
1206:
1201:
1199:
1195:
1191:
1187:
1183:
1175:
1173:
1171:
1167:
1166:formal system
1163:
1159:
1155:
1151:
1147:
1143:
1137:
1129:
1127:
1125:
1117:
1115:
1113:
1109:
1105:
1101:
1097:
1093:
1088:
1086:
1082:
1078:
1074:
1069:
1067:
1063:
1055:
1053:
1051:
1047:
1043:
1035:
1017:
1012:
999:
995:
991:
988:
985:Ω is an
984:
981:
977:
973:
970:
966:
962:
958:
954:
951:
947:
944:
940:
937:
936:normal number
933:
930:
926:
922:
918:
914:
910:
909:
908:
902:
900:
897:
892:
887:
864:
860:
849:
846:
842:
834:
833:
832:
808:
800:
796:
790:
787:
784:
780:
772:
771:
770:
768:
763:
759:
754:
750:
745:
741:
736:
732:
709:
706:
701:
697:
693:
688:
684:
677:
674:
667:
666:
665:
663:
659:
655:
650:
648:
644:
640:
636:
632:
628:
624:
619:
617:
613:
609:
601:
599:
597:
593:
588:
586:
582:
578:
574:
570:
566:
562:
558:
554:
550:
542:
540:
534:
530:
526:
522:
518:
514:
511:. This is an
510:
493:
490:
487:
455:
447:
443:
435:
431:
427:
424:
420:
416:
411:
399:
398:
397:
391:
384:
376:
374:
372:
368:
364:
360:
355:
353:
349:
345:
341:
337:
333:
329:
325:
322:The function
320:
318:
314:
310:
306:
302:
298:
297:
291:
289:
285:
281:
277:
273:
269:
265:
264:concatenation
261:
257:
253:
249:
245:
241:
237:
233:
229:
225:
221:
220:
215:
212:The function
210:
208:
204:
200:
197:
193:
189:
185:
184:
179:
175:
171:
168:Suppose that
166:
164:
156:
154:
152:
148:
144:
140:
136:
131:
129:
123:
121:
117:
113:
109:
105:
101:
97:
93:
82:
79:
71:
68:November 2011
61:
57:
51:
50:
44:
39:
30:
29:
26:
22:
1798:
1794:
1790:Silver ratio
1760:Golden ratio
1709:
1642:
1584:
1544:
1540:
1524:
1515:
1496:
1452:
1448:
1438:
1427:, retrieved
1379:
1366:
1354:
1327:
1315:
1303:. Retrieved
1299:
1287:
1202:
1179:
1161:
1157:
1153:
1139:
1121:
1111:
1107:
1103:
1099:
1095:
1091:
1089:
1080:
1072:
1070:
1065:
1061:
1059:
1045:
1039:
979:
975:
964:
956:
952:
941:It is not a
928:
924:
920:
916:
906:
895:
890:
885:
882:
830:
766:
761:
757:
752:
748:
743:
739:
734:
730:
728:
661:
657:
653:
651:
646:
642:
638:
634:
630:
626:
622:
620:
608:Cantor space
605:
589:
584:
580:
576:
572:
564:
560:
556:
548:
546:
532:
520:
516:
513:infinite sum
508:
478:
389:
382:
380:
361:but never a
356:
347:
343:
339:
335:
331:
327:
323:
321:
316:
312:
308:
304:
300:
294:
292:
287:
283:
279:
275:
271:
267:
259:
255:
251:
247:
243:
239:
235:
231:
227:
223:
217:
213:
211:
206:
202:
198:
195:
191:
181:
177:
169:
167:
162:
160:
132:
127:
124:
107:
103:
99:
94:subfield of
89:
74:
65:
46:
25:
1527:. Springer.
1489:Works cited
1305:3 September
1209:prefix-free
1176:Super Omega
1085:undecidable
948:The set of
569:dovetailing
529:real number
328:prefix-free
116:probability
112:real number
60:introducing
1881:Categories
1690:Lemniscate
1280:References
1186:enumerable
1148:, such as
978:such that
955:such that
903:Properties
598:would be.
377:Definition
326:is called
216:is called
183:computable
180:is called
157:Background
143:computable
43:references
1653:Liouville
1643:Chaitin's
1050:definable
1009:Δ
850:∈
843:⋃
801:−
788:∈
781:∑
710:…
448:−
428:∈
421:∑
408:Ω
219:universal
147:algorithm
1479:22711870
1429:20 March
1420:archived
1381:STACS 98
1263:See also
1144:for the
340:p′
336:p′
254:); here
199:F(x) = y
1819:Euler's
1705:Apéry's
1457:Bibcode
1416:5493426
1386:Bibcode
1222:(i.e.,
1032:of the
996:to the
612:measure
369:to the
90:In the
56:improve
1685:Dottie
1503:
1477:
1414:
1404:
1255:using
1194:random
992:It is
911:It is
479:where
296:domain
286:, and
258:
242:
135:normal
45:, but
1673:Cahen
1668:Omega
1658:Prime
1423:(PDF)
1412:S2CID
1376:(PDF)
1207:of a
1042:finer
350:is a
172:is a
110:is a
106:) or
1501:ISBN
1475:PMID
1431:2022
1402:ISBN
1307:2024
652:Let
606:The
381:Let
293:The
270:and
246:) =
194:and
137:and
98:, a
1712:(3)
1549:doi
1465:doi
1453:370
1394:doi
1259:).
967:in
660:of
299:of
1883::
1829:Pi
1545:13
1543:.
1473:.
1463:.
1451:.
1447:.
1418:,
1410:,
1400:,
1392:,
1378:,
1339:^
1298:.
1098:≤
587:.
373:.
334:,
319:.
234:,
209:.
196:y,
122:.
1835:)
1833:π
1831:(
1825:)
1823:e
1821:(
1803:)
1799:S
1795:δ
1792:(
1781:)
1779:ς
1777:(
1766:)
1764:φ
1762:(
1756:)
1754:E
1752:(
1746:)
1744:ψ
1742:(
1729:)
1727:ρ
1725:(
1714:)
1710:ζ
1707:(
1696:)
1694:ϖ
1692:(
1664:)
1662:ρ
1660:(
1649:)
1647:Ω
1645:(
1624:e
1617:t
1610:v
1555:.
1551::
1532:.
1509:.
1481:.
1467::
1459::
1396::
1388::
1309:.
1241:)
1238:3
1235:(
1231:O
1162:N
1158:N
1154:N
1112:k
1108:k
1104:F
1100:n
1096:k
1092:n
1081:n
1073:n
1066:n
1062:n
1036:.
1018:0
1013:2
989:.
980:q
976:q
971:.
957:q
953:q
931:.
929:n
925:n
921:n
917:n
896:F
891:F
886:F
879:.
865:i
861:S
854:N
847:i
813:|
809:p
805:|
797:2
791:P
785:p
767:P
762:i
758:p
753:i
749:S
744:i
740:S
735:i
731:p
725:.
713:}
707:,
702:2
698:p
694:,
689:1
685:p
681:{
678:=
675:P
662:F
658:P
654:F
647:n
643:n
641:(
639:f
635:f
631:n
627:x
623:x
585:p
581:N
577:p
573:N
565:N
561:p
557:N
549:N
537:F
533:F
521:F
517:p
509:p
494:|
491:p
488:|
475:,
460:|
456:p
452:|
444:2
436:F
432:P
425:p
417:=
412:F
394:F
390:F
386:F
383:P
348:F
344:p
332:p
324:F
317:F
313:p
309:F
305:p
301:F
288:F
284:f
280:w
276:F
272:x
268:w
260:x
256:w
252:x
250:(
248:f
244:x
240:w
238:(
236:F
232:x
228:w
224:f
214:F
207:x
203:y
192:x
178:F
170:F
102:(
81:)
75:(
70:)
66:(
52:.
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.