29:
905:
1717:
629:
1837:
1344:, that is, smaller than the size of the input. Thus, "charging" the algorithm for the size of the input, or for the size of the output, would not truly capture the memory space used. This is solved by defining the
1323:
1345:
1238:
986:
506:
1444:
1348:, which is a standard multi-tape Turing machine, except that the input tape may never be written-to, and the output tape may never be read from. This allows smaller space classes, such as
1066:
1026:
356:
613:
575:
399:
1520:
1482:
1563:
900:{\displaystyle |S|\leq |{\mathcal {C}}|^{|{\mathcal {C}}|}\leq (2^{c.s(n)})^{2^{c.s(n)}}=2^{c.s(n).2^{c.s(n)}}<2^{2^{2c.s(n)}}=2^{2^{o(\log \log n)}}=o(n)}
1352:(logarithmic space), to be defined in terms of the amount of space used by all of the work tapes (excluding the special input and output tapes).
1745:
1248:
1993:
1170:
925:
428:
1927:
1901:
1546:
142:
524:
2355:
112:
93:
1413:
2390:
1986:
1893:
1550:
65:
126:
149:. It represents the total amount of memory space that a "normal" physical computer would need to solve a given
50:
72:
1408:
1337:
1139:
1031:
991:
202:
146:
2395:
1979:
615:
possibilities for every element of a crossing sequence, so the number of different crossing sequences of
79:
2371:
2178:
1919:
333:
584:
546:
380:
2360:
222:
61:
1526:
1404:
1143:
46:
39:
1355:
Since many symbols might be packed into one by taking a suitable power of the alphabet, for all
2314:
138:
2309:
2304:
1487:
1449:
150:
2299:
1712:{\displaystyle {\mathsf {DSPACE}}\subseteq {\mathsf {NSPACE}}\subseteq {\mathsf {DSPACE}}.}
1554:
911:
221:
that can be used, though there may be restrictions on some other complexity measures (like
1937:
8:
1728:
2319:
1923:
1897:
1957:
86:
2283:
2173:
2105:
2095:
2002:
1933:
248:
236:
218:
198:
186:
174:
170:
2278:
2268:
2225:
2215:
2208:
2198:
2188:
2146:
2141:
2136:
2120:
2100:
2078:
2073:
2068:
2056:
2051:
2046:
2041:
2273:
2161:
2083:
2036:
1952:
1392:
1349:
1150:
402:
307:
177:
that can be solved using a certain amount of memory space. For each function
2384:
1911:
373:
be an input of smallest size, denoted by n, that requires more space than
2151:
2061:
1832:{\displaystyle {\mathsf {NTIME}}(t(n))\subseteq {\mathsf {DSPACE}}(t(n))}
1318:{\displaystyle \bigcup _{k\in \mathbb {N} }{\mathsf {DSPACE}}(2^{n^{k}})}
2263:
2088:
577:: if it is longer than that, then some configuration will repeat, and
2258:
1971:
1341:
1233:{\displaystyle \bigcup _{k\in \mathbb {N} }{\mathsf {DSPACE}}(n^{k})}
154:
28:
2253:
2248:
2193:
2016:
1242:
2243:
2350:
2220:
2203:
1541:
1164:
17:
1383:) space is the same as the class of languages recognizable in
2340:
2335:
2156:
1723:
228:
Several important complexity classes are defined in terms of
2168:
2026:
1446:, there exists some language L which is decidable in space
981:{\displaystyle {\mathcal {C}}_{i}(x)={\mathcal {C}}_{j}(x)}
2183:
2115:
2110:
2031:
2021:
1531:
501:{\displaystyle |{\mathcal {C}}|\leq 2^{c.s(n)}=o(\log n)}
267:
space is required to recognize any non-regular language).
581:
will go into an infinite loop. There are also at most
1748:
1566:
1490:
1452:
1416:
1251:
1173:
1034:
994:
928:
632:
587:
549:
431:
383:
336:
1727:is related to DSPACE in the following way. For any
53:. Unsourced material may be challenged and removed.
1831:
1711:
1514:
1476:
1438:
1340:. Several important space complexity classes are
1317:
1232:
1060:
1020:
980:
899:
607:
569:
500:
393:
350:
535:. Note that the length of a crossing sequence of
274:Suppose that there exists a non-regular language
2382:
1868:Space complexity of alternating Turing machines
1346:multi-tape Turing machine with input and output
1138:The above theorem implies the necessity of the
1439:{\displaystyle f:\mathbb {N} \to \mathbb {N} }
1131:. Hence, there does not exist such a language
217:)). There is no restriction on the amount of
1987:
1887:
1099:still behaves exactly the same way on input
1916:Computational complexity. A modern approach
1994:
1980:
1910:
1890:Turing Machines with Sublogarithmic Space
1432:
1424:
1375:, the class of languages recognizable in
1264:
1186:
344:
113:Learn how and when to remove this message
1107:, so it needs the same space to compute
1865:
1068:are the crossing sequences at boundary
2383:
2001:
1806:
1803:
1800:
1797:
1794:
1791:
1763:
1760:
1757:
1754:
1751:
1670:
1667:
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1661:
1658:
1655:
1627:
1624:
1621:
1618:
1615:
1612:
1584:
1581:
1578:
1575:
1572:
1569:
1532:Relation with other complexity classes
1287:
1284:
1281:
1278:
1275:
1272:
1209:
1206:
1203:
1200:
1197:
1194:
1975:
1061:{\displaystyle {\mathcal {C}}_{j}(x)}
1021:{\displaystyle {\mathcal {C}}_{i}(x)}
160:
1539:is the deterministic counterpart of
1398:
51:adding citations to reliable sources
22:
1873:
1850:
13:
1127:, contradicting the definition of
1038:
998:
958:
932:
675:
656:
595:
557:
439:
386:
14:
2407:
2356:Probabilistically checkable proof
1945:
1391:) space. This justifies usage of
1328:
351:{\displaystyle k\in \mathbb {N} }
1551:non-deterministic Turing machine
608:{\displaystyle |{\mathcal {C}}|}
570:{\displaystyle |{\mathcal {C}}|}
27:
1894:Springer Science+Business Media
1336:is traditionally measured on a
523:denote the set of all possible
127:computational complexity theory
38:needs additional citations for
1888:Szepietowski, Andrzej (1994).
1879:Arora & Barak (2009) p. 86
1859:
1826:
1823:
1817:
1811:
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1703:
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1604:
1601:
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1506:
1500:
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1471:
1468:
1462:
1456:
1428:
1312:
1292:
1227:
1214:
1055:
1049:
1015:
1009:
975:
969:
949:
943:
894:
888:
875:
857:
834:
828:
796:
790:
773:
767:
743:
737:
719:
713:
707:
690:
681:
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663:
650:
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634:
601:
589:
563:
551:
495:
483:
472:
466:
445:
433:
394:{\displaystyle {\mathcal {C}}}
330:(1)); thus, for any arbitrary
16:For digital repositories, see
1:
1843:
1409:space-constructible function
1338:deterministic Turing machine
1140:space-constructible function
1083:be the string obtained from
203:deterministic Turing machine
147:deterministic Turing machine
7:
1087:by removing all cells from
512:is a constant depending on
358:, there exists an input of
141:describing the resource of
10:
2412:
2372:List of complexity classes
1920:Cambridge University Press
1524:
362:requiring more space than
15:
2369:
2328:
2292:
2236:
2129:
2009:
1856:Szepietowski (1994) p. 28
2361:Interactive proof system
251:. In fact, REG = DSPACE(
201:that can be solved by a
2391:Computational resources
1866:Alberts, Maris (1985),
1527:space hierarchy theorem
1515:{\displaystyle o(f(n))}
1477:{\displaystyle O(f(n))}
1405:space hierarchy theorem
1144:space hierarchy theorem
2315:Arithmetical hierarchy
1914:; Barak, Boaz (2009).
1833:
1713:
1516:
1478:
1440:
1407:shows that, for every
1319:
1234:
1062:
1022:
982:
914:, there exist indexes
901:
609:
571:
502:
395:
352:
139:computational resource
2310:Grzegorczyk hierarchy
2305:Exponential hierarchy
2237:Considered infeasible
1834:
1714:
1525:Further information:
1517:
1479:
1441:
1320:
1235:
1063:
1023:
983:
902:
610:
572:
503:
396:
353:
322:). By our assumption
173:, sets of all of the
151:computational problem
2300:Polynomial hierarchy
2130:Suspected infeasible
1746:
1564:
1488:
1450:
1414:
1249:
1171:
1032:
992:
926:
912:pigeonhole principle
630:
585:
547:
429:
381:
334:
261:Ω(log log
47:improve this article
2329:Families of classes
2010:Considered feasible
1395:in the definition.
255:(log log
2396:Complexity classes
2003:Complexity classes
1829:
1729:time constructible
1709:
1512:
1474:
1436:
1315:
1269:
1230:
1191:
1142:assumption in the
1058:
1018:
978:
897:
605:
567:
525:crossing sequences
498:
401:be the set of all
391:
348:
171:complexity classes
169:is used to define
161:Complexity classes
2378:
2377:
2320:Boolean hierarchy
2293:Class hierarchies
1929:978-0-521-42426-4
1903:978-3-540-58355-4
1555:Savitch's theorem
1484:but not in space
1399:Hierarchy theorem
1252:
1174:
249:regular languages
232:. These include:
199:decision problems
175:decision problems
123:
122:
115:
97:
2403:
1996:
1989:
1982:
1973:
1972:
1941:
1907:
1880:
1877:
1871:
1870:
1863:
1857:
1854:
1838:
1836:
1835:
1830:
1810:
1809:
1767:
1766:
1718:
1716:
1715:
1710:
1702:
1701:
1674:
1673:
1631:
1630:
1588:
1587:
1521:
1519:
1518:
1513:
1483:
1481:
1480:
1475:
1445:
1443:
1442:
1437:
1435:
1427:
1324:
1322:
1321:
1316:
1311:
1310:
1309:
1308:
1291:
1290:
1268:
1267:
1239:
1237:
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1231:
1226:
1225:
1213:
1212:
1190:
1189:
1134:
1130:
1126:
1114:
1110:
1106:
1102:
1098:
1086:
1082:
1076:, respectively.
1067:
1065:
1064:
1059:
1048:
1047:
1042:
1041:
1027:
1025:
1024:
1019:
1008:
1007:
1002:
1001:
987:
985:
984:
979:
968:
967:
962:
961:
942:
941:
936:
935:
906:
904:
903:
898:
881:
880:
879:
878:
840:
839:
838:
837:
802:
801:
800:
799:
749:
748:
747:
746:
717:
716:
686:
685:
684:
679:
678:
672:
666:
660:
659:
653:
645:
637:
614:
612:
611:
606:
604:
599:
598:
592:
576:
574:
573:
568:
566:
561:
560:
554:
507:
505:
504:
499:
476:
475:
448:
443:
442:
436:
400:
398:
397:
392:
390:
389:
357:
355:
354:
349:
347:
266:
247:is the class of
219:computation time
187:complexity class
118:
111:
107:
104:
98:
96:
55:
31:
23:
2411:
2410:
2406:
2405:
2404:
2402:
2401:
2400:
2381:
2380:
2379:
2374:
2365:
2324:
2288:
2232:
2226:PSPACE-complete
2125:
2005:
2000:
1948:
1930:
1904:
1884:
1883:
1878:
1874:
1864:
1860:
1855:
1851:
1846:
1790:
1789:
1750:
1749:
1747:
1744:
1743:
1697:
1693:
1654:
1653:
1611:
1610:
1568:
1567:
1565:
1562:
1561:
1557:, we have that
1545:, the class of
1534:
1529:
1489:
1486:
1485:
1451:
1448:
1447:
1431:
1423:
1415:
1412:
1411:
1401:
1331:
1304:
1300:
1299:
1295:
1271:
1270:
1263:
1256:
1250:
1247:
1246:
1221:
1217:
1193:
1192:
1185:
1178:
1172:
1169:
1168:
1132:
1128:
1116:
1112:
1108:
1104:
1100:
1096:
1084:
1080:
1043:
1037:
1036:
1035:
1033:
1030:
1029:
1003:
997:
996:
995:
993:
990:
989:
963:
957:
956:
955:
937:
931:
930:
929:
927:
924:
923:
853:
849:
848:
844:
815:
811:
810:
806:
780:
776:
757:
753:
727:
723:
722:
718:
697:
693:
680:
674:
673:
668:
667:
662:
661:
655:
654:
649:
641:
633:
631:
628:
627:
600:
594:
593:
588:
586:
583:
582:
562:
556:
555:
550:
548:
545:
544:
456:
452:
444:
438:
437:
432:
430:
427:
426:
385:
384:
382:
379:
378:
343:
335:
332:
331:
260:
197:)), the set of
163:
119:
108:
102:
99:
56:
54:
44:
32:
21:
12:
11:
5:
2409:
2399:
2398:
2393:
2376:
2375:
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2367:
2366:
2364:
2363:
2358:
2353:
2348:
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2338:
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2325:
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2271:
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2256:
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2246:
2240:
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2234:
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2230:
2229:
2228:
2218:
2213:
2212:
2211:
2201:
2196:
2191:
2186:
2181:
2176:
2171:
2166:
2165:
2164:
2162:co-NP-complete
2159:
2154:
2149:
2139:
2133:
2131:
2127:
2126:
2124:
2123:
2118:
2113:
2108:
2103:
2098:
2093:
2092:
2091:
2081:
2076:
2071:
2066:
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2064:
2054:
2049:
2044:
2039:
2034:
2029:
2024:
2019:
2013:
2011:
2007:
2006:
1999:
1998:
1991:
1984:
1976:
1970:
1969:
1953:Complexity Zoo
1947:
1946:External links
1944:
1943:
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1928:
1912:Arora, Sanjeev
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1397:
1393:big O notation
1330:
1329:Machine models
1327:
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1303:
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1211:
1208:
1205:
1202:
1199:
1196:
1188:
1184:
1181:
1177:
1162:
1135:as assumed. □
1111:as to compute
1095:. The machine
1057:
1054:
1051:
1046:
1040:
1017:
1014:
1011:
1006:
1000:
977:
974:
971:
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830:
827:
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792:
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786:
783:
779:
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772:
769:
766:
763:
760:
756:
752:
745:
742:
739:
736:
733:
730:
726:
721:
715:
712:
709:
706:
703:
700:
696:
692:
689:
683:
677:
671:
665:
658:
652:
648:
644:
640:
636:
603:
597:
591:
565:
559:
553:
497:
494:
491:
488:
485:
482:
479:
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468:
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459:
455:
451:
447:
441:
435:
403:configurations
388:
346:
342:
339:
308:Turing machine
269:
268:
185:), there is a
162:
159:
121:
120:
35:
33:
26:
9:
6:
4:
3:
2:
2408:
2397:
2394:
2392:
2389:
2388:
2386:
2373:
2368:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2333:
2331:
2327:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2301:
2298:
2297:
2295:
2291:
2285:
2282:
2280:
2277:
2275:
2272:
2270:
2267:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2241:
2239:
2235:
2227:
2224:
2223:
2222:
2219:
2217:
2214:
2210:
2207:
2206:
2205:
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2170:
2167:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2144:
2143:
2140:
2138:
2135:
2134:
2132:
2128:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2090:
2087:
2086:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2063:
2060:
2059:
2058:
2055:
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2050:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2014:
2012:
2008:
2004:
1997:
1992:
1990:
1985:
1983:
1978:
1977:
1974:
1967:
1965:
1961:
1955:
1954:
1950:
1949:
1939:
1935:
1931:
1925:
1921:
1917:
1913:
1909:
1905:
1899:
1895:
1891:
1886:
1885:
1876:
1869:
1862:
1853:
1849:
1820:
1814:
1786:
1777:
1771:
1742:
1741:
1740:
1738:
1734:
1730:
1726:
1725:
1706:
1698:
1687:
1681:
1650:
1641:
1635:
1607:
1598:
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1560:
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1548:
1544:
1543:
1538:
1528:
1523:
1503:
1497:
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1453:
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1410:
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1396:
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1358:
1353:
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1296:
1260:
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1160:
1156:
1152:
1149:
1148:
1147:
1145:
1141:
1136:
1124:
1120:
1094:
1090:
1077:
1075:
1071:
1052:
1044:
1012:
1004:
972:
964:
952:
946:
938:
921:
917:
913:
910:According to
891:
885:
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872:
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866:
863:
860:
854:
850:
845:
841:
831:
825:
822:
819:
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807:
803:
793:
787:
784:
781:
777:
770:
764:
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277:
273:
264:
259:)) (that is,
258:
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158:
156:
153:with a given
152:
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117:
114:
106:
95:
92:
88:
85:
81:
78:
74:
71:
67:
64: –
63:
59:
58:Find sources:
52:
48:
42:
41:
36:This article
34:
30:
25:
24:
19:
2345:
1963:
1959:
1951:
1915:
1889:
1875:
1867:
1861:
1852:
1736:
1732:
1722:
1721:
1547:memory space
1540:
1536:
1535:
1402:
1388:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1356:
1354:
1333:
1332:
1158:
1154:
1137:
1122:
1118:
1103:as on input
1092:
1088:
1078:
1073:
1069:
919:
915:
909:
620:
616:
578:
540:
536:
532:
528:
520:
518:
513:
509:
422:
418:
414:
410:
406:
374:
370:
368:
363:
359:
327:
323:
319:
315:
311:
303:
299:
295:
291:
287:
283:
279:
275:
271:
270:
262:
256:
252:
244:
243:(1)), where
240:
229:
227:
214:
210:
206:
205:using space
194:
190:
182:
178:
166:
165:The measure
164:
143:memory space
134:
130:
124:
109:
103:October 2009
100:
90:
83:
76:
69:
57:
45:Please help
40:verification
37:
2209:#P-complete
2147:NP-complete
2062:NL-complete
1739:), we have
1115:. However,
543:is at most
223:alternation
2385:Categories
2264:ELEMENTARY
2089:P-complete
1938:1193.68112
1844:References
1363:such that
1157:(log
922:such that
413:. Because
73:newspapers
2259:2-EXPTIME
1787:⊆
1731:function
1651:⊆
1608:⊆
1429:→
1342:sublinear
1261:∈
1254:⋃
1183:∈
1176:⋃
1153:= DSPACE(
870:
864:
688:≤
647:≤
490:
450:≤
425:)), then
417:∈ DSPACE(
409:on input
341:∈
326:∉ DSPACE(
314:in space
310:deciding
298:(log log
278:∈ DSPACE(
239:= DSPACE(
155:algorithm
2254:EXPSPACE
2249:NEXPTIME
2017:DLOGTIME
1359:≥ 1 and
1243:EXPSPACE
1121:| < |
988:, where
508:, where
286:)), for
62:"DSPACE"
2244:EXPTIME
2152:NP-hard
1958:DSPACE(
1091:+ 1 to
302:). Let
137:is the
87:scholar
2351:NSPACE
2346:DSPACE
2221:PSPACE
1936:
1926:
1900:
1553:. By
1542:NSPACE
1537:DSPACE
1334:DSPACE
1165:PSPACE
377:, and
272:Proof:
230:DSPACE
189:SPACE(
167:DSPACE
145:for a
131:DSPACE
89:
82:
75:
68:
60:
18:DSpace
2341:NTIME
2336:DTIME
2157:co-NP
1724:NTIME
1549:on a
918:<
306:be a
135:SPACE
94:JSTOR
80:books
2169:TFNP
1924:ISBN
1898:ISBN
1403:The
1371:) ≥
1079:Let
1072:and
1028:and
804:<
519:Let
369:Let
294:) =
157:.
66:news
2284:ALL
2184:QMA
2174:FNP
2116:APX
2111:BQP
2106:BPP
2096:ZPP
2027:ACC
1934:Zbl
1377:c f
1119:x'
867:log
861:log
623:is
619:on
539:on
531:on
527:of
487:log
405:of
245:REG
237:REG
225:).
133:or
125:In
49:by
2387::
2279:RE
2269:PR
2216:IP
2204:#P
2199:PP
2194:⊕P
2189:PH
2179:AM
2142:NP
2137:UP
2121:FP
2101:RP
2079:CC
2074:SC
2069:NC
2057:NL
2052:FL
2047:RL
2042:SL
2032:TC
2022:AC
1966:))
1956::
1932:.
1922:.
1918:.
1896:.
1892:.
1522:.
1245:=
1167:=
1161:))
1146:.
1109:x'
1101:x'
1081:x'
778:.2
516:.
366:.
129:,
2274:R
2084:P
2037:L
1995:e
1988:t
1981:v
1968:.
1964:n
1962:(
1960:f
1940:.
1906:.
1839:.
1827:)
1824:)
1821:n
1818:(
1815:t
1812:(
1807:E
1804:C
1801:A
1798:P
1795:S
1792:D
1784:)
1781:)
1778:n
1775:(
1772:t
1769:(
1764:E
1761:M
1758:I
1755:T
1752:N
1737:n
1735:(
1733:t
1707:.
1704:]
1699:2
1695:)
1691:)
1688:n
1685:(
1682:s
1679:(
1676:[
1671:E
1668:C
1665:A
1662:P
1659:S
1656:D
1648:]
1645:)
1642:n
1639:(
1636:s
1633:[
1628:E
1625:C
1622:A
1619:P
1616:S
1613:N
1605:]
1602:)
1599:n
1596:(
1593:s
1590:[
1585:E
1582:C
1579:A
1576:P
1573:S
1570:D
1510:)
1507:)
1504:n
1501:(
1498:f
1495:(
1492:o
1472:)
1469:)
1466:n
1463:(
1460:f
1457:(
1454:O
1433:N
1425:N
1421::
1418:f
1389:n
1387:(
1385:f
1381:n
1379:(
1373:1
1369:n
1367:(
1365:f
1361:f
1357:c
1350:L
1313:)
1306:k
1302:n
1297:2
1293:(
1288:E
1285:C
1282:A
1279:P
1276:S
1273:D
1265:N
1258:k
1228:)
1223:k
1219:n
1215:(
1210:E
1207:C
1204:A
1201:P
1198:S
1195:D
1187:N
1180:k
1159:n
1155:O
1151:L
1133:L
1129:x
1125:|
1123:x
1117:|
1113:x
1105:x
1097:M
1093:j
1089:i
1085:x
1074:j
1070:i
1056:)
1053:x
1050:(
1045:j
1039:C
1016:)
1013:x
1010:(
1005:i
999:C
976:)
973:x
970:(
965:j
959:C
953:=
950:)
947:x
944:(
939:i
933:C
920:j
916:i
895:)
892:n
889:(
886:o
883:=
876:)
873:n
858:(
855:o
851:2
846:2
842:=
835:)
832:n
829:(
826:s
823:.
820:c
817:2
813:2
808:2
797:)
794:n
791:(
788:s
785:.
782:c
774:)
771:n
768:(
765:s
762:.
759:c
755:2
751:=
744:)
741:n
738:(
735:s
732:.
729:c
725:2
720:)
714:)
711:n
708:(
705:s
702:.
699:c
695:2
691:(
682:|
676:C
670:|
664:|
657:C
651:|
643:|
639:S
635:|
621:x
617:M
602:|
596:C
590:|
579:M
564:|
558:C
552:|
541:x
537:M
533:x
529:M
521:S
514:M
510:c
496:)
493:n
484:(
481:o
478:=
473:)
470:n
467:(
464:s
461:.
458:c
454:2
446:|
440:C
434:|
423:n
421:(
419:s
415:M
411:x
407:M
387:C
375:k
371:x
364:k
360:M
345:N
338:k
328:O
324:L
320:n
318:(
316:s
312:L
304:M
300:n
296:o
292:n
290:(
288:s
284:n
282:(
280:s
276:L
265:)
263:n
257:n
253:o
241:O
215:n
213:(
211:f
209:(
207:O
195:n
193:(
191:f
183:n
181:(
179:f
116:)
110:(
105:)
101:(
91:·
84:·
77:·
70:·
43:.
20:.
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