914:
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804:
407:
78:
748:
584:
316:
348:
953:
251:
167:
196:
112:
831:
949:
859:
589:
17:
753:
551:
1086:
362:
1063:
674:
660:
47:
833:
operator yields a "symmetric hierarchy"; the union of the classes produced in this way is equal to the
722:
834:
557:
289:
37:
321:
1030:
209:
125:
175:
1022:
286:
is closed under unions, intersections, and complements. Comparing the definition with that of
1014:
993:
968:
930:
411:
91:
33:
964:
944:
809:
961:
940:
665:
546:
419:
1049:
1080:
1070:
997:
1034:
1054:
1005:
Russell, Alexander; Sundaram, Ravi (1998). "Symmetric alternation captures BPP".
855:
687:
957:
935:
1026:
972:
532:
These straightforward inclusions can be strengthened to show that the class
479:
always answers false. But such a verifier can easily be transformed into an
1018:
984:
Canetti, Ran (1996). "More on BPP and the polynomial-time hierarchy".
670:
909:{\displaystyle \mathrm {S} _{2}^{p}\subseteq \mathrm {{ZPP}^{NP}} }
516:
439:
is in NP, if and only if there exists a polynomial-time verifier
644:{\displaystyle P^{{\mathsf {S}}_{2}^{P}}={\mathsf {S}}_{2}^{P}}
948:. A preliminary version of this paper appeared earlier, in
36:, intermediate between the first and second levels of the
799:{\displaystyle {\mathsf {S}}_{2}P={\mathsf {S}}_{2}^{P}}
862:
812:
756:
725:
592:
560:
365:
324:
292:
212:
178:
128:
94:
50:
908:
825:
798:
742:
643:
578:
401:
342:
310:
245:
190:
161:
106:
72:
409:. This inclusion can in fact be strengthened to
402:{\displaystyle \Sigma _{2}^{P}\cap \Pi _{2}^{P}}
8:
80:if there exists a polynomial-time predicate
750:as an operator on complexity classes; then
719:As an extension, it is possible to define
277:It is immediate from the definition that S
934:
896:
885:
883:
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869:
864:
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817:
811:
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734:
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591:
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393:
388:
375:
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364:
334:
329:
323:
302:
297:
291:
211:
177:
127:
93:
64:
59:
53:
52:
49:
686:. This result yields a strengthening of
273:Relationship to other complexity classes
923:Journal of Computer and System Sciences
846:
780:
760:
729:
625:
601:
467:answers true, and such that for every
54:
507:) for the same language that ignores
73:{\displaystyle {\mathsf {S}}_{2}^{P}}
7:
350:, it also follows immediately that S
900:
897:
892:
889:
886:
865:
562:
511:and otherwise behaves the same as
385:
367:
326:
294:
14:
1013:(2). Birkhäuser Verlag: 152–162.
743:{\displaystyle {\mathsf {S}}_{2}}
663:states that if every language in
1064:Complexity Class of the Week: S
579:{\displaystyle \Delta _{2}^{P}}
311:{\displaystyle \Sigma _{2}^{P}}
18:computational complexity theory
986:Information Processing Letters
690:'s theorem: it is known that S
435:For by definition, a language
234:
216:
150:
132:
1:
998:10.1016/0020-0190(96)00016-6
550:(by a generalization of the
343:{\displaystyle \Pi _{2}^{P}}
1103:
936:10.1016/j.jcss.2003.07.015
246:{\displaystyle P(x,y,z)=0}
162:{\displaystyle P(x,y,z)=1}
675:polynomial time hierarchy
191:{\displaystyle x\notin L}
1007:Computational Complexity
992:(5). Elsevier: 237–241.
973:10.1109/SFCS.2001.959938
671:polynomial size circuits
552:Sipser–Lautemann theorem
451:), such that for every
910:
827:
800:
744:
645:
580:
403:
344:
312:
265:must be polynomial of
247:
198:, then there exists a
192:
163:
114:, then there exists a
108:
107:{\displaystyle x\in L}
74:
1019:10.1007/s000370050007
911:
828:
826:{\displaystyle S_{2}}
801:
745:
707:) for any fixed
646:
581:
515:. By the same token,
404:
345:
313:
248:
193:
164:
109:
75:
860:
854:Cai, Jin-Yi (2007),
835:Polynomial hierarchy
810:
754:
723:
699:is not contained in
590:
558:
363:
322:
290:
210:
176:
126:
92:
48:
38:polynomial hierarchy
879:
795:
715:Symmetric hierarchy
661:Karp–Lipton theorem
655:Karp–Lipton theorem
640:
616:
575:
398:
380:
339:
307:
69:
1087:Complexity classes
1073:, August 28, 2002.
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418:Every language in
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293:
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202:such that for all
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118:such that for all
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586:(more generally,
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434:
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422:also belongs to
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359:is contained in
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34:complexity class
30:
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806:. Iteration of
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208:
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174:
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124:
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90:
89:
46:
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41:
28:
25:
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11:
5:
1100:
1098:
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1050:Complexity Zoo
1044:
1043:External links
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1002:
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731:
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681:
677:collapses to S
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257:where size of
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1088:
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1072:
1071:Lance Fortnow
1068:
1062:
1060:
1052:
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1024:
1020:
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786:
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766:
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714:
712:
710:
706:
689:
676:
672:
668:
667:
662:
659:A version of
654:
652:
636:
631:
619:
612:
607:
594:
571:
566:
553:
549:
548:
518:
514:
510:
506:
502:
498:
494:
478:
474:
470:
466:
462:
459:there exists
458:
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182:
179:
171:
156:
153:
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141:
138:
135:
129:
121:
117:
101:
98:
95:
87:
86:
85:
83:
65:
60:
40:. A language
39:
35:
31:
19:
1048:
1010:
1006:
989:
985:
929:(1): 25–35,
926:
922:
849:
718:
708:
704:
664:
658:
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468:
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417:
410:
276:
266:
262:
258:
256:
203:
199:
119:
115:
81:
21:
15:
519:belongs to
841:References
491:predicate
463:for which
84:such that
1027:1016-3328
881:⊆
673:then the
563:Δ
544:contains
386:Π
382:∩
368:Σ
327:Π
295:Σ
183:∉
99:∈
1081:Category
1035:15331219
958:TR01-030
965:1948751
945:2279029
471:not in
1033:
1025:
956:
952:2001,
943:
688:Kannan
554:) and
44:is in
1031:S2CID
919:(PDF)
517:co-NP
32:is a
1023:ISSN
954:ECCC
950:FOCS
701:SIZE
669:has
318:and
261:and
1015:doi
994:doi
969:doi
931:doi
651:).
455:in
412:ZPP
172:If
88:If
16:In
1083::
1069:,
1053::
1029:.
1021:.
1009:.
990:57
988:.
967:,
962:MR
960:,
941:MR
939:,
927:73
925:,
921:,
837:.
711:.
666:NP
547:MA
475:,
420:NP
415:.
269:.
206:,
122:,
20:,
1066:2
1059:P
1057:2
1055:S
1037:.
1017::
1011:7
1000:.
996::
975:.
971::
933::
916:"
901:P
898:N
893:P
890:P
887:Z
876:p
871:2
866:S
856:"
819:2
815:S
792:P
787:2
781:S
775:=
772:P
767:2
761:S
736:2
730:S
709:k
705:n
703:(
695:2
682:2
637:P
632:2
626:S
620:=
613:P
608:2
602:S
595:P
572:P
567:2
539:2
534:S
530:.
526:2
521:S
513:V
509:z
505:z
503:,
501:y
499:,
497:x
495:(
493:P
486:2
481:S
477:V
473:L
469:x
465:V
461:y
457:L
453:x
449:y
447:,
445:x
443:(
441:V
437:L
433:.
429:2
424:S
395:P
390:2
377:P
372:2
355:2
336:P
331:2
304:P
299:2
282:2
267:x
263:z
259:y
253:,
241:0
238:=
235:)
232:z
229:,
226:y
223:,
220:x
217:(
214:P
204:y
200:z
186:L
180:x
169:,
157:1
154:=
151:)
148:z
145:,
142:y
139:,
136:x
133:(
130:P
120:z
116:y
102:L
96:x
82:P
66:P
61:2
55:S
42:L
27:2
22:S
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