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problem. Because determined effort has yet to find a polynomial-time solution to any of these problems, it is suspected to be difficult to show
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if and only if a given answer can be verified by a deterministic machine in polynomial time. Similarly, a language is in
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if a given answer can be verified in polynomial time, and the verifier machine only accepts at most
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568:"Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets"
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answer for each problem instance. More formally, a language
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if there exists a two-input polynomial-time algorithm
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46:. Unsourced material may be challenged and removed.
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566:Hemaspaandra, Lane A.; Rothe, Jörg (June 1997).
146:with at most one accepting path for each input.
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128:unambiguous non-deterministic polynomial-time
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106:Learn how and when to remove this message
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44:adding citations to reliable sources
551:. UP: Unambiguous Polynomial-Time.
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995:Probabilistically checkable proof
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31:needs additional citations for
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381:{\displaystyle |y|=O(|x|^{c})}
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1011:List of complexity classes
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165:A common reformulation of
144:unambiguous Turing machine
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596:10.1137/S0097539794261970
573:SIAM Journal on Computing
516:is not known to have any
1000:Interactive proof system
490:Valiant–Vazirani theorem
424:{\displaystyle A(x,y)=1}
307:{\displaystyle A(x,y)=1}
954:Arithmetical hierarchy
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949:Grzegorczyk hierarchy
944:Exponential hierarchy
876:Considered infeasible
459:integer factorization
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939:Polynomial hierarchy
769:Suspected infeasible
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156:and is contained in
55:"UP" complexity
40:improve this article
968:Families of classes
649:Considered feasible
457:) contain both the
443:in polynomial time.
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642:Complexity classes
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580:(3): 634–653.
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548:Complexity Zoo
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51:Find sources:
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29:This article
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138:solvable in
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38:Please help
33:verification
30:
848:#P-complete
786:NP-complete
701:NL-complete
463:parity game
197:such that
185:belongs to
96:August 2018
903:ELEMENTARY
728:P-complete
587:cs/9907033
524:References
520:problems.
507:Promise-UP
473:, or even
452:complement
435:algorithm
388:such that
271:such that
66:newspapers
898:2-EXPTIME
604:0097-5397
529:Citations
450:(and its
439:verifies
150:contains
130:) is the
1024:Category
893:EXPSPACE
888:NEXPTIME
656:DLOGTIME
518:complete
201:if x in
883:EXPTIME
791:NP-hard
560:Sources
431:
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80:scholar
990:NSPACE
985:DSPACE
860:PSPACE
602:
142:on an
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980:NTIME
975:DTIME
796:co-NP
582:arXiv
483:co-UP
455:co-UP
326:with
209:with
87:JSTOR
73:books
808:TFNP
600:ISSN
488:The
59:news
923:ALL
823:QMA
813:FNP
755:APX
750:BQP
745:BPP
735:ZPP
666:ACC
592:doi
543:"U"
485:).
179:one
134:of
118:In
42:by
1026::
918:RE
908:PR
855:IP
843:#P
838:PP
833:⊕P
828:PH
818:AM
781:NP
776:UP
760:FP
740:RP
718:CC
713:SC
708:NC
696:NL
691:FL
686:RL
681:SL
671:TC
661:AC
598:.
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578:26
576:.
570:.
545:.
514:UP
510:.
502:NP
498:RP
494:NP
481:∩
479:UP
477:=(
471:UP
448:UP
187:UP
175:UP
171:NP
167:NP
162:.
159:NP
148:UP
124:UP
122:,
913:R
723:P
676:L
634:e
627:t
620:v
606:.
594::
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475:P
469:=
467:P
441:L
437:A
419:1
416:=
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410:y
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404:x
401:(
398:A
376:)
371:c
366:|
361:x
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353:(
350:O
347:=
343:|
339:y
335:|
324:y
320:L
302:1
299:=
296:)
293:y
290:,
287:x
284:(
281:A
259:)
254:c
249:|
244:x
240:|
236:(
233:O
230:=
226:|
222:y
218:|
207:y
203:L
195:c
191:A
183:L
153:P
126:(
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103:(
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94:(
84:·
77:·
70:·
63:·
36:.
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