Outline - Brown Universitycs.brown.edu/courses/csci1010/files/doc/Lecture-22-L and NLcopy.pdf · Outline • The class L and NL • Log space computable functions • NL-complete

Outline

•  TheclassLandNL•  Logspacecomputablefunctions•  NL-completelanguages•  PATHisNL-complete•  ThecoNLclass

11/25/19 TheoryofComputation-Fall'19

LorenzoDeStefani 1

FromSipserChapter8.4-8.6

Chainingpolytimereductions•  Sofar,whendescribingthespacerequirementsofTMwe

haveconsideredatthesametime:–  Thelocationsofthetapeusedtoinitiallystoretheinput–  Thelocationsofthetapeusedtostoretheoutputattheendof

thecomputation–  ThelocationsofthetapeusedbytheTMduringitsexecutionto

maintain“work-in-progress”data

•  Underthischaracterization,alllanguagesareinSPACE(n),wherenistheinputsize!–  Spacerequirementisalwaysatleastlinearw.r.t.theinputsize

•  Insomecircumstanceswewanttofocusonthespaceusedintheexecutionoftheinputofthealgorithmexcludingthespaceforinputandoutput–  Underthisassumption,wecancharacterizealgorithmswhichreq

11/25/19 TheoryofComputation-Fall'19LorenzoDeStefani 2

Logspacetransducer

•  Alog-timetransducerisaTMwith–  Aread-onlyinputtapefortheinput–  Awrite-only,move-to-tight-onlyoutputtape–  Aworktape,whichusesO(logn)space,wherenisthesizeoftheinput–  TheTMalwayshalts

•  Example:Thelanguage{0n1n}canbedecidedbyalogtimetransducer– Weonlyneedtocountthenumberof0sand1s–  Atmostweneedtokeeptrackofanumbern–  Requiredlognspace


TheclassesLandNL

•  L={L|L=L(M)whereMisadeterministicTMwhichusesatmostlog(n)workspace}

•  NL={L|L=L(N)whereNisanon-deterministicTMwhichusesatmostlog(n)workspace}– Thework-spaceofaNon-deterministicTMisthespaceusedatmostforanyinputforanynon-deterministicexecutionbranchoftheTM

•  IsL=NL?Openquestion!– Savitch’sThm.Onlyimpliesthatthereisatmostaquadraticblowup


ThePATHlanguage

•  PATH={<G,s,t>|G=(V,E)isadirectedgraphwithapathfromthevertexstothevertext}–  Alsoreferredasthepath-connectionproblem!

•  PATHNL–  N=“Oninput<G,s,t>”:–  r=s//risareferencetothecurrentstepofthepath–  Fori=2ton=|V|

•  Notdeterministicallypickavertexvamongthesuccessorsofr•  If(r,v)E:

–  Ifv=tthenACCEPT•  Else:REJECT

–  REJECT


2<latexit sha1_base64="Xc4bgFqsWakZpJljp8Fwls1Vmt8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKez6QI9BLx4jmgckS5idzCZDZmeXmV4hLPkELx4U8eoXefNvnCR70MSChqKqm+6uIJHCoOt+O4WV1bX1jeJmaWt7Z3evvH/QNHGqGW+wWMa6HVDDpVC8gQIlbyea0yiQvBWMbqd+64lrI2L1iOOE+xEdKBEKRtFKD12heuWKW3VnIMvEy0kFctR75a9uP2ZpxBUySY3peG6CfkY1Cib5pNRNDU8oG9EB71iqaMSNn81OnZATq/RJGGtbCslM/T2R0ciYcRTYzoji0Cx6U/E/r5NieO1nQiUpcsXmi8JUEozJ9G/SF5ozlGNLKNPC3krYkGrK0KZTsiF4iy8vk+ZZ1TuvXt5fVGo3eRxFOIJjOAUPrqAGd1CHBjAYwDO8wpsjnRfn3fmYtxacfOYQ/sD5/AFO8o3S</latexit>


Analysis

•  Correctness:Ifthereisapathfromstotitwillbeselectedbyonesequenceofnon-deterministicchoices–  Suchpathwillhavelengthatmostn=|V|

•  Analysis– Thealgorithmonlyneedsenoughworkspacetokeeptrackofthepositionofthecurrentvertexintheconstructionofthepath(r)ontheinputtape

– OverallO(logn)space


PAHTL?

•  Weneedtoavoidcycles– OtherwisetheTMisnotguaranteedtohalt!

•  Keepingtrackofthecurrentposition(r)isnotenough

•  Wewouldneedtokeeptrackoftheentirepathsofartoavoidcycles

•  ButapathcanhavelengthO(n)•  WewouldrequireO(n)work-space•  Adeterministicvariationofthediscussedalgorithmdoesnotwork



Logspacereductions•  AlogspacetransducerMcomputesafunctionwheref(w)isthestringremainingontheoutputtapeattheendofthecomputationafterMhaltswithwonitsinputtape

•  Suchafunctionisalogspacecomputablefunction•  AlanguageAislogspacereducibletoalanguageB,ifAismappingreducibletoB,writtenA≤LB,bymeansofalogspacecomputablefunction–  Thatis,itthereisalogspacecomputablefunctionsuchthat

11/25/19 8TheoryofComputation-Fall'19LorenzoDeStefani

f : ⌃⇤ ! ⌃⇤<latexit sha1_base64="9z5zo4YXhhDZgnkLrkqOKs5whTw=">AAACB3icbVDLSgMxFM3UV62vUZeCBIsgLsqMDxRXRTcuK9oHdMaSSTNtaDIZkoxShu7c+CtuXCji1l9w59+YtiNo64ELh3Pu5d57gphRpR3ny8rNzM7NL+QXC0vLK6tr9vpGTYlEYlLFggnZCJAijEakqqlmpBFLgnjASD3oXQz9+h2RioroRvdj4nPUiWhIMdJGatnb4Zl3TTsc3e57kna6Gkkp7uGP1rKLTskZAU4TNyNFkKHSsj+9tsAJJ5HGDCnVdJ1Y+ymSmmJGBgUvUSRGuIc6pGlohDhRfjr6YwB3jdKGoZCmIg1H6u+JFHGl+jwwnRzprpr0huJ/XjPR4amf0ihONInweFGYMKgFHIYC21QSrFnfEIQlNbdC3EUSYW2iK5gQ3MmXp0ntoOQelo6vjorl8yyOPNgCO2APuOAElMElqIAqwOABPIEX8Go9Ws/Wm/U+bs1Z2cwm+APr4xtCIJjt</latexit>

8w 2 ⌃⇤<latexit sha1_base64="Fj3Q/Oqvy0NQMcbsZBwl3O8167s=">AAAB/nicbVDLSsNAFJ34rPUVFVduBosgLkriA10W3bisaB/QxHIznbRDJ5MwM1FKKPgrblwo4tbvcOffOG2z0NYDA4dz7uWeOUHCmdKO823NzS8sLi0XVoqra+sbm/bWdl3FqSS0RmIey2YAinImaE0zzWkzkRSigNNG0L8a+Y0HKhWLxZ0eJNSPoCtYyAhoI7XtXS+MJXCOHz0msHfLuhHcH7XtklN2xsCzxM1JCeWotu0vrxOTNKJCEw5KtVwn0X4GUjPC6bDopYomQPrQpS1DBURU+dk4/hAfGKWDTQ7zhMZj9fdGBpFSgygwkxHonpr2RuJ/XivV4YWfMZGkmgoyORSmHOsYj7rAHSYp0XxgCBDJTFZMeiCBaNNY0ZTgTn95ltSPy+5J+ezmtFS5zOsooD20jw6Ri85RBV2jKqohgjL0jF7Rm/VkvVjv1sdkdM7Kd3bQH1ifP2U5lR8=</latexit>

w 2 A () f(w) 2 B<latexit sha1_base64="Cq0f+xqNgKSmftSL+jVMDjq8DaA=">AAAB/3icbVDLTgIxFL2DL8QXauLGTSMxwQ2Z8RFdIm5cYiKPhJmQTulAQ6czaTsSgiz8FTcuNMatv+HOv7HALBQ8yU1Oz7k3vff4MWdK2/a3lVlaXlldy67nNja3tnfyu3t1FSWS0BqJeCSbPlaUM0FrmmlOm7GkOPQ5bfj9m4nfeKBSsUjc62FMvRB3BQsYwdpI7fzBwGUCXSOXBQEKioMTNHlX2vmCXbKnQIvESUkBUlTb+S+3E5EkpEITjpVqOXasvRGWmhFOxzk3UTTGpI+7tGWowCFV3mi6/xgdG6WDgkiaEhpN1d8TIxwqNQx90xli3VPz3kT8z2slOrjyRkzEiaaCzD4KEo50hCZhoA6TlGg+NAQTycyuiPSwxESbyHImBGf+5EVSPy05Z6WLu/NCuZLGkYVDOIIiOHAJZbiFKtSAwCM8wyu8WU/Wi/VufcxaM1Y6sw9/YH3+ADh2lE0=</latexit>

NL-completelanguages

•  AlanguageLisNL-hardifforanylanguageAinNL,wehaveA≤LL– NL-hardlanguagesareashard(withrespecttospacerequirementasanylanguageinNL)

•  WesaythatalanguageLisNL-completeif–  – LisNL-hard


L 2 NL<latexit sha1_base64="HDrSB58XsF2ARDs0oR2EO49QtiE=">AAAB+nicbVDLSsNAFL2pr1pfqS7dDBbBVUl8oMuiGxdFKtgHNKVMppN26GQSZiZKif0UNy4UceuXuPNvnLRZaPXAwOGce7lnjh9zprTjfFmFpeWV1bXiemljc2t7xy7vtlSUSEKbJOKR7PhYUc4EbWqmOe3EkuLQ57Ttj68yv31PpWKRuNOTmPZCPBQsYARrI/Xtch15TCAvxHrkB+lNfdq3K07VmQH9JW5OKpCj0bc/vUFEkpAKTThWqus6se6lWGpGOJ2WvETRGJMxHtKuoQKHVPXSWfQpOjTKAAWRNE9oNFN/bqQ4VGoS+mYyi6gWvUz8z+smOrjopUzEiaaCzA8FCUc6QlkPaMAkJZpPDMFEMpMVkRGWmGjTVsmU4C5++S9pHVfdk+rZ7WmldpnXUYR9OIAjcOEcanANDWgCgQd4ghd4tR6tZ+vNep+PFqx8Zw9+wfr4Bm7hk3s=</latexit>

Consequences•  Theorem1:ifA≤LBandBL,thenAL– Thereexistalogspacecomputablefunctionsuchthat

– Bydefinition,thereexistsalogspacetransducerthatdecidesB

•  Theorem2:ifanyNL-completelanguageisinL,thenL=NL– LetBbeaNLcompletelanguageinL–  thereexistsalogspacetransducerthatdecidesB– BydefinitionofNL-completeness,foranyAinLwehaveA≤LB




w 2 A () f(w) 2 B<latexit sha1_base64="Cq0f+xqNgKSmftSL+jVMDjq8DaA=">AAAB/3icbVDLTgIxFL2DL8QXauLGTSMxwQ2Z8RFdIm5cYiKPhJmQTulAQ6czaTsSgiz8FTcuNMatv+HOv7HALBQ8yU1Oz7k3vff4MWdK2/a3lVlaXlldy67nNja3tnfyu3t1FSWS0BqJeCSbPlaUM0FrmmlOm7GkOPQ5bfj9m4nfeKBSsUjc62FMvRB3BQsYwdpI7fzBwGUCXSOXBQEKioMTNHlX2vmCXbKnQIvESUkBUlTb+S+3E5EkpEITjpVqOXasvRGWmhFOxzk3UTTGpI+7tGWowCFV3mi6/xgdG6WDgkiaEhpN1d8TIxwqNQx90xli3VPz3kT8z2slOrjyRkzEiaaCzD4KEo50hCZhoA6TlGg+NAQTycyuiPSwxESbyHImBGf+5EVSPy05Z6WLu/NCuZLGkYVDOIIiOHAJZbiFKtSAwCM8wyu8WU/Wi/VufcxaM1Y6sw9/YH3+ADh2lE0=</latexit>

PATHisNL-complete

•  WealreadyprovedthatPATHNL•  WeneedtoarguethatPATHisNL-hard•  Proofidea:ForanylanguageLNLwewanttohaveareductionsuchthatgivenwweconstructaninstance<G,s,t>ofthePATHdecisionproblemsuchthat

•  Recallanalogouspreviousresults:– WeconsiderthecomputationalhistoryofaTMforL– Weconsiderthesequenceofconfigurationsformastartingoneoninputwtoanacceptingone

– Weusesuchsequenceofconfigurationstobuild<G,s,t>




w 2 L () < G, s, t >2 PATH<latexit sha1_base64="tdS/J9GCoe7z8MbqpAc7FqRoe8Q=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBRSmJD3QhUnVhFy4q9AVNKZPppB06mYSZiVJCN278FTcuFHHrP7jzb5y0WWjrgQuHc+7l3nvckFGpLOvbyMzNLywuZZdzK6tr6xvm5lZdBpHApIYDFoimiyRhlJOaooqRZigI8l1GGu7gOvEb90RIGvCqGoak7aMepx7FSGmpY+4+OJTDW+hQz4PnNwVZUBcwkSqX1XLHzFtFaww4S+yU5EGKSsf8croBjnzCFWZIypZthaodI6EoZmSUcyJJQoQHqEdamnLkE9mOx1+M4L5WutALhC6u4Fj9PREjX8qh7+pOH6m+nPYS8T+vFSnvrB1THkaKcDxZ5EUMqgAmkcAuFQQrNtQEYUH1rRD3kUBY6eByOgR7+uVZUj8s2kfFk7vjfOkqjSMLdsAeOAA2OAUlUAYVUAMYPIJn8ArejCfjxXg3PiatGSOd2QZ/YHz+APe7llE=</latexit>

PATHisNL-complete

•  WealreadyprovedthatPATHNL•  WeneedtoarguethatPATHisNL-hard•  Proofidea:ForanylanguageLNLwewanttohaveareductionsuchthatgivenwweconstructaninstance<G,s,t>ofthePATHdecisionproblemsuchthat

•  Recallanalogouspreviousresults:– WeconsiderthecomputationalhistoryofaTMforL– Weconsiderthesequenceofconfigurationsformastartingoneoninputwtoanacceptingone

– Weusesuchsequenceofconfigurationstobuild<G,s,t>




w 2 L () < G, s, t >2 PATH<latexit sha1_base64="tdS/J9GCoe7z8MbqpAc7FqRoe8Q=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBRSmJD3QhUnVhFy4q9AVNKZPppB06mYSZiVJCN278FTcuFHHrP7jzb5y0WWjrgQuHc+7l3nvckFGpLOvbyMzNLywuZZdzK6tr6xvm5lZdBpHApIYDFoimiyRhlJOaooqRZigI8l1GGu7gOvEb90RIGvCqGoak7aMepx7FSGmpY+4+OJTDW+hQz4PnNwVZUBcwkSqX1XLHzFtFaww4S+yU5EGKSsf8croBjnzCFWZIypZthaodI6EoZmSUcyJJQoQHqEdamnLkE9mOx1+M4L5WutALhC6u4Fj9PREjX8qh7+pOH6m+nPYS8T+vFSnvrB1THkaKcDxZ5EUMqgAmkcAuFQQrNtQEYUH1rRD3kUBY6eByOgR7+uVZUj8s2kfFk7vjfOkqjSMLdsAeOAA2OAUlUAYVUAMYPIJn8ArejCfjxXg3PiatGSOd2QZ/YHz+APe7llE=</latexit>

ProvingPATHisNL-complete•  Givenw,L,letNbealogspacenon-deterministictransducersuchthatL(N)=L

•  ConstructadirectedgraphG=(V,E)–  VissuchthateveryvertexcorrespondtoapossibleconfigurationofN

onw–  Eisthesetofedgessuchthatifispossibletoreachthe

configurationcorrespondingtovertexvfromtheconfigurationcorrespondingtovertexuusingasingleapplicationsofthetransitionsfunctions

–  Letvstart(resp.,vaccept)bethevertexcorrespondingtothestarting(resp.,accepting)configuration

•  RuntheprocedureforPATHon<G,vstart,vaccept>•  Wesetthelengthofthepathtodn,wheredisaconstant


(u, v) 2 E<latexit sha1_base64="htikgSAMyV6EF0rV9Hjqrsu9bos=">AAAB83icbVDJSgNBEK2JW4xb1KOXxiBEkDDjgh6DIniMYBbIDKGn05M06ekZegmEIb/hxYMiXv0Zb/6NneWgiQ8KHu9VUVUvTDlT2nW/ndzK6tr6Rn6zsLW9s7tX3D9oqMRIQusk4YlshVhRzgSta6Y5baWS4jjktBkO7iZ+c0ilYol40qOUBjHuCRYxgrWV/LI5Q8NT5DOB7jvFkltxp0DLxJuTEsxR6xS//G5CTEyFJhwr1fbcVAcZlpoRTscF3yiaYjLAPdq2VOCYqiCb3jxGJ1bpoiiRtoRGU/X3RIZjpUZxaDtjrPtq0ZuI/3lto6ObIGMiNZoKMlsUGY50giYBoC6TlGg+sgQTyeytiPSxxETbmAo2BG/x5WXSOK94F5Wrx8tS9XYeRx6O4BjK4ME1VOEBalAHAik8wyu8OcZ5cd6dj1lrzpnPHMIfOJ8/x36QOQ==</latexit>

ProvingPATHisNL-complete•  Correctness:ifinGthereisadirectedpathfromvstarttovaccept,thenthereisacomputationalhistoryonNforwsuchthatwisaccepted.–  Theinversedirectionistruebyconstruction

•  Analysis:Thereductionneedstogive<G,s,t>–  Bydefinition,eachconfigurationofNusesatmostO(logn)space

– Wecangenerateoneatatimethevertices,andoutputthemontheoutputtape

–  FromSavitch’stheoremwehaveSPACE(logn)–  Anacceptingcomputationhistorywillrequireatmostnsteps–  Lognspaceissufficient


✓ TIME(2logn)<latexit sha1_base64="dHqxYqROWN6NBCaNwMSESgr5S7Q=">AAACA3icbVDLSgNBEJz1GeNr1ZteBoMQL2E3KnoMiqAHIUJekI1hdtJJhszOrjOzQlgCXvwVLx4U8epPePNvnDwOmljQUFR1093lR5wp7Tjf1tz8wuLScmolvbq2vrFpb21XVBhLCmUa8lDWfKKAMwFlzTSHWiSBBD6Hqt+7GPrVB5CKhaKk+xE0AtIRrM0o0UZq2ruein0FGu5x6frmMpu/SzwedrAYHDbtjJNzRsCzxJ2QDJqg2LS/vFZI4wCEppwoVXedSDcSIjWjHAZpL1YQEdojHagbKkgAqpGMfhjgA6O0cDuUpoTGI/X3REICpfqBbzoDortq2huK/3n1WLfPGgkTUaxB0PGidsyxDvEwENxiEqjmfUMIlczcimmXSEK1iS1tQnCnX54llXzOPcqd3B5nCueTOFJoD+2jLHLRKSqgK1REZUTRI3pGr+jNerJerHfrY9w6Z01mdtAfWJ8/jmuW0g==</latexit>

RelationbetweenPandNL– Wehave,since:

–  WhataboutNL?•  ATMwhichusesspacef(n)canhaveatmost f(n)2O(f(n))distinctconfigurations

• Wecanboundthetimebythenumberofsuchconfigurations

•  Forf(n)=lognwehavenlognconfigurationsàpolynomialinn

•  Hence,


SPACE(log n) ✓ TIME(2logn) = TIME(n)<latexit sha1_base64="lRG7tT9FOHKag3TBGrYl0wGiHEM=">AAACGnicbVDJSgNBEO2Je9yiHr00BiG5hJmo6EVwQdCDEDEbZMbQ06nExp6esbtHCEO+w4u/4sWDIt7Ei39jZzlo4oOCx3tVVNXzI86Utu1vKzU1PTM7N7+QXlxaXlnNrK1XVRhLChUa8lDWfaKAMwEVzTSHeiSBBD6Hmn932vdrDyAVC0VZdyPwAtIRrM0o0UZqZpzr0vHpWc7lYQeLvKtiX4GGe1y+uDzLFW+SodHL48OhJPLNTNYu2APgSeKMSBaNUGpmPt1WSOMAhKacKNVw7Eh7CZGaUQ69tBsriAi9Ix1oGCpIAMpLBq/18LZRWrgdSlNC44H6eyIhgVLdwDedAdG3atzri/95jVi3D7yEiSjWIOhwUTvmWIe4nxNuMQlU864hhEpmbsX0lkhCtUkzbUJwxl+eJNViwdkp7F3tZo9ORnHMo020hXLIQfvoCJ2jEqogih7RM3pFb9aT9WK9Wx/D1pQ1mtlAf2B9/QCJkJ4M</latexit>

L ✓ P<latexit sha1_base64="KNoWm9HqG95GM6GxVCz0MKwFLAQ=">AAAB83icbVC7SgNBFJ2NrxhfUUubwSBYhV0faBm0sbCIYB6QXWR2cpMMmX04c0cIS37DxkIRW3/Gzr9xkmyhiQcGDuecy71zwlQKja777RSWlldW14rrpY3Nre2d8u5eUydGcWjwRCaqHTINUsTQQIES2qkCFoUSWuHweuK3nkBpkcT3OEohiFg/Fj3BGVrJv/W1CTUgPNL6Q7niVt0p6CLxclIhOWz+y+8m3EQQI5dM647nphhkTKHgEsYl32hIGR+yPnQsjVkEOsimN4/pkVW6tJco+2KkU/X3RMYirUdRaJMRw4Ge9ybif17HYO8yyEScGoSYzxb1jKSY0EkBtCsUcJQjSxhXwt5K+YApxtHWVLIlePNfXiTNk6p3Wj2/O6vUrvI6iuSAHJJj4pELUiM3pE4ahJOUPJNX8uYY58V5dz5m0YKTz+yTP3A+fwC7SJF9</latexit>

NL ✓ P<latexit sha1_base64="fpq4t515gIO8syvrB1MekeYWytw=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsz4QJdFNy5EKtgHtINk0jttaCYzTTKFMvQ73LhQxK0f486/MX0stHogcDjnXO7NCRLBtXHdLye3tLyyupZfL2xsbm3vFHf36jpOFcMai0WsmgHVKLjEmuFGYDNRSKNAYCPoX0/8xhCV5rF8MKME/Yh2JQ85o8ZK/t1tW6eBRoMDUn0sltyyOwX5S7w5KcEcNv/Z7sQsjVAaJqjWLc9NjJ9RZTgTOC60U40JZX3axZalkkao/Wx69JgcWaVDwljZJw2Zqj8nMhppPYoCm4yo6elFbyL+57VSE176GZdJalCy2aIwFcTEZNIA6XCFzIiRJZQpbm8lrEcVZcb2VLAleItf/kvqJ2XvtHx+f1aqXM3ryMMBHMIxeHABFbiBKtSAwQCe4AVenaHz7Lw577NozpnP7MMvOB/fWjGR1Q==</latexit>

TheclasscoNL•  •  – Seesection8.6fordetailedproof

•  Importantconsequence!– SincePATHisalsoNL-completewecanconcludeNL=coNL


coNL = {L|L̄ 2 NL}<latexit sha1_base64="lZckPDTG6ehaDxEH7V22YEugqYs=">AAACA3icbVDLSsNAFJ3UV62vqDvdDBbBVUl8oBuh6MZFKBXsA5pQJtNJO3QyCTMTocSAG3/FjQtF3PoT7vwbp20W2nrgwuGce7n3Hj9mVCrL+jYKC4tLyyvF1dLa+sbmlrm905RRIjBp4IhFou0jSRjlpKGoYqQdC4JCn5GWP7we+617IiSN+J0axcQLUZ/TgGKktNQ193BUc+AldFPnwfWRSJ3MpRzWHDfrmmWrYk0A54mdkzLIUe+aX24vwklIuMIMSdmxrVh5KRKKYkaykptIEiM8RH3S0ZSjkEgvnfyQwUOt9GAQCV1cwYn6eyJFoZSj0NedIVIDOeuNxf+8TqKCCy+lPE4U4Xi6KEgYVBEcBwJ7VBCs2EgThAXVt0I8QAJhpWMr6RDs2ZfnSfO4Yp9Uzm5Py9WrPI4i2AcH4AjY4BxUwQ2ogwbA4BE8g1fwZjwZL8a78TFtLRj5zC74A+PzB5C5ltk=</latexit>

PATH 2 coNL<latexit sha1_base64="17qFA3heq05hS775ZNl+G4PJjRk=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoPgKez6QI9RLzmIRMgLkiXMTibJkNmZdWY2EJZ8hxcPinj1Y7z5N06SPWhiQUNR1U13VxBxpo3rfjuZldW19Y3sZm5re2d3L79/UNcyVoTWiORSNQOsKWeC1gwznDYjRXEYcNoIhndTvzGiSjMpqmYcUT/EfcF6jGBjJb9yUy2jNhOIyIf7Tr7gFt0Z0DLxUlKAFJVO/qvdlSQOqTCEY61bnhsZP8HKMMLpJNeONY0wGeI+bVkqcEi1n8yOnqATq3RRTypbwqCZ+nsiwaHW4zCwnSE2A73oTcX/vFZsetd+wkQUGyrIfFEv5shINE0AdZmixPCxJZgoZm9FZIAVJsbmlLMheIsvL5P6WdE7L14+XhRKt2kcWTiCYzgFD66gBGWoQA0IPMEzvMKbM3JenHfnY96acdKZQ/gD5/MHJ8mRDw==</latexit>

Documents

Outline - Brown Universitycs.brown.edu/courses/csci1010/files/doc/Lecture-22-L and NLcopy.pdf · Outline • The class L and NL • Log space computable functions • NL-complete