More examples of reductions Characterizing Space...

Outline

•  Moreexamplesofreductions•  CharacterizingSpaceComplexity•  SATrequirespolynomialspace•  RelationsbetweenTIMEandSPACE

10/29/19 1 TheoryofComputation-Fall'19

LorenzoDeStefani

Tautologies•  Inlogic,atautologyisaformulaorassertion

thatistrueineverypossibleinterpretation.–  Example:"x=yorx≠y”,ortheballisnotallgreen".

•  –  Consideracertificatewhichisanassignmentof

thevariablessuchthat–  Canbecheckedinpolynomialtime

10/29/19 2TheoryofComputation-Fall'19

LorenzoDeStefani

TAUT = {< ⇡ > |8 assignments a1, a2 . . . , an to �’s

variables, � (a1, a2 . . . , an) = True}

<latexit sha1_base64="qJEGJbDrNwycZp3Olzu8rMIjgPg=">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</latexit>

TAUT 2 NP<latexit sha1_base64="DAO5G6tE+g+FB2hfDyZYx196AR8=">AAAB/3icbVDLSgMxFM34rPVVFdy4CRbBVZnR4mNXdeNKKnTaQmcomTTThmaSIckIZezCX3HjQhG3/oY7/8bMtIhaDwQO55zLvTlBzKjStv1pzc0vLC4tF1aKq2vrG5ulre2mEonExMWCCdkOkCKMcuJqqhlpx5KgKGCkFQyvMr91R6Sigjf0KCZ+hPqchhQjbaRuadcTxs6m08aF2xhDj3J4U++WynbFzgFniTMlZTCFyX94PYGTiHCNGVKq49ix9lMkNcWMjIteokiM8BD1ScdQjiKi/DS/fwwPjNKDoZDmcQ1z9edEiiKlRlFgkhHSA/XXy8T/vE6iwzM/pTxONOF4sihMGNQCZmXAHpUEazYyBGFJza0QD5BEWJvKinkJ5xlOvr88S5pHFee4Ur2tlmuX0zoKYA/sg0PggFNQA9egDlyAwT14BM/gxXqwnqxX620SnbOmMzvgF6z3L1CRlcY=</latexit>

� (a1, a2 . . . , an) = False<latexit sha1_base64="8E0S7ofhfpd7mwGpVXiYOnYeyko=">AAACEXicbZBNS8NAEIY3ftb6VfXoJViEClISFT8OQlEQjwpWhaaEyXbSLm42YXcilOJf8OJf8eJBEa/evPlv3NYifr2w8PDODLPzRpkUhjzv3RkZHRufmCxMFadnZufmSwuL5ybNNcc6T2WqLyMwKIXCOgmSeJlphCSSeBFdHfbrF9eojUjVGXUzbCbQViIWHMhaYakSZB0RSIypAqG/DuFGIFspGUsq0KLdobX9I5AGw1LZq3oDuX/BH0KZDXUSlt6CVsrzBBVxCcY0fC+jZg80CS7xphjkBjPgV9DGhkUFCZpmb3DRjbtqnZYbp9o+Re7A/T7Rg8SYbhLZzgSoY37X+uZ/tUZO8W6zJ1SWEyr+uSjOpUup24/HbQmNnGTXAnAt7F9d3gENnGyIxUEIe31tf538F843qv5mdet0q1w7GMZRYMtshVWYz3ZYjR2zE1ZnnN2ye/bInpw758F5dl4+W0ec4cwS+yHn9QOCC5z1</latexit>

�<latexit sha1_base64="X3M+pAIe1d+FX0EDD3b/sDIWPXI=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0VL0VvXisYD+gDWWz3TRLdzdhdyOU0L/gxYMiXv1D3vw3JmkQtT4YeLw3w8w8L+JMG9v+tEorq2vrG+XNytb2zu5edf+gq8NYEdohIQ9V38OaciZpxzDDaT9SFAuP0543vcn83gNVmoXy3swi6go8kcxnBJtMGkYBG1Vrdt3OgZaJU5AaFGiPqh/DcUhiQaUhHGs9cOzIuAlWhhFO55VhrGmEyRRP6CClEguq3SS/dY5OUmWM/FClJQ3K1Z8TCRZaz4SXdgpsAv3Xy8T/vEFs/Es3YTKKDZVksciPOTIhyh5HY6YoMXyWEkwUS29FJMAKE5PGU8lDuMrQ/H55mXTP6s55vXHXqLWuizjKcATHcAoOXEALbqENHSAQwCM8w4slrCfr1XpbtJasYuYQfsF6/wIsoo53</latexit>

TAUTiscoNP-Complete•  •  iscoNPhard!–  Weshow–  iscoNP-Complete–  “Oninputàoutput”–  Analysis:if,thenisnotsatisfiable•  So•  So• 

–  Reductionrequirespolynomial(linear)time!

10/29/19 3TheoryofComputation-Fall'19LorenzoDeStefani

TAUT 2 NP ! TAUT 2 coNP<latexit sha1_base64="xJvfTsVspN9AzN9cm+JcYtsrsvk=">AAACGHicbVDLSgMxFM34rPVVdekmWARXdarFx67qxlWp0GkLnaFk0kwbmkmGJKOUoZ/hxl9x40IRt935N2amRdR6IHA451xu7vEjRpW27U9rYXFpeWU1t5Zf39jc2i7s7DaViCUmDhZMyLaPFGGUE0dTzUg7kgSFPiMtf3iT+q17IhUVvKFHEfFC1Oc0oBhpI3ULx64wdjqdNK6cxtilHNbq0JW0P9BISvEAUx2mOha1erdQtEt2BjhPyjNSBDOY/MTtCRyHhGvMkFKdsh1pL0FSU8zIOO/GikQID1GfdAzlKCTKS7LDxvDQKD0YCGke1zBTf04kKFRqFPomGSI9UH+9VPzP68Q6uPASyqNYE46ni4KYQS1g2hLsUUmwZiNDEJbU/BXiAZIIa9NlPivhMsXZ98nzpHlSKp+WKneVYvV6VkcO7IMDcATK4BxUwS2oAwdg8AiewSt4s56sF+vd+phGF6zZzB74BWvyBa0En6Y=</latexit>

TAUT<latexit sha1_base64="+exYs16d7hnfT3+Ts0zQxP+lLO8=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KokWP25VLx4rNG2hDWWz3bRLN5uwuxFK6G/w4kERr/4gb/4bN2kQtT4YeLw3w8w8P+ZMadv+tJaWV1bX1ksb5c2t7Z3dyt5+W0WJJNQlEY9k18eKciaoq5nmtBtLikOf044/uc38zgOVikWipacx9UI8EixgBGsjua1rt4UGlapds3OgReIUpAoFmoPKR38YkSSkQhOOleo5dqy9FEvNCKezcj9RNMZkgke0Z6jAIVVemh87Q8dGGaIgkqaERrn6cyLFoVLT0DedIdZj9dfLxP+8XqKDSy9lIk40FWS+KEg40hHKPkdDJinRfGoIJpKZWxEZY4mJNvmU8xCuMpx/v7xI2qc156xWv69XGzdFHCU4hCM4AQcuoAF30AQXCDB4hGd4sYT1ZL1ab/PWJauYOYBfsN6/APK2jkI=</latexit>

TAUT p SAT<latexit sha1_base64="TRQcdLsgQm8vZfBJLLFen+h1gjk=">AAACA3icbVBLS8NAGNz4rPUV9aaXxSJ4KqkWH7dWLx4rJm2hCWGz3bRLNw93N0IJBS/+FS8eFPHqn/Dmv3GTBlHrwMIwM9/ufuPFjAppGJ/a3PzC4tJyaaW8ura+salvbbdFlHBMLByxiHc9JAijIbEklYx0Y05Q4DHS8UaXmd+5I1zQKDTlOCZOgAYh9SlGUkmuvms2LRPajNy6MbQjFc1uSm+a5sTVK0bVyAFnSa0gFVCg5eofdj/CSUBCiRkSolczYumkiEuKGZmU7USQGOERGpCeoiEKiHDSfIcJPFBKH/oRVyeUMFd/TqQoEGIceCoZIDkUf71M/M/rJdI/c1IaxokkIZ4+5CcMyghmhcA+5QRLNlYEYU7VXyEeIo6wVLWV8xLOM5x8rzxL2kfV2nG1fl2vNC6KOkpgD+yDQ1ADp6ABrkALWACDe/AInsGL9qA9aa/a2zQ6pxUzO+AXtPcvSCWXcg==</latexit>

SAT<latexit sha1_base64="KY+CIYarXUXhKt9IE8jRrMrRTD8=">AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsxo8bGrunFZsS9ox5JJM21oZjIkGaUM/Q83LhRx67+482/MTIuo9UDgcM493JvjRZwpbdufVm5hcWl5Jb9aWFvf2Nwqbu80lYgloQ0iuJBtDyvKWUgbmmlO25GkOPA4bXmjq9Rv3VOpmAjrehxRN8CDkPmMYG2ku64wZppNbi/qk16xZJftDGieODNSghlqveJHty9IHNBQE46V6jh2pN0ES80Ip5NCN1Y0wmSEB7RjaIgDqtwku3qCDozSR76Q5oUaZerPRIIDpcaBZyYDrIfqr5eK/3mdWPtnbsLCKNY0JNNFfsyRFiitAPWZpETzsSGYSGZuRWSIJSbaFFXISjhPcfL95XnSPCo7x+XKTaVUvZzVkYc92IdDcOAUqnANNWgAAQmP8Awv1oP1ZL1ab9PRnDXL7MIvWO9f1iaS3g==</latexit>


�<latexit sha1_base64="d1ge7daNkF+17LRjR06FFAqs7/o=">AAAB8XicbVDLSsNAFL2pr1pfVZduBovgqiRafOyKblxWsA9sQplMJ+3QySTMTIQS+hduXCji1r9x5984SYOo9cCFwzn3cu89fsyZ0rb9aZWWlldW18rrlY3Nre2d6u5eR0WJJLRNIh7Jno8V5UzQtmaa014sKQ59Trv+5Drzuw9UKhaJOz2NqRfikWABI1gb6d71sUzdeMxmg2rNrts50CJxClKDAq1B9cMdRiQJqdCEY6X6jh1rL8VSM8LprOImisaYTPCI9g0VOKTKS/OLZ+jIKEMURNKU0ChXf06kOFRqGvqmM8R6rP56mfif1090cOGlTMSJpoLMFwUJRzpC2ftoyCQlmk8NwUQycysiYywx0SakSh7CZYaz75cXSeek7pzWG7eNWvOqiKMMB3AIx+DAOTThBlrQBgICHuEZXixlPVmv1tu8tWQVM/vwC9b7F/G/kTw=</latexit>

� 2 SAT<latexit sha1_base64="cItvS1hNIer6YcmSvqfFPzGTtgQ=">AAACAHicbVA7T8MwGHTKq5RXgIGBxaJCYqoSqHhsBRbGIvqSmqhyXKe16jiR7SBVURb+CgsDCLHyM9j4NzhphIBykqXT3X22v/MiRqWyrE+jtLC4tLxSXq2srW9sbpnbOx0ZxgKTNg5ZKHoekoRRTtqKKkZ6kSAo8BjpepPrzO/eEyFpyFtqGhE3QCNOfYqR0tLA3HOiMYUO5dAJdS67Jrm7bKUDs2rVrBxwntgFqYICzYH54QxDHAeEK8yQlH3bipSbIKEoZiStOLEkEcITNCJ9TTkKiHSTfIEUHmplCP1Q6MMVzNWfEwkKpJwGnk4GSI3lXy8T//P6sfLP3YTyKFaE49lDfsygCmHWBhxSQbBiU00QFlT/FeIxEggr3VklL+Eiw+n3yvOkc1yzT2r123q1cVXUUQb74AAcARucgQa4AU3QBhik4BE8gxfjwXgyXo23WbRkFDO74BeM9y9405Z5</latexit>


8a1, a2, . . . , an,� (a1, a2, . . . , an) = False<latexit sha1_base64="WJnqsLdb7SYlEYidqYyudMN0J/o=">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</latexit>

8a1, a2, . . . , an, � (a1, a2, . . . , an) = True<latexit sha1_base64="OJq8B38cl2By/r3DGxPrgSyfcYw=">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</latexit>

� 2 SAT () � 2 TAUT<latexit sha1_base64="V8XtlqfiIMVtBhek4SE6z5sk/CU=">AAACGHicbVBLSwMxGMzWV62vVY9egkXwVLdafNxavXisuH1At5Rsmm1Ds8mSZIWy7M/w4l/x4kERr735b8y2RdQ6EBhm5kvyjR8xqrTjfFq5peWV1bX8emFjc2t7x97dayoRS0waWDAh2z5ShFFOGppqRtqRJCj0GWn5o5vMbz0Qqajgrh5HpBuiAacBxUgbqWefeNGQQo9y6AmTy65J7mtuaqQggJ6PZJIl0izh1hpuzy46JWcKuEjKc1IEc9R79sTrCxyHhGvMkFKdshPpboKkppiRtODFikQIj9CAdAzlKCSqm0wXS+GRUfowENIcruFU/TmRoFCpceibZIj0UP31MvE/rxPr4LKbUB7FmnA8eyiIGdQCZi3BPpUEazY2BGFJzV8hHiKJsDZdFqYlXGU4/155kTRPS+WzUuWuUqxez+vIgwNwCI5BGVyAKrgFddAAGDyCZ/AK3qwn68V6tz5m0Zw1n9kHv2BNvgAY4p/x</latexit>

Correctionfromclass:denotesthenegationoftheentireexpression,NOTjusttheliterals

�<latexit sha1_base64="d1ge7daNkF+17LRjR06FFAqs7/o=">AAAB8XicbVDLSsNAFL2pr1pfVZduBovgqiRafOyKblxWsA9sQplMJ+3QySTMTIQS+hduXCji1r9x5984SYOo9cCFwzn3cu89fsyZ0rb9aZWWlldW18rrlY3Nre2d6u5eR0WJJLRNIh7Jno8V5UzQtmaa014sKQ59Trv+5Drzuw9UKhaJOz2NqRfikWABI1gb6d71sUzdeMxmg2rNrts50CJxClKDAq1B9cMdRiQJqdCEY6X6jh1rL8VSM8LprOImisaYTPCI9g0VOKTKS/OLZ+jIKEMURNKU0ChXf06kOFRqGvqmM8R6rP56mfif1090cOGlTMSJpoLMFwUJRzpC2ftoyCQlmk8NwUQycysiYywx0SakSh7CZYaz75cXSeek7pzWG7eNWvOqiKMMB3AIx+DAOTThBlrQBgICHuEZXixlPVmv1tu8tWQVM/vwC9b7F/G/kTw=</latexit>


NotAllEquallySatisfiable

NAE-SATisNP-hard•  Reductionto3SAT•  Oninput•  Convertitto

•  Polynomial(linear)runtimereduction


NAE � SAT =

( < C > |C = {C1, C2, . . . , Cm} where each

Ci is a tuple of literals x1, x2, . . . , nn,

s.t. 9 an assignment a1, . . . , ans.t. each Ci

has � 1 satisfied and � 1 unsatisfied literal

)

<latexit sha1_base64="v5FQWNK0Dng+rM+jN6HIVEmgIX0=">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</latexit>

� = (z1,1 _ z1,2 _ z1,3) ^ (z2,1 _ z2,2 _ z2,3) ^ . . .<latexit sha1_base64="+4A+FVBhkh4xxad0Z0osktJ419Q=">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</latexit>

C = (z1,1 _ z1,2 _ z1,3 _ u) ^ (z2,1 _ z2,2 _ z2,3 _ u) ^ . . .<latexit sha1_base64="hR0hLCyqxcIDXVx5QRdIxkMtfWY=">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</latexit>

NotAllEquallySatisfiable•  ObservationifanassignmenttothevariablesinCsatisfiesC,

sodoestheassignmentwithallnegatedvaluesforthevariables

•  Correctness:–  If,thereexistsanassignmentthat

satisfies.ThesameassignmentwillalsosatisfyC.ThusassignCinthesamewayandsetu=False.

–  IfCNAE-SAT,thereexistsanassignmentandusuchthatCissatisfiedandthereisatleastoneTrueandoneFalseliteralineachclause•  Ifu=False,theassignmentsatisfiesà•  Ifu=True,theassignmentsatisfiesà


� 2 3SAT<latexit sha1_base64="22EfBzPB99WgwqvjjFGV/wl/8Cs=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiRafOyqblxW7AuaUCbTSTt0MhlmJkIp/Q03LhRx68+4829M0iBqPXDhcM693HuPLznTxrY/rcLS8srqWnG9tLG5tb1T3t1r6yhWhLZIxCPV9bGmnAnaMsxw2pWK4tDntOOPb1K/80CVZpFomomkXoiHggWMYJNIritHzGUCnd5fNfvlil21M6BF4uSkAjka/fKHO4hIHFJhCMda9xxbGm+KlWGE01nJjTWVmIzxkPYSKnBItTfNbp6ho0QZoCBSSQmDMvXnxBSHWk9CP+kMsRnpv14q/uf1YhNceFMmZGyoIPNFQcyRiVAaABowRYnhk4RgolhyKyIjrDAxSUylLITLFGffLy+S9knVOa3W7mqV+nUeRxEO4BCOwYFzqMMtNKAFBCQ8wjO8WLH1ZL1ab/PWgpXP7MMvWO9fGTWRNQ==</latexit>

a1, a2, . . . , an<latexit sha1_base64="xvHH3lzh9VAsVlYlrQY8wL6qCfE=">AAAB+3icbZDLSgMxFIYz9Vbrrdalm2ARXJQyU4uXXdGNywr2Au0wZDKZNjSTDElGLKWv4saFIm59EXe+jZnpIGr9IfDxn3M4J78fM6q0bX9ahZXVtfWN4mZpa3tnd6+8X+kqkUhMOlgwIfs+UoRRTjqaakb6sSQo8hnp+ZPrtN67J1JRwe/0NCZuhEachhQjbSyvXEGeU0NeozZkgdDKoDGrdt3OBJfByaEKcrW98scwEDiJCNeYIaUGjh1rd4akppiReWmYKBIjPEEjMjDIUUSUO8tun8Nj4wQwFNI8rmHm/pyYoUipaeSbzgjpsfpbS83/aoNEhxfujPI40YTjxaIwYVALmAYBAyoJ1mxqAGFJza0Qj5FEWJu4SlkIl6nOvr+8DN1G3TmtN2+b1dZVHkcRHIIjcAIccA5a4Aa0QQdg8AAewTN4sebWk/VqvS1aC1Y+cwB+yXr/AqHXk64=</latexit>


2<latexit sha1_base64="OGutpjBGqXjN2B7YqOyu423H7bU=">AAAB6nicbVDLSsNAFL2pr1pfUZduBovgqiRafOyKblxWtA9oQ5lMJ+3QySTMTIQS+gluXCji1i9y5984SYOo9cCFwzn3cu89fsyZ0o7zaZWWlldW18rrlY3Nre0de3evraJEEtoiEY9k18eKciZoSzPNaTeWFIc+px1/cp35nQcqFYvEvZ7G1AvxSLCAEayNdNdnYmBXnZqTAy0StyBVKNAc2B/9YUSSkApNOFaq5zqx9lIsNSOczir9RNEYkwke0Z6hAodUeWl+6gwdGWWIgkiaEhrl6s+JFIdKTUPfdIZYj9VfLxP/83qJDi68lIk40VSQ+aIg4UhHKPsbDZmkRPOpIZhIZm5FZIwlJtqkU8lDuMxw9v3yImmf1NzTWv22Xm1cFXGU4QAO4RhcOIcG3EATWkBgBI/wDC8Wt56sV+tt3lqyipl9+AXr/QtlyI4D</latexit>



� 2 3SAT<latexit sha1_base64="C9EC0Xj9I4bEI/ojK964qCi3Qyg=">AAAB9HicbVDLSsNAFL2pr1pfVZduBovgqiRafOyqblxW7AuaUCbTSTt0Mokzk0IJ/Q43LhRx68e4829M0iBqPXDhcM693HuPG3KmtGl+GoWl5ZXVteJ6aWNza3unvLvXVkEkCW2RgAey62JFORO0pZnmtBtKin2X0447vkn9zoRKxQLR1NOQOj4eCuYxgnUiOXY4YshmAp3eXzX75YpZNTOgRWLlpAI5Gv3yhz0ISORToQnHSvUsM9ROjKVmhNNZyY4UDTEZ4yHtJVRgnyonzo6eoaNEGSAvkEkJjTL150SMfaWmvpt0+liP1F8vFf/zepH2LpyYiTDSVJD5Ii/iSAcoTQANmKRE82lCMJEsuRWREZaY6CSnUhbCZYqz75cXSfukap1Wa3e1Sv06j6MIB3AIx2DBOdThFhrQAgIP8AjP8GJMjCfj1XibtxaMfGYffsF4/wJxTJFf</latexit>

a1, a2, . . . , an<latexit sha1_base64="/1EOfd0Mp5cNcv4OUyBmH8SfLxc=">AAACDXicbZDLSsNAFIYnXmu9RV26CVbBhZSkFi+7ohuXFewF2hAmk0k7dDITZiZCCX0BN76KGxeKuHXvzrdxkkZR6w8DH/85hzPn92NKpLLtD2NufmFxabm0Ul5dW9/YNLe225InAuEW4pSLrg8lpoThliKK4m4sMIx8ijv+6DKrd26xkISzGzWOsRvBASMhQVBpyzP3+z4UKfScyVFBNU004Ep+GWzimRW7aueyZsEpoAIKNT3zvR9wlESYKUShlD3HjpWbQqEIonhS7icSxxCN4AD3NDIYYemm+TUT60A7gRVyoR9TVu7+nEhhJOU48nVnBNVQ/q1l5n+1XqLCMzclLE4UZmi6KEyopbiVRWMFRGCk6FgDRILov1poCAVESgdYzkM4z3TyffIstGtV57hav65XGhdFHCWwC/bAIXDAKWiAK9AELYDAHXgAT+DZuDcejRfjddo6ZxQzO+CXjLdPgh+b/Q==</latexit>


� 2 3SAT<latexit sha1_base64="C9EC0Xj9I4bEI/ojK964qCi3Qyg=">AAAB9HicbVDLSsNAFL2pr1pfVZduBovgqiRafOyqblxW7AuaUCbTSTt0Mokzk0IJ/Q43LhRx68e4829M0iBqPXDhcM693HuPG3KmtGl+GoWl5ZXVteJ6aWNza3unvLvXVkEkCW2RgAey62JFORO0pZnmtBtKin2X0447vkn9zoRKxQLR1NOQOj4eCuYxgnUiOXY4YshmAp3eXzX75YpZNTOgRWLlpAI5Gv3yhz0ISORToQnHSvUsM9ROjKVmhNNZyY4UDTEZ4yHtJVRgnyonzo6eoaNEGSAvkEkJjTL150SMfaWmvpt0+liP1F8vFf/zepH2LpyYiTDSVJD5Ii/iSAcoTQANmKRE82lCMJEsuRWREZaY6CSnUhbCZYqz75cXSfukap1Wa3e1Sv06j6MIB3AIx2DBOdThFhrQAgIP8AjP8GJMjCfj1XibtxaMfGYffsF4/wJxTJFf</latexit>

C 2 NAE � SAT () � 2 3SAT<latexit sha1_base64="fNFwAQWLndz5df1P3akd2X31ZXk=">AAACCXicbZDLSsNAFIYn9VbrLerSzWAR3FhSW7zsWovgSir2Bk0ok+mkHTqZhJmJUEK3bnwVNy4UcesbuPNtnKRF1PrDwMd/zuHM+d2QUaks69PILCwuLa9kV3Nr6xubW+b2TksGkcCkiQMWiI6LJGGUk6aiipFOKAjyXUba7qiW1Nt3REga8IYah8Tx0YBTj2KktNUzYQ3alMPr6uXRbbWh2fOgHQ5p6pa01TPzVsFKBeehOIM8mKneMz/sfoAjn3CFGZKyW7RC5cRIKIoZmeTsSJIQ4REakK5GjnwinTi9ZAIPtNOHXiD04wqm7s+JGPlSjn1Xd/pIDeXfWmL+V+tGyjtzYsrDSBGOp4u8iEEVwCQW2KeCYMXGGhAWVP8V4iESCCsdXi4N4TzRyffJ89A6LhRLhfJNOV+5mMWRBXtgHxyCIjgFFXAF6qAJMLgHj+AZvBgPxpPxarxNWzPGbGYX/JLx/gVgLZe+</latexit>


Spacecomplexity•  Sofarwecharacterizedclassesoflanguagesbasedontheirexecutiontimerequirements

•  Wecanintroduceanalogousnotionsbasedontheirmemoryspacerequirement


SPACE(f(n)) =

(L|L = L(n) s.t. M is a canonical TM

which uses Ø(f(n)) space on n-size input

)

<latexit sha1_base64="aNfIAOqNArfLTehLiznxUEJGkJk=">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</latexit>

NSPACE(f(n)) =

(L|L = L(n) s.t. M is a NTM which uses

Ø(f(n)) space on n-size input

)

<latexit sha1_base64="nJhqNYsGsEK3GWpKf9W13khGAsk=">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</latexit>

O

O

•  Consideraverifier:–  Oninput<>,withliterals–  Consider(inaloop)allpossibleassignmentstothevariablesin–  EvaluatebasedonsuchassignmentsandacceptifevaluatestoTrue

–  IfnoassignmentissuchthatevaluatestoTrue,thenreject•  Analysis:–  ExponentiallymanycyclesàexponentialdeterministicruntimeO(2n)

–  Spacerequiredispolynomial,asmanyasthevariable+“index”oftheconfigurationslog2n•  Spaceisreusable


SAT 2 SPACE(N)<latexit sha1_base64="b2rQekMYE4sRsyAUI0LDuKcw+1o=">AAAB+XicbVDLSsNAFJ3UV62vqEs3g0Wom5Jo8bFrLYIrqfQJbSiT6bQdOpmEmUmhhP6JGxeKuPVP3Pk3TtIgaj1w4XDOvdx7jxswKpVlfRqZldW19Y3sZm5re2d3z9w/aEk/FJg0sc980XGRJIxy0lRUMdIJBEGey0jbnVRjvz0lQlKfN9QsII6HRpwOKUZKS33TrFcasEc5rNcq1dvC/WnfzFtFKwFcJnZK8iBFrW9+9AY+Dj3CFWZIyq5tBcqJkFAUMzLP9UJJAoQnaES6mnLkEelEyeVzeKKVARz6QhdXMFF/TkTIk3LmubrTQ2os/3qx+J/XDdXwyokoD0JFOF4sGoYMKh/GMcABFQQrNtMEYUH1rRCPkUBY6bBySQjXMS6+X14mrbOifV4sPZTy5Zs0jiw4AsegAGxwCcrgDtRAE2AwBY/gGbwYkfFkvBpvi9aMkc4cgl8w3r8AuZKR6Q==</latexit>








BNFA<latexit sha1_base64="yXKCNPMD6INkEX+gVSUk2pxvtPs=">AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiRafOxqBXElFewD2lAm00k7dDIJMxOhhH6EGxeKuPV73Pk3TtIgvg5cOJxzL/fe40WcKW3bH1ZhYXFpeaW4Wlpb39jcKm/vtFUYS0JbJOSh7HpYUc4EbWmmOe1GkuLA47TjTS5Tv3NPpWKhuNPTiLoBHgnmM4K1kTqNQXJzdTEblCt21c6A/hInJxXI0RyU3/vDkMQBFZpwrFTPsSPtJlhqRjidlfqxohEmEzyiPUMFDqhyk+zcGTowyhD5oTQlNMrU7xMJDpSaBp7pDLAeq99eKv7n9WLtn7kJE1GsqSDzRX7MkQ5R+jsaMkmJ5lNDMJHM3IrIGEtMtEmolIVwnuLk6+W/pH1UdY6rtdtapd7I4yjCHuzDIThwCnW4hia0gMAEHuAJnq3IerRerNd5a8HKZ3bhB6y3T9znj2Y=</latexit>

•  Howwouldyougoaboutconstructingadeciderforthisalgorithm?–  Inputis<N>–  ItssizeisO(q)– DenoteasFthesetofnon-acceptingstates– DenoteasIthestartingstate

BNFA =

(< N > |N is a q � states NFA and

9w s.t. N does not accept w

)

<latexit sha1_base64="X5fXpKPHR+s304zqxK22hl/jY3M=">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</latexit>

DecidingBNFAinO(q)space

ConsiderthefollowingNTM:–  Oninput<N>:•  LetSibethesetofallpossiblestatesatstepI

–  S0=I•  IfIisnotacceptingthenaccept(εnotaccepted)•  Fori=1to2q:

–  Non-deterministicallypickthei-thsymbolofw(i.e.,wi)–  Si=setofstatesreachablebyanysetofstatesinSi-1upon

readingwiorusinganεtransition–  Ifaccept

•  reject–  Wehave2qiterationsofthecycletoallowthepossibility

ofconsideringanypossiblecompositionofSi


Si \ F = ;<latexit sha1_base64="UXl0IGh+fD2Ar9mbOn+ZX4VgUuo=">AAAB/nicbVDJSgNBEO2JW4zbqHjy0hgET2FGg8tBCAriMaJZIBmGnk4ladKz0F0jhCHgr3jxoIhXv8Obf+MkGUSNDwoe71VRVc+LpNBoWZ9Gbm5+YXEpv1xYWV1b3zA3t+o6jBWHGg9lqJoe0yBFADUUKKEZKWC+J6HhDS7HfuMelBZhcIfDCByf9QLRFZxhKrnmzq0r2pxF9Iqe0zb4EQ41oGsWrZI1AZ0ldkaKJEPVNT/anZDHPgTIJdO6ZVsROglTKLiEUaEda4gYH7AetFIaMB+0k0zOH9H9VOnQbqjSCpBO1J8TCfO1Hvpe2ukz7Ou/3lj8z2vF2D11EhFEMULAp4u6saQY0nEWtCMUcJTDlDCuRHor5X2mGMc0scIkhLMxjr9fniX1w5J9VCrflIuViyyOPNkle+SA2OSEVMg1qZIa4SQhj+SZvBgPxpPxarxNW3NGNrNNfsF4/wIg2pUa</latexit>

DecidingBNFAinO(q)space

Analysis:– Exponentialnon-deterministictime2|q|

– SpacerequirementisthespacerequiredtomaintaintheinformationforSi

– AllpossiblecompositionsofSicanbeencodedasq-bitbinarystring

–  log22|q|=|q|spacerequired


Relationbetweentimeandspace:–  Lemma1:•  Ifanalgorithmusest(n)timestepscanaccessatmost

onememorylocationatatime

–  Lemma2:•  ConsideraTMwhichusest(n)memorycells•  Upto|Σ|t(n)possibleconfigurations•  Theheadofthereadercanbeinatmostt(n)positions•  TheTMcanbeinatmost|q|possiblestates•  Thereareatmost|Σ|t(n)|q|t(n)=O(2t(n))

configurations•  Wecanalwayseliminateredundantcycles•  TheTMmachinecanbeinoneconfigurationineach

timestep


TIME(t(n)) ✓ SPACE(t(n))<latexit sha1_base64="QQJ93hYdfiUNRV7zUSYt94/s7K8=">AAACCHicbVBNS0JBFJ3Xp9mX1bJFQxLYRp4lfewsCWoRGGkKKjJvvOrgvHmvmfsCEZdt+ittWhTRtp/Qrn/TUx9R2YELh3Pu5d57HF8Kg7b9aU1Nz8zOzccW4otLyyuribX1G+MFmkOJe9LTFYcZkEJBCQVKqPgamOtIKDvd/NAv34E2wlNF7PlQd1lbiZbgDEOpkdgqXlyepTCldndrJnAMINzS68JJPhIbiaSdtkegkyQTkSSJUGgkPmpNjwcuKOSSGVPN2D7W+0yj4BIG8VpgwGe8y9pQDaliLph6f/TIgO6ESpO2PB2WQjpSf070mWtMz3XCTpdhx/z1huJ/XjXA1lG9L5QfICg+XtQKJEWPDlOhTaGBo+yFhHEtwlsp7zDNOIbZxUchHA9x8P3yJLnZS2f209mrbDJ3GsURI5tkm6RIhhySHDknBVIinNyTR/JMXqwH68l6td7GrVNWNLNBfsF6/wIkGpek</latexit>

SPACE(t(n)) ✓ TIME(2t(n))<latexit sha1_base64="ChDMeUlLaPi8RFktb52PTwUox0c=">AAACDXicbVDJSgNBEO1xjXGLevTSGIXkEiYaXG4uBPQgREyikMTQ06mYJj09Y3eNEIb8gBd/xYsHRbx69+bfOJMEcXtQ8Hiviqp6ji+FQdv+sMbGJyanphMzydm5+YXF1NJy1XiB5lDhnvT0pcMMSKGgggIlXPoamOtIuHC6R7F/cQvaCE+VsedDw2XXSrQFZxhJzdT6eengqJjBjMpmad0EjgGEG1o+OS1mNq/CWO9nm6m0nbMHoH9JfkTSZIRSM/Veb3k8cEEhl8yYWt72sREyjYJL6CfrgQGf8S67hlpEFXPBNMLBN326ESkt2vZ0VArpQP0+ETLXmJ7rRJ0uw4757cXif14twPZuIxTKDxAUHy5qB5KiR+NoaEto4Ch7EWFci+hWyjtMM45RgMlBCHsxtr9e/kuqm7n8Vq5wVkjvH47iSJBVskYyJE92yD45JiVSIZzckQfyRJ6te+vRerFeh61j1mhmhfyA9fYJjr+Zfg==</latexit>

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