{"paper_meta":{"paper_id":"arxiv:0705.0915","title":"0705.0915","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0705.0915v2 [cs.CC] 9 Jun 2008\nSatis abilit\ny\nP\narsimoniously\nRedu es\nto\nthe\nT\nan\ntrix\nTM\nRotation\nPuzzle\nProblem∗\nDorothea\nBaumeister\nand\nJörg\nRothe\nInstitut\nfür\nInformatik\nHeinri \nh-Heine-Univ\nersität\nDüsseldorf\n40225\nDüsseldorf,\nGerman\ny\nJune\n9,\n2008\nAbstra t\nHolzer\nand\nHolzer\n[HH04℄\npro\nv\ned\nthat\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nNP\n- omplete.\nThey\nalso\nsho\nw\ned\nthat\nfor\nin nite\nrotation\npuzzles,\nthis\nproblem\nb\ne omes\nunde idable.\nW\ne\nstudy\nthe\n oun\nting\nv\nersion\nand\nthe\nunique\nv\nersion\nof\nthis\nproblem.\nW\ne\npro\nv\ne\nthat\nthe\nsatis abilit\ny\nproblem\nparsimoniously\nredu es\nto\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nIn\nparti ular,\nthis\nredu tion\npreserv\nes\nthe\nuniqueness\nof\nthe\nsolution,\nwhi \nh\nimplies\nthat\nthe\nunique\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nas\nhard\nas\nthe\nunique\nsatis abilit\ny\nproblem,\nand\nso\nis\nDP\n- omplete\nunder\np\nolynomial-time\nrandom-\nized\nredu tions,\nwhere\nDP\nis\nthe\nse ond\nlev\nel\nof\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n.\nKey\nw\nords:\n omputational\n omplexit\ny\n,\nrotation\npuzzle,\ntiling\nof\nthe\nplane,\nparsimo-\nnious\nredu tion,\n oun\nting\nproblem.\n1\nIn\ntro\ndu tion\nT\nan\ntrix\nTM\nis\na\npuzzle\ngame\npla\ny\ned\nwith\nhexagonal\ntiles\n rmly\narranged\nin\nthe\nplane\nthat\nea \nh\n an\nb\ne\nrotated\naround\ntheir\naxes.\nThere\nare\nfour\ndi eren\nt\nt\nyp\nes\nof\ntiles\n( alled\nSint,\nBrid,\nChin,\nand\nR\nond,\nsee\nFigure\n1)\nthat\ndi er\nb\ny\nthe\nform\nof\nthe\nthree\n olored\nlines\nthey\nea \nh\nha\nv\ne,\nwhere\nthe\n olors\nare\n \nhosen\namong\nr\ne\nd,\nyel\nlow,\nblue,\nand\ngr\ne\nen.\nThe\nob\nje tiv\ne\nof\nthe\ngame\nis\nto\n nd\na\nrotation\nof\nthe\ngiv\nen\ntiles\nso\nas\nto\n reate\nlong\nlines\nand\nlo\nops\nof\nthe\nsame\n olor.\nSin e\nits\nin\nv\nen\ntion\nin\n1991\nb\ny\nMik\ne\nM Mana\nw\na\ny\nfrom\nNew\nZealand\nand\nits\n ommer ial\nlaun \nh,\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nhas\nb\ne ome\nextremely\np\nopular\nand\n ommer ially\nsu essful.\n∗\nSupp\norted\nin\npart\nb\ny\nDF\nG\ngran\nts\nR\nO\n1202/9-3\nand\nR\nO\n1202/11-1\nand\nb\ny\nthe\nAlexan-\nder\nv\non\nHum\nb\noldt\nF\noundation's\nT\nransCo\nop\nprogram.\nA\npreliminary\nv\nersion\nof\nthis\npap\ner\nap-\np\neared\nin\nthe\nPro\n eedings\nof\nMa hines,\nComputations\nand\nUniversality\n(MCU\n2007).\nURLs:\n . s.uni-duesseldorf.de/∼\n{baumeister,\nrothe}\n(D.\nBaumeister\nand\nJ.\nRothe).\n1\n\nHolzer\nand\nHolzer\n[HH04℄\n onsidered\nt\nw\no\nv\narian\nts\nof\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem,\none\nwith\n nitely\nman\ny\nand\none\nwith\nin nitely\nman\ny\ntiles\nin\na\ngiv\nen\nproblem\ninstan e.\nThey\npro\nv\ned\nthat\nthe\n nite\nv\narian\nt\nof\nthis\nproblem\nis\nNP\n- omplete\nb\ny\nredu ing\nthe\nNP- omplete\nb\no\nolean\n ir uit\nsatis abilit\ny\nproblem\n(restri ted\nto\n ir uits\nwith\nAND\nand\nNOT\ngates\nonly)\nto\nit.\nThey\nalso\nsho\nw\ned\nthat\nthe\nin nite\nv\narian\nt\nof\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nunde idable,\nagain\nemplo\nying\na\n ir uit\n onstru tion.\nF\nor\nother\nresults\non\nthe\n omplexit\ny\nof\nproblems\nrelated\nto\nDomino-lik\ne\nstrategy\ngames,\nw\ne\nrefer\nto\nGrädel\n[Grä90\n℄.\nW\ne\n onsider\nt\nw\no\nv\narian\nts\nof\nthe\n nite\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem,\nits\n oun\nting\nv\nersion\nand\nits\nunique\nv\nersion.\nThe\n oun\nting\nproblem\nasks\nfor\nthe\nn\num\nb\ner\nof\nsolutions\nof\na\ngiv\nen\nrotation\npuzzle\ninstan e.\nThe\nunique\nproblem\nasks\nwhether\na\ngiv\nen\nrotation\npuzzle\ninstan e\nhas\nexa tly\none\nsolution.\nOur\nmain\nresult\nis\nthat\nthe\nsatis abilit\ny\nproblem\nparsimoniously\nredu es\nto\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nThe\n lass #P\nw\nas\nin\ntro\ndu ed\nb\ny\nV\nalian\nt\n[V\nal79\n℄\nto\n apture\nthe\n omplexit\ny\nof\n oun\nting\nthe\nsolutions\nof\nNP\nproblems.\nP\narsimonious\nredu tions\nb\net\nw\neen\nNP\n oun\nting\nproblems \nsu \nh\nas\nours preserv\ne\nthe\npre ise\nn\num\nb\ner\nof\nsolutions.\nThis\nis\nan\nimp\nortan\nt\nprop\nert\ny\nfor\nat\nleast\nt\nw\no\nreasons.\nFirst,\nthe\nstru ture\nof\nthe\nsolution\nspa e\nis\npreserv\ned\nb\ny\na\nparsimonious\nredu tion\nfrom A\nto B\n,\nsin e\nsolutions\nof A\nare\nmapp\ned\nbije tiv\nely\nto\nsolutions\nof B\nin\np\nolynomial\ntime.\nSe ond,\nparsimonious\nredu tions\n an\nb\ne\nused\nto\npro\nv\ne\nlo\nw\ner\nb\nounds\nfor\nthe\nunique\nv\nersions\nof\nNP\nproblems.\nIn\nparti ular,\nw\ne\napply\nour\nab\no\nv\ne-men\ntioned\nparsimonious\nredu tion\nto\npro\nv\ne\nthat\nthe\nunique\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nDP\n- omplete\nunder\np\nolynomial-time\nrandomized\nredu tions\nin\nthe\nsense\nof\nV\nalian\nt\nand\nV\nazirani\n[VV86℄.\nHere,\nDP\nis\nthe\nset\nof\ndi eren es\nof\nan\ny\nt\nw\no\nNP\nsets\n[PY84\n℄;\nso\nNP ⊆\nDP,\nand\nit\nis\n onsidered\nmost\nunlik\nely\nthat\nb\noth\n lasses\nare\nequal.\nNote\nthat\nDP\nis\nthe\nse ond\nlev\nel\nof\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n,\nsee\nCai\net\nal.\n[CGH+\n88\n,\nCGH+\n89℄.\nF\nurther\nresults\non\nDP\nand\n ompletenes\nin\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n an\nb\ne\nfound,\ne.g.,\nin\n[CM87\n,\nRot03℄,\nsee\nalso\nthe\nsurv\ney\n[RR06℄.\nWhile\nman\ny\nstandard\nredu tions\nb\net\nw\neen\nNP- omplete\nproblems\nare\neasily\nseen\nto\nb\ne\nparsimonious,\nthere\nare\na\nn\num\nb\ner\nof\nex eptions.\nF\nor\nexample,\nBarban \nhon\n[Bar04℄\nsho\nw\ned\nthat\nthe\n(planar)\nsatis abilit\ny\nproblem\nis\nparsimoniously\np\nolynomial-time\nredu ible\nto\nthe\n(planar) 3- olorabilit\ny\nproblem\nvia\na\nrather\nsophisti ated\n onstru tion.\nOther\nexamples\nof\nnon\ntrivial\nparsimonious\nredu tions\n an\nb\ne\nfound\nin\n[P\nap94\n℄.\nHolzer\nand\nHolzer's\nredu tion,\nho\nw\nev\ner,\nis\nnot\nparsimonious\n[HH04℄.\nThe\nmain\npurp\nose\nof\nthis\npap\ner\nis\nto\nsho\nw\nho\nw\nto\nmo\ndify\ntheir\nredu tion\nso\nas\nto\nmak\ne\nit\nparsimonious.\nW\ne\nmen\ntion\nin\npassing\nthat\nthis\npap\ner\ndi ers\nfrom\nits\npreliminary\nv\nersion\n[BR07\n℄\nin\nv\narious\nw\na\nys.\nFirst,\nto\nallo\nw\n omparison,\nw\ne\nhere\nexpli itly\nsho\nw\nthe\ndi eren es\nb\net\nw\neen\nHolzer\nand\nHolzer's\noriginal\n onstru tion\n[HH04℄\nand\nour\nmo\ndi ed\n onstru tion\nb\ny\n(a)\npre-\nsen\nting\ntheir\nsubpuzzles\n(mark\ned\nso\nas\nto\n learly\nindi ate\nthe\ntiles\nthat\nrequire\nmo\ndi ation\nif\none\naims\nat\na\nparsimonious\nredu tion),\nand\n(b)\nhighligh\nting\nall\nmo\ndi ed\nor\nadditionally\ninserted\ntiles\nin\nour\nsubpuzzles.\nSe ond,\nunlik\ne\nthe\nredu tions\ngiv\nen\nin\n[HH04,\nBR07℄,\nin\nthe\nredu tion\npresen\nted\nhere\nw\ne\nin\ntro\ndu e\na\nnew\nsubpuzzle\nfor\nsim\nulating\nwire\n rossings\nin\nb\no\nolean\n ir uits.\nThis\nnew\nsubpuzzle,\nwhi \nh\nw\ne\n all\nCR\nOSS,\nwill\nsa\nv\ne\nus\nthe\ne ort\nof\ntransforming\ngeneral\nb\no\nolean\n ir uits\nin\nto\nplanar\nb\no\nolean\n ir uits\n(i.e.,\nin\nto\n ir uits\nwithout\n2\n\nwire\n rossings).\nHen e,\nthe\nredu tion\npro\nvided\nin\nthe\npresen\nt\npap\ners\nis\nmore\ne ien\nt\nand\nthe\ntotal\nn\num\nb\ner\nof\ntiles\nneeded\nto\nsim\nulate\na\ngiv\nen\n ir uit\nis\n onsiderably\nsmaller\nthan\nin\nour\nprevious\n onstru tion\n[BR07℄.\nFinally\n,\nto\npro\nv\ne\n orre tness\nof\nour\nredu tion,\nw\ne\nno\nw unlik\ne\nthe\napproa \nh\ntak\nen\nin\n[BR07℄ argue\nvia\n olor\nsequen es \nof\ntiles,\nwhi \nh\nfa ilitates\nreading\nand\nunderstanding\nthe\nargumen\nts.\nThis\npap\ner\nis\norganized\nas\nfollo\nws.\nIn\nSe tion\n2\n,\nw\ne\nde ne\nsome\n omplexit\ny-theoreti \nnotions\nand\nour\nv\narian\nts\nof\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nIn\nSe tion\n3,\nw\ne\npresen\nt\nour\nparsimonious\nredu tion.\nIn\nSe tion\n4,\n nally\n,\nw\ne\nstudy\nthe\nunique\nv\nersion\nof\nthis\nproblem\nand\nsho\nw\nits\nDP- ompleteness\nunder\nrandomized\nredu tions.\n2\nPreliminaries\n2.1\nDe nition\nof\nSome\nComplexit\ny-Theoreti \nNotions\nFix\nthe\nalphab\net Σ = {0, 1},\nand\nlet Σ∗\ndenote\nthe\nset\nof\nstrings\no\nv\ner Σ .\nAs\nis\n ommon,\nde ision\nproblems\nare\nsuitably\nen o\nded\nas\nlanguages\no\nv\ner Σ .\nF\nor\nan\ny\nlanguage A ⊆Σ∗\n,\nlet ∥A∥\ndenote\nthe\nn\num\nb\ner\nof\nelemen\nts\nin A.\nF\nor\nsome\nba \nkground\non\n omputational\n om-\nplexit\ny\ntheory\n,\nw\ne\nrefer\nto\nan\ny\nstandard\ntextb\no\nok\nof\nthis\n eld,\ne.g.,\n[P\nap94\n,\nRot05℄.\nLet\nNP\ndenote\nthe\n lass\nof\nproblems\nsolv\nable\nin\nnondeterministi \np\nolynomial\ntime.\nGeneralizing\nNP,\nP\napadimitriou\nand\nY\nannak\nakis\n[PY84\n℄\nin\ntro\ndu ed\nthe\n lass\nDP = {A −B | A, B ∈\nNP}\nto\n apture\nthe\n omplexit\ny\nof\nNP-hard\nor\n oNP-hard\nproblems\nthat\nseemingly\nare\nneither\nin\nNP\nnor\nin\n oNP\n.\nIn\nparti ular,\nthey\nsho\nw\ned\nthat\nDP\n on\ntains\na\nn\num\nb\ner\nof\nuniqueness\npr\noblems,\n riti \nal\ngr\naph\npr\noblems,\nand\nexa t\noptimization\npr\noblems,\nand\nthey\nsho\nw\ned\nsome\nof\nthese\nproblems\n omplete\nfor\nDP\n;\nsee\nalso\nthe\nre en\nt\nsurv\ney\n[RR06\n℄.\nDP\nw\nas\nlater\ngeneralized\nb\ny\nCai\net\nal.\n[CGH+\n88,\nCGH+\n89℄,\nwho\nin\ntro\ndu ed\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n.\nNote\nthat\nDP\nis\nthe\nse ond\nlev\nel\nof\nthis\nhierar \nh\ny\n.\nIn\nhis\nseminal\npap\ner,\nV\nalian\nt\n[V\nal79\n℄\ninitiated\nthe\nstudy\nof\n oun\nting\nproblems\nand\nin-\ntro\ndu ed\nthe\nimp\nortan\nt\n oun\nting\n lass #P\n.\nMem\nb\ners\nof #P\nare\nreferred\nto\nas\nNP\n \nounting\npr\noblems.\nA\nw\nell-kno\nwn\nNP\n oun\nting\nproblem\nis # SA\nT,\nthe\n oun\nting\nv\nersion\nof\nthe\nsatis -\nabilit\ny\nproblem:\nGiv\nen\na\nb\no\nolean\nform\nula,\nho\nw\nman\ny\nsatisfying\nassignmen\nts\ndo\nes\nit\nha\nv\ne?\nDe nition\n2.1\n(V\nalian\nt\n[V\nal79\n℄)\nL\net\nNPTM\nb\ne\na\nshorthand\nfor\nnondeterministi \np\nolynomial-time\nT\nuring\nma hine.\nF\nor\nany\nNPTM M\nand\nany\ninput x,\nlet accM(x)\nde-\nnote\nthe\nnumb\ner\nof\na \n \nepting\n \nomputation\np\naths\nof M(x),\ni.e., accM\nis\na\nfun tion\nmapping\nfr\nom Σ∗\nto N .\nDe ne\nthe\nfun tion\n lass #P = {accM | M\nis\nan\nNPTM}.\nW\ne\nno\nw\nde ne\nthe\nnotion\nof\n(p\nolynomial-time)\np\narsimonious\nr\ne\ndu ibility,\nwhi \nh\nwill\nb\ne\nused\nto\n ompare\nthe\nhardness\nof\nsolving\nNP\n oun\nting\nproblems.\nIn\ntuitiv\nely\n,\nan\nNP\n oun\nting\nproblem f\nparsimoniously\nredu es\nto\nan\nNP\n oun\nting\nproblem g\nif\nthe\ninstan es\nof f\n an\nb\ne\ntransformed\nin\nto\ninstan es\nof g\nsu \nh\nthat\nthe\nn\num\nb\ner\nof\nsolutions\nof f\nare\npreserv\ned\nunder\nthis\ntransformation.\nDe nition\n2.2\nL\net f\nand g\nb\ne\nany\ntwo\ngiven\n \nounting\npr\noblems\nmapping\nfr\nom Σ∗\nto N .\nW\ne\nsay f\n(p\nolynomial-time)\nparsimoniously\nredu es\nto g\n(denote\nd\nby f ≤p\npar g\n)\nif\nther\ne\nexists\na\n3\n\n(a)\nSin\nt\n(b)\nBrid\n( )\nChin\n(d)\nRond\n(e)\nred\n(f\n)\ny\nello\nw\n(g)\nblue\n(h)\ngreen\nFigure\n1:\nT\nan\ntrix\nTM\ntiles\nand\n olors\np\nolynomial-time\n \nomputable\nfun tion ρ\nsu h\nthat\nfor\ne\na h x ∈Σ∗\n, f(x) = g(ρ(x)).\nIf F\nand\nG\nar\ne\nthe\nNP\nde\n ision\npr\noblems\n \norr\nesp\nonding\nto\nthe\nNP\n \nounting\npr\noblems f\nand g\nwith\nf ≤p\npar g\n,\nwe\nwil\nl\nalso\nsay\nthat F\np\narsimoniously\nr\ne\ndu \nes\nto G.\n2.2\nV\narian\nts\nof\nthe\nT\nan\ntrix\nTM\nRotation\nPuzzle\nProblem\nThe\nT\nan\ntrix\nTM\nrotation\npuzzle\nhas\nfour\nkinds\nof\nhexagonal\ntiles the\nSint,\nthe\nBrid,\nthe\nChin,\nand\nthe\nR\nond\n ea \nh\nof\nwhi \nh\nhas\nthree\n olored\nlines,\nwhere\nthe\n olors\nare\n \nhosen\namong\nr\ne\nd,\nyel\nlow,\nblue,\nand\ngr\ne\nen,\nsee\nFigure\n1(a) (d).\nThis\ngiv\nes\na\ntotal\nof\n56\ndi eren\nt\ntiles.\nSin e\nw\ne\naren't\nusing\na tually\n olored\n gures,\nw\ne\nen o\nde\nthe\n olors\nas\nsho\nwn\nin\nFigure\n1(e) (h).\nHolzer\nand\nHolzer\n[HH04℄\nsho\nw\ned\nthat\nthe\nde ision\nproblem\nT\nan\ntrix\nTM\nrotation\npuzzle\n(whi \nh\nw\ne\ndenote\nb\ny\nTRP,\nfor\nshort)\nis\nNP- omplete.\nIn\nthis\npap\ner,\nw\ne\nin\ntro\ndu e\nand\nstudy # TRP,\nthe\n oun\nting\nv\nersion\nof\nTRP.\nW\ne\nno\nw\nbrie y\ndes rib\ne\nthe\nformalism\nin\ntro\ndu ed\nb\ny\nHolzer\nand\nHolzer\n[HH04℄\nto\nde-\n ne\nTRP\n,\nsin e\nthe\nsame\nformalism\nis\nuseful\nfor\nde ning # TRP.\nIn\nparti ular,\nto\nrepresen\nt\nthe\ninstan es\nof\nb\noth\nthese\nproblems,\na\nt\nw\no-dimensional\nhexagonal\n o\nordinate\nsystem\nis\nused,\nsee\nFigure\n2.\nIn\nthis\nsystem,\nt\nw\no\ndistin t\npairs a = (u, w)\nand b = (v, x)\nfrom Z2\nare\nadja en\nt\nif\nand\nonly\nif\none\nof\nthe\nfollo\nwing\nfour\n onditions\nis\nsatis ed:\n1. u = v\nand |w −x| = 1,\n2. |u −v| = 1\nand w = x,\n3. u −v = 1\nand w −x = 1,\nand\n4. u −v = −1\nand w −x = −1.\nLet T\nb\ne\nthe\nset\nof\nall\nT\nan\ntrix\nTM\ntiles.\nLet A\nb\ne\na\n(partial)\nfun tion\nmapping\nthe\nelemen\nts\nof Z2\nto T\n,\ni.e.,\nfor\nthose v ∈Z2\non\nwhi \nh A\nis\nde ned, A(v)\nis\nthe\nt\nyp\ne\nof\nthe\ntile\nlo\n ated\nat v\n.\nThe\nset shape(A) = {v ∈Z2 | A(v)\nis\nde ned}\ngiv\nes\nthe\np\nositions\nin Z2\nat\nwhi \nh\ntiles\nare\npla ed.\nF\nor\nall a, b ∈shape(A), A(a)\nis\nadja en\nt\nto A(b)\nif\nand\nonly\nif a\nis\nadja en\nt\nto b.\n4\n\nx\ny\n(1, 1)\n(0, 0)\n(−1, −1)\n(1, 0)\n(0, 1)\n(−1, 0)\n(0, −1)\nFigure\n2:\nA\nt\nw\no-dimensional\nhexagonal\n o\nordinate\nsystem\nTRP\nis\nthen\nde ned\nas\nfollo\nws\n(note\nthat\nthe\ninitial\norien\ntation\nis\nnot\nsp\ne i ed,\nas\nit\ndo\nesn't\nmatter\nfor\nthe\nquestion\nof\nwhether\nthe\nde ision\nproblem\nTRP\nis\nsolv\nable)\n[HH04℄:\n1\nName:\nT\nan\ntrix\nTM\nRotation\nPuzzle\n(TRP,\nfor\nshort).\nGiv\nen:\nA\n nite\nshap\ne\nfun tion A : Z2 →T\n,\nappropriately\nen o\nded\nas\na\nstring.\nQuestion:\nIs\nthe\nrotation\npuzzle\nde ned\nb\ny A\nsolv\nable,\ni.e.,\ndo\nes\nthere\nexist\na\nrotation\nof\nthe\ngiv\nen\ntiles\nat\ntheir\np\nositions\nsu \nh\nthat\nat\nea \nh\njoin\nt\nedge\nof\nt\nw\no\nadja en\nt\ntiles\nthe\n orresp\nonding\n olors\nmat \nh?\nF\nor\nan\ny\ngiv\nen\nTRP\ninstan e A ,\na\nsolution\nof A\nis\na\nsp\ne i ation\n(in\nsome\nappropriate\nen o\nding)\nof\nea \nh\ntile\nin shape(A)\nin\nsome\nparti ular\norien\ntation\nsu \nh\nthat\nfor\nea \nh\njoin\nt\nedge\nof\nt\nw\no\nadja en\nt\ntiles\nthe\n orresp\nonding\n olors\nmat \nh.\nFigure\n3\ngiv\nes\nan\nexample\nof\na\nrotation\npuzzle\ninstan e\nand\nits\nsolution.\nLet\nSol\nTRP(A)\ndenote\nthe\nset\nof\nsolutions\nof\na\ngiv\nen\nTRP\ninstan e A .\nSo A\nis\nin\nTRP\n(view\ned\nas\na\nlanguage)\nif\nand\nonly\nif\nthe\nset\nSol\nTRP(A)\nis\nnonempt\ny\n.\n(a)\nPuzzle\n(b)\nSolution\nFigure\n3:\nAn\nexample\nof\na\nTRP\ninstan e\nand\nits\nsolution\n1\nAs\nnoted\nb\ny\nHolzer\nand\nHolzer\n[HH04℄,\nthere\nis\na\ndi eren e\nb\net\nw\neen\ntheir\nde nition\nof\nTRP,\nwhi \nh\nallo\nws\nholes\nin\nTRP\ninstan es,\nand\nthe\noriginal\nT\nan\ntrix\nTM\ngame,\nwhi \nh\ndo\nes\nnot\nallo\nw\nholes.\nThe\nproblem\nof\nwhether\nthe\nanalog\nof\nTRP\nwithout\nholes\nstill\nis\nNP\n- omplete\nis\nop\nen.\n5\n\nW\ne\nno\nw\nde ne\nthe\n oun\nting\nv\nersion\nand\nthe\nunique\nv\nersion\nof\nTRP,\nwhi \nh\nwill\nb\ne\n on-\nsidered\nin\nSe tions\n3\nand\n4.\nDe nition\n2.3\n1.\nThe\nT\nan\ntrix\nTM\nrotation\npuzzle\n oun\nting\nproblem\nis\nthe\nfun tion\n# TRP : Σ∗→N\nde ne\nd\nby\n# TRP(A) = ∥Sol\nTRP(A)∥,\nwher\ne\nwe\nassume\nthat\ninputs A\nar\ne\nappr\nopriately\nen \no\nde\nd\nas\nstrings\nin Σ∗\nand\nfun tion\nvalues\nar\ne\nnonne\ngative\ninte\ngers\n(r\nepr\nesente\nd\nin\nbinary).\n2.\nThe\nunique\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nde ne\nd\nby\nUnique-\nTRP = {A | # TRP(A) = 1}.\n3\nSatis abilit\ny\nP\narsimoniously\nRedu es\nto\nthe\nT\nan\ntrix\nTM\nRo-\ntation\nPuzzle\nProblem\nOur\nmain\nresult\nis\nTheorem\n3.1\nthe\npro\nof\nof\nwhi \nh\nwill\nb\ne\npresen\nted\nin\nSe tions\n3.1\nthrough\n3.4\n.\nTheorem\n3.1 # SA\nT ≤p\npar # TRP\n.\nT\no\npro\nv\ne\nTRP\nNP- omplete,\nHolzer\nand\nHolzer\n[HH04℄\nga\nv\ne\na\nredu tion\nfrom\nthe\nNP-\n omplete\nproblem\nCir uit∧,¬\n-SA\nT\n(see\nCo\nok\n[Co\no71℄),\nwhi \nh\nis\nde ned\nas\nfollo\nws.\nName:\nCir uit∧,¬\n-SA\nT\n.\nGiv\nen:\nA\nb\no\nolean\n ir uit C\nwith\nAND\nand\nNOT\ngates.\nQuestion:\nDo\nes\nthere\nexist\na\ntruth\nassignmen\nt\nto\nthe\ninput\ngates\nof C\nsu \nh\nthat C\nunder\nthis\nassignmen\nt\nev\naluates\nto\ntrue\n?\nHolzer\nand\nHolzer's\n onstru tion\nsim\nulates\nthe\n omputation\nof\nsu \nh\na\nb\no\nolean\n ir uit C\nb\ny\na\nT\nan\ntrix\nTM\nrotation\npuzzle\nsu \nh\nthat C\nev\naluates\nto\ntrue\nfor\nsome\nassignmen\nt\nto\nits\nv\nariables\nif\nand\nonly\nif\nthe\npuzzle\nhas\na\nsolution.\nOur\nde nition\nof\nb\no\nolean\n ir uits\nfollo\nws\nHolzer\nand\nHolzer\n[HH04℄,\nwho\nview\na\nb\no\nolean\n ir uit C\nwith\ninput\nv\nariables x1, x2, . . . , xn\nas\na\nsequen e (α1, α2, . . . , αm)\nof\nsteps\nsu \nh\nthat\nthe ith\ninstru tion αi\nhas\none\nof\nthe\nfollo\nwing\nforms:\n1.\nfor\nea \nh i\nwith 1 ≤i ≤n , αi = xi\n,\n2.\nfor\nea \nh i\nwith n+1 ≤i ≤m ,\neither αi =\nAND(j, k)\nor αi =\nNOT(j),\nwhere j ≤k < i.\n6\n\nDep\nending\non\nthe\ntruth\nv\nalues\nof\nthe\ninput\nv\nariables\nthe\noutput\ngate\nev\naluates\nto\ntrue\nor\nfalse\nin\nthe\nstandard\nw\na\ny\n.\nIn\ngeneral,\n ir uits\n an\n on\ntain\nan\ny\nn\num\nb\ner\nof\nwire\n rossings,\nwhi \nh\n annot\neasily\nb\ne\nrealized\nb\ny\nT\nan\ntrix\nTM\nsubpuzzles.\nT\no\nbuild\na\n ir uit\n on\ntaining\nonly\n rossings\nof\nt\nw\no\nneigh-\nb\noring\nwires,\nHolzer\nand\nHolzer\nfollo\nw\nGolds \nhlager's\npro\n edure\n[Gol77\n℄:\nIf αi =\nAND(j, k),\nmo\nv\ne\nwire j\nimmediately\nto\nthe\nleft\nof k\n,\nput\nan\nAND\ngate\nin\nfor αi\n,\nmo\nv\ne\nwire j\nba \nk\nto\nits\nstarting\np\noin\nt,\nand\n nally\nmo\nv\ne\nwire i\nto\nthe\nfar\nrigh\nt.\nThe\n ases\nof\ninstru tion αi\nb\neing\neither\nNOT(j)\nor xi\nare\ntreated\nin\na\nsimilar\nw\na\ny\n.\nFigure\n4\nsho\nws\na\npart\nof\na\n ir uit\nwith\nwire\n rossings,\nwhi \nh\n omputes α4 =\nAND(1, 3).\nOb\nviously\n,\nthere\nare\nonly\n rossings\nof\nt\nw\no\ndire tly\nadja en\nt\nwires.\nNote\nthat\nthis\ntransformation\nfrom\ngeneral\nto\nalmost\nplanar\n ir uits\n an\nb\ne\ndone\nin\ndeterministi \nlogarithmi \nspa e.\n2\n1\n3\n1\n1\n1\n2\n2\nAND\n3\n3\n4\n2\n1\n2\n3\n4\n3\n1\n2\n3\nFigure\n4:\nExample\nof\na\n ir uit\nfollo\nwing\nGolds \nhlager's\ntransformation\n[Gol77\n℄\nNo\nw,\nto\nbuild\na\nT\nan\ntrix\nTM\nRotation\nPuzzle\nthat\nsim\nulates\nsu \nh\na\n ir uit,\nHolzer\nand\nHolzer\nuse\na\ntruly\nplanar\n ross-o\nv\ner \ngadget\nthat\nw\nas\nprop\nosed\nb\ny\nM Coll\n[M C81\n℄.\nM -\nColl's\n ir uit\ngadget\nuses\nb\no\nolean\nAND\nand\nNOT\ngates\nto\nsim\nulate\nthe\n rossings\nof\nan\ny\nt\nw\no\nadja en\nt\nwires.\nEa \nh\nsu \nh\n ross-o\nv\ner \ngadget\nneeds\na\ntotal\nof 14\ninstru tion\nsteps\nand\nin\nv\nolv\nes\nt\nw\nelv\ne\nAND\nand\nnine\nNOT\ngates.\nSin e\nman\ny\n rossings\n an\no\n ur\nin\nthe\noriginally\ngiv\nen\n ir uit,\nthis\nma\ny\nlead\nto\na\n onsiderable\n(alb\neit\nstill\np\nolynomial)\nblo\nw-up\nof\nthe\nT\nan\ntrix\nTM\npuzzle\n onstru ted\nin\nour\nredu tion.\nThat\nis\nwh\ny\nw\ne\nprop\nose,\nas\na\nmore\ne ien\nt\nalternativ\ne\nto\nthe\nredu tion\npresen\nted\nin\nour\nprevious\npap\ner\n[BR07℄,\nto\nsim\nulate\nthese\n ross-o\nv\ners\ndire tly\nvia\na\nnew\nT\nan\ntrix\nTM\nsubpuzzle,\nthe\nCR\nOSS\nsubpuzzle\npresen\nted\nin\nFigure\n11\n.\nUnlik\ne\nthe\nredu tions\ngiv\nen\nin\n[HH04,\nBR07℄,\nvia\nsu \nh\nCR\nOSS\nsubpuzzles\nour\nredu tion\ndo\nesn't\nneed\nthe\ntransformation\nfrom\ngeneral\nto\nplanar\n ir uits\nthat\nrequires\nman\ny\nadditional\ngates\nand\ninstru tion\nsteps\n aused\nb\ny\nwire\n rossings.\nF\nurthermore,\nour\n onstru tion\nwill\nmo\ndify\nHolzer\nand\nHolzer's\nredu tion\n[HH04℄\nin\nsu \nh\na\nw\na\ny\nthat\nthere\nis\na\none-to-one\n orresp\nonden e\nb\net\nw\neen\nthe\nsolutions\nof\nthe\ngiv\nen\nCir uit∧,¬\n-SA\nT\ninstan e\nand\nthe\nsolutions\nof\nthe\nresulting\nrotation\npuzzle\ninstan e;\nhen e\n7\n\nour\nredu tion\nis\nparsimonious.\nT\no\nsim\nulate\nthe\n ir uit\nb\ny\na\nrotation\npuzzle,\na\nn\num\nb\ner\nof\nsubpuzzles\nare\nused.\nThe\n olor\nblue\nin\nthese\nsubpuzzles\nwill\nrepresen\nt\nthe\ntruth\nv\nalue\ntrue,\nand\nthe\n olor\nred\nwill\nrepresen\nt\nfalse.\nThis\n olor\nen o\nding\nat\nthe\ninputs\nand\noutputs\nof\nthe\nsubpuzzles\nth\nus\nrepresen\nt\nthe\ntruth\nv\nalues\nof\nthe\n ir uit's\ngates\nand\nwires.\nIn\nthe\nfollo\nwing\nse tions,\nw\ne\npresen\nt\nour\nmo\ndi ed\nsubpuzzles\nand,\nto\nallo\nw\n omparison,\nw\ne\nalso\npresen\nt\nHolzer\nand\nHolzer's\nsubpuzzles\n[HH04℄.\nT\no\nindi ate\nthe\ndi eren es\nb\net\nw\neen\ntheir\noriginal\nand\nour\nmo\ndi ed\nsubpuzzles,\ntiles\nwith\nmore\nthan\none\np\nossible\nsolution\nin\nthe\noriginal\nsubpuzzles\nwill\nha\nv\ne\na\ngrey\ninstead\nof\na\nbla \nk\nb\norder,\nand\nw\ne\nhighligh\nt\nall\nmo\ndi ed\ntiles\nin\nour\nnew\nsubpuzzles\nb\ny\na\ngrey\ninstead\nof\na\nwhite\nba \nkground\n(unless\nstated\notherwise).\nAnother\ndi eren e\nb\net\nw\neen\nour\npro\nof\nhere\nand\nthe\npro\nofs\nof\nHolzer\nand\nHolzer\n[HH04℄\nand\na\npreliminary\nv\nersion\nof\nthis\npap\ner\n[BR07℄\nregards\nthe\nanalysis\nof\nthe\nsubpuzzles.\nIn\nparti ular,\nw\ne\nwill\nhere\nfo\n us\non\nthe\n olor\nsequen es\nof\nthe\nv\narious\nT\nan\ntrix\nTM\ntiles.\nThis\nwill\nallo\nw\nus\nto\ngiv\ne\nthe\nargumen\nts\nmore\ntersely\n.\nF\nor\nexample,\nthe\ntile\nin\nFigure\n1(d)\nhas\nthe\n lo\n \nkwise\n olor\nsequen e yellow-green\n-green-red-red -yellow\n,\nwhi \nh\nwill\nb\ne\nabbreviated\nas yggrry\n.\n3.1\nWire\nSubpuzzles\nWires\nof\nthe\n ir uit\nare\nsim\nulated\nb\ny\nthe\nsubpuzzles\nWIRE,\nMO\nVE,\nCOPY,\nand\nCR\nOSS.\nT\no\nsim\nulate\nsimple\nv\nerti al\nwires,\nthe\nWIRE\nsubpuzzle\nis\nused.\nThe\noriginal\nv\nersion\nof\nHolzer\nand\nHolzer\n[HH04℄\nis\npresen\nted\nin\nFigure\n5.\nIt\nis\neasy\nto\nsee\nthat\nb\noth\ntiles\nha\nv\ne\nt\nw\no\np\nossible\norien\ntations\nfor\nea \nh\ninput\n olor.\nW\ne\nmo\ndify\nthis\nsubpuzzle\nas\nsho\nwn\nin\nFigure\n6,\ninserting\na\nnew\nRond\nat\np\nosition x.\nWithout\nthis\ntile,\nthe\np\nossible\n olor\nsequen es\nfor\nthe\nedges\nof\ntile x\njoin\nt\nwith a\nand b\nare rr , ry , yr ,\nand yy\nif\nthe\ninput\n olor\nis\nblue,\nand\nare bb ,\nby, yb ,\nand yy\nif\nthe\ninput\n olor\nis\nr\ne\nd.\nHo\nw\nev\ner,\nsin e\nthe\nnew\ntile x\ndo\nes\nnot\n on\ntain\nthe\n olor\nyel\nlow,\nthe\nsolutions\nare\n xed\nwith,\nresp\ne tiv\nely\n, bb\nand rr\nat\nthe\njoin\nt\nedges\nof\ntile x\nwith a\nand b,\nand\nw\ne\nobtain\nunique\nsolutions\nfor\nb\noth\ninput\n olors.\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nIN\na\nb\nOUT\n( )\nS \nheme\nFigure\n5:\nOriginal\nsubpuzzle\nWIRE,\nsee\n[HH04℄\nT\no\nbuild\nlonger\nwires,\nsev\neral\nWIRE\nsubpuzzles\nare\n onne ted.\nNote\nthat\nthis\nWIRE\nsubpuzzle\nhas\nheigh\nt\nt\nw\no.\nThis\nfor es\nall\nother\nsubpuzzles\nto\nha\nv\ne\nev\nen\nheigh\nt,\nb\ne ause\nthey\nm\nust\nb\ne\n onne ted\nb\ny\nWIRE\nsubpuzzles.\n8\n\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nx\nIN\na\nb\nOUT\n( )\nS \nheme\nFigure\n6:\nMo\ndi ed\nsubpuzzle\nWIRE\nBy\nthe\nMO\nVE\nsubpuzzle\nthe\n ir uit's\nwires\n an\nb\ne\nmo\nv\ned\none\np\nosition\nto\nthe\nleft\nor\none\np\nosition\nto\nthe\nrigh\nt.\nW\ne\ndis uss\na\nmo\nv\ne\nto\nthe\nrigh\nt\nin\ndetail\nand\nmen\ntion\nthat\na\nmo\nv\ne\nto\nthe\nleft\n an\nb\ne\nhandled\nanalogously\n.\nThe\noriginal\nv\nersion\nof\nHolzer\nand\nHolzer\n[HH04℄\nis\nsho\nwn\nin\nFigure\n7,\nand\nour\nmo\ndi ed\nsubpuzzle\nis\nsho\nwn\nin\nFigure\n8.\nIn\nthe\noriginal\nMO\nVE\nsubpuzzle,\ntiles a\nand i\nare\nmark\ned\nb\ne ause\nthey\nha\nv\ne\nt\nw\no\norien\nta-\ntions\nfor\nthe\nsolutions\nsho\nwn\nin\nFigure\n7.\nNote,\nho\nw\nev\ner,\nthat\nin\naddition\nto\nthe\nam\nbiguit\ny\n aused\nb\ny\ntiles a\nand i,\nthere\ndo\nes\nexist\nstill\nanother\nsolution\nfor\nea \nh\ninput\n olor.\nIn\nparti -\nular,\nif\nthe\ninput\n olor\nis\nblue\nthen\none\n an\nsimply\nsw\nap\nthe\n olors\nr\ne\nd\nand\nyel\nlow\nto\nobtain\nanother\nsolution,\nand\nif\nthe\ninput\n olor\nis\nr\ne\nd\nthen\ntiles b\nand d\n an\nb\ne\nrotated\nb\ny 60\nand\nall\nother\ntiles\nb\ny 240\ndegrees\nin\n lo\n \nkwise\ndire tion.\nW\ne\n x\nthe\nsolution\nb\ny\ninserting\ntiles x\nand y\n.\nFirst,\n onsider\nthe\n ase\nthat\nthe\ninput\n olor\nis\nblue.\nIt\nis\n lear\nthat\nthe\nedge\nof\ntile e\njoin\nt\nwith\ntile x\nm\nust\nb\ne\nblue,\nand\nsin e\nthe\nedges\nof\ntiles x\njoin\nt\nwith\ntiles b\nand f\nm\nust\nha\nv\ne\nthe\nsame\n olor,\nthe\norien\ntation\nof\ntile x\nis\n xed.\nThis\n xes\nalso\nthe\norien\ntation\nof\nall\nother\ntiles\nex ept a\nand i.\nThe\norien\ntation\nof\ntile a\nis\n xed\nwith\nr\ne\nd\nat\nthe\nedge\njoin\nt\nwith x.\nTile y\n xes\nthe\norien\ntation\nof\ntile i,\nsin e\nit\ndo\nes\nnot\n on\ntain\nthe\n olor\nsequen e byy ,\nbut\nthe\n olor\nsequen e byr\nfor\nthe\nedges\njoin\nt\nwith\ntiles g\n, h,\nand i.\nThe\n ase\nof\nr\ne\nd\nb\neing\nthe\ninput\n olor\n an\nb\ne\nhandled\nsimilarly\n.\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nc\nIN\na\nb\nd\ne\nf\nh\ni\nOUT\ng\n( )\nS \nheme\nFigure\n7:\nOriginal\nsubpuzzle\nMO\nVE,\nsee\n[HH04℄\nThe\nCOPY\nsubpuzzle\nis\nused\nto\n split \na\nwire\nin\nto\nt\nw\no\n opies.\nIts\noriginal\nv\nersion\nfrom\n[HH04℄\nis\nsho\nwn\nin\nFigure\n9\n,\nand\nthe\nmo\ndi ed\nv\nersion\nis\nsho\nwn\nin\nFigure\n10\n.\nIn\nits\noriginal\nv\nersion,\ntiles a\nthrough i\nare\na\nmo\nv\ne\nto\nthe\nrigh\nt,\nmerged\nwith\na\nmo\nv\ne\nto\nthe\nleft\n onsisting\nof\ntiles a\nthrough d\nand\ntiles i\nthrough m .\nT\no\nobtain\na\nunique\nsolution,\nw\ne\ninsert\n9\n\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nc\nIN\na\nb\nd\nx\ne\nf\nh\ni\nOUT\ng\ny\n( )\nS \nheme\nFigure\n8:\nMo\ndi ed\nsubpuzzle\nMO\nVE\nthe\nsame\ntiles x\nand y\nas\nfor\nthe\nMO\nVE\nsubpuzzle.\nBy\nthe\nsame\nargumen\nt\nas\nab\no\nv\ne,\nthis\n xes\nthe\norien\ntation\nof\nall\nother\ntiles,\nex ept\ntile m .\nBut\ninserting\ntile z\n,\nwhi \nh\nis\nof\nthe\nsame\nt\nyp\ne\nas\ntile y\n,\nalso\n xes\nthe\norien\ntation\nof m .\nTh\nus,\nthe\nsolution\nfor\nthis\nsubpuzzle\nis\nalso\nunique\nfor\nb\noth\ninput\n olors.\nOUT\nIN\nOUT\n(a)\nIn:\ntrue\nOUT\nIN\nOUT\n(b)\nIn:\nfalse\nk\nj\nl\nm\nOUT\ni\nIN\na\nb\nc\nd\ne\ng\nh\nOUT\nf\n( )\nS \nheme\nFigure\n9:\nOriginal\nsubpuzzle\nCOPY,\nsee\n[HH04℄\nOUT\nIN\nOUT\n(a)\nIn:\ntrue\nOUT\nIN\nOUT\n(b)\nIn:\nfalse\nk\nz\nj\nl\nm\nOUT\ni\nIN\na\nb\nc\nx\nd\ne\ng\nh\nOUT\nf\ny\n( )\nS \nheme\nFigure\n10:\nMo\ndi ed\nsubpuzzle\nCOPY\nT\no\nrealize\nwire\n rossings,\nHolzer\nand\nHolzer\n[HH04℄\nused\na\n ir uit\n onstru tion\n onsisting\n10\n\nof\nAND\nand\nNOT\ngates\nthat\nis\nbased\non\nGolds \nhlager's\npro\n edure\n[Gol77\n℄\nand\nM Coll's\n ross-o\nv\ner \ngadget\n[M C81\n℄.\nThis\napproa \nh\nw\nas\nalso\ntak\nen\nin\na\npreliminary\nv\nersion\nof\nthis\npap\ner\n[BR07℄.\nAs\nmen\ntioned\nab\no\nv\ne,\nho\nw\nev\ner,\nw\ne\nhere\nsimplify\nthe\n onstru tion\nb\ny\nsim\nulating\nwire\n rossings\ndire tly\n.\nT\no\nthis\nend,\nw\ne\nin\ntro\ndu e\na\nnew\nT\nan\ntrix\nTM\nsubpuzzle\n alled\nCR\nOSS,\nwhi \nh\nis\npresen\nted\nin\nFigure\n11\n.\nThis\nsubpuzzle\nhas\nt\nw\no\ninputs\nand\nt\nw\no\noutputs,\nwhere\nthe\nleft\noutput\nwill\nb\ne\nthe\nsame\nas\nthe\nrigh\nt\ninput\nand\nvi e\nv\nersa.\nJust\nas\nall\nour\nmo\ndi ed\nsubpuzzles,\nour\nno\nv\nel\nCR\nOSS\nsubpuzzle\nhas\nunique\nsolutions\nfor\nea \nh\n om\nbination\nof\np\nossible\ninput\n olors.\nT\no\nanalyze\nthis\nsubpuzzle,\nw\ne\nsub\ndivide\nit\nin\nto\nthree\ndi eren\nt\nparts.\nThe\nlo\nw\ner\npart\n onsists\nof\ntiles a\nthrough k\n,\nand\nthe\nupp\ner\npart\nof\ntiles l\nthrough t .\nLet\nus\n onsider\nthe\nupp\ner\npart\nof\nthe\nCR\nOSS\nsubpuzzle\n rst.\nTiles l\nthrough o\nform\nthe\nleft\noutput.\nSin e\ntile j\ndo\nes\nnot\n on\ntain\ngr\ne\nen,\nthe\ninput\n olor\nto\nthis\npart\nwill\nb\ne\neither\nblue\nor\nyel\nlow.\nIf\nthe\ninput\n olor\nis\nblue,\nall\ntiles\nm\nust\nha\nv\ne\nv\nerti al\nblue\nlines,\nand\nsin e\ntile\no\ndo\nes\nnot\n on\ntain\nyel\nlow\nlines,\nthe\norien\ntation\nof\nall\ntiles\nis\n xed\nwith\ngr\ne\nen\nat\nthe\njoin\nt\nedges\nof,\nresp\ne tiv\nely\n,\ntiles o\nand n ,\ntiles n\nand m ,\nand\ntiles m\nand l\n.\nThe\n ase\nof\nyel\nlow\nb\neing\nthe\ninput\n olor\nfor\nthis\npart\n an\nb\ne\nhandled\nsimilarly\nand\nyields\nthe\noutput\n olor\nr\ne\nd.\nThe\nrigh\nt\noutput\n onsists\nof\ntiles p\nthrough t .\nHere,\nthe\ninput\n olors\nr\ne\nd,\nblue,\nand\nyel\nlow\nare\np\nossible,\nwhere\nyel\nlow\nand\nr\ne\nd\nb\noth\nlead\nto\noutput\n olor\nr\ne\nd.\nSin e\ntile s\n on\ntains\nno\nr\ne\nd\nlines,\nw\ne\nobtain\nunique\nsolutions\nfor\nall\np\nossible\n om\nbinations\nof\ninput\n olors,\nb\ny\na\nsimilar\nargumen\nt\nas\nfor\nthe\nleft\noutput.\nW\ne\nno\nw\nturn\nto\nthe\nmore\n ompli ated\nlo\nw\ner\npart\nof\nthis\nsubpuzzle.\nTiles a, b,\nand c\nmo\nv\ne\nthe\nleft\ninput\n olor\nto\nthe\njoin\nt\nedge\nof\ntiles b\nand g\n.\nRegarding\ntiles a, b,\nand c,\nthere\nare\nt\nw\no\np\nossible\nsolutions\nfor\nea \nh\ninput\n olor.\nThe\np\nossible\n olor\nsequen es\nfor\nthe\nedges\nof\ntiles c\nand b\njoin\nt\nwith\ntile g\nare by\nand br\nfor\ninput\n olor\nblue,\nand\nare ry\nand rb\nfor\ninput\n olor\nr\ne\nd.\nSin e g\n on\ntains\nexa tly\none\nof\nthese\n olor\nsequen es\nfor\nea \nh\ninput\n olor,\nthe\norien\ntation\nof\ntiles a, b,\nand c\nis\n xed,\nand\nthe\ninput\n olor\nis\npassed\non\nto\nthe\njoin\nt\nedge\nof\ntiles b\nand g\n.\nThe\nsame\nargumen\nt\napplies\nto\nthe\nrigh\nt\ninput,\nwhi \nh\n onsists\nof\ntiles d,\ne, f\n,\nand h,\nsin e\nthey\nare\nmirror-symmetri al.\nIf\nb\noth\ninputs\nare\nblue,\nthe\norien\ntations\nof\ntiles g\nand h\nis\n xed\nb\ny\ntiles b, c, e,\nand f\n,\nresp\ne tiv\nely\n.\nBoth g\nand h\nwill\nha\nv\ne\nyel\nlow\nat\ntheir\nedges\njoin\nt\nwith\ntile i,\nwhi \nh\nleads\nto\nr\ne\nd\nat\nthe\nedges\nof\ntile i\njoin\nt\nwith\ntiles j\nand k\n,\nresp\ne tiv\nely\n,\nand\nthe\norien\ntation\nof\nall\ntiles\nin\nthe\nlo\nw\ner\npart\nis\n xed.\nThe\n onne tions\nb\net\nw\neen\nthe\nlo\nw\ner\nand\nthe\nupp\ner\npart\nare\nthe\njoin\nt\nedges\nof\ntiles j\nand l\nand\nof\ntiles k\nand p ,\nresp\ne tiv\nely\n.\nThese\nedges\nare\nb\noth\nblue\nif\nb\noth\ninput\n olors\nare\nblue.\nNo\nw\n onsider\nthe\n ase\nthat\nb\noth\ninput\n olors\nare\nr\ne\nd.\nThe\np\nossible\n olors\nfor\nthe\njoin\nt\nedges\nof\ntiles g\nand h\nwith\ntile i\nare\nr\ne\nd\nand\nblue.\nClearly\n,\nit\nis\nnot\np\nossible\nthat\nthey\nb\noth\nare\nblue.\nF\nurthermore,\nit\nis\nnot\np\nossible\nthat\nthe\njoin\nt\nedge\nof\ntiles g\nand i\nis\nblue,\nsin e\nin\nthis\n ase\nthe\njoin\nt\nedge\nof\ntiles j\nand l\nw\nould\nha\nv\ne\nto\nb\ne\nr\ne\nd,\nwhi \nh\nis\nnot\np\nossible\nb\ne ause l\ndo\nes\nnot\n on\ntain\nr\ne\nd.\nF\nor\nthe\nsame\nreason\nit\nis\nnot\np\nossible\nthat\nthe\njoin\nt\nedges\nof\ntiles g\nand h\nwith i\nare\nb\noth\nr\ne\nd,\nso\nthe\nonly\np\nossible\nsolution\nis\nthat\nthe\njoin\nt\nedge\nof\ntiles g\nand i\nis\nr\ne\nd\nand\nthat\nthe\njoin\nt\nedge\nof\ntiles h\nand i\nis\nblue.\nThis\nalso\nuniquely\ndetermines\nthe\norien\ntation\nof\ntiles j\nand k\nwith\nyel\nlow\nat\nthe\njoin\nt\nedge\nof\ntiles j\nand l\nand\nr\ne\nd\nat\nthe\njoin\nt\nedge\nof\ntiles k\nand p .\nThe\n ases\nthat\none\ninput\nis\nr\ne\nd\nand\nthe\nother\none\nis\n11\n\nblue\n an\nb\ne\nhandled\nb\ny\nsimilar\nargumen\nts\nto\nsho\nw\nthat\nw\ne\nthen\nha\nv\ne\nunique\nsolutions\nas\nw\nell.\nIN\nOUT\nIN\nOUT\n(a)\nIn:\ntrue,\ntrue\nIN\nOUT\nIN\nOUT\n(b)\nIn:\ntrue,\nfalse\nIN\nOUT\nIN\nOUT\n( )\nIn:\nfalse,\ntrue\nIN\nOUT\nIN\nOUT\n(d)\nIn:\nfalse,\nfalse\nIN\na\nc\nm\no\nOUT\nb\ng\nj\nl\nn\ni\ne\nh\nk\np\nr\nIN\nd\nf\nq\nt\nOUT\ns\n(e)\nS \nheme\nFigure\n11:\nSubpuzzle\nCR\nOSS\n3.2\nGate\nSubpuzzles\nThe\ngates\nof\nthe\nb\no\nolean\n ir uit\nare\nsim\nulated\nb\ny\nthe\n orresp\nonding\nAND\nand\nNOT\nsub-\npuzzles.\nThe\noriginal\nv\nersion\nof\nthe\nNOT\nsubpuzzle\nfrom\n[HH04℄\nis\npresen\nted\nin\nFigure\n12\n,\nand\nour\nnew\nv\nersion\nis\nsho\nwn\nin\nFigure\n13\n.\nThe\npurp\nose\nof\nthis\nsubpuzzle\nis\nto\n negate \nthe\ninput\n olor,\ni.e.,\nif\nthe\ninput\n olor\nis\nr\ne\nd\nthen\nthe\noutput\n olor\nwill\nb\ne\nblue,\nand\nvi e\nv\nersa.\nIn\nthe\noriginal\nsubpuzzle,\nthere\nis\nonly\none\np\nossible\nsolution\nto\nthe\nthree\nRonds e, f\n,\nand g\n:\nSin e\ntiles c, b,\nand f\ndo\nnot\n on\ntain\ngr\ne\nen,\nthe\njoin\nt\nedge\nof\ntiles e\nand g\nm\nust\nb\ne\ngr\ne\nen,\nwhi \nh\nfor es\nthe\njoin\nt\nedge\nof\ntiles g\nand f\nto\nb\ne\nyel\nlow.\nSo\nthe\norien\ntation\nof\ntiles\ne, f\n,\nand g\nis\n xed,\nwith\nr\ne\nd\nat\nthe\nedges\njoin\nt\nwith\ntiles b\nand c,\nresp\ne tiv\nely\n.\nThere\nare\nonly\nt\nw\no\np\nossible\norien\ntations\nleft\nfor\nthe\ntiles b\nand c,\none\nfor\ninput\n olor\nblue\nand\none\nfor\ninput\n olor\nr\ne\nd.\nThe\nonly\ntiles\nstill\nha\nving\nmore\nthan\none\np\nossible\norien\ntation\nare a\nand d.\n12\n\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nf\ng\ne\nIN\na\nb\nc\nd\nOUT\n( )\nS \nheme\nFigure\n12:\nOriginal\nsubpuzzle\nNOT,\nsee\n[HH04℄\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nf\ng\ne\nIN\na\nb\nc\nd\nOUT\nx\ny\n( )\nS \nheme\nFigure\n13:\nMo\ndi ed\nsubpuzzle\nNOT\nThey\nwill\nb\ne\n xed\nb\ny\ninserting\ntiles x\nand y\n.\nThe\np\nossible\n olor\nsequen es\nfor\nthe\nedges\nof\ntiles a\nand b\njoin\nt\nwith\ntile x\nare yy\nor ry\nif\nthe\ninput\n olor\nis\nblue,\nand\nthey\nare bb\nor\nyb\nif\nthe\ninput\n olor\nis\nr\ne\nd.\nTile x,\nho\nw\nev\ner,\n on\ntains\nonly\nthe\nsequen es yy\nand bb,\nso\nthe\norien\ntation\nis\n xed.\nNote\nthat\ntiles c\nand d\nb\neha\nv\ne\njust\nlik\ne\na\nWIRE\nsubpuzzle\nand\nsin e\nthe\norien\ntation\nof\ntile c\nis\n xed\nwith\nr\ne\nd\nat\nthe\nedge\njoin\nt\nwith\ntile e,\nw\ne\ninsert\na\nRond\nat\np\nosition y\n on\ntaining\nthe\n olor\nsequen e yy\nbut\nnone\nof\nthe\nsequen es yb\nand yr .\nWith\nthese\nt\nw\no\nnew\ntiles,\nunique\nsolutions\nare\nenfor ed\nfor\nea \nh\ninput\n olor.\nThe\nsomewhat\nmore\n ompli ated\nAND\nsubpuzzle\nis\nsho\nwn\nin\nFigure\n14\nin\nits\noriginal\nv\nersion\nfrom\n[HH04℄,\nwhile\nFigure\n15\npresen\nts\nour\nmo\ndi ed\nv\nersion.\nThis\nsubpuzzle\n an\nagain\nb\ne\nsub\ndivided\nin\nto\nt\nw\no\ndi eren\nt\nparts,\nan\nupp\ner\npart\nand\na\nlo\nw\ner\npart\nthat\nare\n onne ted\nat\nthe\njoin\nt\nedge\nof\ntiles c\nand j\n.\nFirst,\nw\ne\n onsider\nthe\nupp\ner\npart\nof\nthe\nmo\ndi ed\nAND\nsubpuzzle.\nW\ne\nsho\nw\nthat\nits\noutput\n olor\nis\nr\ne\nd\nif\nthe\njoin\nt\nedge\nof\ntiles j\nand c\nis\nyel\nlow,\nand\nthat\nits\noutput\n olor\nis\nblue\nif\nthis\nedge\nis\nblue.\nNote\nthat,\njust\nas\nfor\nthe\nNOT\nsubpuzzle,\nthe\nRonds o, p ,\nand q\nha\nv\ne\nonly\none\np\nossible\norien\ntation,\nth\nus\nfor ing\nthe\nedges\nof\ntiles n\nand m\njoin\nt\nwith\ntile o\nto\nb\ne\nyel\nlow.\nIf\nthe\ninput\nto\nthis\npart\nof\nthe\nsubpuzzle\nis\nblue\nthen\ntiles j\nand k\nm\nust\nha\nv\ne\na\nv\nerti al\nblue\nline,\nand\nsin e\nthe\nedges\nof\ntile l\njoin\nt\nwith j\nand k\n annot\nb\ne\nblue,\ntiles l\n, m ,\nand n\nm\nust\nha\nv\ne\nv\nerti al\nblue\nlines,\nto\no.\nSin e\nthe\nedges\nof\ntiles n\nand m\njoin\nt\nwith\ntile o\n13\n\nIN\nOUT\nIN\n(a)\nIn:\ntrue,\ntrue\nIN\nOUT\nIN\n(b)\nIn:\ntrue,\nfalse\nIN\nOUT\nIN\n( )\nIn:\nfalse,\ntrue\nIN\nOUT\nIN\n(d)\nIn:\nfalse,\nfalse\nh\nIN\na\nb\nc\nj\nk\ng\nl\nm\nn\nOUT\nf\no\nIN\nd\ne\np\nq\ni\n(e)\nS \nheme\nFigure\n14:\nOriginal\nsubpuzzle\nAND,\nsee\n[HH04℄\nare\nyel\nlow,\nthe\norien\ntation\nof\nall\nother\ntiles\nis\nuniquely\ndetermined\nin\nthe\nupp\ner\npart.\nThe\n ase\nof\nyel\nlow\nb\neing\nthe\ninput\n olor\nfor\nthe\nupp\ner\npart\n an\nb\ne\nhandeld\nsimilarly:\nWithout\nan\ny\nmo\ndi ation,\nw\ne\nha\nv\ne\nunique\nsolutions\nfor\nthe\nupp\ner\npart\nof\nthe\nsubpuzzle.\nNo\nw\nw\ne\n onsider\nthe\nlo\nw\ner\npart\nof\nthis\nsubpuzzle.\nF\nor\nea \nh\n om\nbination\nof\ninput\n olors,\ntiles a\nand d\n(whi \nh\nare\nadja en\nt\nto\nthe\ninput\ntiles\nof\nthe\nAND\nsubpuzzle)\nand\ntiles h\nand i\n(ha\nving\nonly\none\n onne ting\nedge\nto\nthe\nrest\nof\nthe\noriginal\nAND\nsubpuzzle\nin\nFigure\n14)\nea \nh\nha\nv\ne\nt\nw\no\np\nossible\norien\ntations.\nExamining\nall\np\nossible\n olor\nsequen es\nfor\nea \nh\n om\nbination\nof\np\nossible\ninput\n olors,\ntiles x\nand y\n an\nb\ne\ndetermined\nas\nsho\nwn\nin\nFigure\n15\nto\n x\nthe\norien\ntation\nof\ntiles a\nand h\nand\nof\ntiles d\nand i,\nresp\ne tiv\nely\n.\nThen,\nthe\ninput\n olor\nis\npassed\non\nto\nthe\njoin\nt\nedge\nof\ntiles b\nand c,\nand\nto\nthe\njoin\nt\nedge\nof\ntiles e\nand f\n.\nTile g\nin\nthe\n en\nter\nof\nthe\nsubpuzzle\nhas\nagain\nt\nw\no\np\nossible\norien\ntations\nfor\nea \nh\n om\nbination\nof\ninput\n olors.\nNote\nthat\nour\nmo\ndi ations\nmade\nso\nfar namely\n,\ninserting\nnew\ntiles\nin\nto\nthe\nsubpuzzles do\nnot\nw\nork\nhere,\nas\nit\nis\nnot\np\nossible\nto\ninsert\na\nnew\ntile\nin\nthe\nneigh\nb\norho\no\nd\nof\ntile g\n,\nand\nth\nus\nw\ne\nha\nv\ne\nto\nrepla e\nit.\nAs\nw\ne\nha\nv\ne\nseen\nin\nthe\nanalysis\nof\nthe\nupp\ner\npart,\nthe\n olor\nyel\nlow\nof\ntile j\nis\nonly\nused\nfor\nthe\nedge\njoin\nt\nwith\ntile c.\nW\ne\nwill\nrepla e\nthe\n olor\nyel\nlow\nof\nb\noth\ntiles\nwith\nthe\n olor\ngr\ne\nen.\nThis\nis\np\nossible,\n14\n\nIN\nOUT\nIN\n(a)\nIn:\ntrue,\ntrue\nIN\nOUT\nIN\n(b)\nIn:\ntrue,\nfalse\nIN\nOUT\nIN\n( )\nIn:\nfalse,\ntrue\nIN\nOUT\nIN\n(d)\nIn:\nfalse,\nfalse\nx\nh\nIN\na\nb\nc\nj\nk\ng\nl\nm\nn\nOUT\nf\no\nIN\nd\ne\np\nq\ny\ni\n(e)\nS \nheme\nFigure\n15:\nMo\ndi ed\nsubpuzzle\nAND\nb\ne ause\nthe\njoin\nt\nedge\nof\ntiles c\nand b\nwill\nnev\ner\nb\ne\nyel\nlow,\nand\ntile g\nwill\nb\ne\nrepla ed\nb\ny\na\nnew\none.\nT\no\ndo\nthis,\nw\ne\n onsider\nall\np\nossible\n olors\nat\nthe\nedges\nof\ntiles c\nand f\njoin\nt\nwith\ntile g\n,\nwith\nthe\nrestri tion\nthat\nthe\njoin\nt\nedge\nof\ntiles c\nand j\nm\nust\nb\ne\neither\nblue\nor\ngr\ne\nen.\nIf\nthe\nrigh\nt\ninput\n olor\nis\nblue\nthen\nthe\njoin\nt\nedge\nof\ntiles f\nand g\n an\nb\ne\neither\nblue\nor\nyel\nlow,\nand\nif\nthe\nrigh\nt\ninput\n olor\nis\nr\ne\nd\nthen\nthis\nedge\n an\nb\ne\neither\nr\ne\nd\nor\nyel\nlow.\nF\nor\nthe\nleft\ninput,\nthe\njoin\nt\nedge\nof\ntiles c\nand g\nm\nust\nb\ne\nr\ne\nd\nif\nthe\ninput\nis\nr\ne\nd,\nb\ne ause\nthen\nthe\njoin\nt\nedge\nof\ntiles c\nand j\n annot\nb\ne\nr\ne\nd.\nIf\nb\noth\ninputs\nare\nblue\nthen\nthe\noutput\n olor\nof\nthe\nlo\nw\ner\npart\nm\nust\nb\ne\nblue\nas\nw\nell,\nand\nthen\nthe\njoin\nt\nedge\nof\ntiles c\nand g\nis\ngr\ne\nen.\nIf\nonly\nthe\nleft\ninput\nis\nblue,\nthe\noutput\n olor\nof\nthe\nlo\nw\ner\npart\nm\nust\nb\ne\ngr\ne\nen,\nand\nthe\njoin\nt\nedge\nof\ntiles c\nand g\nis\nblue.\nThis\nleads\nto\nthe\nfollo\nwing\nrestri tions\nfor\ntile g\n:\n1.\nIf\nthe\ninput\n olors\nare\nb\noth\nblue\nthen g\nm\nust\n on\ntain\nexa tly\none\nof\nthe\n olor\nsequen es\ngxb\nor gxy,\nwhere x\nstands\nfor\nan\narbitrary\n olor.\n2.\nIf\nthe\nrigh\nt\ninput\n olor\nis\nr\ne\nd\nand\nthe\nleft\ninput\n olor\nis\nblue, g\nm\nust\n on\ntain\neither\nbxr\nor bxy.\n15\n\n3.\nIf\nthe\nrigh\nt\ninput\n olor\nis\nblue\nand\nand\nthe\nleft\ninput\n olor\nis\nr\ne\nd\nthen\nthe\np\nossible\n olor\nsequen es\nfor g\nare\neither rxb\nor rxy.\n4.\nF\nor\nthe\nlast\n om\nbination\nof\nb\noth\ninput\n olors\nb\neing\nr\ne\nd,\ntile g\nm\nust\n on\ntain\neither rxr\nor rxy.\nIn\nour\nmo\ndi ed\nAND\nsubpuzzle,\nw\ne\ninsert\na\nSin\nt\ninstead\nof\na\nRond,\nwhi \nh\n on\ntains\nea \nh\nof\nthe\n olor\nsequen es gxb , bxr, rxb,\nand rxr\nexa tly\non e.\nThis\nleads\nto\nunique\nsolutions\nfor\nthe\nlo\nw\ner\npart\nand\nth\nus\nfor\nthe\nwhole\nsubpuzzle,\nfor\nea \nh\np\nossible\n om\nbination\nof\ninput\n olors.\n3.3\nInput\nand\nOutput\nSubpuzzles\nThe\nv\nariables\nof\nthe\nb\no\nolean\n ir uit\nare\nrepresen\nted\nb\ny\nthe\nsubpuzzle\nBOOL.\nHolzer\nand\nHolzer\n[HH04℄\nsho\nw\ned\nthat\nthis\nsubpuzzle\nhas\nonly\nt\nw\no\nsolutions,\none\nwith\noutput\n olor\nblue\nand\nthe\nother\none\nwith\noutput\n olor\nr\ne\nd.\nThis\noutput\n olor\n orresp\nonds\nto\nthe\nb\no\nolean\nv\nalue\nof\nthe\n orresp\nonding\ninput\nv\nariable,\nwhere\nblue\nstands\nfor\nthe\ntruth\nv\nalue\ntrue,\nand\nr\ne\nd\nstands\nfor\nfalse.\nThe\nrestri tion\nthat\nyel\nlow\nand\ngr\ne\nen\nare\nnot\np\nossible\nas\noutput\n olors\nfor\nthis\nsubpuzzle,\nensures\nthat\nthe\nfollo\nwing\nsubpuzzles\nwill\nalw\na\nys\nha\nv\ne\na\nv\nalid\ninput\n olor,\nnamely\nblue\nor\nr\ne\nd.\nThe\nlast\nsubpuzzle\nneeded\nto\nsim\nulate\na\nb\no\nolean\n ir uit\nis\nthe\nsubpuzzle\nTEST.\nIt\nis\npla ed\nat\nthe\noutput\ngate\nof\nthe\n ir uit,\nand\nits\npurp\nose\nis\nto\nv\nerify\nthat\nthe\n ir uit\nev\nalutes\nto\ntrue.\nHolzer\nand\nHolzer\n[HH04℄\nmen\ntion\nthat\nthis\nsubpuzzle\nhas\nonly\none\nv\nalid\nsolution\nwith\nblue\nas\nthe\ninput\n olor.\nThis\nensures\nthat\nthe\noutput\nof\nthe\nwhole\n ir uit\nwill\nb\ne\ntrue\nif\nand\nonly\nif\nthe\nT\nan\ntrix\nTM\npuzzle\n onstru ted\nhas\na\nsolution.\nLik\ne\nthe\noriginal\nsubpuzzle\nBOOL,\nthe\noriginal\nTEST\nsubpuzzle\nalready\nhas\na\nunique\nsolution,\nso\nit\nis\nnot\nmo\ndi ed.\nThese\nt\nw\no\nsubpuzzles\nare\nthe\nonly\nones\nfrom\n[HH04℄\nthat\nare\nnot\nmo\ndi ed.\nF\nor\nthe\nsak\ne\nof\n ompleteness,\nthey\nare\npresen\nted\nin\nFigure\n16.\nOUT\n(a)\nBOOL\nOut:\ntrue\nOUT\n(b)\nBOOL\nOut:\nfalse\nIN\n( )\nTEST\nFigure\n16:\nSubpuzzles\nBOOL\nand\nTEST,\nsee\n[HH04℄\nThe\nshap\nes\nof\nour\nmo\ndi ed\nsubpuzzles\nha\nv\ne\n \nhanged\nsligh\ntly\n,\nso\nit\nmigh\nt\nb\ne\np\nossible\nthat\nunin\ntended\nin\ntera tions\nb\net\nw\neen\nt\nw\no\nneigh\nb\noring\nsubpuzzles\ndo\no\n ur.\nHo\nw\nev\ner,\nas\nnoted\nb\ny\nHolzer\nand\nHolzer\n[HH04℄,\nthe\nminimal\nhorizon\ntal\ndistan e\nb\net\nw\neen\nt\nw\no\nwires\nand/or\n16\n\ngates\nis\nat\nleast\nfour,\nand\nthis\nis\nstill\nenough\nto\nprev\nen\nt\nan\ny\nunin\ntended\nin\ntera tions\nb\net\nw\neen\nour\nmo\ndi ed\nsubpuzzles.\n3.4\nPro\nof\nof\nTheorem\n3.1\nW\ne\nare\nno\nw\nready\nto\npro\nv\ne\nTheorem\n3.1.\nLet\nSA\nT\ndenote\nthe\nsatis abilit\ny\nproblem.\nLemma\n3.2\nSA\nT\np\narsimoniously\nr\ne\ndu \nes\nto\nCir\n uit∧,¬\n-\nSA\nT\n.\nPro\nof.\nNote\nthat\nthe\nproblems\nSA\nT\nand\nCir uit\n-SA\nT\n(whi \nh\nis\nthe\nsame\nas\nCir uit∧,¬\n-SA\nT\nex ept\nwith\nOR\ngates\nallo\nw\ned\nas\nw\nell)\nare\nequiv\nalen\nt\nunder\nparsimonious\nredu tions\n[P\nap94\n℄.\nSin e\nOR\ngates\n an\nb\ne\nexpressed\nb\ny\nAND\nand\nNOT\ngates\nwithout\n \nhanging\nthe\nn\num\nb\ner\nof\nsolutions,\nthis\ngiv\nes\na\nparsimonious\nredu tion\nfrom\nSA\nT\nto\nCir uit∧,¬\n-SA\nT\n.\n❑\nNo\nw,\nthe\nparsimonious\nredu tion\nfrom\nSA\nT\nto\nTRP\nimmediately\nfollo\nws\nfrom\nLemma\n3.2\nand\nthe\n onstru tion\nand\nthe\nargumen\nts\npresen\nted\nin\nSe tions\n3.1,\n3.2,\nand\n3.3.\nThat\nis,\nb\ny\nour\nmo\ndi ations,\nfor\nea \nh\nsatisfying\nassignmen\nt\nto\nthe\n ir uit\nthere\nis\nexa tly\none\nsolution\nto\nthe\nT\nan\ntrix\nTM\npuzzle\n onstru ted.\n4\nThe\nUnique\nT\nan\ntrix\nTM\nRotation\nPuzzle\nProblem\nis\nDP-\nComplete\nunder\nRandomized\nRedu tions\nV\nalian\nt\nand\nV\nazirani\nin\ntro\ndu ed\nr\nandomize\nd\np\nolynomial-time\nr\ne\ndu tions\nin\ntheir\nw\nork\nsho\nw-\ning\nthat\nNP\nis\nas\neasy\nas\ndete ting\nunique\nsolutions\n[VV86℄.\nW\ne\nwill\nuse ≤p\nran\nto\nde-\nnote\ntheir\nt\nyp\ne\nof\nredu tions.\nIn\nparti ular,\nV\nalian\nt\nand\nV\nazirani\n[VV86℄\npro\nv\ned\nthat\nUnique-SA\nT\n,\nthe\nunique\nv\nersion\nof\nSA\nT\n,\nis ≤p\nran\n- omplete\nin\nDP\n(see\nalso\nChang,\nKadin,\nand\nRohatgi\n[CKR95℄).\nTheorem\n4.1\n1.\nUnique\n-SA\nT\np\narsimoniously\nr\ne\ndu \nes\nto\nUnique\n-TRP.\n2.\nUnique\n-TRP\nis\nDP- \nomplete\nunder ≤p\nran\n-r\ne\ndu tions.\nPro\nof.\nT\no\npro\nv\ne\nthe\n rst\npart,\nnote\nthat\nb\ny\nLemma\n3.2\nand\nTheorem\n3.1\n,\nw\ne\nobtain\na\nparsimonious\nredu tion\nfrom\nSA\nT\nto\nTRP.\nIt\nfollo\nws\nthat\nUnique-SA\nT\nparsimoniously\nredu es\nto\nUnique\n-TRP.\nThe\nse ond\npart\nfollo\nws\nfrom\nthe\n rst\npart\nand\nV\nalian\nt\nand\nV\nazirani's\nab\no\nv\ne-men\ntioned\nresult\nthat\nUnique-SA\nT\nis ≤p\nran\n- omplete\nin\nDP,\nand\nfrom\nthe\nob\nvious\nfa t\nthat\nUnique\n-TRP\nis\nin\nDP\n.\n❑\nA\n \nkno\nwledgmen\nts:\nW\ne\nthank\nthe\nanon\nymous\nMCU\n2007\nreferees\nfor\ntheir\nhelpful\n om-\nmen\nts.\n17\n\nReferen es\n[Bar04℄\nR.\nBarban \nhon.\nOn\nunique\ngraph\n3- olorabilit\ny\nand\nparsimonious\nredu tions\nin\nthe\nplane.\nThe\nor\neti \nal\nComputer\nS ien \ne,\n319(1 3):455 482,\n2004.\n[BR07℄\nD.\nBaumeister\nand\nJ.\nRothe.\nSatis abilit\ny\nparsimoniously\nredu es\nto\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nIn\nPr\no\n \ne\ne\ndings\nof\nthe\n5th\nConfer\nen \ne\non\nMa hines,\nComputations\nand\nUniversality,\npages\n134 145.\nSpringer-V\nerlag\nL\ne\n -\ntur\ne\nNotes\nin\nComputer\nS ien \ne\n#4664,\nSeptem\nb\ner\n2007.\n[CGH+\n88℄\nJ.\nCai,\nT.\nGundermann,\nJ.\nHartmanis,\nL.\nHema \nhandra,\nV.\nSew\nelson,\nK.\nW\nagner,\nand\nG.\nW\ne \nhsung.\nThe\nb\no\nolean\nhierar \nh\ny\nI:\nStru tural\nprop\nerties.\nSIAM\nJournal\non\nComputing,\n17(6):1232 1252,\n1988.\n[CGH+\n89℄\nJ.\nCai,\nT.\nGundermann,\nJ.\nHartmanis,\nL.\nHema \nhandra,\nV.\nSew\nelson,\nK.\nW\nagner,\nand\nG.\nW\ne \nhsung.\nThe\nb\no\nolean\nhierar \nh\ny\nI\nI:\nAppli ations.\nSIAM\nJournal\non\nComputing,\n18(1):95 111,\n1989.\n[CKR95℄\nR.\nChang,\nJ.\nKadin,\nand\nP\n.\nRohatgi.\nOn\nunique\nsatis abilit\ny\nand\nthe\nthreshold\nb\neha\nvior\nof\nrandomized\nredu tions.\nJournal\nof\nComputer\nand\nSystem\nS ien \nes,\n50(3):359 373,\n1995.\n[CM87℄\nJ.\nCai\nand\nG.\nMey\ner.\nGraph\nminimal\nun olorabilit\ny\nis DP\n- omplete.\nSIAM\nJournal\non\nComputing,\n16(2):259 277,\nApril\n1987.\n[Co\no71℄\nS.\nCo\nok.\nThe\n omplexit\ny\nof\ntheorem-pro\nving\npro\n edures.\nIn\nPr\no\n \ne\ne\ndings\nof\nthe\n3r\nd\nA\nCM\nSymp\nosium\non\nThe\nory\nof\nComputing,\npages\n151 158.\nA\nCM\nPress,\n1971.\n[Gol77℄\nL.\nGolds \nhlager.\nThe\nmonotone\nand\nplanar\n ir uit\nv\nalue\nproblems\nare\nlog\nspa e\n omplete\nfor\nP.\nSIGA\nCT\nNews,\n9(2):25 29,\n1977.\n[Grä90℄\nE.\nGrädel.\nDomino\ngames\nand\n omplexit\ny\n.\nSIAM\nJournal\non\nComputing,\n19(5):787 804,\n1990.\n[HH04℄\nM.\nHolzer\nand\nW.\nHolzer.\nT\nan\ntrix\nTM\nrotation\npuzzles\nare\nin\ntra table.\nDis r\nete\nApplie\nd\nMathemati s,\n144(3):345 358,\n2004.\n[M C81℄\nW.\nM Coll.\nPlanar\n rosso\nv\ners.\nIEEE\nT\nr\nansa tions\non\nComputers,\nC-30(3):223 \n225,\n1981.\n[P\nap94℄\nC.\nP\napadimitriou.\nComputational\nComplexity.\nA\nddison-W\nesley\n,\n1994.\n[PY84℄\nC.\nP\napadimitriou\nand\nM.\nY\nannak\nakis.\nThe\n omplexit\ny\nof\nfa ets\n(and\nsome\nfa ets\nof\n omplexit\ny).\nJournal\nof\nComputer\nand\nSystem\nS ien \nes,\n28(2):244 259,\n1984.\n[Rot03℄\nJ.\nRothe.\nExa t\n omplexit\ny\nof\nExa t-Four-Colorabilit\ny\n.\nInformation\nPr\no\n \nessing\nL\netters,\n87(1):7 12,\nJuly\n2003.\n18\n\n[Rot05℄\nJ.\nRothe.\nComplexity\nThe\nory\nand\nCryptolo\ngy.\nA\nn\nIntr\no\ndu tion\nto\nCrypto\n \nom-\nplexity.\nEA\nTCS\nT\nexts\nin\nTheoreti al\nComputer\nS ien e.\nSpringer-V\nerlag,\nBerlin,\nHeidelb\nerg,\nNew\nY\nork,\n2005.\n[RR06℄\nT.\nRiege\nand\nJ.\nRothe.\nCompleteness\nin\nthe\nb\no\nolean\nhierar \nh\ny:\nExa t-Four-\nColorabilit\ny\n,\nminimal\ngraph\nun olorabilit\ny\n,\nand\nexa t\ndomati \nn\num\nb\ner\nproblems\n \na\nsurv\ney\n.\nJournal\nof\nUniversal\nComputer\nS ien \ne,\n12(5):551 578,\n2006.\n[V\nal79℄\nL.\nV\nalian\nt.\nThe\n omplexit\ny\nof\n omputing\nthe\np\nermanen\nt.\nThe\nor\neti \nal\nComputer\nS ien \ne,\n8(2):189 201,\n1979.\n[VV86℄\nL.\nV\nalian\nt\nand\nV.\nV\nazirani.\nNP\nis\nas\neasy\nas\ndete ting\nunique\nsolutions.\nThe\no-\nr\neti \nal\nComputer\nS ien \ne,\n47:85 93,\n1986.\n19","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0705.0915v2 [cs.CC] 9 Jun 2008\nSatis abilit\ny\nP\narsimoniously\nRedu es\nto\nthe\nT\nan\ntrix\nTM\nRotation\nPuzzle\nProblem∗\nDorothea\nBaumeister\nand\nJörg\nRothe\nInstitut\nfür\nInformatik\nHeinri \nh-Heine-Univ\nersität\nDüsseldorf\n40225\nDüsseldorf,\nGerman\ny\nJune\n9,\n2008\nAbstra t\nHolzer\nand\nHolzer\n[HH04℄\npro\nv\ned\nthat\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nNP\n- omplete.\nThey\nalso\nsho\nw\ned\nthat\nfor\nin nite\nrotation\npuzzles,\nthis\nproblem\nb\ne omes\nunde idable.\nW\ne\nstudy\nthe\n oun\nting\nv\nersion\nand\nthe\nunique\nv\nersion\nof\nthis\nproblem.\nW\ne\npro\nv\ne\nthat\nthe\nsatis abilit\ny\nproblem\nparsimoniously\nredu es\nto\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nIn\nparti ular,\nthis\nredu tion\npreserv\nes\nthe\nuniqueness\nof\nthe\nsolution,\nwhi \nh\nimplies\nthat\nthe\nunique\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nas\nhard\nas\nthe\nunique\nsatis abilit\ny\nproblem,\nand\nso\nis\nDP\n- omplete\nunder\np\nolynomial-time\nrandom-\nized\nredu tions,\nwhere\nDP\nis\nthe\nse ond\nlev\nel\nof\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n.\nKey\nw\nords:\n omputational\n omplexit\ny\n,\nrotation\npuzzle,\ntiling\nof\nthe\nplane,\nparsimo-\nnious\nredu tion,\n oun\nting\nproblem.\n1\nIn\ntro\ndu tion\nT\nan\ntrix\nTM\nis\na\npuzzle\ngame\npla\ny\ned\nwith\nhexagonal\ntiles\n rmly\narranged\nin\nthe\nplane\nthat\nea \nh\n an\nb\ne\nrotated\naround\ntheir\naxes.\nThere\nare\nfour\ndi eren\nt\nt\nyp\nes\nof\ntiles\n( alled\nSint,\nBrid,\nChin,\nand\nR\nond,\nsee\nFigure\n1)\nthat\ndi er\nb\ny\nthe\nform\nof\nthe\nthree\n olored\nlines\nthey\nea \nh\nha\nv\ne,\nwhere\nthe\n olors\nare"},{"paragraph_id":"p2","order":2,"text":"hosen\namong\nr\ne\nd,\nyel\nlow,\nblue,\nand\ngr\ne\nen.\nThe\nob\nje tiv\ne\nof\nthe\ngame\nis\nto\n nd\na\nrotation\nof\nthe\ngiv\nen\ntiles\nso\nas\nto\n reate\nlong\nlines\nand\nlo\nops\nof\nthe\nsame\n olor.\nSin e\nits\nin\nv\nen\ntion\nin\n1991\nb\ny\nMik\ne\nM Mana\nw\na\ny\nfrom\nNew\nZealand\nand\nits\n ommer ial\nlaun \nh,\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nhas\nb\ne ome\nextremely\np\nopular\nand\n ommer ially\nsu essful.\n∗\nSupp\norted\nin\npart\nb\ny\nDF\nG\ngran\nts\nR\nO\n1202/9-3\nand\nR\nO\n1202/11-1\nand\nb\ny\nthe\nAlexan-\nder\nv\non\nHum\nb\noldt\nF\noundation's\nT\nransCo\nop\nprogram.\nA\npreliminary\nv\nersion\nof\nthis\npap\ner\nap-\np\neared\nin\nthe\nPro\n eedings\nof\nMa hines,\nComputations\nand\nUniversality\n(MCU\n2007).\nURLs:\n . s.uni-duesseldorf.de/∼\n{baumeister,\nrothe}\n(D.\nBaumeister\nand\nJ.\nRothe).\n1"},{"paragraph_id":"p3","order":3,"text":"Holzer\nand\nHolzer\n[HH04℄\n onsidered\nt\nw\no\nv\narian\nts\nof\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem,\none\nwith\n nitely\nman\ny\nand\none\nwith\nin nitely\nman\ny\ntiles\nin\na\ngiv\nen\nproblem\ninstan e.\nThey\npro\nv\ned\nthat\nthe\n nite\nv\narian\nt\nof\nthis\nproblem\nis\nNP\n- omplete\nb\ny\nredu ing\nthe\nNP- omplete\nb\no\nolean\n ir uit\nsatis abilit\ny\nproblem\n(restri ted\nto\n ir uits\nwith\nAND\nand\nNOT\ngates\nonly)\nto\nit.\nThey\nalso\nsho\nw\ned\nthat\nthe\nin nite\nv\narian\nt\nof\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nunde idable,\nagain\nemplo\nying\na\n ir uit\n onstru tion.\nF\nor\nother\nresults\non\nthe\n omplexit\ny\nof\nproblems\nrelated\nto\nDomino-lik\ne\nstrategy\ngames,\nw\ne\nrefer\nto\nGrädel\n[Grä90\n℄.\nW\ne\n onsider\nt\nw\no\nv\narian\nts\nof\nthe\n nite\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem,\nits\n oun\nting\nv\nersion\nand\nits\nunique\nv\nersion.\nThe\n oun\nting\nproblem\nasks\nfor\nthe\nn\num\nb\ner\nof\nsolutions\nof\na\ngiv\nen\nrotation\npuzzle\ninstan e.\nThe\nunique\nproblem\nasks\nwhether\na\ngiv\nen\nrotation\npuzzle\ninstan e\nhas\nexa tly\none\nsolution.\nOur\nmain\nresult\nis\nthat\nthe\nsatis abilit\ny\nproblem\nparsimoniously\nredu es\nto\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nThe\n lass #P\nw\nas\nin\ntro\ndu ed\nb\ny\nV\nalian\nt\n[V\nal79\n℄\nto\n apture\nthe\n omplexit\ny\nof\n oun\nting\nthe\nsolutions\nof\nNP\nproblems.\nP\narsimonious\nredu tions\nb\net\nw\neen\nNP\n oun\nting\nproblems \nsu \nh\nas\nours preserv\ne\nthe\npre ise\nn\num\nb\ner\nof\nsolutions.\nThis\nis\nan\nimp\nortan\nt\nprop\nert\ny\nfor\nat\nleast\nt\nw\no\nreasons.\nFirst,\nthe\nstru ture\nof\nthe\nsolution\nspa e\nis\npreserv\ned\nb\ny\na\nparsimonious\nredu tion\nfrom A\nto B\n,\nsin e\nsolutions\nof A\nare\nmapp\ned\nbije tiv\nely\nto\nsolutions\nof B\nin\np\nolynomial\ntime.\nSe ond,\nparsimonious\nredu tions\n an\nb\ne\nused\nto\npro\nv\ne\nlo\nw\ner\nb\nounds\nfor\nthe\nunique\nv\nersions\nof\nNP\nproblems.\nIn\nparti ular,\nw\ne\napply\nour\nab\no\nv\ne-men\ntioned\nparsimonious\nredu tion\nto\npro\nv\ne\nthat\nthe\nunique\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nDP\n- omplete\nunder\np\nolynomial-time\nrandomized\nredu tions\nin\nthe\nsense\nof\nV\nalian\nt\nand\nV\nazirani\n[VV86℄.\nHere,\nDP\nis\nthe\nset\nof\ndi eren es\nof\nan\ny\nt\nw\no\nNP\nsets\n[PY84\n℄;\nso\nNP ⊆\nDP,\nand\nit\nis\n onsidered\nmost\nunlik\nely\nthat\nb\noth\n lasses\nare\nequal.\nNote\nthat\nDP\nis\nthe\nse ond\nlev\nel\nof\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n,\nsee\nCai\net\nal.\n[CGH+\n88\n,\nCGH+\n89℄.\nF\nurther\nresults\non\nDP\nand\n ompletenes\nin\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n an\nb\ne\nfound,\ne.g.,\nin\n[CM87\n,\nRot03℄,\nsee\nalso\nthe\nsurv\ney\n[RR06℄.\nWhile\nman\ny\nstandard\nredu tions\nb\net\nw\neen\nNP- omplete\nproblems\nare\neasily\nseen\nto\nb\ne\nparsimonious,\nthere\nare\na\nn\num\nb\ner\nof\nex eptions.\nF\nor\nexample,\nBarban \nhon\n[Bar04℄\nsho\nw\ned\nthat\nthe\n(planar)\nsatis abilit\ny\nproblem\nis\nparsimoniously\np\nolynomial-time\nredu ible\nto\nthe\n(planar) 3- olorabilit\ny\nproblem\nvia\na\nrather\nsophisti ated\n onstru tion.\nOther\nexamples\nof\nnon\ntrivial\nparsimonious\nredu tions\n an\nb\ne\nfound\nin\n[P\nap94\n℄.\nHolzer\nand\nHolzer's\nredu tion,\nho\nw\nev\ner,\nis\nnot\nparsimonious\n[HH04℄.\nThe\nmain\npurp\nose\nof\nthis\npap\ner\nis\nto\nsho\nw\nho\nw\nto\nmo\ndify\ntheir\nredu tion\nso\nas\nto\nmak\ne\nit\nparsimonious.\nW\ne\nmen\ntion\nin\npassing\nthat\nthis\npap\ner\ndi ers\nfrom\nits\npreliminary\nv\nersion\n[BR07\n℄\nin\nv\narious\nw\na\nys.\nFirst,\nto\nallo\nw\n omparison,\nw\ne\nhere\nexpli itly\nsho\nw\nthe\ndi eren es\nb\net\nw\neen\nHolzer\nand\nHolzer's\noriginal\n onstru tion\n[HH04℄\nand\nour\nmo\ndi ed\n onstru tion\nb\ny\n(a)\npre-\nsen\nting\ntheir\nsubpuzzles\n(mark\ned\nso\nas\nto\n learly\nindi ate\nthe\ntiles\nthat\nrequire\nmo\ndi ation\nif\none\naims\nat\na\nparsimonious\nredu tion),\nand\n(b)\nhighligh\nting\nall\nmo\ndi ed\nor\nadditionally\ninserted\ntiles\nin\nour\nsubpuzzles.\nSe ond,\nunlik\ne\nthe\nredu tions\ngiv\nen\nin\n[HH04,\nBR07℄,\nin\nthe\nredu tion\npresen\nted\nhere\nw\ne\nin\ntro\ndu e\na\nnew\nsubpuzzle\nfor\nsim\nulating\nwire\n rossings\nin\nb\no\nolean\n ir uits.\nThis\nnew\nsubpuzzle,\nwhi \nh\nw\ne\n all\nCR\nOSS,\nwill\nsa\nv\ne\nus\nthe\ne ort\nof\ntransforming\ngeneral\nb\no\nolean\n ir uits\nin\nto\nplanar\nb\no\nolean\n ir uits\n(i.e.,\nin\nto\n ir uits\nwithout\n2"},{"paragraph_id":"p4","order":4,"text":"wire\n rossings).\nHen e,\nthe\nredu tion\npro\nvided\nin\nthe\npresen\nt\npap\ners\nis\nmore\ne ien\nt\nand\nthe\ntotal\nn\num\nb\ner\nof\ntiles\nneeded\nto\nsim\nulate\na\ngiv\nen\n ir uit\nis\n onsiderably\nsmaller\nthan\nin\nour\nprevious\n onstru tion\n[BR07℄.\nFinally\n,\nto\npro\nv\ne\n orre tness\nof\nour\nredu tion,\nw\ne\nno\nw unlik\ne\nthe\napproa \nh\ntak\nen\nin\n[BR07℄ argue\nvia\n olor\nsequen es \nof\ntiles,\nwhi \nh\nfa ilitates\nreading\nand\nunderstanding\nthe\nargumen\nts.\nThis\npap\ner\nis\norganized\nas\nfollo\nws.\nIn\nSe tion\n2\n,\nw\ne\nde ne\nsome\n omplexit\ny-theoreti \nnotions\nand\nour\nv\narian\nts\nof\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nIn\nSe tion\n3,\nw\ne\npresen\nt\nour\nparsimonious\nredu tion.\nIn\nSe tion\n4,\n nally\n,\nw\ne\nstudy\nthe\nunique\nv\nersion\nof\nthis\nproblem\nand\nsho\nw\nits\nDP- ompleteness\nunder\nrandomized\nredu tions.\n2\nPreliminaries\n2.1\nDe nition\nof\nSome\nComplexit\ny-Theoreti \nNotions\nFix\nthe\nalphab\net Σ = {0, 1},\nand\nlet Σ∗\ndenote\nthe\nset\nof\nstrings\no\nv\ner Σ .\nAs\nis\n ommon,\nde ision\nproblems\nare\nsuitably\nen o\nded\nas\nlanguages\no\nv\ner Σ .\nF\nor\nan\ny\nlanguage A ⊆Σ∗\n,\nlet ∥A∥\ndenote\nthe\nn\num\nb\ner\nof\nelemen\nts\nin A.\nF\nor\nsome\nba \nkground\non\n omputational\n om-\nplexit\ny\ntheory\n,\nw\ne\nrefer\nto\nan\ny\nstandard\ntextb\no\nok\nof\nthis\n eld,\ne.g.,\n[P\nap94\n,\nRot05℄.\nLet\nNP\ndenote\nthe\n lass\nof\nproblems\nsolv\nable\nin\nnondeterministi \np\nolynomial\ntime.\nGeneralizing\nNP,\nP\napadimitriou\nand\nY\nannak\nakis\n[PY84\n℄\nin\ntro\ndu ed\nthe\n lass\nDP = {A −B | A, B ∈\nNP}\nto\n apture\nthe\n omplexit\ny\nof\nNP-hard\nor\n oNP-hard\nproblems\nthat\nseemingly\nare\nneither\nin\nNP\nnor\nin\n oNP\n.\nIn\nparti ular,\nthey\nsho\nw\ned\nthat\nDP\n on\ntains\na\nn\num\nb\ner\nof\nuniqueness\npr\noblems,\n riti \nal\ngr\naph\npr\noblems,\nand\nexa t\noptimization\npr\noblems,\nand\nthey\nsho\nw\ned\nsome\nof\nthese\nproblems\n omplete\nfor\nDP\n;\nsee\nalso\nthe\nre en\nt\nsurv\ney\n[RR06\n℄.\nDP\nw\nas\nlater\ngeneralized\nb\ny\nCai\net\nal.\n[CGH+\n88,\nCGH+\n89℄,\nwho\nin\ntro\ndu ed\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n.\nNote\nthat\nDP\nis\nthe\nse ond\nlev\nel\nof\nthis\nhierar \nh\ny\n.\nIn\nhis\nseminal\npap\ner,\nV\nalian\nt\n[V\nal79\n℄\ninitiated\nthe\nstudy\nof\n oun\nting\nproblems\nand\nin-\ntro\ndu ed\nthe\nimp\nortan\nt\n oun\nting\n lass #P\n.\nMem\nb\ners\nof #P\nare\nreferred\nto\nas\nNP"},{"paragraph_id":"p5","order":5,"text":"ounting\npr\noblems.\nA\nw\nell-kno\nwn\nNP\n oun\nting\nproblem\nis # SA\nT,\nthe\n oun\nting\nv\nersion\nof\nthe\nsatis -\nabilit\ny\nproblem:\nGiv\nen\na\nb\no\nolean\nform\nula,\nho\nw\nman\ny\nsatisfying\nassignmen\nts\ndo\nes\nit\nha\nv\ne?\nDe nition\n2.1\n(V\nalian\nt\n[V\nal79\n℄)\nL\net\nNPTM\nb\ne\na\nshorthand\nfor\nnondeterministi \np\nolynomial-time\nT\nuring\nma hine.\nF\nor\nany\nNPTM M\nand\nany\ninput x,\nlet accM(x)\nde-\nnote\nthe\nnumb\ner\nof\na"},{"paragraph_id":"p6","order":6,"text":"epting"},{"paragraph_id":"p7","order":7,"text":"omputation\np\naths\nof M(x),\ni.e., accM\nis\na\nfun tion\nmapping\nfr\nom Σ∗\nto N .\nDe ne\nthe\nfun tion\n lass #P = {accM | M\nis\nan\nNPTM}.\nW\ne\nno\nw\nde ne\nthe\nnotion\nof\n(p\nolynomial-time)\np\narsimonious\nr\ne\ndu ibility,\nwhi \nh\nwill\nb\ne\nused\nto\n ompare\nthe\nhardness\nof\nsolving\nNP\n oun\nting\nproblems.\nIn\ntuitiv\nely\n,\nan\nNP\n oun\nting\nproblem f\nparsimoniously\nredu es\nto\nan\nNP\n oun\nting\nproblem g\nif\nthe\ninstan es\nof f\n an\nb\ne\ntransformed\nin\nto\ninstan es\nof g\nsu \nh\nthat\nthe\nn\num\nb\ner\nof\nsolutions\nof f\nare\npreserv\ned\nunder\nthis\ntransformation.\nDe nition\n2.2\nL\net f\nand g\nb\ne\nany\ntwo\ngiven"},{"paragraph_id":"p8","order":8,"text":"ounting\npr\noblems\nmapping\nfr\nom Σ∗\nto N .\nW\ne\nsay f\n(p\nolynomial-time)\nparsimoniously\nredu es\nto g\n(denote\nd\nby f ≤p\npar g\n)\nif\nther\ne\nexists\na\n3"},{"paragraph_id":"p9","order":9,"text":"(a)\nSin\nt\n(b)\nBrid\n( )\nChin\n(d)\nRond\n(e)\nred\n(f\n)\ny\nello\nw\n(g)\nblue\n(h)\ngreen\nFigure\n1:\nT\nan\ntrix\nTM\ntiles\nand\n olors\np\nolynomial-time"},{"paragraph_id":"p10","order":10,"text":"omputable\nfun tion ρ\nsu h\nthat\nfor\ne\na h x ∈Σ∗\n, f(x) = g(ρ(x)).\nIf F\nand\nG\nar\ne\nthe\nNP\nde\n ision\npr\noblems"},{"paragraph_id":"p11","order":11,"text":"orr\nesp\nonding\nto\nthe\nNP"},{"paragraph_id":"p12","order":12,"text":"ounting\npr\noblems f\nand g\nwith\nf ≤p\npar g\n,\nwe\nwil\nl\nalso\nsay\nthat F\np\narsimoniously\nr\ne\ndu \nes\nto G.\n2.2\nV\narian\nts\nof\nthe\nT\nan\ntrix\nTM\nRotation\nPuzzle\nProblem\nThe\nT\nan\ntrix\nTM\nrotation\npuzzle\nhas\nfour\nkinds\nof\nhexagonal\ntiles the\nSint,\nthe\nBrid,\nthe\nChin,\nand\nthe\nR\nond\n ea \nh\nof\nwhi \nh\nhas\nthree\n olored\nlines,\nwhere\nthe\n olors\nare"},{"paragraph_id":"p13","order":13,"text":"hosen\namong\nr\ne\nd,\nyel\nlow,\nblue,\nand\ngr\ne\nen,\nsee\nFigure\n1(a) (d).\nThis\ngiv\nes\na\ntotal\nof\n56\ndi eren\nt\ntiles.\nSin e\nw\ne\naren't\nusing\na tually\n olored\n gures,\nw\ne\nen o\nde\nthe\n olors\nas\nsho\nwn\nin\nFigure\n1(e) (h).\nHolzer\nand\nHolzer\n[HH04℄\nsho\nw\ned\nthat\nthe\nde ision\nproblem\nT\nan\ntrix\nTM\nrotation\npuzzle\n(whi \nh\nw\ne\ndenote\nb\ny\nTRP,\nfor\nshort)\nis\nNP- omplete.\nIn\nthis\npap\ner,\nw\ne\nin\ntro\ndu e\nand\nstudy # TRP,\nthe\n oun\nting\nv\nersion\nof\nTRP.\nW\ne\nno\nw\nbrie y\ndes rib\ne\nthe\nformalism\nin\ntro\ndu ed\nb\ny\nHolzer\nand\nHolzer\n[HH04℄\nto\nde-\n ne\nTRP\n,\nsin e\nthe\nsame\nformalism\nis\nuseful\nfor\nde ning # TRP.\nIn\nparti ular,\nto\nrepresen\nt\nthe\ninstan es\nof\nb\noth\nthese\nproblems,\na\nt\nw\no-dimensional\nhexagonal\n o\nordinate\nsystem\nis\nused,\nsee\nFigure\n2.\nIn\nthis\nsystem,\nt\nw\no\ndistin t\npairs a = (u, w)\nand b = (v, x)\nfrom Z2\nare\nadja en\nt\nif\nand\nonly\nif\none\nof\nthe\nfollo\nwing\nfour\n onditions\nis\nsatis ed:\n1. u = v\nand |w −x| = 1,\n2. |u −v| = 1\nand w = x,\n3. u −v = 1\nand w −x = 1,\nand\n4. u −v = −1\nand w −x = −1.\nLet T\nb\ne\nthe\nset\nof\nall\nT\nan\ntrix\nTM\ntiles.\nLet A\nb\ne\na\n(partial)\nfun tion\nmapping\nthe\nelemen\nts\nof Z2\nto T\n,\ni.e.,\nfor\nthose v ∈Z2\non\nwhi \nh A\nis\nde ned, A(v)\nis\nthe\nt\nyp\ne\nof\nthe\ntile\nlo\n ated\nat v\n.\nThe\nset shape(A) = {v ∈Z2 | A(v)\nis\nde ned}\ngiv\nes\nthe\np\nositions\nin Z2\nat\nwhi \nh\ntiles\nare\npla ed.\nF\nor\nall a, b ∈shape(A), A(a)\nis\nadja en\nt\nto A(b)\nif\nand\nonly\nif a\nis\nadja en\nt\nto b.\n4"},{"paragraph_id":"p14","order":14,"text":"x\ny\n(1, 1)\n(0, 0)\n(−1, −1)\n(1, 0)\n(0, 1)\n(−1, 0)\n(0, −1)\nFigure\n2:\nA\nt\nw\no-dimensional\nhexagonal\n o\nordinate\nsystem\nTRP\nis\nthen\nde ned\nas\nfollo\nws\n(note\nthat\nthe\ninitial\norien\ntation\nis\nnot\nsp\ne i ed,\nas\nit\ndo\nesn't\nmatter\nfor\nthe\nquestion\nof\nwhether\nthe\nde ision\nproblem\nTRP\nis\nsolv\nable)\n[HH04℄:\n1\nName:\nT\nan\ntrix\nTM\nRotation\nPuzzle\n(TRP,\nfor\nshort).\nGiv\nen:\nA\n nite\nshap\ne\nfun tion A : Z2 →T\n,\nappropriately\nen o\nded\nas\na\nstring.\nQuestion:\nIs\nthe\nrotation\npuzzle\nde ned\nb\ny A\nsolv\nable,\ni.e.,\ndo\nes\nthere\nexist\na\nrotation\nof\nthe\ngiv\nen\ntiles\nat\ntheir\np\nositions\nsu \nh\nthat\nat\nea \nh\njoin\nt\nedge\nof\nt\nw\no\nadja en\nt\ntiles\nthe\n orresp\nonding\n olors\nmat \nh?\nF\nor\nan\ny\ngiv\nen\nTRP\ninstan e A ,\na\nsolution\nof A\nis\na\nsp\ne i ation\n(in\nsome\nappropriate\nen o\nding)\nof\nea \nh\ntile\nin shape(A)\nin\nsome\nparti ular\norien\ntation\nsu \nh\nthat\nfor\nea \nh\njoin\nt\nedge\nof\nt\nw\no\nadja en\nt\ntiles\nthe\n orresp\nonding\n olors\nmat \nh.\nFigure\n3\ngiv\nes\nan\nexample\nof\na\nrotation\npuzzle\ninstan e\nand\nits\nsolution.\nLet\nSol\nTRP(A)\ndenote\nthe\nset\nof\nsolutions\nof\na\ngiv\nen\nTRP\ninstan e A .\nSo A\nis\nin\nTRP\n(view\ned\nas\na\nlanguage)\nif\nand\nonly\nif\nthe\nset\nSol\nTRP(A)\nis\nnonempt\ny\n.\n(a)\nPuzzle\n(b)\nSolution\nFigure\n3:\nAn\nexample\nof\na\nTRP\ninstan e\nand\nits\nsolution\n1\nAs\nnoted\nb\ny\nHolzer\nand\nHolzer\n[HH04℄,\nthere\nis\na\ndi eren e\nb\net\nw\neen\ntheir\nde nition\nof\nTRP,\nwhi \nh\nallo\nws\nholes\nin\nTRP\ninstan es,\nand\nthe\noriginal\nT\nan\ntrix\nTM\ngame,\nwhi \nh\ndo\nes\nnot\nallo\nw\nholes.\nThe\nproblem\nof\nwhether\nthe\nanalog\nof\nTRP\nwithout\nholes\nstill\nis\nNP\n- omplete\nis\nop\nen.\n5"},{"paragraph_id":"p15","order":15,"text":"W\ne\nno\nw\nde ne\nthe\n oun\nting\nv\nersion\nand\nthe\nunique\nv\nersion\nof\nTRP,\nwhi \nh\nwill\nb\ne\n on-\nsidered\nin\nSe tions\n3\nand\n4.\nDe nition\n2.3\n1.\nThe\nT\nan\ntrix\nTM\nrotation\npuzzle\n oun\nting\nproblem\nis\nthe\nfun tion\n# TRP : Σ∗→N\nde ne\nd\nby\n# TRP(A) = ∥Sol\nTRP(A)∥,\nwher\ne\nwe\nassume\nthat\ninputs A\nar\ne\nappr\nopriately\nen \no\nde\nd\nas\nstrings\nin Σ∗\nand\nfun tion\nvalues\nar\ne\nnonne\ngative\ninte\ngers\n(r\nepr\nesente\nd\nin\nbinary).\n2.\nThe\nunique\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nde ne\nd\nby\nUnique-\nTRP = {A | # TRP(A) = 1}.\n3\nSatis abilit\ny\nP\narsimoniously\nRedu es\nto\nthe\nT\nan\ntrix\nTM\nRo-\ntation\nPuzzle\nProblem\nOur\nmain\nresult\nis\nTheorem\n3.1\nthe\npro\nof\nof\nwhi \nh\nwill\nb\ne\npresen\nted\nin\nSe tions\n3.1\nthrough\n3.4\n.\nTheorem\n3.1 # SA\nT ≤p\npar # TRP\n.\nT\no\npro\nv\ne\nTRP\nNP- omplete,\nHolzer\nand\nHolzer\n[HH04℄\nga\nv\ne\na\nredu tion\nfrom\nthe\nNP-\n omplete\nproblem\nCir uit∧,¬\n-SA\nT\n(see\nCo\nok\n[Co\no71℄),\nwhi \nh\nis\nde ned\nas\nfollo\nws.\nName:\nCir uit∧,¬\n-SA\nT\n.\nGiv\nen:\nA\nb\no\nolean\n ir uit C\nwith\nAND\nand\nNOT\ngates.\nQuestion:\nDo\nes\nthere\nexist\na\ntruth\nassignmen\nt\nto\nthe\ninput\ngates\nof C\nsu \nh\nthat C\nunder\nthis\nassignmen\nt\nev\naluates\nto\ntrue\n?\nHolzer\nand\nHolzer's\n onstru tion\nsim\nulates\nthe\n omputation\nof\nsu \nh\na\nb\no\nolean\n ir uit C\nb\ny\na\nT\nan\ntrix\nTM\nrotation\npuzzle\nsu \nh\nthat C\nev\naluates\nto\ntrue\nfor\nsome\nassignmen\nt\nto\nits\nv\nariables\nif\nand\nonly\nif\nthe\npuzzle\nhas\na\nsolution.\nOur\nde nition\nof\nb\no\nolean\n ir uits\nfollo\nws\nHolzer\nand\nHolzer\n[HH04℄,\nwho\nview\na\nb\no\nolean\n ir uit C\nwith\ninput\nv\nariables x1, x2, . . . , xn\nas\na\nsequen e (α1, α2, . . . , αm)\nof\nsteps\nsu \nh\nthat\nthe ith\ninstru tion αi\nhas\none\nof\nthe\nfollo\nwing\nforms:\n1.\nfor\nea \nh i\nwith 1 ≤i ≤n , αi = xi\n,\n2.\nfor\nea \nh i\nwith n+1 ≤i ≤m ,\neither αi =\nAND(j, k)\nor αi =\nNOT(j),\nwhere j ≤k < i.\n6"},{"paragraph_id":"p16","order":16,"text":"Dep\nending\non\nthe\ntruth\nv\nalues\nof\nthe\ninput\nv\nariables\nthe\noutput\ngate\nev\naluates\nto\ntrue\nor\nfalse\nin\nthe\nstandard\nw\na\ny\n.\nIn\ngeneral,\n ir uits\n an\n on\ntain\nan\ny\nn\num\nb\ner\nof\nwire\n rossings,\nwhi \nh\n annot\neasily\nb\ne\nrealized\nb\ny\nT\nan\ntrix\nTM\nsubpuzzles.\nT\no\nbuild\na\n ir uit\n on\ntaining\nonly\n rossings\nof\nt\nw\no\nneigh-\nb\noring\nwires,\nHolzer\nand\nHolzer\nfollo\nw\nGolds \nhlager's\npro\n edure\n[Gol77\n℄:\nIf αi =\nAND(j, k),\nmo\nv\ne\nwire j\nimmediately\nto\nthe\nleft\nof k\n,\nput\nan\nAND\ngate\nin\nfor αi\n,\nmo\nv\ne\nwire j\nba \nk\nto\nits\nstarting\np\noin\nt,\nand\n nally\nmo\nv\ne\nwire i\nto\nthe\nfar\nrigh\nt.\nThe\n ases\nof\ninstru tion αi\nb\neing\neither\nNOT(j)\nor xi\nare\ntreated\nin\na\nsimilar\nw\na\ny\n.\nFigure\n4\nsho\nws\na\npart\nof\na\n ir uit\nwith\nwire\n rossings,\nwhi \nh\n omputes α4 =\nAND(1, 3).\nOb\nviously\n,\nthere\nare\nonly\n rossings\nof\nt\nw\no\ndire tly\nadja en\nt\nwires.\nNote\nthat\nthis\ntransformation\nfrom\ngeneral\nto\nalmost\nplanar\n ir uits\n an\nb\ne\ndone\nin\ndeterministi \nlogarithmi \nspa e.\n2\n1\n3\n1\n1\n1\n2\n2\nAND\n3\n3\n4\n2\n1\n2\n3\n4\n3\n1\n2\n3\nFigure\n4:\nExample\nof\na\n ir uit\nfollo\nwing\nGolds \nhlager's\ntransformation\n[Gol77\n℄\nNo\nw,\nto\nbuild\na\nT\nan\ntrix\nTM\nRotation\nPuzzle\nthat\nsim\nulates\nsu \nh\na\n ir uit,\nHolzer\nand\nHolzer\nuse\na\ntruly\nplanar\n ross-o\nv\ner \ngadget\nthat\nw\nas\nprop\nosed\nb\ny\nM Coll\n[M C81\n℄.\nM -\nColl's\n ir uit\ngadget\nuses\nb\no\nolean\nAND\nand\nNOT\ngates\nto\nsim\nulate\nthe\n rossings\nof\nan\ny\nt\nw\no\nadja en\nt\nwires.\nEa \nh\nsu \nh\n ross-o\nv\ner \ngadget\nneeds\na\ntotal\nof 14\ninstru tion\nsteps\nand\nin\nv\nolv\nes\nt\nw\nelv\ne\nAND\nand\nnine\nNOT\ngates.\nSin e\nman\ny\n rossings\n an\no\n ur\nin\nthe\noriginally\ngiv\nen\n ir uit,\nthis\nma\ny\nlead\nto\na\n onsiderable\n(alb\neit\nstill\np\nolynomial)\nblo\nw-up\nof\nthe\nT\nan\ntrix\nTM\npuzzle\n onstru ted\nin\nour\nredu tion.\nThat\nis\nwh\ny\nw\ne\nprop\nose,\nas\na\nmore\ne ien\nt\nalternativ\ne\nto\nthe\nredu tion\npresen\nted\nin\nour\nprevious\npap\ner\n[BR07℄,\nto\nsim\nulate\nthese\n ross-o\nv\ners\ndire tly\nvia\na\nnew\nT\nan\ntrix\nTM\nsubpuzzle,\nthe\nCR\nOSS\nsubpuzzle\npresen\nted\nin\nFigure\n11\n.\nUnlik\ne\nthe\nredu tions\ngiv\nen\nin\n[HH04,\nBR07℄,\nvia\nsu \nh\nCR\nOSS\nsubpuzzles\nour\nredu tion\ndo\nesn't\nneed\nthe\ntransformation\nfrom\ngeneral\nto\nplanar\n ir uits\nthat\nrequires\nman\ny\nadditional\ngates\nand\ninstru tion\nsteps\n aused\nb\ny\nwire\n rossings.\nF\nurthermore,\nour\n onstru tion\nwill\nmo\ndify\nHolzer\nand\nHolzer's\nredu tion\n[HH04℄\nin\nsu \nh\na\nw\na\ny\nthat\nthere\nis\na\none-to-one\n orresp\nonden e\nb\net\nw\neen\nthe\nsolutions\nof\nthe\ngiv\nen\nCir uit∧,¬\n-SA\nT\ninstan e\nand\nthe\nsolutions\nof\nthe\nresulting\nrotation\npuzzle\ninstan e;\nhen e\n7"},{"paragraph_id":"p17","order":17,"text":"our\nredu tion\nis\nparsimonious.\nT\no\nsim\nulate\nthe\n ir uit\nb\ny\na\nrotation\npuzzle,\na\nn\num\nb\ner\nof\nsubpuzzles\nare\nused.\nThe\n olor\nblue\nin\nthese\nsubpuzzles\nwill\nrepresen\nt\nthe\ntruth\nv\nalue\ntrue,\nand\nthe\n olor\nred\nwill\nrepresen\nt\nfalse.\nThis\n olor\nen o\nding\nat\nthe\ninputs\nand\noutputs\nof\nthe\nsubpuzzles\nth\nus\nrepresen\nt\nthe\ntruth\nv\nalues\nof\nthe\n ir uit's\ngates\nand\nwires.\nIn\nthe\nfollo\nwing\nse tions,\nw\ne\npresen\nt\nour\nmo\ndi ed\nsubpuzzles\nand,\nto\nallo\nw\n omparison,\nw\ne\nalso\npresen\nt\nHolzer\nand\nHolzer's\nsubpuzzles\n[HH04℄.\nT\no\nindi ate\nthe\ndi eren es\nb\net\nw\neen\ntheir\noriginal\nand\nour\nmo\ndi ed\nsubpuzzles,\ntiles\nwith\nmore\nthan\none\np\nossible\nsolution\nin\nthe\noriginal\nsubpuzzles\nwill\nha\nv\ne\na\ngrey\ninstead\nof\na\nbla \nk\nb\norder,\nand\nw\ne\nhighligh\nt\nall\nmo\ndi ed\ntiles\nin\nour\nnew\nsubpuzzles\nb\ny\na\ngrey\ninstead\nof\na\nwhite\nba \nkground\n(unless\nstated\notherwise).\nAnother\ndi eren e\nb\net\nw\neen\nour\npro\nof\nhere\nand\nthe\npro\nofs\nof\nHolzer\nand\nHolzer\n[HH04℄\nand\na\npreliminary\nv\nersion\nof\nthis\npap\ner\n[BR07℄\nregards\nthe\nanalysis\nof\nthe\nsubpuzzles.\nIn\nparti ular,\nw\ne\nwill\nhere\nfo\n us\non\nthe\n olor\nsequen es\nof\nthe\nv\narious\nT\nan\ntrix\nTM\ntiles.\nThis\nwill\nallo\nw\nus\nto\ngiv\ne\nthe\nargumen\nts\nmore\ntersely\n.\nF\nor\nexample,\nthe\ntile\nin\nFigure\n1(d)\nhas\nthe\n lo"},{"paragraph_id":"p18","order":18,"text":"kwise\n olor\nsequen e yellow-green\n-green-red-red -yellow\n,\nwhi \nh\nwill\nb\ne\nabbreviated\nas yggrry\n.\n3.1\nWire\nSubpuzzles\nWires\nof\nthe\n ir uit\nare\nsim\nulated\nb\ny\nthe\nsubpuzzles\nWIRE,\nMO\nVE,\nCOPY,\nand\nCR\nOSS.\nT\no\nsim\nulate\nsimple\nv\nerti al\nwires,\nthe\nWIRE\nsubpuzzle\nis\nused.\nThe\noriginal\nv\nersion\nof\nHolzer\nand\nHolzer\n[HH04℄\nis\npresen\nted\nin\nFigure\n5.\nIt\nis\neasy\nto\nsee\nthat\nb\noth\ntiles\nha\nv\ne\nt\nw\no\np\nossible\norien\ntations\nfor\nea \nh\ninput\n olor.\nW\ne\nmo\ndify\nthis\nsubpuzzle\nas\nsho\nwn\nin\nFigure\n6,\ninserting\na\nnew\nRond\nat\np\nosition x.\nWithout\nthis\ntile,\nthe\np\nossible\n olor\nsequen es\nfor\nthe\nedges\nof\ntile x\njoin\nt\nwith a\nand b\nare rr , ry , yr ,\nand yy\nif\nthe\ninput\n olor\nis\nblue,\nand\nare bb ,\nby, yb ,\nand yy\nif\nthe\ninput\n olor\nis\nr\ne\nd.\nHo\nw\nev\ner,\nsin e\nthe\nnew\ntile x\ndo\nes\nnot\n on\ntain\nthe\n olor\nyel\nlow,\nthe\nsolutions\nare\n xed\nwith,\nresp\ne tiv\nely\n, bb\nand rr\nat\nthe\njoin\nt\nedges\nof\ntile x\nwith a\nand b,\nand\nw\ne\nobtain\nunique\nsolutions\nfor\nb\noth\ninput\n olors.\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nIN\na\nb\nOUT\n( )\nS \nheme\nFigure\n5:\nOriginal\nsubpuzzle\nWIRE,\nsee\n[HH04℄\nT\no\nbuild\nlonger\nwires,\nsev\neral\nWIRE\nsubpuzzles\nare\n onne ted.\nNote\nthat\nthis\nWIRE\nsubpuzzle\nhas\nheigh\nt\nt\nw\no.\nThis\nfor es\nall\nother\nsubpuzzles\nto\nha\nv\ne\nev\nen\nheigh\nt,\nb\ne ause\nthey\nm\nust\nb\ne\n onne ted\nb\ny\nWIRE\nsubpuzzles.\n8"},{"paragraph_id":"p19","order":19,"text":"IN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nx\nIN\na\nb\nOUT\n( )\nS \nheme\nFigure\n6:\nMo\ndi ed\nsubpuzzle\nWIRE\nBy\nthe\nMO\nVE\nsubpuzzle\nthe\n ir uit's\nwires\n an\nb\ne\nmo\nv\ned\none\np\nosition\nto\nthe\nleft\nor\none\np\nosition\nto\nthe\nrigh\nt.\nW\ne\ndis uss\na\nmo\nv\ne\nto\nthe\nrigh\nt\nin\ndetail\nand\nmen\ntion\nthat\na\nmo\nv\ne\nto\nthe\nleft\n an\nb\ne\nhandled\nanalogously\n.\nThe\noriginal\nv\nersion\nof\nHolzer\nand\nHolzer\n[HH04℄\nis\nsho\nwn\nin\nFigure\n7,\nand\nour\nmo\ndi ed\nsubpuzzle\nis\nsho\nwn\nin\nFigure\n8.\nIn\nthe\noriginal\nMO\nVE\nsubpuzzle,\ntiles a\nand i\nare\nmark\ned\nb\ne ause\nthey\nha\nv\ne\nt\nw\no\norien\nta-\ntions\nfor\nthe\nsolutions\nsho\nwn\nin\nFigure\n7.\nNote,\nho\nw\nev\ner,\nthat\nin\naddition\nto\nthe\nam\nbiguit\ny\n aused\nb\ny\ntiles a\nand i,\nthere\ndo\nes\nexist\nstill\nanother\nsolution\nfor\nea \nh\ninput\n olor.\nIn\nparti -\nular,\nif\nthe\ninput\n olor\nis\nblue\nthen\none\n an\nsimply\nsw\nap\nthe\n olors\nr\ne\nd\nand\nyel\nlow\nto\nobtain\nanother\nsolution,\nand\nif\nthe\ninput\n olor\nis\nr\ne\nd\nthen\ntiles b\nand d\n an\nb\ne\nrotated\nb\ny 60\nand\nall\nother\ntiles\nb\ny 240\ndegrees\nin\n lo"},{"paragraph_id":"p20","order":20,"text":"kwise\ndire tion.\nW\ne\n x\nthe\nsolution\nb\ny\ninserting\ntiles x\nand y\n.\nFirst,\n onsider\nthe\n ase\nthat\nthe\ninput\n olor\nis\nblue.\nIt\nis\n lear\nthat\nthe\nedge\nof\ntile e\njoin\nt\nwith\ntile x\nm\nust\nb\ne\nblue,\nand\nsin e\nthe\nedges\nof\ntiles x\njoin\nt\nwith\ntiles b\nand f\nm\nust\nha\nv\ne\nthe\nsame\n olor,\nthe\norien\ntation\nof\ntile x\nis\n xed.\nThis\n xes\nalso\nthe\norien\ntation\nof\nall\nother\ntiles\nex ept a\nand i.\nThe\norien\ntation\nof\ntile a\nis\n xed\nwith\nr\ne\nd\nat\nthe\nedge\njoin\nt\nwith x.\nTile y\n xes\nthe\norien\ntation\nof\ntile i,\nsin e\nit\ndo\nes\nnot\n on\ntain\nthe\n olor\nsequen e byy ,\nbut\nthe\n olor\nsequen e byr\nfor\nthe\nedges\njoin\nt\nwith\ntiles g\n, h,\nand i.\nThe\n ase\nof\nr\ne\nd\nb\neing\nthe\ninput\n olor\n an\nb\ne\nhandled\nsimilarly\n.\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nc\nIN\na\nb\nd\ne\nf\nh\ni\nOUT\ng\n( )\nS \nheme\nFigure\n7:\nOriginal\nsubpuzzle\nMO\nVE,\nsee\n[HH04℄\nThe\nCOPY\nsubpuzzle\nis\nused\nto\n split \na\nwire\nin\nto\nt\nw\no\n opies.\nIts\noriginal\nv\nersion\nfrom\n[HH04℄\nis\nsho\nwn\nin\nFigure\n9\n,\nand\nthe\nmo\ndi ed\nv\nersion\nis\nsho\nwn\nin\nFigure\n10\n.\nIn\nits\noriginal\nv\nersion,\ntiles a\nthrough i\nare\na\nmo\nv\ne\nto\nthe\nrigh\nt,\nmerged\nwith\na\nmo\nv\ne\nto\nthe\nleft\n onsisting\nof\ntiles a\nthrough d\nand\ntiles i\nthrough m .\nT\no\nobtain\na\nunique\nsolution,\nw\ne\ninsert\n9"},{"paragraph_id":"p21","order":21,"text":"IN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nc\nIN\na\nb\nd\nx\ne\nf\nh\ni\nOUT\ng\ny\n( )\nS \nheme\nFigure\n8:\nMo\ndi ed\nsubpuzzle\nMO\nVE\nthe\nsame\ntiles x\nand y\nas\nfor\nthe\nMO\nVE\nsubpuzzle.\nBy\nthe\nsame\nargumen\nt\nas\nab\no\nv\ne,\nthis\n xes\nthe\norien\ntation\nof\nall\nother\ntiles,\nex ept\ntile m .\nBut\ninserting\ntile z\n,\nwhi \nh\nis\nof\nthe\nsame\nt\nyp\ne\nas\ntile y\n,\nalso\n xes\nthe\norien\ntation\nof m .\nTh\nus,\nthe\nsolution\nfor\nthis\nsubpuzzle\nis\nalso\nunique\nfor\nb\noth\ninput\n olors.\nOUT\nIN\nOUT\n(a)\nIn:\ntrue\nOUT\nIN\nOUT\n(b)\nIn:\nfalse\nk\nj\nl\nm\nOUT\ni\nIN\na\nb\nc\nd\ne\ng\nh\nOUT\nf\n( )\nS \nheme\nFigure\n9:\nOriginal\nsubpuzzle\nCOPY,\nsee\n[HH04℄\nOUT\nIN\nOUT\n(a)\nIn:\ntrue\nOUT\nIN\nOUT\n(b)\nIn:\nfalse\nk\nz\nj\nl\nm\nOUT\ni\nIN\na\nb\nc\nx\nd\ne\ng\nh\nOUT\nf\ny\n( )\nS \nheme\nFigure\n10:\nMo\ndi ed\nsubpuzzle\nCOPY\nT\no\nrealize\nwire\n rossings,\nHolzer\nand\nHolzer\n[HH04℄\nused\na\n ir uit\n onstru tion\n onsisting\n10"},{"paragraph_id":"p22","order":22,"text":"of\nAND\nand\nNOT\ngates\nthat\nis\nbased\non\nGolds \nhlager's\npro\n edure\n[Gol77\n℄\nand\nM Coll's\n ross-o\nv\ner \ngadget\n[M C81\n℄.\nThis\napproa \nh\nw\nas\nalso\ntak\nen\nin\na\npreliminary\nv\nersion\nof\nthis\npap\ner\n[BR07℄.\nAs\nmen\ntioned\nab\no\nv\ne,\nho\nw\nev\ner,\nw\ne\nhere\nsimplify\nthe\n onstru tion\nb\ny\nsim\nulating\nwire\n rossings\ndire tly\n.\nT\no\nthis\nend,\nw\ne\nin\ntro\ndu e\na\nnew\nT\nan\ntrix\nTM\nsubpuzzle\n alled\nCR\nOSS,\nwhi \nh\nis\npresen\nted\nin\nFigure\n11\n.\nThis\nsubpuzzle\nhas\nt\nw\no\ninputs\nand\nt\nw\no\noutputs,\nwhere\nthe\nleft\noutput\nwill\nb\ne\nthe\nsame\nas\nthe\nrigh\nt\ninput\nand\nvi e\nv\nersa.\nJust\nas\nall\nour\nmo\ndi ed\nsubpuzzles,\nour\nno\nv\nel\nCR\nOSS\nsubpuzzle\nhas\nunique\nsolutions\nfor\nea \nh\n om\nbination\nof\np\nossible\ninput\n olors.\nT\no\nanalyze\nthis\nsubpuzzle,\nw\ne\nsub\ndivide\nit\nin\nto\nthree\ndi eren\nt\nparts.\nThe\nlo\nw\ner\npart\n onsists\nof\ntiles a\nthrough k\n,\nand\nthe\nupp\ner\npart\nof\ntiles l\nthrough t .\nLet\nus\n onsider\nthe\nupp\ner\npart\nof\nthe\nCR\nOSS\nsubpuzzle\n rst.\nTiles l\nthrough o\nform\nthe\nleft\noutput.\nSin e\ntile j\ndo\nes\nnot\n on\ntain\ngr\ne\nen,\nthe\ninput\n olor\nto\nthis\npart\nwill\nb\ne\neither\nblue\nor\nyel\nlow.\nIf\nthe\ninput\n olor\nis\nblue,\nall\ntiles\nm\nust\nha\nv\ne\nv\nerti al\nblue\nlines,\nand\nsin e\ntile\no\ndo\nes\nnot\n on\ntain\nyel\nlow\nlines,\nthe\norien\ntation\nof\nall\ntiles\nis\n xed\nwith\ngr\ne\nen\nat\nthe\njoin\nt\nedges\nof,\nresp\ne tiv\nely\n,\ntiles o\nand n ,\ntiles n\nand m ,\nand\ntiles m\nand l\n.\nThe\n ase\nof\nyel\nlow\nb\neing\nthe\ninput\n olor\nfor\nthis\npart\n an\nb\ne\nhandled\nsimilarly\nand\nyields\nthe\noutput\n olor\nr\ne\nd.\nThe\nrigh\nt\noutput\n onsists\nof\ntiles p\nthrough t .\nHere,\nthe\ninput\n olors\nr\ne\nd,\nblue,\nand\nyel\nlow\nare\np\nossible,\nwhere\nyel\nlow\nand\nr\ne\nd\nb\noth\nlead\nto\noutput\n olor\nr\ne\nd.\nSin e\ntile s\n on\ntains\nno\nr\ne\nd\nlines,\nw\ne\nobtain\nunique\nsolutions\nfor\nall\np\nossible\n om\nbinations\nof\ninput\n olors,\nb\ny\na\nsimilar\nargumen\nt\nas\nfor\nthe\nleft\noutput.\nW\ne\nno\nw\nturn\nto\nthe\nmore\n ompli ated\nlo\nw\ner\npart\nof\nthis\nsubpuzzle.\nTiles a, b,\nand c\nmo\nv\ne\nthe\nleft\ninput\n olor\nto\nthe\njoin\nt\nedge\nof\ntiles b\nand g\n.\nRegarding\ntiles a, b,\nand c,\nthere\nare\nt\nw\no\np\nossible\nsolutions\nfor\nea \nh\ninput\n olor.\nThe\np\nossible\n olor\nsequen es\nfor\nthe\nedges\nof\ntiles c\nand b\njoin\nt\nwith\ntile g\nare by\nand br\nfor\ninput\n olor\nblue,\nand\nare ry\nand rb\nfor\ninput\n olor\nr\ne\nd.\nSin e g\n on\ntains\nexa tly\none\nof\nthese\n olor\nsequen es\nfor\nea \nh\ninput\n olor,\nthe\norien\ntation\nof\ntiles a, b,\nand c\nis\n xed,\nand\nthe\ninput\n olor\nis\npassed\non\nto\nthe\njoin\nt\nedge\nof\ntiles b\nand g\n.\nThe\nsame\nargumen\nt\napplies\nto\nthe\nrigh\nt\ninput,\nwhi \nh\n onsists\nof\ntiles d,\ne, f\n,\nand h,\nsin e\nthey\nare\nmirror-symmetri al.\nIf\nb\noth\ninputs\nare\nblue,\nthe\norien\ntations\nof\ntiles g\nand h\nis\n xed\nb\ny\ntiles b, c, e,\nand f\n,\nresp\ne tiv\nely\n.\nBoth g\nand h\nwill\nha\nv\ne\nyel\nlow\nat\ntheir\nedges\njoin\nt\nwith\ntile i,\nwhi \nh\nleads\nto\nr\ne\nd\nat\nthe\nedges\nof\ntile i\njoin\nt\nwith\ntiles j\nand k\n,\nresp\ne tiv\nely\n,\nand\nthe\norien\ntation\nof\nall\ntiles\nin\nthe\nlo\nw\ner\npart\nis\n xed.\nThe\n onne tions\nb\net\nw\neen\nthe\nlo\nw\ner\nand\nthe\nupp\ner\npart\nare\nthe\njoin\nt\nedges\nof\ntiles j\nand l\nand\nof\ntiles k\nand p ,\nresp\ne tiv\nely\n.\nThese\nedges\nare\nb\noth\nblue\nif\nb\noth\ninput\n olors\nare\nblue.\nNo\nw\n onsider\nthe\n ase\nthat\nb\noth\ninput\n olors\nare\nr\ne\nd.\nThe\np\nossible\n olors\nfor\nthe\njoin\nt\nedges\nof\ntiles g\nand h\nwith\ntile i\nare\nr\ne\nd\nand\nblue.\nClearly\n,\nit\nis\nnot\np\nossible\nthat\nthey\nb\noth\nare\nblue.\nF\nurthermore,\nit\nis\nnot\np\nossible\nthat\nthe\njoin\nt\nedge\nof\ntiles g\nand i\nis\nblue,\nsin e\nin\nthis\n ase\nthe\njoin\nt\nedge\nof\ntiles j\nand l\nw\nould\nha\nv\ne\nto\nb\ne\nr\ne\nd,\nwhi \nh\nis\nnot\np\nossible\nb\ne ause l\ndo\nes\nnot\n on\ntain\nr\ne\nd.\nF\nor\nthe\nsame\nreason\nit\nis\nnot\np\nossible\nthat\nthe\njoin\nt\nedges\nof\ntiles g\nand h\nwith i\nare\nb\noth\nr\ne\nd,\nso\nthe\nonly\np\nossible\nsolution\nis\nthat\nthe\njoin\nt\nedge\nof\ntiles g\nand i\nis\nr\ne\nd\nand\nthat\nthe\njoin\nt\nedge\nof\ntiles h\nand i\nis\nblue.\nThis\nalso\nuniquely\ndetermines\nthe\norien\ntation\nof\ntiles j\nand k\nwith\nyel\nlow\nat\nthe\njoin\nt\nedge\nof\ntiles j\nand l\nand\nr\ne\nd\nat\nthe\njoin\nt\nedge\nof\ntiles k\nand p .\nThe\n ases\nthat\none\ninput\nis\nr\ne\nd\nand\nthe\nother\none\nis\n11"},{"paragraph_id":"p23","order":23,"text":"blue\n an\nb\ne\nhandled\nb\ny\nsimilar\nargumen\nts\nto\nsho\nw\nthat\nw\ne\nthen\nha\nv\ne\nunique\nsolutions\nas\nw\nell.\nIN\nOUT\nIN\nOUT\n(a)\nIn:\ntrue,\ntrue\nIN\nOUT\nIN\nOUT\n(b)\nIn:\ntrue,\nfalse\nIN\nOUT\nIN\nOUT\n( )\nIn:\nfalse,\ntrue\nIN\nOUT\nIN\nOUT\n(d)\nIn:\nfalse,\nfalse\nIN\na\nc\nm\no\nOUT\nb\ng\nj\nl\nn\ni\ne\nh\nk\np\nr\nIN\nd\nf\nq\nt\nOUT\ns\n(e)\nS \nheme\nFigure\n11:\nSubpuzzle\nCR\nOSS\n3.2\nGate\nSubpuzzles\nThe\ngates\nof\nthe\nb\no\nolean\n ir uit\nare\nsim\nulated\nb\ny\nthe\n orresp\nonding\nAND\nand\nNOT\nsub-\npuzzles.\nThe\noriginal\nv\nersion\nof\nthe\nNOT\nsubpuzzle\nfrom\n[HH04℄\nis\npresen\nted\nin\nFigure\n12\n,\nand\nour\nnew\nv\nersion\nis\nsho\nwn\nin\nFigure\n13\n.\nThe\npurp\nose\nof\nthis\nsubpuzzle\nis\nto\n negate \nthe\ninput\n olor,\ni.e.,\nif\nthe\ninput\n olor\nis\nr\ne\nd\nthen\nthe\noutput\n olor\nwill\nb\ne\nblue,\nand\nvi e\nv\nersa.\nIn\nthe\noriginal\nsubpuzzle,\nthere\nis\nonly\none\np\nossible\nsolution\nto\nthe\nthree\nRonds e, f\n,\nand g\n:\nSin e\ntiles c, b,\nand f\ndo\nnot\n on\ntain\ngr\ne\nen,\nthe\njoin\nt\nedge\nof\ntiles e\nand g\nm\nust\nb\ne\ngr\ne\nen,\nwhi \nh\nfor es\nthe\njoin\nt\nedge\nof\ntiles g\nand f\nto\nb\ne\nyel\nlow.\nSo\nthe\norien\ntation\nof\ntiles\ne, f\n,\nand g\nis\n xed,\nwith\nr\ne\nd\nat\nthe\nedges\njoin\nt\nwith\ntiles b\nand c,\nresp\ne tiv\nely\n.\nThere\nare\nonly\nt\nw\no\np\nossible\norien\ntations\nleft\nfor\nthe\ntiles b\nand c,\none\nfor\ninput\n olor\nblue\nand\none\nfor\ninput\n olor\nr\ne\nd.\nThe\nonly\ntiles\nstill\nha\nving\nmore\nthan\none\np\nossible\norien\ntation\nare a\nand d.\n12"},{"paragraph_id":"p24","order":24,"text":"IN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nf\ng\ne\nIN\na\nb\nc\nd\nOUT\n( )\nS \nheme\nFigure\n12:\nOriginal\nsubpuzzle\nNOT,\nsee\n[HH04℄\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nf\ng\ne\nIN\na\nb\nc\nd\nOUT\nx\ny\n( )\nS \nheme\nFigure\n13:\nMo\ndi ed\nsubpuzzle\nNOT\nThey\nwill\nb\ne\n xed\nb\ny\ninserting\ntiles x\nand y\n.\nThe\np\nossible\n olor\nsequen es\nfor\nthe\nedges\nof\ntiles a\nand b\njoin\nt\nwith\ntile x\nare yy\nor ry\nif\nthe\ninput\n olor\nis\nblue,\nand\nthey\nare bb\nor\nyb\nif\nthe\ninput\n olor\nis\nr\ne\nd.\nTile x,\nho\nw\nev\ner,\n on\ntains\nonly\nthe\nsequen es yy\nand bb,\nso\nthe\norien\ntation\nis\n xed.\nNote\nthat\ntiles c\nand d\nb\neha\nv\ne\njust\nlik\ne\na\nWIRE\nsubpuzzle\nand\nsin e\nthe\norien\ntation\nof\ntile c\nis\n xed\nwith\nr\ne\nd\nat\nthe\nedge\njoin\nt\nwith\ntile e,\nw\ne\ninsert\na\nRond\nat\np\nosition y\n on\ntaining\nthe\n olor\nsequen e yy\nbut\nnone\nof\nthe\nsequen es yb\nand yr .\nWith\nthese\nt\nw\no\nnew\ntiles,\nunique\nsolutions\nare\nenfor ed\nfor\nea \nh\ninput\n olor.\nThe\nsomewhat\nmore\n ompli ated\nAND\nsubpuzzle\nis\nsho\nwn\nin\nFigure\n14\nin\nits\noriginal\nv\nersion\nfrom\n[HH04℄,\nwhile\nFigure\n15\npresen\nts\nour\nmo\ndi ed\nv\nersion.\nThis\nsubpuzzle\n an\nagain\nb\ne\nsub\ndivided\nin\nto\nt\nw\no\ndi eren\nt\nparts,\nan\nupp\ner\npart\nand\na\nlo\nw\ner\npart\nthat\nare\n onne ted\nat\nthe\njoin\nt\nedge\nof\ntiles c\nand j\n.\nFirst,\nw\ne\n onsider\nthe\nupp\ner\npart\nof\nthe\nmo\ndi ed\nAND\nsubpuzzle.\nW\ne\nsho\nw\nthat\nits\noutput\n olor\nis\nr\ne\nd\nif\nthe\njoin\nt\nedge\nof\ntiles j\nand c\nis\nyel\nlow,\nand\nthat\nits\noutput\n olor\nis\nblue\nif\nthis\nedge\nis\nblue.\nNote\nthat,\njust\nas\nfor\nthe\nNOT\nsubpuzzle,\nthe\nRonds o, p ,\nand q\nha\nv\ne\nonly\none\np\nossible\norien\ntation,\nth\nus\nfor ing\nthe\nedges\nof\ntiles n\nand m\njoin\nt\nwith\ntile o\nto\nb\ne\nyel\nlow.\nIf\nthe\ninput\nto\nthis\npart\nof\nthe\nsubpuzzle\nis\nblue\nthen\ntiles j\nand k\nm\nust\nha\nv\ne\na\nv\nerti al\nblue\nline,\nand\nsin e\nthe\nedges\nof\ntile l\njoin\nt\nwith j\nand k\n annot\nb\ne\nblue,\ntiles l\n, m ,\nand n\nm\nust\nha\nv\ne\nv\nerti al\nblue\nlines,\nto\no.\nSin e\nthe\nedges\nof\ntiles n\nand m\njoin\nt\nwith\ntile o\n13"},{"paragraph_id":"p25","order":25,"text":"IN\nOUT\nIN\n(a)\nIn:\ntrue,\ntrue\nIN\nOUT\nIN\n(b)\nIn:\ntrue,\nfalse\nIN\nOUT\nIN\n( )\nIn:\nfalse,\ntrue\nIN\nOUT\nIN\n(d)\nIn:\nfalse,\nfalse\nh\nIN\na\nb\nc\nj\nk\ng\nl\nm\nn\nOUT\nf\no\nIN\nd\ne\np\nq\ni\n(e)\nS \nheme\nFigure\n14:\nOriginal\nsubpuzzle\nAND,\nsee\n[HH04℄\nare\nyel\nlow,\nthe\norien\ntation\nof\nall\nother\ntiles\nis\nuniquely\ndetermined\nin\nthe\nupp\ner\npart.\nThe\n ase\nof\nyel\nlow\nb\neing\nthe\ninput\n olor\nfor\nthe\nupp\ner\npart\n an\nb\ne\nhandeld\nsimilarly:\nWithout\nan\ny\nmo\ndi ation,\nw\ne\nha\nv\ne\nunique\nsolutions\nfor\nthe\nupp\ner\npart\nof\nthe\nsubpuzzle.\nNo\nw\nw\ne\n onsider\nthe\nlo\nw\ner\npart\nof\nthis\nsubpuzzle.\nF\nor\nea \nh\n om\nbination\nof\ninput\n olors,\ntiles a\nand d\n(whi \nh\nare\nadja en\nt\nto\nthe\ninput\ntiles\nof\nthe\nAND\nsubpuzzle)\nand\ntiles h\nand i\n(ha\nving\nonly\none\n onne ting\nedge\nto\nthe\nrest\nof\nthe\noriginal\nAND\nsubpuzzle\nin\nFigure\n14)\nea \nh\nha\nv\ne\nt\nw\no\np\nossible\norien\ntations.\nExamining\nall\np\nossible\n olor\nsequen es\nfor\nea \nh\n om\nbination\nof\np\nossible\ninput\n olors,\ntiles x\nand y\n an\nb\ne\ndetermined\nas\nsho\nwn\nin\nFigure\n15\nto\n x\nthe\norien\ntation\nof\ntiles a\nand h\nand\nof\ntiles d\nand i,\nresp\ne tiv\nely\n.\nThen,\nthe\ninput\n olor\nis\npassed\non\nto\nthe\njoin\nt\nedge\nof\ntiles b\nand c,\nand\nto\nthe\njoin\nt\nedge\nof\ntiles e\nand f\n.\nTile g\nin\nthe\n en\nter\nof\nthe\nsubpuzzle\nhas\nagain\nt\nw\no\np\nossible\norien\ntations\nfor\nea \nh\n om\nbination\nof\ninput\n olors.\nNote\nthat\nour\nmo\ndi ations\nmade\nso\nfar namely\n,\ninserting\nnew\ntiles\nin\nto\nthe\nsubpuzzles do\nnot\nw\nork\nhere,\nas\nit\nis\nnot\np\nossible\nto\ninsert\na\nnew\ntile\nin\nthe\nneigh\nb\norho\no\nd\nof\ntile g\n,\nand\nth\nus\nw\ne\nha\nv\ne\nto\nrepla e\nit.\nAs\nw\ne\nha\nv\ne\nseen\nin\nthe\nanalysis\nof\nthe\nupp\ner\npart,\nthe\n olor\nyel\nlow\nof\ntile j\nis\nonly\nused\nfor\nthe\nedge\njoin\nt\nwith\ntile c.\nW\ne\nwill\nrepla e\nthe\n olor\nyel\nlow\nof\nb\noth\ntiles\nwith\nthe\n olor\ngr\ne\nen.\nThis\nis\np\nossible,\n14"},{"paragraph_id":"p26","order":26,"text":"IN\nOUT\nIN\n(a)\nIn:\ntrue,\ntrue\nIN\nOUT\nIN\n(b)\nIn:\ntrue,\nfalse\nIN\nOUT\nIN\n( )\nIn:\nfalse,\ntrue\nIN\nOUT\nIN\n(d)\nIn:\nfalse,\nfalse\nx\nh\nIN\na\nb\nc\nj\nk\ng\nl\nm\nn\nOUT\nf\no\nIN\nd\ne\np\nq\ny\ni\n(e)\nS \nheme\nFigure\n15:\nMo\ndi ed\nsubpuzzle\nAND\nb\ne ause\nthe\njoin\nt\nedge\nof\ntiles c\nand b\nwill\nnev\ner\nb\ne\nyel\nlow,\nand\ntile g\nwill\nb\ne\nrepla ed\nb\ny\na\nnew\none.\nT\no\ndo\nthis,\nw\ne\n onsider\nall\np\nossible\n olors\nat\nthe\nedges\nof\ntiles c\nand f\njoin\nt\nwith\ntile g\n,\nwith\nthe\nrestri tion\nthat\nthe\njoin\nt\nedge\nof\ntiles c\nand j\nm\nust\nb\ne\neither\nblue\nor\ngr\ne\nen.\nIf\nthe\nrigh\nt\ninput\n olor\nis\nblue\nthen\nthe\njoin\nt\nedge\nof\ntiles f\nand g\n an\nb\ne\neither\nblue\nor\nyel\nlow,\nand\nif\nthe\nrigh\nt\ninput\n olor\nis\nr\ne\nd\nthen\nthis\nedge\n an\nb\ne\neither\nr\ne\nd\nor\nyel\nlow.\nF\nor\nthe\nleft\ninput,\nthe\njoin\nt\nedge\nof\ntiles c\nand g\nm\nust\nb\ne\nr\ne\nd\nif\nthe\ninput\nis\nr\ne\nd,\nb\ne ause\nthen\nthe\njoin\nt\nedge\nof\ntiles c\nand j\n annot\nb\ne\nr\ne\nd.\nIf\nb\noth\ninputs\nare\nblue\nthen\nthe\noutput\n olor\nof\nthe\nlo\nw\ner\npart\nm\nust\nb\ne\nblue\nas\nw\nell,\nand\nthen\nthe\njoin\nt\nedge\nof\ntiles c\nand g\nis\ngr\ne\nen.\nIf\nonly\nthe\nleft\ninput\nis\nblue,\nthe\noutput\n olor\nof\nthe\nlo\nw\ner\npart\nm\nust\nb\ne\ngr\ne\nen,\nand\nthe\njoin\nt\nedge\nof\ntiles c\nand g\nis\nblue.\nThis\nleads\nto\nthe\nfollo\nwing\nrestri tions\nfor\ntile g\n:\n1.\nIf\nthe\ninput\n olors\nare\nb\noth\nblue\nthen g\nm\nust\n on\ntain\nexa tly\none\nof\nthe\n olor\nsequen es\ngxb\nor gxy,\nwhere x\nstands\nfor\nan\narbitrary\n olor.\n2.\nIf\nthe\nrigh\nt\ninput\n olor\nis\nr\ne\nd\nand\nthe\nleft\ninput\n olor\nis\nblue, g\nm\nust\n on\ntain\neither\nbxr\nor bxy.\n15"},{"paragraph_id":"p27","order":27,"text":"3.\nIf\nthe\nrigh\nt\ninput\n olor\nis\nblue\nand\nand\nthe\nleft\ninput\n olor\nis\nr\ne\nd\nthen\nthe\np\nossible\n olor\nsequen es\nfor g\nare\neither rxb\nor rxy.\n4.\nF\nor\nthe\nlast\n om\nbination\nof\nb\noth\ninput\n olors\nb\neing\nr\ne\nd,\ntile g\nm\nust\n on\ntain\neither rxr\nor rxy.\nIn\nour\nmo\ndi ed\nAND\nsubpuzzle,\nw\ne\ninsert\na\nSin\nt\ninstead\nof\na\nRond,\nwhi \nh\n on\ntains\nea \nh\nof\nthe\n olor\nsequen es gxb , bxr, rxb,\nand rxr\nexa tly\non e.\nThis\nleads\nto\nunique\nsolutions\nfor\nthe\nlo\nw\ner\npart\nand\nth\nus\nfor\nthe\nwhole\nsubpuzzle,\nfor\nea \nh\np\nossible\n om\nbination\nof\ninput\n olors.\n3.3\nInput\nand\nOutput\nSubpuzzles\nThe\nv\nariables\nof\nthe\nb\no\nolean\n ir uit\nare\nrepresen\nted\nb\ny\nthe\nsubpuzzle\nBOOL.\nHolzer\nand\nHolzer\n[HH04℄\nsho\nw\ned\nthat\nthis\nsubpuzzle\nhas\nonly\nt\nw\no\nsolutions,\none\nwith\noutput\n olor\nblue\nand\nthe\nother\none\nwith\noutput\n olor\nr\ne\nd.\nThis\noutput\n olor\n orresp\nonds\nto\nthe\nb\no\nolean\nv\nalue\nof\nthe\n orresp\nonding\ninput\nv\nariable,\nwhere\nblue\nstands\nfor\nthe\ntruth\nv\nalue\ntrue,\nand\nr\ne\nd\nstands\nfor\nfalse.\nThe\nrestri tion\nthat\nyel\nlow\nand\ngr\ne\nen\nare\nnot\np\nossible\nas\noutput\n olors\nfor\nthis\nsubpuzzle,\nensures\nthat\nthe\nfollo\nwing\nsubpuzzles\nwill\nalw\na\nys\nha\nv\ne\na\nv\nalid\ninput\n olor,\nnamely\nblue\nor\nr\ne\nd.\nThe\nlast\nsubpuzzle\nneeded\nto\nsim\nulate\na\nb\no\nolean\n ir uit\nis\nthe\nsubpuzzle\nTEST.\nIt\nis\npla ed\nat\nthe\noutput\ngate\nof\nthe\n ir uit,\nand\nits\npurp\nose\nis\nto\nv\nerify\nthat\nthe\n ir uit\nev\nalutes\nto\ntrue.\nHolzer\nand\nHolzer\n[HH04℄\nmen\ntion\nthat\nthis\nsubpuzzle\nhas\nonly\none\nv\nalid\nsolution\nwith\nblue\nas\nthe\ninput\n olor.\nThis\nensures\nthat\nthe\noutput\nof\nthe\nwhole\n ir uit\nwill\nb\ne\ntrue\nif\nand\nonly\nif\nthe\nT\nan\ntrix\nTM\npuzzle\n onstru ted\nhas\na\nsolution.\nLik\ne\nthe\noriginal\nsubpuzzle\nBOOL,\nthe\noriginal\nTEST\nsubpuzzle\nalready\nhas\na\nunique\nsolution,\nso\nit\nis\nnot\nmo\ndi ed.\nThese\nt\nw\no\nsubpuzzles\nare\nthe\nonly\nones\nfrom\n[HH04℄\nthat\nare\nnot\nmo\ndi ed.\nF\nor\nthe\nsak\ne\nof\n ompleteness,\nthey\nare\npresen\nted\nin\nFigure\n16.\nOUT\n(a)\nBOOL\nOut:\ntrue\nOUT\n(b)\nBOOL\nOut:\nfalse\nIN\n( )\nTEST\nFigure\n16:\nSubpuzzles\nBOOL\nand\nTEST,\nsee\n[HH04℄\nThe\nshap\nes\nof\nour\nmo\ndi ed\nsubpuzzles\nha\nv\ne"},{"paragraph_id":"p28","order":28,"text":"hanged\nsligh\ntly\n,\nso\nit\nmigh\nt\nb\ne\np\nossible\nthat\nunin\ntended\nin\ntera tions\nb\net\nw\neen\nt\nw\no\nneigh\nb\noring\nsubpuzzles\ndo\no\n ur.\nHo\nw\nev\ner,\nas\nnoted\nb\ny\nHolzer\nand\nHolzer\n[HH04℄,\nthe\nminimal\nhorizon\ntal\ndistan e\nb\net\nw\neen\nt\nw\no\nwires\nand/or\n16"},{"paragraph_id":"p29","order":29,"text":"gates\nis\nat\nleast\nfour,\nand\nthis\nis\nstill\nenough\nto\nprev\nen\nt\nan\ny\nunin\ntended\nin\ntera tions\nb\net\nw\neen\nour\nmo\ndi ed\nsubpuzzles.\n3.4\nPro\nof\nof\nTheorem\n3.1\nW\ne\nare\nno\nw\nready\nto\npro\nv\ne\nTheorem\n3.1.\nLet\nSA\nT\ndenote\nthe\nsatis abilit\ny\nproblem.\nLemma\n3.2\nSA\nT\np\narsimoniously\nr\ne\ndu \nes\nto\nCir\n uit∧,¬\n-\nSA\nT\n.\nPro\nof.\nNote\nthat\nthe\nproblems\nSA\nT\nand\nCir uit\n-SA\nT\n(whi \nh\nis\nthe\nsame\nas\nCir uit∧,¬\n-SA\nT\nex ept\nwith\nOR\ngates\nallo\nw\ned\nas\nw\nell)\nare\nequiv\nalen\nt\nunder\nparsimonious\nredu tions\n[P\nap94\n℄.\nSin e\nOR\ngates\n an\nb\ne\nexpressed\nb\ny\nAND\nand\nNOT\ngates\nwithout"},{"paragraph_id":"p30","order":30,"text":"hanging\nthe\nn\num\nb\ner\nof\nsolutions,\nthis\ngiv\nes\na\nparsimonious\nredu tion\nfrom\nSA\nT\nto\nCir uit∧,¬\n-SA\nT\n.\n❑\nNo\nw,\nthe\nparsimonious\nredu tion\nfrom\nSA\nT\nto\nTRP\nimmediately\nfollo\nws\nfrom\nLemma\n3.2\nand\nthe\n onstru tion\nand\nthe\nargumen\nts\npresen\nted\nin\nSe tions\n3.1,\n3.2,\nand\n3.3.\nThat\nis,\nb\ny\nour\nmo\ndi ations,\nfor\nea \nh\nsatisfying\nassignmen\nt\nto\nthe\n ir uit\nthere\nis\nexa tly\none\nsolution\nto\nthe\nT\nan\ntrix\nTM\npuzzle\n onstru ted.\n4\nThe\nUnique\nT\nan\ntrix\nTM\nRotation\nPuzzle\nProblem\nis\nDP-\nComplete\nunder\nRandomized\nRedu tions\nV\nalian\nt\nand\nV\nazirani\nin\ntro\ndu ed\nr\nandomize\nd\np\nolynomial-time\nr\ne\ndu tions\nin\ntheir\nw\nork\nsho\nw-\ning\nthat\nNP\nis\nas\neasy\nas\ndete ting\nunique\nsolutions\n[VV86℄.\nW\ne\nwill\nuse ≤p\nran\nto\nde-\nnote\ntheir\nt\nyp\ne\nof\nredu tions.\nIn\nparti ular,\nV\nalian\nt\nand\nV\nazirani\n[VV86℄\npro\nv\ned\nthat\nUnique-SA\nT\n,\nthe\nunique\nv\nersion\nof\nSA\nT\n,\nis ≤p\nran\n- omplete\nin\nDP\n(see\nalso\nChang,\nKadin,\nand\nRohatgi\n[CKR95℄).\nTheorem\n4.1\n1.\nUnique\n-SA\nT\np\narsimoniously\nr\ne\ndu \nes\nto\nUnique\n-TRP.\n2.\nUnique\n-TRP\nis\nDP- \nomplete\nunder ≤p\nran\n-r\ne\ndu tions.\nPro\nof.\nT\no\npro\nv\ne\nthe\n rst\npart,\nnote\nthat\nb\ny\nLemma\n3.2\nand\nTheorem\n3.1\n,\nw\ne\nobtain\na\nparsimonious\nredu tion\nfrom\nSA\nT\nto\nTRP.\nIt\nfollo\nws\nthat\nUnique-SA\nT\nparsimoniously\nredu es\nto\nUnique\n-TRP.\nThe\nse ond\npart\nfollo\nws\nfrom\nthe\n rst\npart\nand\nV\nalian\nt\nand\nV\nazirani's\nab\no\nv\ne-men\ntioned\nresult\nthat\nUnique-SA\nT\nis ≤p\nran\n- omplete\nin\nDP,\nand\nfrom\nthe\nob\nvious\nfa t\nthat\nUnique\n-TRP\nis\nin\nDP\n.\n❑\nA"},{"paragraph_id":"p31","order":31,"text":"kno\nwledgmen\nts:\nW\ne\nthank\nthe\nanon\nymous\nMCU\n2007\nreferees\nfor\ntheir\nhelpful\n om-\nmen\nts.\n17"},{"paragraph_id":"p32","order":32,"text":"Referen es\n[Bar04℄\nR.\nBarban \nhon.\nOn\nunique\ngraph\n3- olorabilit\ny\nand\nparsimonious\nredu tions\nin\nthe\nplane.\nThe\nor\neti \nal\nComputer\nS ien \ne,\n319(1 3):455 482,\n2004.\n[BR07℄\nD.\nBaumeister\nand\nJ.\nRothe.\nSatis abilit\ny\nparsimoniously\nredu es\nto\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nIn\nPr\no"},{"paragraph_id":"p33","order":33,"text":"e\ne\ndings\nof\nthe\n5th\nConfer\nen \ne\non\nMa hines,\nComputations\nand\nUniversality,\npages\n134 145.\nSpringer-V\nerlag\nL\ne\n -\ntur\ne\nNotes\nin\nComputer\nS ien \ne\n#4664,\nSeptem\nb\ner\n2007.\n[CGH+\n88℄\nJ.\nCai,\nT.\nGundermann,\nJ.\nHartmanis,\nL.\nHema \nhandra,\nV.\nSew\nelson,\nK.\nW\nagner,\nand\nG.\nW\ne \nhsung.\nThe\nb\no\nolean\nhierar \nh\ny\nI:\nStru tural\nprop\nerties.\nSIAM\nJournal\non\nComputing,\n17(6):1232 1252,\n1988.\n[CGH+\n89℄\nJ.\nCai,\nT.\nGundermann,\nJ.\nHartmanis,\nL.\nHema \nhandra,\nV.\nSew\nelson,\nK.\nW\nagner,\nand\nG.\nW\ne \nhsung.\nThe\nb\no\nolean\nhierar \nh\ny\nI\nI:\nAppli ations.\nSIAM\nJournal\non\nComputing,\n18(1):95 111,\n1989.\n[CKR95℄\nR.\nChang,\nJ.\nKadin,\nand\nP\n.\nRohatgi.\nOn\nunique\nsatis abilit\ny\nand\nthe\nthreshold\nb\neha\nvior\nof\nrandomized\nredu tions.\nJournal\nof\nComputer\nand\nSystem\nS ien \nes,\n50(3):359 373,\n1995.\n[CM87℄\nJ.\nCai\nand\nG.\nMey\ner.\nGraph\nminimal\nun olorabilit\ny\nis DP\n- omplete.\nSIAM\nJournal\non\nComputing,\n16(2):259 277,\nApril\n1987.\n[Co\no71℄\nS.\nCo\nok.\nThe\n omplexit\ny\nof\ntheorem-pro\nving\npro\n edures.\nIn\nPr\no"},{"paragraph_id":"p34","order":34,"text":"e\ne\ndings\nof\nthe\n3r\nd\nA\nCM\nSymp\nosium\non\nThe\nory\nof\nComputing,\npages\n151 158.\nA\nCM\nPress,\n1971.\n[Gol77℄\nL.\nGolds \nhlager.\nThe\nmonotone\nand\nplanar\n ir uit\nv\nalue\nproblems\nare\nlog\nspa e\n omplete\nfor\nP.\nSIGA\nCT\nNews,\n9(2):25 29,\n1977.\n[Grä90℄\nE.\nGrädel.\nDomino\ngames\nand\n omplexit\ny\n.\nSIAM\nJournal\non\nComputing,\n19(5):787 804,\n1990.\n[HH04℄\nM.\nHolzer\nand\nW.\nHolzer.\nT\nan\ntrix\nTM\nrotation\npuzzles\nare\nin\ntra table.\nDis r\nete\nApplie\nd\nMathemati s,\n144(3):345 358,\n2004.\n[M C81℄\nW.\nM Coll.\nPlanar\n rosso\nv\ners.\nIEEE\nT\nr\nansa tions\non\nComputers,\nC-30(3):223 \n225,\n1981.\n[P\nap94℄\nC.\nP\napadimitriou.\nComputational\nComplexity.\nA\nddison-W\nesley\n,\n1994.\n[PY84℄\nC.\nP\napadimitriou\nand\nM.\nY\nannak\nakis.\nThe\n omplexit\ny\nof\nfa ets\n(and\nsome\nfa ets\nof\n omplexit\ny).\nJournal\nof\nComputer\nand\nSystem\nS ien \nes,\n28(2):244 259,\n1984.\n[Rot03℄\nJ.\nRothe.\nExa t\n omplexit\ny\nof\nExa t-Four-Colorabilit\ny\n.\nInformation\nPr\no"},{"paragraph_id":"p35","order":35,"text":"essing\nL\netters,\n87(1):7 12,\nJuly\n2003.\n18"},{"paragraph_id":"p36","order":36,"text":"[Rot05℄\nJ.\nRothe.\nComplexity\nThe\nory\nand\nCryptolo\ngy.\nA\nn\nIntr\no\ndu tion\nto\nCrypto"},{"paragraph_id":"p37","order":37,"text":"om-\nplexity.\nEA\nTCS\nT\nexts\nin\nTheoreti al\nComputer\nS ien e.\nSpringer-V\nerlag,\nBerlin,\nHeidelb\nerg,\nNew\nY\nork,\n2005.\n[RR06℄\nT.\nRiege\nand\nJ.\nRothe.\nCompleteness\nin\nthe\nb\no\nolean\nhierar \nh\ny:\nExa t-Four-\nColorabilit\ny\n,\nminimal\ngraph\nun olorabilit\ny\n,\nand\nexa t\ndomati \nn\num\nb\ner\nproblems"},{"paragraph_id":"p38","order":38,"text":"a\nsurv\ney\n.\nJournal\nof\nUniversal\nComputer\nS ien \ne,\n12(5):551 578,\n2006.\n[V\nal79℄\nL.\nV\nalian\nt.\nThe\n omplexit\ny\nof\n omputing\nthe\np\nermanen\nt.\nThe\nor\neti \nal\nComputer\nS ien \ne,\n8(2):189 201,\n1979.\n[VV86℄\nL.\nV\nalian\nt\nand\nV.\nV\nazirani.\nNP\nis\nas\neasy\nas\ndete ting\nunique\nsolutions.\nThe\no-\nr\neti \nal\nComputer\nS ien \ne,\n47:85 93,\n1986.\n19"}],"pages":[{"page":1,"text":"arXiv:0705.0915v2 [cs.CC] 9 Jun 2008\nSatis abilit\ny\nP\narsimoniously\nRedu es\nto\nthe\nT\nan\ntrix\nTM\nRotation\nPuzzle\nProblem∗\nDorothea\nBaumeister\nand\nJörg\nRothe\nInstitut\nfür\nInformatik\nHeinri \nh-Heine-Univ\nersität\nDüsseldorf\n40225\nDüsseldorf,\nGerman\ny\nJune\n9,\n2008\nAbstra t\nHolzer\nand\nHolzer\n[HH04℄\npro\nv\ned\nthat\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nNP\n- omplete.\nThey\nalso\nsho\nw\ned\nthat\nfor\nin nite\nrotation\npuzzles,\nthis\nproblem\nb\ne omes\nunde idable.\nW\ne\nstudy\nthe\n oun\nting\nv\nersion\nand\nthe\nunique\nv\nersion\nof\nthis\nproblem.\nW\ne\npro\nv\ne\nthat\nthe\nsatis abilit\ny\nproblem\nparsimoniously\nredu es\nto\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nIn\nparti ular,\nthis\nredu tion\npreserv\nes\nthe\nuniqueness\nof\nthe\nsolution,\nwhi \nh\nimplies\nthat\nthe\nunique\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nas\nhard\nas\nthe\nunique\nsatis abilit\ny\nproblem,\nand\nso\nis\nDP\n- omplete\nunder\np\nolynomial-time\nrandom-\nized\nredu tions,\nwhere\nDP\nis\nthe\nse ond\nlev\nel\nof\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n.\nKey\nw\nords:\n omputational\n omplexit\ny\n,\nrotation\npuzzle,\ntiling\nof\nthe\nplane,\nparsimo-\nnious\nredu tion,\n oun\nting\nproblem.\n1\nIn\ntro\ndu tion\nT\nan\ntrix\nTM\nis\na\npuzzle\ngame\npla\ny\ned\nwith\nhexagonal\ntiles\n rmly\narranged\nin\nthe\nplane\nthat\nea \nh\n an\nb\ne\nrotated\naround\ntheir\naxes.\nThere\nare\nfour\ndi eren\nt\nt\nyp\nes\nof\ntiles\n( alled\nSint,\nBrid,\nChin,\nand\nR\nond,\nsee\nFigure\n1)\nthat\ndi er\nb\ny\nthe\nform\nof\nthe\nthree\n olored\nlines\nthey\nea \nh\nha\nv\ne,\nwhere\nthe\n olors\nare\n \nhosen\namong\nr\ne\nd,\nyel\nlow,\nblue,\nand\ngr\ne\nen.\nThe\nob\nje tiv\ne\nof\nthe\ngame\nis\nto\n nd\na\nrotation\nof\nthe\ngiv\nen\ntiles\nso\nas\nto\n reate\nlong\nlines\nand\nlo\nops\nof\nthe\nsame\n olor.\nSin e\nits\nin\nv\nen\ntion\nin\n1991\nb\ny\nMik\ne\nM Mana\nw\na\ny\nfrom\nNew\nZealand\nand\nits\n ommer ial\nlaun \nh,\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nhas\nb\ne ome\nextremely\np\nopular\nand\n ommer ially\nsu essful.\n∗\nSupp\norted\nin\npart\nb\ny\nDF\nG\ngran\nts\nR\nO\n1202/9-3\nand\nR\nO\n1202/11-1\nand\nb\ny\nthe\nAlexan-\nder\nv\non\nHum\nb\noldt\nF\noundation's\nT\nransCo\nop\nprogram.\nA\npreliminary\nv\nersion\nof\nthis\npap\ner\nap-\np\neared\nin\nthe\nPro\n eedings\nof\nMa hines,\nComputations\nand\nUniversality\n(MCU\n2007).\nURLs:\n . s.uni-duesseldorf.de/∼\n{baumeister,\nrothe}\n(D.\nBaumeister\nand\nJ.\nRothe).\n1"},{"page":2,"text":"Holzer\nand\nHolzer\n[HH04℄\n onsidered\nt\nw\no\nv\narian\nts\nof\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem,\none\nwith\n nitely\nman\ny\nand\none\nwith\nin nitely\nman\ny\ntiles\nin\na\ngiv\nen\nproblem\ninstan e.\nThey\npro\nv\ned\nthat\nthe\n nite\nv\narian\nt\nof\nthis\nproblem\nis\nNP\n- omplete\nb\ny\nredu ing\nthe\nNP- omplete\nb\no\nolean\n ir uit\nsatis abilit\ny\nproblem\n(restri ted\nto\n ir uits\nwith\nAND\nand\nNOT\ngates\nonly)\nto\nit.\nThey\nalso\nsho\nw\ned\nthat\nthe\nin nite\nv\narian\nt\nof\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nunde idable,\nagain\nemplo\nying\na\n ir uit\n onstru tion.\nF\nor\nother\nresults\non\nthe\n omplexit\ny\nof\nproblems\nrelated\nto\nDomino-lik\ne\nstrategy\ngames,\nw\ne\nrefer\nto\nGrädel\n[Grä90\n℄.\nW\ne\n onsider\nt\nw\no\nv\narian\nts\nof\nthe\n nite\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem,\nits\n oun\nting\nv\nersion\nand\nits\nunique\nv\nersion.\nThe\n oun\nting\nproblem\nasks\nfor\nthe\nn\num\nb\ner\nof\nsolutions\nof\na\ngiv\nen\nrotation\npuzzle\ninstan e.\nThe\nunique\nproblem\nasks\nwhether\na\ngiv\nen\nrotation\npuzzle\ninstan e\nhas\nexa tly\none\nsolution.\nOur\nmain\nresult\nis\nthat\nthe\nsatis abilit\ny\nproblem\nparsimoniously\nredu es\nto\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nThe\n lass #P\nw\nas\nin\ntro\ndu ed\nb\ny\nV\nalian\nt\n[V\nal79\n℄\nto\n apture\nthe\n omplexit\ny\nof\n oun\nting\nthe\nsolutions\nof\nNP\nproblems.\nP\narsimonious\nredu tions\nb\net\nw\neen\nNP\n oun\nting\nproblems \nsu \nh\nas\nours preserv\ne\nthe\npre ise\nn\num\nb\ner\nof\nsolutions.\nThis\nis\nan\nimp\nortan\nt\nprop\nert\ny\nfor\nat\nleast\nt\nw\no\nreasons.\nFirst,\nthe\nstru ture\nof\nthe\nsolution\nspa e\nis\npreserv\ned\nb\ny\na\nparsimonious\nredu tion\nfrom A\nto B\n,\nsin e\nsolutions\nof A\nare\nmapp\ned\nbije tiv\nely\nto\nsolutions\nof B\nin\np\nolynomial\ntime.\nSe ond,\nparsimonious\nredu tions\n an\nb\ne\nused\nto\npro\nv\ne\nlo\nw\ner\nb\nounds\nfor\nthe\nunique\nv\nersions\nof\nNP\nproblems.\nIn\nparti ular,\nw\ne\napply\nour\nab\no\nv\ne-men\ntioned\nparsimonious\nredu tion\nto\npro\nv\ne\nthat\nthe\nunique\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nDP\n- omplete\nunder\np\nolynomial-time\nrandomized\nredu tions\nin\nthe\nsense\nof\nV\nalian\nt\nand\nV\nazirani\n[VV86℄.\nHere,\nDP\nis\nthe\nset\nof\ndi eren es\nof\nan\ny\nt\nw\no\nNP\nsets\n[PY84\n℄;\nso\nNP ⊆\nDP,\nand\nit\nis\n onsidered\nmost\nunlik\nely\nthat\nb\noth\n lasses\nare\nequal.\nNote\nthat\nDP\nis\nthe\nse ond\nlev\nel\nof\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n,\nsee\nCai\net\nal.\n[CGH+\n88\n,\nCGH+\n89℄.\nF\nurther\nresults\non\nDP\nand\n ompletenes\nin\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n an\nb\ne\nfound,\ne.g.,\nin\n[CM87\n,\nRot03℄,\nsee\nalso\nthe\nsurv\ney\n[RR06℄.\nWhile\nman\ny\nstandard\nredu tions\nb\net\nw\neen\nNP- omplete\nproblems\nare\neasily\nseen\nto\nb\ne\nparsimonious,\nthere\nare\na\nn\num\nb\ner\nof\nex eptions.\nF\nor\nexample,\nBarban \nhon\n[Bar04℄\nsho\nw\ned\nthat\nthe\n(planar)\nsatis abilit\ny\nproblem\nis\nparsimoniously\np\nolynomial-time\nredu ible\nto\nthe\n(planar) 3- olorabilit\ny\nproblem\nvia\na\nrather\nsophisti ated\n onstru tion.\nOther\nexamples\nof\nnon\ntrivial\nparsimonious\nredu tions\n an\nb\ne\nfound\nin\n[P\nap94\n℄.\nHolzer\nand\nHolzer's\nredu tion,\nho\nw\nev\ner,\nis\nnot\nparsimonious\n[HH04℄.\nThe\nmain\npurp\nose\nof\nthis\npap\ner\nis\nto\nsho\nw\nho\nw\nto\nmo\ndify\ntheir\nredu tion\nso\nas\nto\nmak\ne\nit\nparsimonious.\nW\ne\nmen\ntion\nin\npassing\nthat\nthis\npap\ner\ndi ers\nfrom\nits\npreliminary\nv\nersion\n[BR07\n℄\nin\nv\narious\nw\na\nys.\nFirst,\nto\nallo\nw\n omparison,\nw\ne\nhere\nexpli itly\nsho\nw\nthe\ndi eren es\nb\net\nw\neen\nHolzer\nand\nHolzer's\noriginal\n onstru tion\n[HH04℄\nand\nour\nmo\ndi ed\n onstru tion\nb\ny\n(a)\npre-\nsen\nting\ntheir\nsubpuzzles\n(mark\ned\nso\nas\nto\n learly\nindi ate\nthe\ntiles\nthat\nrequire\nmo\ndi ation\nif\none\naims\nat\na\nparsimonious\nredu tion),\nand\n(b)\nhighligh\nting\nall\nmo\ndi ed\nor\nadditionally\ninserted\ntiles\nin\nour\nsubpuzzles.\nSe ond,\nunlik\ne\nthe\nredu tions\ngiv\nen\nin\n[HH04,\nBR07℄,\nin\nthe\nredu tion\npresen\nted\nhere\nw\ne\nin\ntro\ndu e\na\nnew\nsubpuzzle\nfor\nsim\nulating\nwire\n rossings\nin\nb\no\nolean\n ir uits.\nThis\nnew\nsubpuzzle,\nwhi \nh\nw\ne\n all\nCR\nOSS,\nwill\nsa\nv\ne\nus\nthe\ne ort\nof\ntransforming\ngeneral\nb\no\nolean\n ir uits\nin\nto\nplanar\nb\no\nolean\n ir uits\n(i.e.,\nin\nto\n ir uits\nwithout\n2"},{"page":3,"text":"wire\n rossings).\nHen e,\nthe\nredu tion\npro\nvided\nin\nthe\npresen\nt\npap\ners\nis\nmore\ne ien\nt\nand\nthe\ntotal\nn\num\nb\ner\nof\ntiles\nneeded\nto\nsim\nulate\na\ngiv\nen\n ir uit\nis\n onsiderably\nsmaller\nthan\nin\nour\nprevious\n onstru tion\n[BR07℄.\nFinally\n,\nto\npro\nv\ne\n orre tness\nof\nour\nredu tion,\nw\ne\nno\nw unlik\ne\nthe\napproa \nh\ntak\nen\nin\n[BR07℄ argue\nvia\n olor\nsequen es \nof\ntiles,\nwhi \nh\nfa ilitates\nreading\nand\nunderstanding\nthe\nargumen\nts.\nThis\npap\ner\nis\norganized\nas\nfollo\nws.\nIn\nSe tion\n2\n,\nw\ne\nde ne\nsome\n omplexit\ny-theoreti \nnotions\nand\nour\nv\narian\nts\nof\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nIn\nSe tion\n3,\nw\ne\npresen\nt\nour\nparsimonious\nredu tion.\nIn\nSe tion\n4,\n nally\n,\nw\ne\nstudy\nthe\nunique\nv\nersion\nof\nthis\nproblem\nand\nsho\nw\nits\nDP- ompleteness\nunder\nrandomized\nredu tions.\n2\nPreliminaries\n2.1\nDe nition\nof\nSome\nComplexit\ny-Theoreti \nNotions\nFix\nthe\nalphab\net Σ = {0, 1},\nand\nlet Σ∗\ndenote\nthe\nset\nof\nstrings\no\nv\ner Σ .\nAs\nis\n ommon,\nde ision\nproblems\nare\nsuitably\nen o\nded\nas\nlanguages\no\nv\ner Σ .\nF\nor\nan\ny\nlanguage A ⊆Σ∗\n,\nlet ∥A∥\ndenote\nthe\nn\num\nb\ner\nof\nelemen\nts\nin A.\nF\nor\nsome\nba \nkground\non\n omputational\n om-\nplexit\ny\ntheory\n,\nw\ne\nrefer\nto\nan\ny\nstandard\ntextb\no\nok\nof\nthis\n eld,\ne.g.,\n[P\nap94\n,\nRot05℄.\nLet\nNP\ndenote\nthe\n lass\nof\nproblems\nsolv\nable\nin\nnondeterministi \np\nolynomial\ntime.\nGeneralizing\nNP,\nP\napadimitriou\nand\nY\nannak\nakis\n[PY84\n℄\nin\ntro\ndu ed\nthe\n lass\nDP = {A −B | A, B ∈\nNP}\nto\n apture\nthe\n omplexit\ny\nof\nNP-hard\nor\n oNP-hard\nproblems\nthat\nseemingly\nare\nneither\nin\nNP\nnor\nin\n oNP\n.\nIn\nparti ular,\nthey\nsho\nw\ned\nthat\nDP\n on\ntains\na\nn\num\nb\ner\nof\nuniqueness\npr\noblems,\n riti \nal\ngr\naph\npr\noblems,\nand\nexa t\noptimization\npr\noblems,\nand\nthey\nsho\nw\ned\nsome\nof\nthese\nproblems\n omplete\nfor\nDP\n;\nsee\nalso\nthe\nre en\nt\nsurv\ney\n[RR06\n℄.\nDP\nw\nas\nlater\ngeneralized\nb\ny\nCai\net\nal.\n[CGH+\n88,\nCGH+\n89℄,\nwho\nin\ntro\ndu ed\nthe\nb\no\nolean\nhierar \nh\ny\no\nv\ner\nNP\n.\nNote\nthat\nDP\nis\nthe\nse ond\nlev\nel\nof\nthis\nhierar \nh\ny\n.\nIn\nhis\nseminal\npap\ner,\nV\nalian\nt\n[V\nal79\n℄\ninitiated\nthe\nstudy\nof\n oun\nting\nproblems\nand\nin-\ntro\ndu ed\nthe\nimp\nortan\nt\n oun\nting\n lass #P\n.\nMem\nb\ners\nof #P\nare\nreferred\nto\nas\nNP\n \nounting\npr\noblems.\nA\nw\nell-kno\nwn\nNP\n oun\nting\nproblem\nis # SA\nT,\nthe\n oun\nting\nv\nersion\nof\nthe\nsatis -\nabilit\ny\nproblem:\nGiv\nen\na\nb\no\nolean\nform\nula,\nho\nw\nman\ny\nsatisfying\nassignmen\nts\ndo\nes\nit\nha\nv\ne?\nDe nition\n2.1\n(V\nalian\nt\n[V\nal79\n℄)\nL\net\nNPTM\nb\ne\na\nshorthand\nfor\nnondeterministi \np\nolynomial-time\nT\nuring\nma hine.\nF\nor\nany\nNPTM M\nand\nany\ninput x,\nlet accM(x)\nde-\nnote\nthe\nnumb\ner\nof\na \n \nepting\n \nomputation\np\naths\nof M(x),\ni.e., accM\nis\na\nfun tion\nmapping\nfr\nom Σ∗\nto N .\nDe ne\nthe\nfun tion\n lass #P = {accM | M\nis\nan\nNPTM}.\nW\ne\nno\nw\nde ne\nthe\nnotion\nof\n(p\nolynomial-time)\np\narsimonious\nr\ne\ndu ibility,\nwhi \nh\nwill\nb\ne\nused\nto\n ompare\nthe\nhardness\nof\nsolving\nNP\n oun\nting\nproblems.\nIn\ntuitiv\nely\n,\nan\nNP\n oun\nting\nproblem f\nparsimoniously\nredu es\nto\nan\nNP\n oun\nting\nproblem g\nif\nthe\ninstan es\nof f\n an\nb\ne\ntransformed\nin\nto\ninstan es\nof g\nsu \nh\nthat\nthe\nn\num\nb\ner\nof\nsolutions\nof f\nare\npreserv\ned\nunder\nthis\ntransformation.\nDe nition\n2.2\nL\net f\nand g\nb\ne\nany\ntwo\ngiven\n \nounting\npr\noblems\nmapping\nfr\nom Σ∗\nto N .\nW\ne\nsay f\n(p\nolynomial-time)\nparsimoniously\nredu es\nto g\n(denote\nd\nby f ≤p\npar g\n)\nif\nther\ne\nexists\na\n3"},{"page":4,"text":"(a)\nSin\nt\n(b)\nBrid\n( )\nChin\n(d)\nRond\n(e)\nred\n(f\n)\ny\nello\nw\n(g)\nblue\n(h)\ngreen\nFigure\n1:\nT\nan\ntrix\nTM\ntiles\nand\n olors\np\nolynomial-time\n \nomputable\nfun tion ρ\nsu h\nthat\nfor\ne\na h x ∈Σ∗\n, f(x) = g(ρ(x)).\nIf F\nand\nG\nar\ne\nthe\nNP\nde\n ision\npr\noblems\n \norr\nesp\nonding\nto\nthe\nNP\n \nounting\npr\noblems f\nand g\nwith\nf ≤p\npar g\n,\nwe\nwil\nl\nalso\nsay\nthat F\np\narsimoniously\nr\ne\ndu \nes\nto G.\n2.2\nV\narian\nts\nof\nthe\nT\nan\ntrix\nTM\nRotation\nPuzzle\nProblem\nThe\nT\nan\ntrix\nTM\nrotation\npuzzle\nhas\nfour\nkinds\nof\nhexagonal\ntiles the\nSint,\nthe\nBrid,\nthe\nChin,\nand\nthe\nR\nond\n ea \nh\nof\nwhi \nh\nhas\nthree\n olored\nlines,\nwhere\nthe\n olors\nare\n \nhosen\namong\nr\ne\nd,\nyel\nlow,\nblue,\nand\ngr\ne\nen,\nsee\nFigure\n1(a) (d).\nThis\ngiv\nes\na\ntotal\nof\n56\ndi eren\nt\ntiles.\nSin e\nw\ne\naren't\nusing\na tually\n olored\n gures,\nw\ne\nen o\nde\nthe\n olors\nas\nsho\nwn\nin\nFigure\n1(e) (h).\nHolzer\nand\nHolzer\n[HH04℄\nsho\nw\ned\nthat\nthe\nde ision\nproblem\nT\nan\ntrix\nTM\nrotation\npuzzle\n(whi \nh\nw\ne\ndenote\nb\ny\nTRP,\nfor\nshort)\nis\nNP- omplete.\nIn\nthis\npap\ner,\nw\ne\nin\ntro\ndu e\nand\nstudy # TRP,\nthe\n oun\nting\nv\nersion\nof\nTRP.\nW\ne\nno\nw\nbrie y\ndes rib\ne\nthe\nformalism\nin\ntro\ndu ed\nb\ny\nHolzer\nand\nHolzer\n[HH04℄\nto\nde-\n ne\nTRP\n,\nsin e\nthe\nsame\nformalism\nis\nuseful\nfor\nde ning # TRP.\nIn\nparti ular,\nto\nrepresen\nt\nthe\ninstan es\nof\nb\noth\nthese\nproblems,\na\nt\nw\no-dimensional\nhexagonal\n o\nordinate\nsystem\nis\nused,\nsee\nFigure\n2.\nIn\nthis\nsystem,\nt\nw\no\ndistin t\npairs a = (u, w)\nand b = (v, x)\nfrom Z2\nare\nadja en\nt\nif\nand\nonly\nif\none\nof\nthe\nfollo\nwing\nfour\n onditions\nis\nsatis ed:\n1. u = v\nand |w −x| = 1,\n2. |u −v| = 1\nand w = x,\n3. u −v = 1\nand w −x = 1,\nand\n4. u −v = −1\nand w −x = −1.\nLet T\nb\ne\nthe\nset\nof\nall\nT\nan\ntrix\nTM\ntiles.\nLet A\nb\ne\na\n(partial)\nfun tion\nmapping\nthe\nelemen\nts\nof Z2\nto T\n,\ni.e.,\nfor\nthose v ∈Z2\non\nwhi \nh A\nis\nde ned, A(v)\nis\nthe\nt\nyp\ne\nof\nthe\ntile\nlo\n ated\nat v\n.\nThe\nset shape(A) = {v ∈Z2 | A(v)\nis\nde ned}\ngiv\nes\nthe\np\nositions\nin Z2\nat\nwhi \nh\ntiles\nare\npla ed.\nF\nor\nall a, b ∈shape(A), A(a)\nis\nadja en\nt\nto A(b)\nif\nand\nonly\nif a\nis\nadja en\nt\nto b.\n4"},{"page":5,"text":"x\ny\n(1, 1)\n(0, 0)\n(−1, −1)\n(1, 0)\n(0, 1)\n(−1, 0)\n(0, −1)\nFigure\n2:\nA\nt\nw\no-dimensional\nhexagonal\n o\nordinate\nsystem\nTRP\nis\nthen\nde ned\nas\nfollo\nws\n(note\nthat\nthe\ninitial\norien\ntation\nis\nnot\nsp\ne i ed,\nas\nit\ndo\nesn't\nmatter\nfor\nthe\nquestion\nof\nwhether\nthe\nde ision\nproblem\nTRP\nis\nsolv\nable)\n[HH04℄:\n1\nName:\nT\nan\ntrix\nTM\nRotation\nPuzzle\n(TRP,\nfor\nshort).\nGiv\nen:\nA\n nite\nshap\ne\nfun tion A : Z2 →T\n,\nappropriately\nen o\nded\nas\na\nstring.\nQuestion:\nIs\nthe\nrotation\npuzzle\nde ned\nb\ny A\nsolv\nable,\ni.e.,\ndo\nes\nthere\nexist\na\nrotation\nof\nthe\ngiv\nen\ntiles\nat\ntheir\np\nositions\nsu \nh\nthat\nat\nea \nh\njoin\nt\nedge\nof\nt\nw\no\nadja en\nt\ntiles\nthe\n orresp\nonding\n olors\nmat \nh?\nF\nor\nan\ny\ngiv\nen\nTRP\ninstan e A ,\na\nsolution\nof A\nis\na\nsp\ne i ation\n(in\nsome\nappropriate\nen o\nding)\nof\nea \nh\ntile\nin shape(A)\nin\nsome\nparti ular\norien\ntation\nsu \nh\nthat\nfor\nea \nh\njoin\nt\nedge\nof\nt\nw\no\nadja en\nt\ntiles\nthe\n orresp\nonding\n olors\nmat \nh.\nFigure\n3\ngiv\nes\nan\nexample\nof\na\nrotation\npuzzle\ninstan e\nand\nits\nsolution.\nLet\nSol\nTRP(A)\ndenote\nthe\nset\nof\nsolutions\nof\na\ngiv\nen\nTRP\ninstan e A .\nSo A\nis\nin\nTRP\n(view\ned\nas\na\nlanguage)\nif\nand\nonly\nif\nthe\nset\nSol\nTRP(A)\nis\nnonempt\ny\n.\n(a)\nPuzzle\n(b)\nSolution\nFigure\n3:\nAn\nexample\nof\na\nTRP\ninstan e\nand\nits\nsolution\n1\nAs\nnoted\nb\ny\nHolzer\nand\nHolzer\n[HH04℄,\nthere\nis\na\ndi eren e\nb\net\nw\neen\ntheir\nde nition\nof\nTRP,\nwhi \nh\nallo\nws\nholes\nin\nTRP\ninstan es,\nand\nthe\noriginal\nT\nan\ntrix\nTM\ngame,\nwhi \nh\ndo\nes\nnot\nallo\nw\nholes.\nThe\nproblem\nof\nwhether\nthe\nanalog\nof\nTRP\nwithout\nholes\nstill\nis\nNP\n- omplete\nis\nop\nen.\n5"},{"page":6,"text":"W\ne\nno\nw\nde ne\nthe\n oun\nting\nv\nersion\nand\nthe\nunique\nv\nersion\nof\nTRP,\nwhi \nh\nwill\nb\ne\n on-\nsidered\nin\nSe tions\n3\nand\n4.\nDe nition\n2.3\n1.\nThe\nT\nan\ntrix\nTM\nrotation\npuzzle\n oun\nting\nproblem\nis\nthe\nfun tion\n# TRP : Σ∗→N\nde ne\nd\nby\n# TRP(A) = ∥Sol\nTRP(A)∥,\nwher\ne\nwe\nassume\nthat\ninputs A\nar\ne\nappr\nopriately\nen \no\nde\nd\nas\nstrings\nin Σ∗\nand\nfun tion\nvalues\nar\ne\nnonne\ngative\ninte\ngers\n(r\nepr\nesente\nd\nin\nbinary).\n2.\nThe\nunique\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem\nis\nde ne\nd\nby\nUnique-\nTRP = {A | # TRP(A) = 1}.\n3\nSatis abilit\ny\nP\narsimoniously\nRedu es\nto\nthe\nT\nan\ntrix\nTM\nRo-\ntation\nPuzzle\nProblem\nOur\nmain\nresult\nis\nTheorem\n3.1\nthe\npro\nof\nof\nwhi \nh\nwill\nb\ne\npresen\nted\nin\nSe tions\n3.1\nthrough\n3.4\n.\nTheorem\n3.1 # SA\nT ≤p\npar # TRP\n.\nT\no\npro\nv\ne\nTRP\nNP- omplete,\nHolzer\nand\nHolzer\n[HH04℄\nga\nv\ne\na\nredu tion\nfrom\nthe\nNP-\n omplete\nproblem\nCir uit∧,¬\n-SA\nT\n(see\nCo\nok\n[Co\no71℄),\nwhi \nh\nis\nde ned\nas\nfollo\nws.\nName:\nCir uit∧,¬\n-SA\nT\n.\nGiv\nen:\nA\nb\no\nolean\n ir uit C\nwith\nAND\nand\nNOT\ngates.\nQuestion:\nDo\nes\nthere\nexist\na\ntruth\nassignmen\nt\nto\nthe\ninput\ngates\nof C\nsu \nh\nthat C\nunder\nthis\nassignmen\nt\nev\naluates\nto\ntrue\n?\nHolzer\nand\nHolzer's\n onstru tion\nsim\nulates\nthe\n omputation\nof\nsu \nh\na\nb\no\nolean\n ir uit C\nb\ny\na\nT\nan\ntrix\nTM\nrotation\npuzzle\nsu \nh\nthat C\nev\naluates\nto\ntrue\nfor\nsome\nassignmen\nt\nto\nits\nv\nariables\nif\nand\nonly\nif\nthe\npuzzle\nhas\na\nsolution.\nOur\nde nition\nof\nb\no\nolean\n ir uits\nfollo\nws\nHolzer\nand\nHolzer\n[HH04℄,\nwho\nview\na\nb\no\nolean\n ir uit C\nwith\ninput\nv\nariables x1, x2, . . . , xn\nas\na\nsequen e (α1, α2, . . . , αm)\nof\nsteps\nsu \nh\nthat\nthe ith\ninstru tion αi\nhas\none\nof\nthe\nfollo\nwing\nforms:\n1.\nfor\nea \nh i\nwith 1 ≤i ≤n , αi = xi\n,\n2.\nfor\nea \nh i\nwith n+1 ≤i ≤m ,\neither αi =\nAND(j, k)\nor αi =\nNOT(j),\nwhere j ≤k < i.\n6"},{"page":7,"text":"Dep\nending\non\nthe\ntruth\nv\nalues\nof\nthe\ninput\nv\nariables\nthe\noutput\ngate\nev\naluates\nto\ntrue\nor\nfalse\nin\nthe\nstandard\nw\na\ny\n.\nIn\ngeneral,\n ir uits\n an\n on\ntain\nan\ny\nn\num\nb\ner\nof\nwire\n rossings,\nwhi \nh\n annot\neasily\nb\ne\nrealized\nb\ny\nT\nan\ntrix\nTM\nsubpuzzles.\nT\no\nbuild\na\n ir uit\n on\ntaining\nonly\n rossings\nof\nt\nw\no\nneigh-\nb\noring\nwires,\nHolzer\nand\nHolzer\nfollo\nw\nGolds \nhlager's\npro\n edure\n[Gol77\n℄:\nIf αi =\nAND(j, k),\nmo\nv\ne\nwire j\nimmediately\nto\nthe\nleft\nof k\n,\nput\nan\nAND\ngate\nin\nfor αi\n,\nmo\nv\ne\nwire j\nba \nk\nto\nits\nstarting\np\noin\nt,\nand\n nally\nmo\nv\ne\nwire i\nto\nthe\nfar\nrigh\nt.\nThe\n ases\nof\ninstru tion αi\nb\neing\neither\nNOT(j)\nor xi\nare\ntreated\nin\na\nsimilar\nw\na\ny\n.\nFigure\n4\nsho\nws\na\npart\nof\na\n ir uit\nwith\nwire\n rossings,\nwhi \nh\n omputes α4 =\nAND(1, 3).\nOb\nviously\n,\nthere\nare\nonly\n rossings\nof\nt\nw\no\ndire tly\nadja en\nt\nwires.\nNote\nthat\nthis\ntransformation\nfrom\ngeneral\nto\nalmost\nplanar\n ir uits\n an\nb\ne\ndone\nin\ndeterministi \nlogarithmi \nspa e.\n2\n1\n3\n1\n1\n1\n2\n2\nAND\n3\n3\n4\n2\n1\n2\n3\n4\n3\n1\n2\n3\nFigure\n4:\nExample\nof\na\n ir uit\nfollo\nwing\nGolds \nhlager's\ntransformation\n[Gol77\n℄\nNo\nw,\nto\nbuild\na\nT\nan\ntrix\nTM\nRotation\nPuzzle\nthat\nsim\nulates\nsu \nh\na\n ir uit,\nHolzer\nand\nHolzer\nuse\na\ntruly\nplanar\n ross-o\nv\ner \ngadget\nthat\nw\nas\nprop\nosed\nb\ny\nM Coll\n[M C81\n℄.\nM -\nColl's\n ir uit\ngadget\nuses\nb\no\nolean\nAND\nand\nNOT\ngates\nto\nsim\nulate\nthe\n rossings\nof\nan\ny\nt\nw\no\nadja en\nt\nwires.\nEa \nh\nsu \nh\n ross-o\nv\ner \ngadget\nneeds\na\ntotal\nof 14\ninstru tion\nsteps\nand\nin\nv\nolv\nes\nt\nw\nelv\ne\nAND\nand\nnine\nNOT\ngates.\nSin e\nman\ny\n rossings\n an\no\n ur\nin\nthe\noriginally\ngiv\nen\n ir uit,\nthis\nma\ny\nlead\nto\na\n onsiderable\n(alb\neit\nstill\np\nolynomial)\nblo\nw-up\nof\nthe\nT\nan\ntrix\nTM\npuzzle\n onstru ted\nin\nour\nredu tion.\nThat\nis\nwh\ny\nw\ne\nprop\nose,\nas\na\nmore\ne ien\nt\nalternativ\ne\nto\nthe\nredu tion\npresen\nted\nin\nour\nprevious\npap\ner\n[BR07℄,\nto\nsim\nulate\nthese\n ross-o\nv\ners\ndire tly\nvia\na\nnew\nT\nan\ntrix\nTM\nsubpuzzle,\nthe\nCR\nOSS\nsubpuzzle\npresen\nted\nin\nFigure\n11\n.\nUnlik\ne\nthe\nredu tions\ngiv\nen\nin\n[HH04,\nBR07℄,\nvia\nsu \nh\nCR\nOSS\nsubpuzzles\nour\nredu tion\ndo\nesn't\nneed\nthe\ntransformation\nfrom\ngeneral\nto\nplanar\n ir uits\nthat\nrequires\nman\ny\nadditional\ngates\nand\ninstru tion\nsteps\n aused\nb\ny\nwire\n rossings.\nF\nurthermore,\nour\n onstru tion\nwill\nmo\ndify\nHolzer\nand\nHolzer's\nredu tion\n[HH04℄\nin\nsu \nh\na\nw\na\ny\nthat\nthere\nis\na\none-to-one\n orresp\nonden e\nb\net\nw\neen\nthe\nsolutions\nof\nthe\ngiv\nen\nCir uit∧,¬\n-SA\nT\ninstan e\nand\nthe\nsolutions\nof\nthe\nresulting\nrotation\npuzzle\ninstan e;\nhen e\n7"},{"page":8,"text":"our\nredu tion\nis\nparsimonious.\nT\no\nsim\nulate\nthe\n ir uit\nb\ny\na\nrotation\npuzzle,\na\nn\num\nb\ner\nof\nsubpuzzles\nare\nused.\nThe\n olor\nblue\nin\nthese\nsubpuzzles\nwill\nrepresen\nt\nthe\ntruth\nv\nalue\ntrue,\nand\nthe\n olor\nred\nwill\nrepresen\nt\nfalse.\nThis\n olor\nen o\nding\nat\nthe\ninputs\nand\noutputs\nof\nthe\nsubpuzzles\nth\nus\nrepresen\nt\nthe\ntruth\nv\nalues\nof\nthe\n ir uit's\ngates\nand\nwires.\nIn\nthe\nfollo\nwing\nse tions,\nw\ne\npresen\nt\nour\nmo\ndi ed\nsubpuzzles\nand,\nto\nallo\nw\n omparison,\nw\ne\nalso\npresen\nt\nHolzer\nand\nHolzer's\nsubpuzzles\n[HH04℄.\nT\no\nindi ate\nthe\ndi eren es\nb\net\nw\neen\ntheir\noriginal\nand\nour\nmo\ndi ed\nsubpuzzles,\ntiles\nwith\nmore\nthan\none\np\nossible\nsolution\nin\nthe\noriginal\nsubpuzzles\nwill\nha\nv\ne\na\ngrey\ninstead\nof\na\nbla \nk\nb\norder,\nand\nw\ne\nhighligh\nt\nall\nmo\ndi ed\ntiles\nin\nour\nnew\nsubpuzzles\nb\ny\na\ngrey\ninstead\nof\na\nwhite\nba \nkground\n(unless\nstated\notherwise).\nAnother\ndi eren e\nb\net\nw\neen\nour\npro\nof\nhere\nand\nthe\npro\nofs\nof\nHolzer\nand\nHolzer\n[HH04℄\nand\na\npreliminary\nv\nersion\nof\nthis\npap\ner\n[BR07℄\nregards\nthe\nanalysis\nof\nthe\nsubpuzzles.\nIn\nparti ular,\nw\ne\nwill\nhere\nfo\n us\non\nthe\n olor\nsequen es\nof\nthe\nv\narious\nT\nan\ntrix\nTM\ntiles.\nThis\nwill\nallo\nw\nus\nto\ngiv\ne\nthe\nargumen\nts\nmore\ntersely\n.\nF\nor\nexample,\nthe\ntile\nin\nFigure\n1(d)\nhas\nthe\n lo\n \nkwise\n olor\nsequen e yellow-green\n-green-red-red -yellow\n,\nwhi \nh\nwill\nb\ne\nabbreviated\nas yggrry\n.\n3.1\nWire\nSubpuzzles\nWires\nof\nthe\n ir uit\nare\nsim\nulated\nb\ny\nthe\nsubpuzzles\nWIRE,\nMO\nVE,\nCOPY,\nand\nCR\nOSS.\nT\no\nsim\nulate\nsimple\nv\nerti al\nwires,\nthe\nWIRE\nsubpuzzle\nis\nused.\nThe\noriginal\nv\nersion\nof\nHolzer\nand\nHolzer\n[HH04℄\nis\npresen\nted\nin\nFigure\n5.\nIt\nis\neasy\nto\nsee\nthat\nb\noth\ntiles\nha\nv\ne\nt\nw\no\np\nossible\norien\ntations\nfor\nea \nh\ninput\n olor.\nW\ne\nmo\ndify\nthis\nsubpuzzle\nas\nsho\nwn\nin\nFigure\n6,\ninserting\na\nnew\nRond\nat\np\nosition x.\nWithout\nthis\ntile,\nthe\np\nossible\n olor\nsequen es\nfor\nthe\nedges\nof\ntile x\njoin\nt\nwith a\nand b\nare rr , ry , yr ,\nand yy\nif\nthe\ninput\n olor\nis\nblue,\nand\nare bb ,\nby, yb ,\nand yy\nif\nthe\ninput\n olor\nis\nr\ne\nd.\nHo\nw\nev\ner,\nsin e\nthe\nnew\ntile x\ndo\nes\nnot\n on\ntain\nthe\n olor\nyel\nlow,\nthe\nsolutions\nare\n xed\nwith,\nresp\ne tiv\nely\n, bb\nand rr\nat\nthe\njoin\nt\nedges\nof\ntile x\nwith a\nand b,\nand\nw\ne\nobtain\nunique\nsolutions\nfor\nb\noth\ninput\n olors.\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nIN\na\nb\nOUT\n( )\nS \nheme\nFigure\n5:\nOriginal\nsubpuzzle\nWIRE,\nsee\n[HH04℄\nT\no\nbuild\nlonger\nwires,\nsev\neral\nWIRE\nsubpuzzles\nare\n onne ted.\nNote\nthat\nthis\nWIRE\nsubpuzzle\nhas\nheigh\nt\nt\nw\no.\nThis\nfor es\nall\nother\nsubpuzzles\nto\nha\nv\ne\nev\nen\nheigh\nt,\nb\ne ause\nthey\nm\nust\nb\ne\n onne ted\nb\ny\nWIRE\nsubpuzzles.\n8"},{"page":9,"text":"IN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nx\nIN\na\nb\nOUT\n( )\nS \nheme\nFigure\n6:\nMo\ndi ed\nsubpuzzle\nWIRE\nBy\nthe\nMO\nVE\nsubpuzzle\nthe\n ir uit's\nwires\n an\nb\ne\nmo\nv\ned\none\np\nosition\nto\nthe\nleft\nor\none\np\nosition\nto\nthe\nrigh\nt.\nW\ne\ndis uss\na\nmo\nv\ne\nto\nthe\nrigh\nt\nin\ndetail\nand\nmen\ntion\nthat\na\nmo\nv\ne\nto\nthe\nleft\n an\nb\ne\nhandled\nanalogously\n.\nThe\noriginal\nv\nersion\nof\nHolzer\nand\nHolzer\n[HH04℄\nis\nsho\nwn\nin\nFigure\n7,\nand\nour\nmo\ndi ed\nsubpuzzle\nis\nsho\nwn\nin\nFigure\n8.\nIn\nthe\noriginal\nMO\nVE\nsubpuzzle,\ntiles a\nand i\nare\nmark\ned\nb\ne ause\nthey\nha\nv\ne\nt\nw\no\norien\nta-\ntions\nfor\nthe\nsolutions\nsho\nwn\nin\nFigure\n7.\nNote,\nho\nw\nev\ner,\nthat\nin\naddition\nto\nthe\nam\nbiguit\ny\n aused\nb\ny\ntiles a\nand i,\nthere\ndo\nes\nexist\nstill\nanother\nsolution\nfor\nea \nh\ninput\n olor.\nIn\nparti -\nular,\nif\nthe\ninput\n olor\nis\nblue\nthen\none\n an\nsimply\nsw\nap\nthe\n olors\nr\ne\nd\nand\nyel\nlow\nto\nobtain\nanother\nsolution,\nand\nif\nthe\ninput\n olor\nis\nr\ne\nd\nthen\ntiles b\nand d\n an\nb\ne\nrotated\nb\ny 60\nand\nall\nother\ntiles\nb\ny 240\ndegrees\nin\n lo\n \nkwise\ndire tion.\nW\ne\n x\nthe\nsolution\nb\ny\ninserting\ntiles x\nand y\n.\nFirst,\n onsider\nthe\n ase\nthat\nthe\ninput\n olor\nis\nblue.\nIt\nis\n lear\nthat\nthe\nedge\nof\ntile e\njoin\nt\nwith\ntile x\nm\nust\nb\ne\nblue,\nand\nsin e\nthe\nedges\nof\ntiles x\njoin\nt\nwith\ntiles b\nand f\nm\nust\nha\nv\ne\nthe\nsame\n olor,\nthe\norien\ntation\nof\ntile x\nis\n xed.\nThis\n xes\nalso\nthe\norien\ntation\nof\nall\nother\ntiles\nex ept a\nand i.\nThe\norien\ntation\nof\ntile a\nis\n xed\nwith\nr\ne\nd\nat\nthe\nedge\njoin\nt\nwith x.\nTile y\n xes\nthe\norien\ntation\nof\ntile i,\nsin e\nit\ndo\nes\nnot\n on\ntain\nthe\n olor\nsequen e byy ,\nbut\nthe\n olor\nsequen e byr\nfor\nthe\nedges\njoin\nt\nwith\ntiles g\n, h,\nand i.\nThe\n ase\nof\nr\ne\nd\nb\neing\nthe\ninput\n olor\n an\nb\ne\nhandled\nsimilarly\n.\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nc\nIN\na\nb\nd\ne\nf\nh\ni\nOUT\ng\n( )\nS \nheme\nFigure\n7:\nOriginal\nsubpuzzle\nMO\nVE,\nsee\n[HH04℄\nThe\nCOPY\nsubpuzzle\nis\nused\nto\n split \na\nwire\nin\nto\nt\nw\no\n opies.\nIts\noriginal\nv\nersion\nfrom\n[HH04℄\nis\nsho\nwn\nin\nFigure\n9\n,\nand\nthe\nmo\ndi ed\nv\nersion\nis\nsho\nwn\nin\nFigure\n10\n.\nIn\nits\noriginal\nv\nersion,\ntiles a\nthrough i\nare\na\nmo\nv\ne\nto\nthe\nrigh\nt,\nmerged\nwith\na\nmo\nv\ne\nto\nthe\nleft\n onsisting\nof\ntiles a\nthrough d\nand\ntiles i\nthrough m .\nT\no\nobtain\na\nunique\nsolution,\nw\ne\ninsert\n9"},{"page":10,"text":"IN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nc\nIN\na\nb\nd\nx\ne\nf\nh\ni\nOUT\ng\ny\n( )\nS \nheme\nFigure\n8:\nMo\ndi ed\nsubpuzzle\nMO\nVE\nthe\nsame\ntiles x\nand y\nas\nfor\nthe\nMO\nVE\nsubpuzzle.\nBy\nthe\nsame\nargumen\nt\nas\nab\no\nv\ne,\nthis\n xes\nthe\norien\ntation\nof\nall\nother\ntiles,\nex ept\ntile m .\nBut\ninserting\ntile z\n,\nwhi \nh\nis\nof\nthe\nsame\nt\nyp\ne\nas\ntile y\n,\nalso\n xes\nthe\norien\ntation\nof m .\nTh\nus,\nthe\nsolution\nfor\nthis\nsubpuzzle\nis\nalso\nunique\nfor\nb\noth\ninput\n olors.\nOUT\nIN\nOUT\n(a)\nIn:\ntrue\nOUT\nIN\nOUT\n(b)\nIn:\nfalse\nk\nj\nl\nm\nOUT\ni\nIN\na\nb\nc\nd\ne\ng\nh\nOUT\nf\n( )\nS \nheme\nFigure\n9:\nOriginal\nsubpuzzle\nCOPY,\nsee\n[HH04℄\nOUT\nIN\nOUT\n(a)\nIn:\ntrue\nOUT\nIN\nOUT\n(b)\nIn:\nfalse\nk\nz\nj\nl\nm\nOUT\ni\nIN\na\nb\nc\nx\nd\ne\ng\nh\nOUT\nf\ny\n( )\nS \nheme\nFigure\n10:\nMo\ndi ed\nsubpuzzle\nCOPY\nT\no\nrealize\nwire\n rossings,\nHolzer\nand\nHolzer\n[HH04℄\nused\na\n ir uit\n onstru tion\n onsisting\n10"},{"page":11,"text":"of\nAND\nand\nNOT\ngates\nthat\nis\nbased\non\nGolds \nhlager's\npro\n edure\n[Gol77\n℄\nand\nM Coll's\n ross-o\nv\ner \ngadget\n[M C81\n℄.\nThis\napproa \nh\nw\nas\nalso\ntak\nen\nin\na\npreliminary\nv\nersion\nof\nthis\npap\ner\n[BR07℄.\nAs\nmen\ntioned\nab\no\nv\ne,\nho\nw\nev\ner,\nw\ne\nhere\nsimplify\nthe\n onstru tion\nb\ny\nsim\nulating\nwire\n rossings\ndire tly\n.\nT\no\nthis\nend,\nw\ne\nin\ntro\ndu e\na\nnew\nT\nan\ntrix\nTM\nsubpuzzle\n alled\nCR\nOSS,\nwhi \nh\nis\npresen\nted\nin\nFigure\n11\n.\nThis\nsubpuzzle\nhas\nt\nw\no\ninputs\nand\nt\nw\no\noutputs,\nwhere\nthe\nleft\noutput\nwill\nb\ne\nthe\nsame\nas\nthe\nrigh\nt\ninput\nand\nvi e\nv\nersa.\nJust\nas\nall\nour\nmo\ndi ed\nsubpuzzles,\nour\nno\nv\nel\nCR\nOSS\nsubpuzzle\nhas\nunique\nsolutions\nfor\nea \nh\n om\nbination\nof\np\nossible\ninput\n olors.\nT\no\nanalyze\nthis\nsubpuzzle,\nw\ne\nsub\ndivide\nit\nin\nto\nthree\ndi eren\nt\nparts.\nThe\nlo\nw\ner\npart\n onsists\nof\ntiles a\nthrough k\n,\nand\nthe\nupp\ner\npart\nof\ntiles l\nthrough t .\nLet\nus\n onsider\nthe\nupp\ner\npart\nof\nthe\nCR\nOSS\nsubpuzzle\n rst.\nTiles l\nthrough o\nform\nthe\nleft\noutput.\nSin e\ntile j\ndo\nes\nnot\n on\ntain\ngr\ne\nen,\nthe\ninput\n olor\nto\nthis\npart\nwill\nb\ne\neither\nblue\nor\nyel\nlow.\nIf\nthe\ninput\n olor\nis\nblue,\nall\ntiles\nm\nust\nha\nv\ne\nv\nerti al\nblue\nlines,\nand\nsin e\ntile\no\ndo\nes\nnot\n on\ntain\nyel\nlow\nlines,\nthe\norien\ntation\nof\nall\ntiles\nis\n xed\nwith\ngr\ne\nen\nat\nthe\njoin\nt\nedges\nof,\nresp\ne tiv\nely\n,\ntiles o\nand n ,\ntiles n\nand m ,\nand\ntiles m\nand l\n.\nThe\n ase\nof\nyel\nlow\nb\neing\nthe\ninput\n olor\nfor\nthis\npart\n an\nb\ne\nhandled\nsimilarly\nand\nyields\nthe\noutput\n olor\nr\ne\nd.\nThe\nrigh\nt\noutput\n onsists\nof\ntiles p\nthrough t .\nHere,\nthe\ninput\n olors\nr\ne\nd,\nblue,\nand\nyel\nlow\nare\np\nossible,\nwhere\nyel\nlow\nand\nr\ne\nd\nb\noth\nlead\nto\noutput\n olor\nr\ne\nd.\nSin e\ntile s\n on\ntains\nno\nr\ne\nd\nlines,\nw\ne\nobtain\nunique\nsolutions\nfor\nall\np\nossible\n om\nbinations\nof\ninput\n olors,\nb\ny\na\nsimilar\nargumen\nt\nas\nfor\nthe\nleft\noutput.\nW\ne\nno\nw\nturn\nto\nthe\nmore\n ompli ated\nlo\nw\ner\npart\nof\nthis\nsubpuzzle.\nTiles a, b,\nand c\nmo\nv\ne\nthe\nleft\ninput\n olor\nto\nthe\njoin\nt\nedge\nof\ntiles b\nand g\n.\nRegarding\ntiles a, b,\nand c,\nthere\nare\nt\nw\no\np\nossible\nsolutions\nfor\nea \nh\ninput\n olor.\nThe\np\nossible\n olor\nsequen es\nfor\nthe\nedges\nof\ntiles c\nand b\njoin\nt\nwith\ntile g\nare by\nand br\nfor\ninput\n olor\nblue,\nand\nare ry\nand rb\nfor\ninput\n olor\nr\ne\nd.\nSin e g\n on\ntains\nexa tly\none\nof\nthese\n olor\nsequen es\nfor\nea \nh\ninput\n olor,\nthe\norien\ntation\nof\ntiles a, b,\nand c\nis\n xed,\nand\nthe\ninput\n olor\nis\npassed\non\nto\nthe\njoin\nt\nedge\nof\ntiles b\nand g\n.\nThe\nsame\nargumen\nt\napplies\nto\nthe\nrigh\nt\ninput,\nwhi \nh\n onsists\nof\ntiles d,\ne, f\n,\nand h,\nsin e\nthey\nare\nmirror-symmetri al.\nIf\nb\noth\ninputs\nare\nblue,\nthe\norien\ntations\nof\ntiles g\nand h\nis\n xed\nb\ny\ntiles b, c, e,\nand f\n,\nresp\ne tiv\nely\n.\nBoth g\nand h\nwill\nha\nv\ne\nyel\nlow\nat\ntheir\nedges\njoin\nt\nwith\ntile i,\nwhi \nh\nleads\nto\nr\ne\nd\nat\nthe\nedges\nof\ntile i\njoin\nt\nwith\ntiles j\nand k\n,\nresp\ne tiv\nely\n,\nand\nthe\norien\ntation\nof\nall\ntiles\nin\nthe\nlo\nw\ner\npart\nis\n xed.\nThe\n onne tions\nb\net\nw\neen\nthe\nlo\nw\ner\nand\nthe\nupp\ner\npart\nare\nthe\njoin\nt\nedges\nof\ntiles j\nand l\nand\nof\ntiles k\nand p ,\nresp\ne tiv\nely\n.\nThese\nedges\nare\nb\noth\nblue\nif\nb\noth\ninput\n olors\nare\nblue.\nNo\nw\n onsider\nthe\n ase\nthat\nb\noth\ninput\n olors\nare\nr\ne\nd.\nThe\np\nossible\n olors\nfor\nthe\njoin\nt\nedges\nof\ntiles g\nand h\nwith\ntile i\nare\nr\ne\nd\nand\nblue.\nClearly\n,\nit\nis\nnot\np\nossible\nthat\nthey\nb\noth\nare\nblue.\nF\nurthermore,\nit\nis\nnot\np\nossible\nthat\nthe\njoin\nt\nedge\nof\ntiles g\nand i\nis\nblue,\nsin e\nin\nthis\n ase\nthe\njoin\nt\nedge\nof\ntiles j\nand l\nw\nould\nha\nv\ne\nto\nb\ne\nr\ne\nd,\nwhi \nh\nis\nnot\np\nossible\nb\ne ause l\ndo\nes\nnot\n on\ntain\nr\ne\nd.\nF\nor\nthe\nsame\nreason\nit\nis\nnot\np\nossible\nthat\nthe\njoin\nt\nedges\nof\ntiles g\nand h\nwith i\nare\nb\noth\nr\ne\nd,\nso\nthe\nonly\np\nossible\nsolution\nis\nthat\nthe\njoin\nt\nedge\nof\ntiles g\nand i\nis\nr\ne\nd\nand\nthat\nthe\njoin\nt\nedge\nof\ntiles h\nand i\nis\nblue.\nThis\nalso\nuniquely\ndetermines\nthe\norien\ntation\nof\ntiles j\nand k\nwith\nyel\nlow\nat\nthe\njoin\nt\nedge\nof\ntiles j\nand l\nand\nr\ne\nd\nat\nthe\njoin\nt\nedge\nof\ntiles k\nand p .\nThe\n ases\nthat\none\ninput\nis\nr\ne\nd\nand\nthe\nother\none\nis\n11"},{"page":12,"text":"blue\n an\nb\ne\nhandled\nb\ny\nsimilar\nargumen\nts\nto\nsho\nw\nthat\nw\ne\nthen\nha\nv\ne\nunique\nsolutions\nas\nw\nell.\nIN\nOUT\nIN\nOUT\n(a)\nIn:\ntrue,\ntrue\nIN\nOUT\nIN\nOUT\n(b)\nIn:\ntrue,\nfalse\nIN\nOUT\nIN\nOUT\n( )\nIn:\nfalse,\ntrue\nIN\nOUT\nIN\nOUT\n(d)\nIn:\nfalse,\nfalse\nIN\na\nc\nm\no\nOUT\nb\ng\nj\nl\nn\ni\ne\nh\nk\np\nr\nIN\nd\nf\nq\nt\nOUT\ns\n(e)\nS \nheme\nFigure\n11:\nSubpuzzle\nCR\nOSS\n3.2\nGate\nSubpuzzles\nThe\ngates\nof\nthe\nb\no\nolean\n ir uit\nare\nsim\nulated\nb\ny\nthe\n orresp\nonding\nAND\nand\nNOT\nsub-\npuzzles.\nThe\noriginal\nv\nersion\nof\nthe\nNOT\nsubpuzzle\nfrom\n[HH04℄\nis\npresen\nted\nin\nFigure\n12\n,\nand\nour\nnew\nv\nersion\nis\nsho\nwn\nin\nFigure\n13\n.\nThe\npurp\nose\nof\nthis\nsubpuzzle\nis\nto\n negate \nthe\ninput\n olor,\ni.e.,\nif\nthe\ninput\n olor\nis\nr\ne\nd\nthen\nthe\noutput\n olor\nwill\nb\ne\nblue,\nand\nvi e\nv\nersa.\nIn\nthe\noriginal\nsubpuzzle,\nthere\nis\nonly\none\np\nossible\nsolution\nto\nthe\nthree\nRonds e, f\n,\nand g\n:\nSin e\ntiles c, b,\nand f\ndo\nnot\n on\ntain\ngr\ne\nen,\nthe\njoin\nt\nedge\nof\ntiles e\nand g\nm\nust\nb\ne\ngr\ne\nen,\nwhi \nh\nfor es\nthe\njoin\nt\nedge\nof\ntiles g\nand f\nto\nb\ne\nyel\nlow.\nSo\nthe\norien\ntation\nof\ntiles\ne, f\n,\nand g\nis\n xed,\nwith\nr\ne\nd\nat\nthe\nedges\njoin\nt\nwith\ntiles b\nand c,\nresp\ne tiv\nely\n.\nThere\nare\nonly\nt\nw\no\np\nossible\norien\ntations\nleft\nfor\nthe\ntiles b\nand c,\none\nfor\ninput\n olor\nblue\nand\none\nfor\ninput\n olor\nr\ne\nd.\nThe\nonly\ntiles\nstill\nha\nving\nmore\nthan\none\np\nossible\norien\ntation\nare a\nand d.\n12"},{"page":13,"text":"IN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nf\ng\ne\nIN\na\nb\nc\nd\nOUT\n( )\nS \nheme\nFigure\n12:\nOriginal\nsubpuzzle\nNOT,\nsee\n[HH04℄\nIN\nOUT\n(a)\nIn:\ntrue\nIN\nOUT\n(b)\nIn:\nfalse\nf\ng\ne\nIN\na\nb\nc\nd\nOUT\nx\ny\n( )\nS \nheme\nFigure\n13:\nMo\ndi ed\nsubpuzzle\nNOT\nThey\nwill\nb\ne\n xed\nb\ny\ninserting\ntiles x\nand y\n.\nThe\np\nossible\n olor\nsequen es\nfor\nthe\nedges\nof\ntiles a\nand b\njoin\nt\nwith\ntile x\nare yy\nor ry\nif\nthe\ninput\n olor\nis\nblue,\nand\nthey\nare bb\nor\nyb\nif\nthe\ninput\n olor\nis\nr\ne\nd.\nTile x,\nho\nw\nev\ner,\n on\ntains\nonly\nthe\nsequen es yy\nand bb,\nso\nthe\norien\ntation\nis\n xed.\nNote\nthat\ntiles c\nand d\nb\neha\nv\ne\njust\nlik\ne\na\nWIRE\nsubpuzzle\nand\nsin e\nthe\norien\ntation\nof\ntile c\nis\n xed\nwith\nr\ne\nd\nat\nthe\nedge\njoin\nt\nwith\ntile e,\nw\ne\ninsert\na\nRond\nat\np\nosition y\n on\ntaining\nthe\n olor\nsequen e yy\nbut\nnone\nof\nthe\nsequen es yb\nand yr .\nWith\nthese\nt\nw\no\nnew\ntiles,\nunique\nsolutions\nare\nenfor ed\nfor\nea \nh\ninput\n olor.\nThe\nsomewhat\nmore\n ompli ated\nAND\nsubpuzzle\nis\nsho\nwn\nin\nFigure\n14\nin\nits\noriginal\nv\nersion\nfrom\n[HH04℄,\nwhile\nFigure\n15\npresen\nts\nour\nmo\ndi ed\nv\nersion.\nThis\nsubpuzzle\n an\nagain\nb\ne\nsub\ndivided\nin\nto\nt\nw\no\ndi eren\nt\nparts,\nan\nupp\ner\npart\nand\na\nlo\nw\ner\npart\nthat\nare\n onne ted\nat\nthe\njoin\nt\nedge\nof\ntiles c\nand j\n.\nFirst,\nw\ne\n onsider\nthe\nupp\ner\npart\nof\nthe\nmo\ndi ed\nAND\nsubpuzzle.\nW\ne\nsho\nw\nthat\nits\noutput\n olor\nis\nr\ne\nd\nif\nthe\njoin\nt\nedge\nof\ntiles j\nand c\nis\nyel\nlow,\nand\nthat\nits\noutput\n olor\nis\nblue\nif\nthis\nedge\nis\nblue.\nNote\nthat,\njust\nas\nfor\nthe\nNOT\nsubpuzzle,\nthe\nRonds o, p ,\nand q\nha\nv\ne\nonly\none\np\nossible\norien\ntation,\nth\nus\nfor ing\nthe\nedges\nof\ntiles n\nand m\njoin\nt\nwith\ntile o\nto\nb\ne\nyel\nlow.\nIf\nthe\ninput\nto\nthis\npart\nof\nthe\nsubpuzzle\nis\nblue\nthen\ntiles j\nand k\nm\nust\nha\nv\ne\na\nv\nerti al\nblue\nline,\nand\nsin e\nthe\nedges\nof\ntile l\njoin\nt\nwith j\nand k\n annot\nb\ne\nblue,\ntiles l\n, m ,\nand n\nm\nust\nha\nv\ne\nv\nerti al\nblue\nlines,\nto\no.\nSin e\nthe\nedges\nof\ntiles n\nand m\njoin\nt\nwith\ntile o\n13"},{"page":14,"text":"IN\nOUT\nIN\n(a)\nIn:\ntrue,\ntrue\nIN\nOUT\nIN\n(b)\nIn:\ntrue,\nfalse\nIN\nOUT\nIN\n( )\nIn:\nfalse,\ntrue\nIN\nOUT\nIN\n(d)\nIn:\nfalse,\nfalse\nh\nIN\na\nb\nc\nj\nk\ng\nl\nm\nn\nOUT\nf\no\nIN\nd\ne\np\nq\ni\n(e)\nS \nheme\nFigure\n14:\nOriginal\nsubpuzzle\nAND,\nsee\n[HH04℄\nare\nyel\nlow,\nthe\norien\ntation\nof\nall\nother\ntiles\nis\nuniquely\ndetermined\nin\nthe\nupp\ner\npart.\nThe\n ase\nof\nyel\nlow\nb\neing\nthe\ninput\n olor\nfor\nthe\nupp\ner\npart\n an\nb\ne\nhandeld\nsimilarly:\nWithout\nan\ny\nmo\ndi ation,\nw\ne\nha\nv\ne\nunique\nsolutions\nfor\nthe\nupp\ner\npart\nof\nthe\nsubpuzzle.\nNo\nw\nw\ne\n onsider\nthe\nlo\nw\ner\npart\nof\nthis\nsubpuzzle.\nF\nor\nea \nh\n om\nbination\nof\ninput\n olors,\ntiles a\nand d\n(whi \nh\nare\nadja en\nt\nto\nthe\ninput\ntiles\nof\nthe\nAND\nsubpuzzle)\nand\ntiles h\nand i\n(ha\nving\nonly\none\n onne ting\nedge\nto\nthe\nrest\nof\nthe\noriginal\nAND\nsubpuzzle\nin\nFigure\n14)\nea \nh\nha\nv\ne\nt\nw\no\np\nossible\norien\ntations.\nExamining\nall\np\nossible\n olor\nsequen es\nfor\nea \nh\n om\nbination\nof\np\nossible\ninput\n olors,\ntiles x\nand y\n an\nb\ne\ndetermined\nas\nsho\nwn\nin\nFigure\n15\nto\n x\nthe\norien\ntation\nof\ntiles a\nand h\nand\nof\ntiles d\nand i,\nresp\ne tiv\nely\n.\nThen,\nthe\ninput\n olor\nis\npassed\non\nto\nthe\njoin\nt\nedge\nof\ntiles b\nand c,\nand\nto\nthe\njoin\nt\nedge\nof\ntiles e\nand f\n.\nTile g\nin\nthe\n en\nter\nof\nthe\nsubpuzzle\nhas\nagain\nt\nw\no\np\nossible\norien\ntations\nfor\nea \nh\n om\nbination\nof\ninput\n olors.\nNote\nthat\nour\nmo\ndi ations\nmade\nso\nfar namely\n,\ninserting\nnew\ntiles\nin\nto\nthe\nsubpuzzles do\nnot\nw\nork\nhere,\nas\nit\nis\nnot\np\nossible\nto\ninsert\na\nnew\ntile\nin\nthe\nneigh\nb\norho\no\nd\nof\ntile g\n,\nand\nth\nus\nw\ne\nha\nv\ne\nto\nrepla e\nit.\nAs\nw\ne\nha\nv\ne\nseen\nin\nthe\nanalysis\nof\nthe\nupp\ner\npart,\nthe\n olor\nyel\nlow\nof\ntile j\nis\nonly\nused\nfor\nthe\nedge\njoin\nt\nwith\ntile c.\nW\ne\nwill\nrepla e\nthe\n olor\nyel\nlow\nof\nb\noth\ntiles\nwith\nthe\n olor\ngr\ne\nen.\nThis\nis\np\nossible,\n14"},{"page":15,"text":"IN\nOUT\nIN\n(a)\nIn:\ntrue,\ntrue\nIN\nOUT\nIN\n(b)\nIn:\ntrue,\nfalse\nIN\nOUT\nIN\n( )\nIn:\nfalse,\ntrue\nIN\nOUT\nIN\n(d)\nIn:\nfalse,\nfalse\nx\nh\nIN\na\nb\nc\nj\nk\ng\nl\nm\nn\nOUT\nf\no\nIN\nd\ne\np\nq\ny\ni\n(e)\nS \nheme\nFigure\n15:\nMo\ndi ed\nsubpuzzle\nAND\nb\ne ause\nthe\njoin\nt\nedge\nof\ntiles c\nand b\nwill\nnev\ner\nb\ne\nyel\nlow,\nand\ntile g\nwill\nb\ne\nrepla ed\nb\ny\na\nnew\none.\nT\no\ndo\nthis,\nw\ne\n onsider\nall\np\nossible\n olors\nat\nthe\nedges\nof\ntiles c\nand f\njoin\nt\nwith\ntile g\n,\nwith\nthe\nrestri tion\nthat\nthe\njoin\nt\nedge\nof\ntiles c\nand j\nm\nust\nb\ne\neither\nblue\nor\ngr\ne\nen.\nIf\nthe\nrigh\nt\ninput\n olor\nis\nblue\nthen\nthe\njoin\nt\nedge\nof\ntiles f\nand g\n an\nb\ne\neither\nblue\nor\nyel\nlow,\nand\nif\nthe\nrigh\nt\ninput\n olor\nis\nr\ne\nd\nthen\nthis\nedge\n an\nb\ne\neither\nr\ne\nd\nor\nyel\nlow.\nF\nor\nthe\nleft\ninput,\nthe\njoin\nt\nedge\nof\ntiles c\nand g\nm\nust\nb\ne\nr\ne\nd\nif\nthe\ninput\nis\nr\ne\nd,\nb\ne ause\nthen\nthe\njoin\nt\nedge\nof\ntiles c\nand j\n annot\nb\ne\nr\ne\nd.\nIf\nb\noth\ninputs\nare\nblue\nthen\nthe\noutput\n olor\nof\nthe\nlo\nw\ner\npart\nm\nust\nb\ne\nblue\nas\nw\nell,\nand\nthen\nthe\njoin\nt\nedge\nof\ntiles c\nand g\nis\ngr\ne\nen.\nIf\nonly\nthe\nleft\ninput\nis\nblue,\nthe\noutput\n olor\nof\nthe\nlo\nw\ner\npart\nm\nust\nb\ne\ngr\ne\nen,\nand\nthe\njoin\nt\nedge\nof\ntiles c\nand g\nis\nblue.\nThis\nleads\nto\nthe\nfollo\nwing\nrestri tions\nfor\ntile g\n:\n1.\nIf\nthe\ninput\n olors\nare\nb\noth\nblue\nthen g\nm\nust\n on\ntain\nexa tly\none\nof\nthe\n olor\nsequen es\ngxb\nor gxy,\nwhere x\nstands\nfor\nan\narbitrary\n olor.\n2.\nIf\nthe\nrigh\nt\ninput\n olor\nis\nr\ne\nd\nand\nthe\nleft\ninput\n olor\nis\nblue, g\nm\nust\n on\ntain\neither\nbxr\nor bxy.\n15"},{"page":16,"text":"3.\nIf\nthe\nrigh\nt\ninput\n olor\nis\nblue\nand\nand\nthe\nleft\ninput\n olor\nis\nr\ne\nd\nthen\nthe\np\nossible\n olor\nsequen es\nfor g\nare\neither rxb\nor rxy.\n4.\nF\nor\nthe\nlast\n om\nbination\nof\nb\noth\ninput\n olors\nb\neing\nr\ne\nd,\ntile g\nm\nust\n on\ntain\neither rxr\nor rxy.\nIn\nour\nmo\ndi ed\nAND\nsubpuzzle,\nw\ne\ninsert\na\nSin\nt\ninstead\nof\na\nRond,\nwhi \nh\n on\ntains\nea \nh\nof\nthe\n olor\nsequen es gxb , bxr, rxb,\nand rxr\nexa tly\non e.\nThis\nleads\nto\nunique\nsolutions\nfor\nthe\nlo\nw\ner\npart\nand\nth\nus\nfor\nthe\nwhole\nsubpuzzle,\nfor\nea \nh\np\nossible\n om\nbination\nof\ninput\n olors.\n3.3\nInput\nand\nOutput\nSubpuzzles\nThe\nv\nariables\nof\nthe\nb\no\nolean\n ir uit\nare\nrepresen\nted\nb\ny\nthe\nsubpuzzle\nBOOL.\nHolzer\nand\nHolzer\n[HH04℄\nsho\nw\ned\nthat\nthis\nsubpuzzle\nhas\nonly\nt\nw\no\nsolutions,\none\nwith\noutput\n olor\nblue\nand\nthe\nother\none\nwith\noutput\n olor\nr\ne\nd.\nThis\noutput\n olor\n orresp\nonds\nto\nthe\nb\no\nolean\nv\nalue\nof\nthe\n orresp\nonding\ninput\nv\nariable,\nwhere\nblue\nstands\nfor\nthe\ntruth\nv\nalue\ntrue,\nand\nr\ne\nd\nstands\nfor\nfalse.\nThe\nrestri tion\nthat\nyel\nlow\nand\ngr\ne\nen\nare\nnot\np\nossible\nas\noutput\n olors\nfor\nthis\nsubpuzzle,\nensures\nthat\nthe\nfollo\nwing\nsubpuzzles\nwill\nalw\na\nys\nha\nv\ne\na\nv\nalid\ninput\n olor,\nnamely\nblue\nor\nr\ne\nd.\nThe\nlast\nsubpuzzle\nneeded\nto\nsim\nulate\na\nb\no\nolean\n ir uit\nis\nthe\nsubpuzzle\nTEST.\nIt\nis\npla ed\nat\nthe\noutput\ngate\nof\nthe\n ir uit,\nand\nits\npurp\nose\nis\nto\nv\nerify\nthat\nthe\n ir uit\nev\nalutes\nto\ntrue.\nHolzer\nand\nHolzer\n[HH04℄\nmen\ntion\nthat\nthis\nsubpuzzle\nhas\nonly\none\nv\nalid\nsolution\nwith\nblue\nas\nthe\ninput\n olor.\nThis\nensures\nthat\nthe\noutput\nof\nthe\nwhole\n ir uit\nwill\nb\ne\ntrue\nif\nand\nonly\nif\nthe\nT\nan\ntrix\nTM\npuzzle\n onstru ted\nhas\na\nsolution.\nLik\ne\nthe\noriginal\nsubpuzzle\nBOOL,\nthe\noriginal\nTEST\nsubpuzzle\nalready\nhas\na\nunique\nsolution,\nso\nit\nis\nnot\nmo\ndi ed.\nThese\nt\nw\no\nsubpuzzles\nare\nthe\nonly\nones\nfrom\n[HH04℄\nthat\nare\nnot\nmo\ndi ed.\nF\nor\nthe\nsak\ne\nof\n ompleteness,\nthey\nare\npresen\nted\nin\nFigure\n16.\nOUT\n(a)\nBOOL\nOut:\ntrue\nOUT\n(b)\nBOOL\nOut:\nfalse\nIN\n( )\nTEST\nFigure\n16:\nSubpuzzles\nBOOL\nand\nTEST,\nsee\n[HH04℄\nThe\nshap\nes\nof\nour\nmo\ndi ed\nsubpuzzles\nha\nv\ne\n \nhanged\nsligh\ntly\n,\nso\nit\nmigh\nt\nb\ne\np\nossible\nthat\nunin\ntended\nin\ntera tions\nb\net\nw\neen\nt\nw\no\nneigh\nb\noring\nsubpuzzles\ndo\no\n ur.\nHo\nw\nev\ner,\nas\nnoted\nb\ny\nHolzer\nand\nHolzer\n[HH04℄,\nthe\nminimal\nhorizon\ntal\ndistan e\nb\net\nw\neen\nt\nw\no\nwires\nand/or\n16"},{"page":17,"text":"gates\nis\nat\nleast\nfour,\nand\nthis\nis\nstill\nenough\nto\nprev\nen\nt\nan\ny\nunin\ntended\nin\ntera tions\nb\net\nw\neen\nour\nmo\ndi ed\nsubpuzzles.\n3.4\nPro\nof\nof\nTheorem\n3.1\nW\ne\nare\nno\nw\nready\nto\npro\nv\ne\nTheorem\n3.1.\nLet\nSA\nT\ndenote\nthe\nsatis abilit\ny\nproblem.\nLemma\n3.2\nSA\nT\np\narsimoniously\nr\ne\ndu \nes\nto\nCir\n uit∧,¬\n-\nSA\nT\n.\nPro\nof.\nNote\nthat\nthe\nproblems\nSA\nT\nand\nCir uit\n-SA\nT\n(whi \nh\nis\nthe\nsame\nas\nCir uit∧,¬\n-SA\nT\nex ept\nwith\nOR\ngates\nallo\nw\ned\nas\nw\nell)\nare\nequiv\nalen\nt\nunder\nparsimonious\nredu tions\n[P\nap94\n℄.\nSin e\nOR\ngates\n an\nb\ne\nexpressed\nb\ny\nAND\nand\nNOT\ngates\nwithout\n \nhanging\nthe\nn\num\nb\ner\nof\nsolutions,\nthis\ngiv\nes\na\nparsimonious\nredu tion\nfrom\nSA\nT\nto\nCir uit∧,¬\n-SA\nT\n.\n❑\nNo\nw,\nthe\nparsimonious\nredu tion\nfrom\nSA\nT\nto\nTRP\nimmediately\nfollo\nws\nfrom\nLemma\n3.2\nand\nthe\n onstru tion\nand\nthe\nargumen\nts\npresen\nted\nin\nSe tions\n3.1,\n3.2,\nand\n3.3.\nThat\nis,\nb\ny\nour\nmo\ndi ations,\nfor\nea \nh\nsatisfying\nassignmen\nt\nto\nthe\n ir uit\nthere\nis\nexa tly\none\nsolution\nto\nthe\nT\nan\ntrix\nTM\npuzzle\n onstru ted.\n4\nThe\nUnique\nT\nan\ntrix\nTM\nRotation\nPuzzle\nProblem\nis\nDP-\nComplete\nunder\nRandomized\nRedu tions\nV\nalian\nt\nand\nV\nazirani\nin\ntro\ndu ed\nr\nandomize\nd\np\nolynomial-time\nr\ne\ndu tions\nin\ntheir\nw\nork\nsho\nw-\ning\nthat\nNP\nis\nas\neasy\nas\ndete ting\nunique\nsolutions\n[VV86℄.\nW\ne\nwill\nuse ≤p\nran\nto\nde-\nnote\ntheir\nt\nyp\ne\nof\nredu tions.\nIn\nparti ular,\nV\nalian\nt\nand\nV\nazirani\n[VV86℄\npro\nv\ned\nthat\nUnique-SA\nT\n,\nthe\nunique\nv\nersion\nof\nSA\nT\n,\nis ≤p\nran\n- omplete\nin\nDP\n(see\nalso\nChang,\nKadin,\nand\nRohatgi\n[CKR95℄).\nTheorem\n4.1\n1.\nUnique\n-SA\nT\np\narsimoniously\nr\ne\ndu \nes\nto\nUnique\n-TRP.\n2.\nUnique\n-TRP\nis\nDP- \nomplete\nunder ≤p\nran\n-r\ne\ndu tions.\nPro\nof.\nT\no\npro\nv\ne\nthe\n rst\npart,\nnote\nthat\nb\ny\nLemma\n3.2\nand\nTheorem\n3.1\n,\nw\ne\nobtain\na\nparsimonious\nredu tion\nfrom\nSA\nT\nto\nTRP.\nIt\nfollo\nws\nthat\nUnique-SA\nT\nparsimoniously\nredu es\nto\nUnique\n-TRP.\nThe\nse ond\npart\nfollo\nws\nfrom\nthe\n rst\npart\nand\nV\nalian\nt\nand\nV\nazirani's\nab\no\nv\ne-men\ntioned\nresult\nthat\nUnique-SA\nT\nis ≤p\nran\n- omplete\nin\nDP,\nand\nfrom\nthe\nob\nvious\nfa t\nthat\nUnique\n-TRP\nis\nin\nDP\n.\n❑\nA\n \nkno\nwledgmen\nts:\nW\ne\nthank\nthe\nanon\nymous\nMCU\n2007\nreferees\nfor\ntheir\nhelpful\n om-\nmen\nts.\n17"},{"page":18,"text":"Referen es\n[Bar04℄\nR.\nBarban \nhon.\nOn\nunique\ngraph\n3- olorabilit\ny\nand\nparsimonious\nredu tions\nin\nthe\nplane.\nThe\nor\neti \nal\nComputer\nS ien \ne,\n319(1 3):455 482,\n2004.\n[BR07℄\nD.\nBaumeister\nand\nJ.\nRothe.\nSatis abilit\ny\nparsimoniously\nredu es\nto\nthe\nT\nan\ntrix\nTM\nrotation\npuzzle\nproblem.\nIn\nPr\no\n \ne\ne\ndings\nof\nthe\n5th\nConfer\nen \ne\non\nMa hines,\nComputations\nand\nUniversality,\npages\n134 145.\nSpringer-V\nerlag\nL\ne\n -\ntur\ne\nNotes\nin\nComputer\nS ien \ne\n#4664,\nSeptem\nb\ner\n2007.\n[CGH+\n88℄\nJ.\nCai,\nT.\nGundermann,\nJ.\nHartmanis,\nL.\nHema \nhandra,\nV.\nSew\nelson,\nK.\nW\nagner,\nand\nG.\nW\ne \nhsung.\nThe\nb\no\nolean\nhierar \nh\ny\nI:\nStru tural\nprop\nerties.\nSIAM\nJournal\non\nComputing,\n17(6):1232 1252,\n1988.\n[CGH+\n89℄\nJ.\nCai,\nT.\nGundermann,\nJ.\nHartmanis,\nL.\nHema \nhandra,\nV.\nSew\nelson,\nK.\nW\nagner,\nand\nG.\nW\ne \nhsung.\nThe\nb\no\nolean\nhierar \nh\ny\nI\nI:\nAppli ations.\nSIAM\nJournal\non\nComputing,\n18(1):95 111,\n1989.\n[CKR95℄\nR.\nChang,\nJ.\nKadin,\nand\nP\n.\nRohatgi.\nOn\nunique\nsatis abilit\ny\nand\nthe\nthreshold\nb\neha\nvior\nof\nrandomized\nredu tions.\nJournal\nof\nComputer\nand\nSystem\nS ien \nes,\n50(3):359 373,\n1995.\n[CM87℄\nJ.\nCai\nand\nG.\nMey\ner.\nGraph\nminimal\nun olorabilit\ny\nis DP\n- omplete.\nSIAM\nJournal\non\nComputing,\n16(2):259 277,\nApril\n1987.\n[Co\no71℄\nS.\nCo\nok.\nThe\n omplexit\ny\nof\ntheorem-pro\nving\npro\n edures.\nIn\nPr\no\n \ne\ne\ndings\nof\nthe\n3r\nd\nA\nCM\nSymp\nosium\non\nThe\nory\nof\nComputing,\npages\n151 158.\nA\nCM\nPress,\n1971.\n[Gol77℄\nL.\nGolds \nhlager.\nThe\nmonotone\nand\nplanar\n ir uit\nv\nalue\nproblems\nare\nlog\nspa e\n omplete\nfor\nP.\nSIGA\nCT\nNews,\n9(2):25 29,\n1977.\n[Grä90℄\nE.\nGrädel.\nDomino\ngames\nand\n omplexit\ny\n.\nSIAM\nJournal\non\nComputing,\n19(5):787 804,\n1990.\n[HH04℄\nM.\nHolzer\nand\nW.\nHolzer.\nT\nan\ntrix\nTM\nrotation\npuzzles\nare\nin\ntra table.\nDis r\nete\nApplie\nd\nMathemati s,\n144(3):345 358,\n2004.\n[M C81℄\nW.\nM Coll.\nPlanar\n rosso\nv\ners.\nIEEE\nT\nr\nansa tions\non\nComputers,\nC-30(3):223 \n225,\n1981.\n[P\nap94℄\nC.\nP\napadimitriou.\nComputational\nComplexity.\nA\nddison-W\nesley\n,\n1994.\n[PY84℄\nC.\nP\napadimitriou\nand\nM.\nY\nannak\nakis.\nThe\n omplexit\ny\nof\nfa ets\n(and\nsome\nfa ets\nof\n omplexit\ny).\nJournal\nof\nComputer\nand\nSystem\nS ien \nes,\n28(2):244 259,\n1984.\n[Rot03℄\nJ.\nRothe.\nExa t\n omplexit\ny\nof\nExa t-Four-Colorabilit\ny\n.\nInformation\nPr\no\n \nessing\nL\netters,\n87(1):7 12,\nJuly\n2003.\n18"},{"page":19,"text":"[Rot05℄\nJ.\nRothe.\nComplexity\nThe\nory\nand\nCryptolo\ngy.\nA\nn\nIntr\no\ndu tion\nto\nCrypto\n \nom-\nplexity.\nEA\nTCS\nT\nexts\nin\nTheoreti al\nComputer\nS ien e.\nSpringer-V\nerlag,\nBerlin,\nHeidelb\nerg,\nNew\nY\nork,\n2005.\n[RR06℄\nT.\nRiege\nand\nJ.\nRothe.\nCompleteness\nin\nthe\nb\no\nolean\nhierar \nh\ny:\nExa t-Four-\nColorabilit\ny\n,\nminimal\ngraph\nun olorabilit\ny\n,\nand\nexa t\ndomati \nn\num\nb\ner\nproblems\n \na\nsurv\ney\n.\nJournal\nof\nUniversal\nComputer\nS ien \ne,\n12(5):551 578,\n2006.\n[V\nal79℄\nL.\nV\nalian\nt.\nThe\n omplexit\ny\nof\n omputing\nthe\np\nermanen\nt.\nThe\nor\neti \nal\nComputer\nS ien \ne,\n8(2):189 201,\n1979.\n[VV86℄\nL.\nV\nalian\nt\nand\nV.\nV\nazirani.\nNP\nis\nas\neasy\nas\ndete ting\nunique\nsolutions.\nThe\no-\nr\neti \nal\nComputer\nS ien \ne,\n47:85 93,\n1986.\n19"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"et Σ = {0, 1},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"DP = {A −B | A, B ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"lass #P = {accM | M","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":", f(x) = g(ρ(x)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"pairs a = (u, w)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"and b = (v, x)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"1. u = v","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"and |w −x| = 1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"2. |u −v| = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"and w = x,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"3. u −v = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"and w −x = 1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"4. u −v = −1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"and w −x = −1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"set shape(A) = {v ∈Z2 | A(v)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"# TRP(A) = ∥Sol","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"TRP = {A | # TRP(A) = 1}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"with 1 ≤i ≤n , αi = xi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"either αi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"or αi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"If αi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"omputes α4 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":39979,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}