A continuous-time approach to constraint satisfaction: Optimization hardness as transient chaos

Transcription

1 A coninuou-ime pproch o conrin ifcion: Opimizion hrdne rnien cho PN-II-RU-TE--- Finl Sineic Repor Generl im nd objecive of he projec Conrin ifcion problem (uch Boolen ifibiliy) coniue one of he hrde cle of opimizion problem wih mny pplicion in echnology nd indury. Undernding why hee problem re hrd i crucilly imporn for he developmen of efficien lgorihm. Before ring hi projec we provided n exc mpping of Boolen ifibiliy (k-sat) ino deerminiic coninuouime dynmicl yem (CTDS) wih unique correpondence beween i e of rcor nd he k-sat oluion []. We hve hown h opimizion hrdne i fundmenlly equivlen o he phenomenon of cho nd urbulence: fer criicl conrin deniy i reched, he rjecorie become rnienly choic before finding he oluion, ignling he ppernce of opimizion hrdne. In hi projec we propoed o explore hi revoluionry new connecion beween rdiionlly epre field. The reerch propoed hd wo min pr wih everl objecive: Aim I: Undernding opimizion hrdne by udying choic properie of he CTDS. Aim II: Developing Cellulr Nonliner Nework (CNN) model for olving ifibiliy problem. In Aim I we propoed o nlyze he choic properie of he CTDS preened in [] rying o revel why cerin problem re hrd for ll known lgorihm. Our gol w o inveige hree mjor queion: A) The choic phe rniion ppering in NP-complee problem. B) he choic behvior of locked occupion problem, hee being conidered he hrde cl of conrin ifcion problem. C) Diinguihing hrd-sat formul chrcerized by long choic rnien from permnenly choic UNSAT formul. In Aim II we propoed A) o develop CNN model for olving SAT problem nd B) udy he effec of noie on hee nlog lgorihm (boh he originl nd he new one), ince noie i he min concern in he implemenion of ll nlog device. Objecive chieved The Gn chr below how he evolvemen of our projec. Even if he order of olving he propoed k h chnged, ll im nd objecive were uccefully chieved. The fir reul cme reled o Aim IB. We udied he properie of Sudoku, which i locked occupion problem, bu being well known o he generl public we ued i n exmple o preen our more generl reul. We developed mehod o meure he hrdne of individul problem. Thi reul w publihed in he open cce journl of Nure Publihing Group, he Scienific Repor [] nd i genered lrge echo in he medi (ee dieminion of reul he end of he repor). Uing he hrdne meure developed we were ble o chieve Aim IA. We proved he exience of choic phe rniion in SAT nd we udied i properie in deil. A pper bou hee reul i under preprion. In he l yer of he projec we worked minly on Aim IC. We developed n lgorihm for olving he mx-sat problem, which prciclly men idenifying he minimum energy level in unifible SAT innce. Thee reul were preened he CNNA conference in, Nore Dme, USA. In prllel we hve lo been working on Aim IIA ince he beginning of he projec. We developed he CNN model, i w publihed nd preened on he CNNA conference in Turin []. In

2 we lo red o work on Aim IIB, udying he effec of noie on hee nlog SAT olver yem. Thee reul hve been finlized nd publihed in Europhyic Leer in. Below we will ummrize hee reul nd will lo menion ome exr reerch civiie nd collborion wih foreign reercher, which hve no been originlly plnned, bu reuled in excellen publicion uch pper in Phyicl Review E, Nure Communicion, PLoS One, nd mo impornly Science pper. A he end we will li ll dieminion civiie, uch publicion, conference, preenion ec. In ol he projec reuled in ISI publicion, one book chper, conference pricipion ( ricle nd orl preenion), invied lk foreign univeriie nd mny echoe in he medi. Tk% $ $ $ $ Aim%I.%Undernding%opmizon%hrdne%by%udying%choc%propere%of%he%CTDS.%% A)$Cho-c$phe$rni-on$(SR/MER)$ B)$Proper-e$of$LOP$(MER)$ C)$Di-nguihing$hrdESAT$form$UNSAT$ (MER/BM/RS)$ Aim%II.%Developing%%CNN%model% A)$Deigning$he$model$(BM/MER)$ B)$Sudying$effec$of$noie$(RS/BM/MER)$ Figure : Gn chr of he projec evoluion. Iniil indice he em member priciping in ech pr of he projec. MER: Dr. Mári Ercey-Rvz, projec direcor. RS: Dr. Rober Sumi, podocorl reercher. BM: Boond Molnár, PhD uden. Summry of Reul Conrin ifcion problem (CSP) rie in mny domin of compuer cience, informion heory nd iicl phyic. In hee problem we re given e of conrin nd we re ked o find n ignmen of he vrible which ifie ll conrin. One of he mo udied CSP i Boolen ifibiliy, epecilly he rndom k-sat problem, which involve N Boolen vrible (x i {, }, i =,..., N) nd M = αn conrin or clue. A k-clue i n OR operion ( ) beween k rndomly choen lierl, mening i i ified if nd only if le one of he lierl i rue. A lierl cn be he norml or neged form (mrked wih overline) of Boolen vrible. A differen phring could be h k-clue forbid one rndom ignmen (ou of he k poibiliie) of he k rndomly included vrible. For exmple x x x forbid (x =, x =, x = ). k-sat (for k ) w he fir CSP proved o be NP-complee [, ] nd h mjor pplicion in rificil inelligence, elecronic deign, uomion, error-correcion, bio-compuionl pplicion ec. Rndom k-sat h become ndrd wy of eing lgorihmic performnce nd evluing complexiy nd hrdne of NP-complee problem. For olving k-sat wih coninuou-ime dynmicl yem [] we reformule he problem on he pce of N coninuou vrible i (i =,..., N)) which cn ke vlue in he inervl i [, ]. When defining correpondence wih he dicree SAT problem, i = would correpond o he Boolen vrible being fle nd i = + o he vrible being rue. Similrly o he rnformion of SAT o pin-gl model, here we lo define n energy funcion which h o be minimized, i ground e (E( ) = ) correponding o he oluion of he SAT innce. We need ech conrin o be chrcerized by co funcion which i if nd only if he Boolen verion of hee rel vrible would ify he clue. The OR operion which pper in ech clue cn be mpped ino muliplicion in he co funcion.

3 For clue like x x x o be ified i i enough h one of he lierl o be rue, o in he co funcion ech lierl will be fcor which i zero if nd only if he lierl i rue. For hi clue hi funcion would look like K() = ( )( )(+ )/, where we inroduce he fcor o normlize he co funcion o he inervl K() [, ]. The energy funcion of he whole SAT innce will be defined he um of he qure of co funcion for ll M clue: E() = M m= K m(). Serching for he minimum hrough coninuou-ime dynmicl yem would nurlly cll for grdien decen dynmic. However, hi energy funcion h mny locl minim nd direc minimizion will innly drive he dynmic in one if hee locl well, non-oluion rcor, where he energy i no zero. When deigning he dynmic our gol w o chieve he following properie []: ) Deerminiic dynmic. ) The dynmic hould y confined in pce [, ] N. ) Every oluion hould be n rcive fixed poin. ) The only fixed poin of ifible formule hould be he oluion. ) Limi cycle hould no exi. We could chieve ll hee by inroducing uxiliry vrible wih imilr role Lgrnge muliplier. Ech conrin i ocied wih n uxiliry vrible m (, ) (m =,..., M), weigh which chrcerize i impornce given momen. The new energy funcion will be defined : V (, ) = M m= mk m (). We cn ee h V (, ) nd V ( ) = if nd only if i oluion. The yem of ordinry differenil equion (ODE) which chieve ll he deired properie i he following: d i d = V (, ) i () d m = m K m (), d () nd we mu hve i () [, ], i nd m () >, m. The dynmic of he vrible i now defined grdien decen on he V (, ) energy lndcpe nd he uxiliry vrible how n exponenil incree whenever he given conrin i no ified (K m () > ). The uxiliry vrible of he lrger K m () vlue grow he fe, hey rpidly domine he V (, ) energy nd he dynmic of move in direcion reducing he K m () co funcion of hee clue. While ech oluion i ble fixed poin of he yem, he rcor re in fc much lrger ubpce correponding o he oluion cluer (for more deiled explnion nd illurion ee he upplemenry informion of []). Aim I. B) Meuring hrdne of conrin ifcion problem Becue of he evolvemen of he reerch we fir were ble o chieve Aim IB, nd only conequenly Aim IA. We preen he reul in hi order. The mhemicl rucure of Sudoku puzzle i kin o ypicl hrd conrin ifcion problem lying he bi of mny pplicion, including proein folding nd he ground-e problem of gly pin yem. Vi n exc mpping of Sudoku ino deerminiic, coninuou-ime dynmicl yem, we hve hown h he difficuly of Sudoku rnle ino rnien choic behvior exhibied by he dynmicl yem. We hve lo hown h he ecpe re κ, n invrin of rnien cho, provide clr meure of he puzzle hrdne h correle well wih humn difficuly ring. Accordingly, η = log κ cn be ued o define Richer -ype cle for puzzle hrdne, wih ey puzzle hving < η, medium one < η, hrd wih < η nd ulr-hrd wih η >. To our be knowledge, here re no known puzzle wih η >. Sudoku k-sat. Becue our coninuou-ime dynmicl yem [] w deigned o olve k-sat formule in conjuncive norml form (CNF), we fir briefly decribe how Sudoku cn be inerpreed +-in--sat formul, nd hen how i i rnformed ino he ndrd CNF form. In Sudoku puzzle we re given qure grid wih = cell, ech o be filled wih one of nine ymbol (digi) D ij {,..., }, i, j =,..., (wih he upper-lef corner of he puzzle correponding

4 o i =, j = ). When he puzzle i compleed ech of he column, row nd ub-grid (block priioned by bold line, Fig. ) mu conin ll he ymbol. Equivlenly, ll ymbol mu pper once nd only once in ech row, column nd ub-grid. To formule Sudoku conrin ifcion problem (CSP) uing boolen vrible, we ocie o ech ymbol (digi) n ordered e of boolen vrible (TRUE=, FALSE= ). The digi D ij in cell (i, j) will be repreened he ordered e (x ij,..., x ij ) wih x ij {, }, =,...,, uch h lwy one nd only one of hem i (TRUE). Thu D ij = i equivlen o wriing x b ij = δ,b, where δ,b i he Kronecker del funcion. Thi wy we hve in ol = boolen vrible x ij, which we cn picure being plced on D grid (Fig. b), wih correponding o he grid index long he vericl direcion, nd hence i he digi h i filling he correponding (i, j) cell in he originl puzzle. The correponding horizonl D lyer heigh will be denoed by L. Inroducing he noion of uch (horizonl) lyer mke i eier o expre he conrin of he Sudoku rule on i repreenion by - nd - decribed below. For exmple in he puzzle hown in Fig., we hve D, =. In he given vericl column he vrible in he h cell (h i in lyer L ) i x, = δ, (hown he boolen vrible filling he cell nex o he digi hown in red, in Fig b). Thi eup llow u o encode he Sudoku conrin in imple mnner. They come from: ) uniquene of he digi in ll he (i, j) Sudoku cell, ) digi mu occur once nd only once in ech row, column nd in ech of he nine ubgrid, nd ) obeying he clue. Conrin ype ) w lredy expreed bove, nmely h for every cell (i, j), in he e (x ij,..., x ij ) one nd only one vrible i TRUE, ll oher mu be FALSE. Type ) conrin re imilr, e.g., in row i nd lyer L he e (x i,..., x i ) mu conin one nd only one TRUE vrible, ll oher mu be fle nd hi mu hold for ll row nd lyer, ec. Oberve h ll conrin re in he form of e of boolen vrible of which we demnd h one nd only one of hem be TRUE, ll oher FALSE. When hi i ified, we y h he conrin ielf (or clue ) i ified, or TRUE. Such CSP re clled +-in-k-sat nd hey re pr of o-clled locked occupion problem, which i cl of excepionlly hrd CSP [, ]. Type ) conrin re genered by he clue (or given) which re ymbol lredy filled in ome of he cell nd heir number nd poiioning deermine he difficuly of he puzzle. They re lo e in wy o gurnee unique oluion o he whole puzzle. If here re given d clue, hen hi implie eing d boolen vrible o TRUE, which men elimining excly d conrin of ype ) nd ) (one vericl or uniquene conrin, one row, one column nd one ubgrid conrin). Thu, Sudoku i +-in--sat ype CSP wih N boolen vrible nd d conrin of +-in-k-sat ype (k ). N i compliced funcion of he poiioning of he clue. The exmple in Fig. h d = clue wih N = unknown boolen vrible. In lyer L illured in Fig. c here re unknown boolen vrible (whie cell). Thee vrible pper in ol of conrin of +-in-k-sat ype. More preciely here i one +-in--sat, ix +-in--sat, four +-in--sat nd ix +-in--sat ype of conrin reled o L. The oher lyer genere he remining d = of +-in-k-sat ype conrin (wih k ). Since our coninuou-ime SAT olver h been deigned o olve boolen ifibiliy problem in conjuncive norml form (CNF), we need o bring he +-in-k-sat ype problem bove ino hi form, nd hu formule i k-sat CNF problem. The CNF i conjuncion (AND, denoed by ) of clue ech clue expreed he dijuncion (OR, denoed by ) of lierl. For k-sat in CNF here re N boolen vrible x i = {, } nd n innce i given propoiionl formul F, which i he conjuncion of M clue C m : F = C C m C M, wih C m = z m... z mkm, k m k nd z j {x j, x j }. According o well known heorem of propoiionl clculu, ll boolen propoiion cn be formuled in CNF uing De Morgn lw nd he diribuive lw, nd hu ny +-in-k-sat ype conrin well (for deil ee []). Once he rnformion o CNF i compleed we re lef wih N vrible nd M clue of he ype decribed bove, clled CNF clue from here on. In our ce he number of vrible ppering in CNF conrin h he propery k m. The prmeer N, M nd {k m } M m= depend on he clue h re difficul o expre nlyiclly, bu ey

5 () (b) (c) L Figure : Sudoku nd i boolen repreenion. () ypicl puzzle wih bold digi clue (given). (b) Seup of he boolen repreenion in grid. (c) Lyer L of he puzzle (he one conining he digi ) wih - in he locion of he clue nd he region blocked ou for digi by he preence of he clue (hded re). o deermine compuionlly. The puzzle from Fig. i ulimely formuled CNF SAT problem wih N = vrible nd M = CNF clue. An ofen ued prmeer of ifibiliy problem i he number of CNF conrin per vrible, or conrin deniy, α = M/N, lo ued ypicl hrdne indicor, however, we how below, hi i no n ccure meure of hrdne. Puzzle hrdne rnien choic dynmic. Since Sudoku puzzle lwy hve oluion, he correponding boolen SAT CNF formulion lo h oluion, nd yem (-) will lwy find i. The nure of he dynmic, however, will depend on he hrdne of he puzzle we decribe nex. In Fig. we how n ey puzzle wih clue (blck number). Afer rnforming hi problem ino SAT CNF, we obin N = nd M =, wih conrin deniy of α = M/N =.. A decribed bove, in our implemenion here i coninuou pin vrible ij ocied o every boolen vrible x ij in every D cell (i, j, ). In he righ pnel of Fig. we how he dynmic of he pin vrible in he cell of he grid formed by row - nd column -. The ij () curve re colored by he digi hey repreen ( =,..., ) indiced in he color legend of Fig. The dynmic w red from rndom iniil condiion. Indeed, our olver find he oluion very quickly, in bou ime uni, for he ey puzzle in Fig.. In Fig. b we how he dynmicl evoluion of vrible for n ulr-hrd Sudoku innce wih only clue. Thi puzzle h been lied one of he world hrde Sudoku, nd even h pecil nme: Plinum Blonde [ ], nd i w he mo difficul for our olver mong ll he puzzle we ried. Afer rnforming i ino SAT CNF, we obin N = vrible nd M = conrin. No only h we hve wice mny unknown vrible bu he conrin deniy α = M/N =. i lo lrger hn in he previou ce, ignling he hrdne of he correponding SAT innce. The complexiy of he dynmic in hi ce i een in he righ pnel of Fig. b, exhibiing long choic rnien before he oluion i found round. For n nimion of he dynmic for imilrly hrd puzzle [] ee Ref []. A Richer-ype cle for Sudoku hrdne. A uggeed by he wo exmple in Fig, he hrdne of Sudoku puzzle correle wih he lengh of choic rnien. A conien wy o chrcerize hee choic rnien i o plo he diribuion of heir lifeime. Sring rjecorie from mny rndom iniil condiion, le p() indice he probbiliy h he dynmic h no found he oluion by ime. A chrceriic propery of rnien cho [, ] in hyperbolic dynmicl yem i h p() how n ympoic exponenil decy: p() e κ, where κ i clled he ecpe re. The ecpe re i n invrin meure of he dynmic in he ene h i chrcerize olely he choic non-rcing e in he phe pce of he yem, nd i doe no depend on he diribuion of he

6 - - i i - Ey% Hrd% ij ij () (b) Figure : Solving Sudoku puzzle wih he deerminiic coninuou-ime olver (-). () preen n ey puzzle wih he evoluion of he coninuou-ime dynmic hown wihin grid (row -, column -). (b) how he me, bu for known, ulr-hrd puzzle clled Plinum Blonde [ ]. The rjecorie in he righ pnel how he evoluion of he nlog vrible ij () colored by he correponding digi. Thu for ech cell (i, j) we hve uch running rjecorie, bu hey cnno lwy be dicerned mny of hem re running on op of one noher, cloe o zero in ().

7 iniil condiion, i uppor, or he deil of he region from where he ecpe i meured ( long i conin he non-rcing e) []. In Fig. we plo he diribuion p() in log-liner cle for everl puzzle ghered form he lierure. The diribuion were obined from over rndom iniil condiion. The decy how wide rnge of vriion beween he puzzle. For ey puzzle he rnien re very hor, p() decy f reuling in lrge ecpe re bu for hrd puzzle κ cn be very mll. Fig. b how zoom ono he p() of hrd puzzle. For exmple, for he puzzle in Fig. we obin κ =., where for Fig.b (Plinum Blonde) he ecpe re i κ =.. In pie of he lrge vribiliy of he decy re, we ee h in ll ce he ecpe i exponenilly f. The everl order of mgniude vribiliy of κ nurlly ugge he ue of logrihmic meure of κ for puzzle hrdne, ee Fig.c, which how he ecpe re on emilog cle funcion of he number of clue, d. Thu, he ecpe re cn be ued o define kind of Richer -ype cle for Sudoku hrdne: η = log (κ) () wih ey puzzle flling in he rnge < η, medium one in < η, hrd one in < η nd for ulr-hrd puzzle η >. We choe everl innce from he Sudoku of he Dy webie [] in four of he cegorie defined here: ey (blck qure), medium (red circle), hrd (green x) nd burd (blue r). Thee ring on he webie ry o eime he hrdne of puzzle when olved by humn. Thee ring correle very well wih our hrdne meure η. Occionlly, dily newpper preen puzzle climed o be he hrde Sudoku puzzle of he yer. In priculr, he ecpe re for he Cvemn Circu winner [] (urquoie dimond) nd he Gurdin hrde puzzle [] (mroon dimond) re indeed one order of mgniude mller hn he hrde puzzle on he dily Sudoku webie, plcing hem η =. nd η =. on he hrdne cle. The USA Tody hrde puzzle [], however, doe no eem o be h hrd for our lgorihm hving η =. (mgen dimond). Eppein [] give wo Sudoku exmple (ornge lef-poining ringle) while decribing hi lgorihm, one wih η =. nd much hrder one wih η =.. Eler e l. [] preen n exremely hrd Sudoku (blck filled circle), which h n ecpe re of κ =. reuling in η =.. The mlle ecpe re we hve found re for he Sudoku lied he hrde on Wikipedi [, ] (red ringle). While he ecpe re correle urpriingly well wih humn ring of Sudoku hrdne, i i nurl o expec correlion wih he number of clue, d. Indeed, generl rule of humb, he fewer clue re given, he hrder he puzzle, however, hi i no univerlly rue []. In Fig.d we hen plo he ecpe re funcion of he conrin deniy α = M/N, leding o prciclly he me concluion. Thi i becue he conrin deniy α i eenilly linerly correled wih he number of given d, hown in Fig.e. The ppren non-monoonic behvior of puzzle hrdne wih he number of given, (or conrin deniy) i due o he fc h hrdne cnno imply be chrcerized by globl, ic vrible uch d or α, bu i lo depend on he poiioning pern of he clue, lo hown by concree exmple in Ref []. Uing he world of Sudoku puzzle, we hve preened furher evidence h opimizion hrdne rnle ino complex dynmicl behvior by n lgorihm erching for oluion in n opiml fhion. Nmely, here eem o be rde-off beween lgorihmic performnce nd he complexiy of he lgorihm nd/or i behvior []. Simple, equenil erch lgorihm hve rivil decripion nd imple dynmic, bu n byml wor-ce performnce ( N ), where lgorihm h re mong he be performer re complex in heir decripion (inrucion-li) nd/or behvior (dynmic). Thi hppen becue in order o improve performnce, lgorihm hve o exploi he rucure of he problem one wy or noher. A hrd problem hve complex rucure, he dynmic of he lgorihm hould be indicive of he problem hrdne. The coninuou-ime dynmicl yem [] (-) deerminiic lgorihm doe hve hee feure: ) he erch hppen on n energy lndcpe V = m mkm h incorpore imulneouly ll he conrin (problem rucure) ) i olve ey problem efficienly

8 () (b) p.. p (c) κ (e). Ey) < pple. = Medium). < pple. = Hrd). < pple. = Ulr&Hrd). <.. d = d κ log pple... ey Ref. [] medium Ref. [] hrd Ref. [] burd Ref. [] Cvemn Circu Ref. [] Gurdin Ref. [] USA Tody Ref. [] exreme Ref. [] very few clue Ref. [-] Eppein Ref. [] Eler e l. Ref. [] Wikipedi Ref. [] (d) κ. d (number of clue) Figure : Ecpe re hrdne indicor. () how he diribuion in log-liner cle of he frcion p() of rndomly red rjecorie of (-) h hve no ye found oluion by nlog ime for number of Sudoku puzzle ken from he lierure (ee legend nd ex) wih wide rnge of humn difficuly ring. The ecpe re i obined from he be fi o he il of he diribuion. (b) i mgnificion of () for hrd puzzle. (c) nd (d) how he ecpe re κ in emilog cle v he number of clue d nd conrin deniy α indicing good correlion wih humn ring (color bnd). (e) how he relionhip beween he number of clue d nd α for he puzzle conidered.

9 (polynomil ime, boh nlog nd dicree) nd ) i gurnee o find oluion o hrd problem even for olvble ce where mny oher lgorihm fil. Alhough i i no polynomil co lgorihm, i eem o find oluion in coninuou-ime h cle polynomilly wih N []. Thee feure nd he fc h he lgorihm i formuled deerminiic dynmicl yem wih coninuou vrible, llow u o pply he heory of nonliner dynmicl yem on CTDS (-) o chrcerize he hrdne of boolen ifibiliy problem. In priculr, vi he meurble ecpe re κ, or i negive logvlue η, we cn provide ingle-clr meure of hrdne, well defined for ny finie innce. We hve illured hi here on Sudoku puzzle, bu he nlyi cn be repeed on ny oher enemble from NP. Hving mhemiclly well-defined number o chrcerize opimizion hrdne for pecific problem in NP provide more informion hn he polynomil/exponenil-ime olvbiliy clificion, or knowing wh he conrin deniy α = M/N i (he ler being non-dynmic/ic meure). Moreover, wihin he frmework of CTDS (-), dynmicl yem nd cho heory mehod cn now be brough forh o help develop novel undernding of opimizion hrdne. Thee reul hve been publihed in Nure Scienific Repor [] nd genered lrge echo in he cienific communiy nd even he medi. In he fir week fer publicion, people cceed our pper, nd everl journl wroe new ricle bou our reul, uch Dily Mil, Huffingon Po ec. Aim I. A) Choic phe rniion Aim IA h now been compleed, bu reul re ill under publicion. For hi reon he ummry of reul i no included in hi online verion of our repor. Aim I. C) Diinguihing hrd-sat problem wih long choic rnien from unifible innce Thi Aim i prciclly equivlen wih olving he mx-sat problem, which k one o find he mximum poible number of ifible conrin in given SAT innce. Thi men h even in unifible (UNSAT) innce we hve o find he be oluion, o we need o be ble o idenify UNSAT innce nd diinguih hem from ifible bu hrd innce. We gin ued he me dynmic preened nd dicued bove. We hve hown, h even if rjecorie never ge rpped, he mo efficien regy i no o r one ingle rjecory nd wi unil i find oluion, bu o r more rjecorie nd op hem fer relively mll moun of ime. To how hi, here we will ue new dicree energy funcion of he yem which decribe he exc number of ol conrin remined unolved fer running he imulion for ime ( [, mx ]). Tking mx-sat problem nd ring N ini rjecorie from rndom iniil condiion, one cn meure for ech ime nd dicree energy E d he number of rjecorie p(, E d ), which hve found e wih energy E d or mller up o ime. Then we define he verge ime-co of finding n energy level E d by running rjecorie up o ime k(, E d ) = N ini /p(, E d ) (Fig. ). We cn ee here i n opiml vlue for ech energy level (red curve). To efficienly olve mx-sat problem mll lerning lgorihm wih he following min ep w developed: ) running he dynmic from differen iniil condiion wih mll mx ; ) recording which energy level were reched in every ime ep, before reching mx (p(, E d )); ) uing he formul k(, E d ) = N ini /p(, E d ), he co of reching energy level E d before ime i clculed (Fig. ); ) he minimum co for every energy level i eleced; ) uing he Levenberg-Mrqurd non-liner curve fiing mehod he energy curve funcion of he co i fied in rel-ime (Fig. ) o pproxime he curve prmeer; ) hee ep re repeed unil good pproximion i obined, increing he number of innce if needed. On Fig. he men co (blue color) of reching he energy level E d by running N ini rjecorie up o ime i repreened. Unil now relively mll innce hve been conidered wih N = vrible nd M = conrin. Afer elecing he lowe co for every energy level (red line in Fig. ) one cn

10 Figure : The verge co k(, E d ) (blue colour cle) of reching energy E d by running rjecorie up o ime in mx-sat problem wih N = lierl nd M = conrin. Drker colour repreen increed co, blck colour correponding o infiniely lrge co (here i no e wih h low energy). The red curve how he opiml ime limi unil rjecorie hould run o rech given energy. Figure : The reched energy level funcion of he necery (opiml) co i repreened of mx-sat yem wih N = lierl nd M = conrin wih he red line repreening he Levenberg-Mrqurd fi. The prmeer of he non-liner curve fiing give u he expeced energy minimum nd he hrdne of he problem. ee h here i no need for running he imulion longer hn mx =, bu increing he iic lower energy level become rechble. The energy level funcion of he co of reching h level h power-lw decy, which converge o conn digi, decribed by: f(x) = A + A x A. The Levenberg-Mrqurd non-liner curve fiing lgorihm w ued in order o clcule he prmeer of he power-lw decy. We oberved h he prediced lowe poible energy e of he yem i repreened by he conn o which our yem i converging (A ) nd he exponen (A ) of he power-lw cn be conidered hrdne of he problem, providing poenil ool for clificion of NP-hrd problem ccording o heir difficuly. In order o verify our finding he reul were compred o he one obined by he mxz lgorihm, he winner of he h Inernionl mx-sat Solving Compeiion (). Thi lgorihm i n exc olver. The comprion cn be een in he following ble:

11 problem mxz Our predicion Our reched Thee were benchmrk mx-sat innce were ued nd rn from round..m epre iniil condiion, depending on he hrdne of he problem. We developed mehod bed on he nlog dynmicl yem preened in [] o pproxime he globl minim of NP-hrd mx-sat problem. Reul how h he opiml regy i o run mny rjecorie up o relively mll ime. By exrpoling he prmeer of he power-lw decy of he reched energy level funcion of he needed co we cn pproxime he globl minimum of he yem nd he hrdne of he problem. Even if hi pproximion mehod require fine uning, he reul re in good greemen wih he one obined uing he mxz exc olver. We preened nd publihed hee reul he CNNA conference. We re ill working on refining he mehod nd our fuure gol i o pply i lo on he CNN SAT-olver preened in [, ]. Aim II. A) CNN model for k-sat One of he min bck drw of our nlog yem i h i cnno be implemened by nlog circui (he uxiliry vrible re unbounded). For h reon in Aim IIA we propoed o develop deerminiic coninuou-ime recurren neurl nework imilr o CNN model, which cn olve Boolen ifibiliy (k-sat) problem wihou geing rpped in non-oluion fixed poin. The model cn be implemened by nlog circui, in which ce he lgorihm would ke ingle operion: he emple (connecion weigh) i e by he k-sat innce nd ring from ny iniil condiion he yem converge o oluion. We proved h here i one-o-one correpondence beween he ble fixed poin of he model nd he k-sat oluion nd preen numericl evidence h limi cycle my lo be voided by ppropriely chooing he prmeer of he model []. The coninuou-ime recurren neurl nework model for olving k-sat wih he following key properie: I h deerminiic coninuo-ime dynmic. All vrible remin bounded. The dynmic cn be implemened wih nlog circui (h lmo he me form ued in CNN compuer). There i one-o-one correpondence beween he ble fixed-poin of he yem nd he oluion of he k-sat problem. On n nlog device hi would be one-operion (one ingle emple) lgorihm for olving k-sat: he connecion weigh (emple) re defined by he k-sat innce nd ring from ny rbirry iniil condiion he yem converge o oluion wihou geing rpped. The model. The imple form of dynmic ued in CTRNN i: dx i () d = x i () + j w ij f(x j ()) + u i ()

12 where x i i he e vlue, or civion poenil of he cell, f(x) i he oupu funcion of he neuron (uully igmoid), u i i he inpu, or bi of he neuron nd w ij re connecion weigh. There re mny vrin ued in he lierure, where in ome ce he inpu my be ime dependen u i (); or i i llowed for cell (in CNN) o be influenced by he inpu of neighboring cell, ec. Here we ue hi imple form. Figure : ) The biprie grph repreenion of k-sat: vrible repreen he vrible of he SAT formul, cell correpond o he conrin. The ign of connecion i given by he mrix c mi encoding he k-sat innce. b) The oupu funcion of -ype cell. c) he oupu funcion of -ype cell. We define CTRNN on biprie grph wih wo ype of node (cell) (Fig.). One ype ( ype ) repreen he vrible of k-sat, whoe e vlue will be denoed by i, i =,..., N nd heir oupu funcion defined vi (ee Fig.b): f( i ) = ( i + i ). () Here we ign o f( i ) = he Boolen vrible x i being rue (x i = ) nd o f( i ) = he vrible being fle (x i = ). However, in he dynmic we lo llow ny coninuou vlue f( i ) [, ]. For impliciy we y h f() i oluion of k-sat, whenever x = [f() + ]/ i oluion. The elf-coupling prmeer will be fixed vlue w ii = A nd he inpu o hee -ype cell i u i = i. The econd ype of cell repreen he clue wih e vlue m, m =,..., M nd oupu funcion (Fig.c): g( m ) = ( + m m ). () Thee vrible, or -ype cell will ply imilr role he Lgrnge muliplier in [ ] or he uxiliry vrible in []. They deermine he impc clue h given momen on he dynmic of he vrible. For hi reon g( m ) = will correpond o he clue being rue, nd g( m ) = o he clue being fle. For hee cell he elf-coupling i w mm = B nd he inpu i u m = u = k where k repreen he number of vrible in he clue (k = for -SAT). Thi i needed in order o chieve he correpondence beween k-sat oluion nd ble fixed poin. The connecion weigh beween he cell re deermined by he c mi mrix elemen of he given k-sat problem. The dynmicl yem

13 i defined vi: ṡ i () = d i() d ȧ m () = d m() d = i () + Af( i ()) + c mi g( m ()) m () = m () + Bg( m ()) c mi f( i ()) + k i () Properie of he CTRNN. We proved he following heorem []: Theorem. Vrible remin bounded: If iniilly i () nd m (), hen he e vlue of cell i () nd m () remin bounded for ll >, i, m. The following bound re igh: i () + A + m c mi () k m () + B () Theorem. Every k-sat oluion h correponding ble fixed poin: Given k-sat formul F, if f( i ) = ±, i =,..., N i oluion of F nd A >, B > hen he (, ) poin: i = Af( i ), m = j c mj f( j ) + k () i =,..., N, m =,..., M i ble fixed poin of he yem (-). Theorem. A ble fixed poin lwy correpond o oluion. If < A <, < B < nd (, ) i ble fixed poin, hen f( ) mu be oluion of he k-sat formul. Numericl reul. While we proved nlyiclly h ll ble fixed poin of he yem correpond o k-sat oluion, i doe no gurnee h here re no oher rcor (uch limi cycle or choic rcor) in he yem. The exience or non-exience of uch rcor i very difficul o how nlyiclly, bu here we preen numericl evidence, howing h by ppropriely chooing he prmeer A, B he dynmic void geing rpped in non-oluion rcor nd converge o k-sat oluion. We inveiged in more deph how he efficiency of he yem in finding oluion depend on he A, B prmeer. In Fig. we how wo mp covering he A (, ), B (, ) prmeer region, depicing he performnce of he yem. For ech poin of he mp we rn differen -SAT problem (we ue only ifible innce) wih N = nd α =., hi being in he hrde region for -SAT. A prepring uch mp re compuionlly coly, we hd o ue mll innce. On Fig he color give he frcion of olved problem in he given ime mx = nd on Figb we how he verge coninuou-ime (no he imulion running ime) he yem ke o olve hem (ee color br). There i lrge prmeer region where he oluion i found efficienly. I i inereing h he opiml prmeer eing eem o be round A =., B =. nd i i firly independen from he ize of he yem N (no hown). Thi ugge h Theorem could be exended for lrger vlue of B >, emen h we will inveige in fuure udie. Conrry o he yem preened in [] we expec n exponenil ime complexiy, becue he vrible remin bounded nd here i no exr energy inroduced ino he yem. On Fig we cn ee he frcion p() of problem which remin unolved fer ime for vriou ize (N =,,..., ) of rndomly choen -SAT innce he opiml prmeer vlue A =., B =.. The diribuion re decreing power lw (p() β(n) ), where he power β(n) depend on he ize of he k-sat innce. Becue β(n) i gin power lw (very cloe o β(n) /N) i cn be hown h indeed, he ime complexiy of he model i exponenil for olving fixed frcion of problem. Thi powerlw decree of p() how h he probbiliy of no finding he oluion goe o zero (no poiive conn), upporing he clim h he dynmic doe no ge rpped in limi cycle.

14 %.. B.. mx %.... B.. A.. A Figure : For ech poin A (, ), B (, ) on he mp we olve rndomly choen ifible -SAT innce, wih N =, α =., mx =. ) The frcion of olved problem, b) he verge coninuo-ime (ee color br). p.. -. β e+. N Figure : The number of -SAT problem p() which remin unolved funcion of he coninuou-ime of he yem, for differen vlue of N (ee he legend). The diribuion re fied wih power lw p() β(n ), nd he righ pnel how he dependence of he exponen β on N. Thi cn be fied wih power lw β(n ) N..

15 Dicuion. We preened n implemenion friendly dynmicl yem imilr o CNN model, which olve Boolen ifibiliy problem by converging o ble fixed poin of he dynmicl yem. On n nlog device hi lgorihm would ke ingle operion: he connecion weigh (emple) re bed on he c mi mrix correponding o he given k-sat innce nd ring from ny iniil condiion he yem converge o oluion, wihou he need of ny furher inervenion by he uer. We hve hown numericlly h here i lrge nd robu prmeer region where limi cycle do no pper, nd he oluion i lwy found even if he dynmic goe hrough rnienly choic phe. Alhough hi CTDS i very differen from he one preened in [], hi model lo how choic behvior, epecilly in he hrd-sat phe. Thi confirm he quliive equivlence beween opimizion hrdne nd choic behvior exhibied by nlog erch lgorihm. The reul h been publihed in he conference proceeding of he Cellulr Nnocle Nework nd Applicion Conference in, Turin []. I genered gre inere, he projec direcor h been ler invied o he Péer Pázmny Cholic Univeriy in Budpe o give lk o he engineer, in Februry. Ler in we lo publihed more deiled udy in journl PLoS One. Aim II. B) Robune o noie of he nlog SAT olver When implemening hee nlog SAT olver wih nlog circui, he gre queion come if hee will be robu o noie. Noie i chrceriic of ll elecronic circui, nd i cued by mll flucuion of curren nd volge. In elecronic circui differen noie ype cn be idenified: ho, herml, flicker, bur, vlnche noie ec. []. The effec of differen noie ource i inroduced by dding rndom erm on he righ hnd ide of equion nd, rnforming hem ino ochic differenil equion (SDE). Simuling SDE need compleely new echnique, i i pr of relively new evolving field, which we needed o ge fmilir wih. We conidered wo ype of noie: whie nd colored noie. Whie noie rndom vrible i defined he limi: ξ() = lim N(, /d), () d where N(, /d) denoe norml rndom vrible wih men nd vrince /d []. In order o hve he noie ineniy equl o D we muliply ξ() by D. Equion nd become: i = V (, ) + Dξ i (), i =,..., N () i m = m K m + Dξ m (), m =,..., M () Whie noie h he following iicl properie: ξ() =, ξ()ξ( ) = δ( ), () where δ denoe he Dirc-del funcion. In conr wih whie noie, which i uncorrelled in ime, colored noie i correled proce. I cn be obined by olving he Lngevin equion [ ]: ɛ() = γɛ() + Dξ(), () The oluion ɛ() i exponenilly colored noie (Ornein-Uhlenbeck proce) wih properie: ɛ() =, ɛ()ɛ( ) = D γ e γ ) ()

16 τ = /γ i he correlion ime of he colored noie, nd Dτ i he ineniy. Thu he colored noie driven yem i decribed by he following SDE yem: i = M m c mi K mi K m + ɛ i (), i =,..., N () m= ɛ i = γɛ i + Dξ(), i =,..., N () m = m K m + ɛ m (), m =,..., M () ɛ m = γɛ m + Dξ(), m =,..., M () Numericl oluion. To olve our equion we ued he Euler-Mruym mehod, which i one of he imple numericl mehod for olving SDE []. Conider he ochic proce X on he ime inervl [, T ] nd he following SDE: Fir rewrie hi equion o X = (, X ) + b(, X )ξ () dx = (, X )d + b(, X ) dn(, ), () where N(, ) denoe he ndrd norml rndom vrible. The Euler-Mruym pproximion conider he dicreizion = τ < τ < < τ n < < τ N = T of he ime inervl, nd pproxime X by coninuou ime ochic proce Y ifying: Y n+ = Y n + (τ n, Y n ) n + b(τ n, Y n ) n N(, ), () where Y n = Y (τ n ) nd n = τ n+ τ n. Simulion reul. We hve imuled he effec of noie on lrge number of rndom SAT innce wih differen number of vrible nd conrin deniy prmeer, α = M/N. To illure how noie cn ffec rjecorie in Fig. we how he dynmic of one choen i nd m vrible in - SAT problem wih N = vrible nd α =. conrin deniy (which fll in he hrd-sat region of -SAT). We plo he rjecory of he vrible for hree differen noie ineniie I =.,.,. nd everl correlion ime. In Fig. for exmple we how he rjecory of vrible noie ineniy I =.. Zooming on he rjecorie (ee ine) we ee h whie noie how he lrge flucuion (red). Increing he correlion ime he curve eem o be much mooher (D i mller), however he effec of noie i ill here becue of he correlion (I = Dτ for colored noie). The figure how h rjecorie follow he me ph up o relively long ime ( even lrge ineniy Fig. e), when ome of hem uddenly ge divered. Neverhele, he oluion i ill found, omeime even much erlier hn wihou noie. I i inereing h even if curve re ploed for differen prmeer eing in ol, here pper only chrceriic ph. I eem h noie i ju wiching beween everl rongly rcing rjecorie. Anoher wy o inveige he enibiliy of rjecorie o noie i o plo he rcor bin mp. Fixing rndom iniil condiion in -SAT problem wih N =, α =. we vry only wo rndomly choen vrible ( nd ) long grid nd colour he poin ccording o he oluion hey flow o up o ime mx =. Blck colour indice h no oluion h been found. In Fig. we how hi mp wih differen noie-prmeer eing. The ineniy of noie clerly ffec he rjecorie. The mp how ignificn chnge nd become more nd more blurry, bu he oluion re ill found wih high probbiliy. Eiming he ize of rcor bin (probbiliy of finding oluion) in he whole phe pce (no only long he mp hown) indice h noie doe no induce ignificn chnge. For he eing ued in Fig. hee how very mll vribiliy:.,.,.,. (rcor bin mrked wih green),.,.,.,. (ornge),.,.,

17 no noie no noie... no = noie= f d e c b =. = ==. whie noie =. noie =no. = =. =whie =. no noie. no noie whie noie =noie = = = ==. = =. == noie whie noie whie noie =. =.whie Figure : Time evoluion of rndomly choen (lef column) nd vrible (righ column) in problem wih N = vrible nd α =.. Color correpond o differen τ correlion ime (ee legend). The ineniy i, b) I =., c, d) I =., e, f) I =.. Trjecorie ermine when oluion i found. A mgnified porion i hown in he ine. Figure : Arcor bin in -SAT problem wih N =, α =.. Fixing rndom iniil condiion we chnge nd long grid. Ech poin i colored ccording o he oluion he dynmic flow o, nd blck when no oluion h been found up o ime mx =. Noie-prmeer eing: ) No noie, b) τ =, I =., c) τ =., I =., d) τ =., I =.. Lrger ineniy nd mller correlion ime mke he mp blurrier.

18 .,. (nvy blue),.,.,.,. (red),.,.,.,. (royl blue). Concerning he vibiliy of our SAT olving mehod in preence of noie he mo imporn queion i no wheher rjecorie ge divered by noie, bu wheher he probbiliy of finding oluion chnge. When here re more oluion no lgorihm cn conrol which oluion i found, i will depend on iniil condiion or oher prmeer. In our ce here i mpping beween he iniil condiion nd he oluion found (Fig. ). Noie chnge hi mpping, bu if he expeced ime for finding oluion i no chnged he gol of he lgorihm i chieved. For hi reon we meured he diribuion of rnien ime. Tking, rndom -SAT problem boh in he ey- (α =.) nd in he hrd-sat phe (α =.), ech ime we r he dynmic from rndom iniil condiion nd we meure wh i he probbiliy p() h he oluion h no been found up o ime. In Fig.,b we plo hi diribuion in ce of whie noie for everl prmeer eing. Wihou noie hi diribuion follow n exponenil decy p() e κ, chrceriic o rnien cho, κ denoing he ecpe re [, ]. In he preence of whie noie minly he end of he diribuion chnge. Fiing he curve (no hown) indice h hey follow p() = c + be κ, mening hey ure o conn c in. The fied prmeer κ nd c re hown funcion of he noie ineniy in Fig. c,d. Up o I =. lmo no chnge cn be noiced, c. Above hi hrehold, which i firly independen of N or α, boh c nd κ uddenly incree. Neverhele, even lrger ineniie he fir pr of he diribuion ( < ) i unffeced, mening here re ill mny hor rjecorie finding oluion no diurbed by noie. c men here will be rjecorie never finding oluion. Thi hppen, becue noie cn omeime puh ou rjecorie from he H = [, ] N hypercube, from where hey cnno reurn. The effec of colored noie i very imilr (Fig.,c), he ineniy hrehold i pproximely he me. Sudying he effec of correlion ime he iic w mde on rndom -SAT problem wih N = vrible wo differen conrin deniie nd noie ineniy I =. (Fig. b,d). The effec i lrge when τ i mll, he hrehold being omewhere round τ. Robune of he CNN model Even if he olered noie level i relively high, implemening hi yem in elecronic circui poe ome chllenge, mo imporn being he unbounded nure of he uxiliry vrible. Subequenly he polynomil-ime complexiy cn be chieved only up o limi of problem ize, depending on he poible rnge of uxiliry vrible ured by he implemenion. In hope of more righforwrd implemenion we recenly developed cellulr neurl nework ype model, which cn lo olve SAT problem [, ]. CNN model hve lredy been implemened nd ued in engineering pplicion []. In hi ce ll vrible re bounded, no exr energy cn be inroduced ino he yem o ure he polynomil efficiency, bu he peed of compuing wih cul nlog circui could ill offer unuully high compuionl efficiency. In he CNN model we gin hve wo ype of vrible, lo clled cell [?], repreening he boolen vrible (-ype) nd clue (-ype) repecively. A in ny recurren neurl nework model, ech cell h e vlue nd n oupu vlue. The oupu funcion of -ype vrible i f( i ) = ( i + i ) uring h f( i ) [, ], nd he oupu of -ype vrible will now be limied o [, ] inervl hrough he oupu funcion g( m ) = ( + m m ). The connecion beween cell re bed on he SAT problem nd re gin deermined by he c mi vlue defined bove (e.g. if vrible i pper in i norml form in clue m here will be connecion beween hee wo cell wih weigh c mi = +). The CNN olving k-sat problem i: ṡ i = i + Af( i ) + m c mi g( m ) () ȧ m = m + Bg( m ) i c mi f( i ) + k () where A, B re elf-connecing prmeer of he wo ype of cell. In [?, ] we hve hown h hi model lo preen one-o-one correpondence beween i ble fixed poin nd he SAT oluion.

19 b p() α=. p() α=. c. κ. d. c.. I x I x N= N= I = x N= Figure : Frcion of unolved problem p() funcion of ime in he preence of whie noie wih vrying ineniie I (ee legend) for yem ize N = (circle), N = (ringle) nd N = (r) nd conrin deniie ) α =. b) α =.. Afer fiing p() = c + be κ we plo c) κ nd d) c funcion of he ineniy for he wo differen α:. (blue),. (red). While limi cycle do exi, hee cn be voided by properly chooing he vlue of A nd B, which hve n opiml region firly independen of he priculr SAT problem (he vlue of N, M, nd even k) []. To udy he effec of noie in hi dynmicl yem, we ued he me mehod decribed for he previou model. Ech vrible i igned correponding rndom vrible ξ () i () nd ξ m () () nd he erm Dξ () i () nd Dξ m () () re dded o he righ hnd ide of equion. For coloured noie he Lngevin equion mu lo be included he me wy in Eq. (??-??). Similr nlyi hve been performed in he previou ce. The rjecorie being choic re nurlly eniive o noie, bu ploing he p() diribuion of rnien ime i remin firly unffeced up o lrge ineniy. Here he oupu funcion of cell do no llow he yem o ecpe from he erching pce nd diribuion do no ure, hey keep power-lw decy p() β []. Thi i differen compred o he exponenil decy een in he previou model nd i indice in fc he quliive difference beween he wo model, hi i why he CNN model cnno chieve polynomil coninuou-ime efficiency (ee []). In Fig., b we plo hee diribuion wo differen A, B prmeer eing, one of hem being in he opiml region (where he yem work mo efficienly, hown in []), nd he oher being nonopiml. Only reul for whie noie re hown, bu colored noie h very imilr effec. The ine how he power β, which how liner decy funcion of he ineniy. I i inereing h in ce of he non-opiml prmeer eing up o urpriingly lrge noie ineniy I he yem become more efficien. Thi i promiing for cul implemenion, nd decree even more he eniiviy of he model o he prmeer A, B. Thi ineniy i well olered lo in ce of he opiml prmeer. In circui implemenion of CNN model noher concern i he preciion of connecion weigh. When producing circui elemen (reior, cpcior, ec.) he prmeer of hee elemen will how error compred o he heoreiclly propoed vlue. For h reon we lo inveiged he effec

20 p(). κ c. c. α=. x I p().. c. κ α=.. τ I x. τ N= N= I = x N= α=. α=. τ =. b d α=. α=. α=. Figure : Frcion of unolved problem p() for ) α =., τ = nd vrying N nd I (ee legend); b) N =, α =.,., I =. nd vrying correlion ime τ (legend). Ine how he vriion of κ. c,d) how he vriion of he conn c funcion of I nd τ repecively.

21 A=. B=. i b A=. B=. i connec i onwei gher r or whi enoi e c ε d ε Figure : Frcion of unolved problem for N =, α =. wih prmeer,c) A =., B =. nd b,d) A =., B =. in preence of,b) whie noie wih differen ineniie (ee legend) nd c,d) connecion weigh error of differen mgniude (ee legend). Ine how he vriion of β funcion of noie ineniy.

22 of error in he c mi connecion weigh. Ech connecion weigh i igned rndom deviion vlue beween [ ɛ, ɛ], conn in ime. In Fig. c, d we plo he p() diribuion for he wo A, B prmeer eing ued bove, for differen vlue of ɛ. A very imilr behvior cn be oberved. Surpriingly lrge error up o ɛ. ill llow he yem o funcion. Dicuion I i necery o k how he noie ineniie hown here o be ccepble compre o cul noie oberved in circui nd nlog compuer device. In CNN compuer he mpliude of noie h been hown o be le wo order of mgniude mller hn he vlue of volge (hi h even improved in ler implemenion). For exmple in he Q-eye chip [, ] he mximl ignl-o-noie A rio (SNR) i db. By definiion SNR= log ignl A noie, mening h he mpliude of noie on he chip i le. ime mller hn he mpliude of he ignl. In ce of whie noie nd for vlue chnging beween [, ] hi would men flucuion A noie = D <., correponding o whie noie ineniy I <.. In our CNN model he ccepble ineniy i round I =, nd even in he fir, more eniive model i i round I =.. The gree problem in nlog compuing h lwy been he preence of noie. Unil now developing robu dynmicl lgorihm hve been poible only for imple converging yem wih Lypunov dynmic. Solving complex opimizion problem uch imple dynmic cn minimize n energy funcion only by converging o he cloe locl minimum [?, ]. In order o find he globl opimum hi Lypunov dynmic w ued bic ep of more complex lgorihm imilr o hoe ued in digil compuing (uch imuled nneling ec.) [ ]. Thi ignificnly reduced he power of nlog compuing. Our model offer wo gre dvnge: ) he opimizion i relized hrough one ingle dynmicl proce: he dynmic i red from rndom iniil condiion nd erche unil finding he oluion, wihou furher inervenion needed by he uer; ) In pie of he fc h hrd opimizion problem necerily preen rnienly choic dynmic, he probbiliy of finding oluion i robu o noie. The lrge noie ineniy level olered promie he poibiliy for highly robu nd efficien phyicl implemenion. Thee reul were publihed in Europhyic Leer (). Reerch nd collborion beide he originlly plnned civiie During hee hree yer ome gre opporuniie cme o collbore wih foreign reercher. A he direcor of young Romnin reerch em I conidered i imporn o minin hee relion nd pricipe in hee reerch projec even if hee were ou of plnning. Thee were exr civiie nd did no ler he coure of he originlly plnned projec. Collboring wih my previou upervior, Prof. Zolán Toroczki, he Univeriy of Nore Dme, IN, USA we finlized ome publicion, which were hen cceped in Phyicl Review E () nd Nure Communicion (ee dieminion civiie). Thee re heoreicl work reled o complex nework nd grph heory. Alo in, collboring wih Prof. Toroczki, Prof. Józef Brnyi from UK, nd Zolán Lkner form Hungry, we publihed pper in PLoS One bou he complexiy of he inernionl gro-food rde nework. The gree opporuniy cme collborion wih he em of Dr. Henry Kennedy, neurocieni form he Sem Cell nd Brin Reerch Iniue from Lyon. While I previouly collbored wih hem pr of oher projec, now new opporuniy cme. They ked me o pricipe in wriing review pper in he neurocience pecil iue of Science. Thi i reled o he iner-rel nework of he brin, which i highly complex yem. I priciped in nlyzing heir d form phyici poin of view. Thi pper ppered in November (ee dieminion of reul).

23 Dieminion of reul Alo ee our webpge: hp://iriu.phy.ubbcluj.ro:/ercey-rvz/turbcomp/ Publicion ISI Journl ricle: - R. Sumi, M. Vrg, Z. Toroczki, M. Ercey-Rvz, From order o cho in rndom ifibiliy problem, o be ubmied oon - Y. Ren, M. Ercey-Rvz, P. Wng, M.C. Gonzlez, Z. Toroczki, Predicing commuer flow in pil nework uing rdiion model bed on emporl rnge, Nure Communicion,, () (impc fcor:.) -R. Sumi, B. Molnár, M. Ercey-Rvz, Robu opimizion wih rnienly choic dynmicl yem, Europhyic Leer,,, () (impc fcor., relive infl. core.) - N.T. Mrkov, M. Ercey-Rvz, D.C. Vn Een, K. Knobluch, Z. Toroczki, H. Kennedy, Coricl High-deniy Couner-rem Archiecure, Science,,, (impc fcor., relive influence core.) - M. Ercey-Rvz, Z. Toroczki, The Cho Wihin Sudoku, Nure Scienific Repor, () doi:./rep (impc fcor:.) - M. Ercey-Rvz, R. Lichenwler, N.W. Chwl, Z. Toroczki, Rnge-limied Cenrliy Meure in Non-weighed nd Weighed Complex Nework, Phyicl Review E, (). rxiv:. (impc fcor., relive influence core.) - M. Ercey-Rvz, Z. Toroczki, Z. Lkner, J. Brnyi, Complexiy of he Inernionl Agro-Food Trde Nework, PLoS ONE (), e (). doi:./journl.pone. (impc fcor., relive influence core.) ISI Conference Proceeding: - B. Molnár, M. Ercey-Rvz, Anlog dynmic for olving mx-sat problem, Proceeding of CNNA, Nore Dme, IN, USA () - B. Molnár, R. Sumi, M. Ercey-Rvz, A CNN SAT-olver robu o noie, Proceeding of CNNA, Nore Dme, IN, USA () - K. Knobluch, M. Ercey-Rvz, H. Kennedy, Z. Toroczki, The Brin in Spce, Proc. of IPSEN, Pri, My (). - B. Molnár, M. Ercey-Rvz, Z. Toroczki, Coninuou-ime Neurl Nework Wihou Locl Trp for Solving Boolen Sifibiliy, Proceeding of CNNA, p., Torino, Ily () Book chper: - M. Ercey-Rvz, Z. Toroczki, Döneek fizikáj é rejvények káoz ( Phyic of deciion mking nd cho of puzzle ) in A fizik, memik é művéze lálkozá z okábn, kuábn (Phyic, mhemic nd r in educion nd reerch), Ed.: A. Juház, T. Tél, Publiher: Science Deprmen of he Eövö Lóránd Univeriy, Hungry,. Oher publicion: - M. Ercey-Rvz, Z. Toroczki, A dönéhozl é Sudoku káoz ( The Cho Wihin Sudoku nd Deciion Mking ), Terméze Világ (World of Nure), invied pper in he pecil iue Káoz, Környeze, Komplexiá ( Cho, Environmen, Complexiy ), Budpe, Hungry, Ocober