Monte Carlo Tree Search in Hex
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- Tyrone Barrett
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1 IEEE Trns. on Comput l Intl. n AI n Gms (spl ssu: Mont Crlo Tnqus n Computr Go) vol no D 00 - Mont Crlo Tr Sr n Hx Bror Arnson, Ryn B. Hywr, Plp Hnrson Astrt Hx, t lss or m nvnt y Pt Hn n n npnntly y Jon Ns n, s n omn o rtl ntlln rsr sn Clu Snnon s smnl wor n t 0s. Untl t Mont Crlo Go rvoluton w yrs o, t st omputr Hx plyrs us nowl-ntnsv lp-t sr. Sn tt tm, stron Mont Crlo Hx plyrs v ppr tt r on pr wt t st lp-t Hx plyrs. In ts ppr w sr MoHx, t Mont Crlo Tr Sr Hx plyr tt won ol n t 00 Computr Olymp. Our mn ontrutons to Mont Crlo Tr Sr nlu usn nror ll nlyss n onnton strty omputton to prun t sr tr. In prtulr, w run our rnom m smultons not on t tul m poston, ut on ru quvlnt or. I. INTRODUCTION Mont Crlo Tr Sr (MCTS) s n xtn nw vlopmnt n t l o utomt m plyrs, ttnn rt suss n vrty o ms, most notly Go [0], [], [], []. Hx s t lss or m nvnt y Pt Hn n n npnntly y Jon Ns n [], []. Hx s n omn o rtl ntlln rsr sn Clu Snnon s smnl wor n t 0s []. L Go, Hx s lr rnn tor tt lmts utomt plyrs to sllow or sltv sr. Unl Go prorms, Hx prorms v rsonly stron vluton untons, n tror strtorwr lp-t tnqus v n sussul [], [], []. W n xprmntn wt t posslty o n tv utomt Hx plyr s on t MCTS rmwor n rly 00. T rsultn plyr, nm MoHx, won slvr n ol rsptvly n t 00 n 00 Hx Computr Olymps [], []. In t lttr tournmnt MoHx ws unt. In ts ppr w sr t rmwor o MoHx, wt prtulr mpss on ts lortm ontrutons w xplot mtmtl proprts o Hx: MoHx uss provn nowl n t MCTS tr to prun nror lrn, n MoHx omputs Mont Crlo smultons rom ru quvlnt or, rtr tn t tul m or. In II w rvw t ruls o Hx, n lortms or nn nror lls n onnton strts. In III w suss prvous utomt Hx plyrs n rvw t suss o lp-t s prorms. In IV w sr t s rmwor o MoHx. In V w sr our nnmnts o MoHx, splly Hx-sp tnqus or prunn lrn n t MCTS tr, n or onstrutn ru ut quvlnt ors w mprov n lrt MoHx s m smultons. In VI w nlyz xprmntl t w msurs t tvnss o our tnqus n t sln o MoHx s strnt wt rtr omputn powr. In VII w nlyz MoHx s prormn nst otr utomt Hx plyrs. In VIII w suss t potntl pplton o ts tnqus to rlt ms, su s Y n Hvnn. In IX w rvw MoHx s urrnt lmttons, n vnus or utur rsr. A. Ruls o Hx II. RULES AND ALGORITHMS OF HEX Hx s smpl ruls: Bl n Wt ltrnt turns, n on turn plyr pls snl ston o tr olour on ny unoup ll. T wnnr s t plyr wo orms n o tr stons onntn tr two opposn or ss. S Fur. F.. An mpty Hx or n m won y Wt. It s nvr svntous or plyrs to v n xtr ston o tr olour on t or, n Hx n nvr n n rw, so y Ns s strty-stln rumnt Hx s rstplyr wn on ll n n ors []. T rst-plyr vnt s onsrl n wn unrstrt usully ls to n sy wn, so Hx s otn ply wt t swp rul: t rst plyr ms Bl s rst mov, n t son plyr tn ooss wtr to ply s Bl or Wt. Wt ms t nxt mov, n t plyrs ltrnt turns trtr. Ts vrton s son-plyr wn, ut n prt prous losr ms tn wtout t swp rul. L Go, Hx n ply on ny n n or. T Computr Olymp Hx tournmnts us ors. Humns typlly ply on or szs rnn rom to [], [], wt nnrs otn strtn out on smllr ors. Automt Hx solvrs v solv ll opnn movs on ll or szs up to n nlun []. B. Inror Cll Anlyss Tr r two mn tnqus n Hx nror ll nlyss: lln n mov prunn. Flln s t pross o n to Hx poston st o stons tt s urnt not to ltr t poston s mnmx
2 A vrtul onnton (VC) s son-plyr strty or onntn two nponts, n vrtul sm onnton (SC) s rst-plyr strty or onntn two nponts. A wnnn onnton s strty wos two nponts r opposn or ss. T rrr o onnton strty s t (mnml) st o mpty lls rqur to rry out ts strty. S Fur. F.. Lol lln pttrns. Top tr pttrns nty lls, n n ll n wt stons o tr plyr. Bottom tr pttrns nty Bl-ptur rons, n n ll n wt Bl stons. + vlu. Tr r two mn tors o lln: lls n ptur rons. A ll s ll tt s provly uslss or ot plyrs. A ptur ron s st o lls on w on plyr s son-plyr strty tt nts ny nt tr opponnt mt n rom plyn n t ron. MoHx uss nntn lol pttrns or ntyn lls n ptur rons; som r sown n Fur. MoHx lso uss rp-tort lortms n or ompostons to n lrr lsss o lln onurtons []. Mov prunn s t pross o omttn rom onsrton ll mov n t urrnt m poston. Comntorl m tory llows rvrsl movs to ypss, n omnt movs to prun s lon s som omntn mov s onsr []. Ts tory orms t ss or prunn vrous orms o Hx ll, nlun vulnrl, ptur-rvrsl, lln-omnt, n nu-pt omnt lls [], [], [], [], []. As wt lln, MoHx uss (out 0) lol pttrns to n su lls. S Fur. F.. A Bl VC n Bl SC. Crrrs r s, nponts r ott, n t rst mov o t SC strty s. I H-sr ns wnnn VC or tr plyr, or wnnn SC or t plyr to mov, tn ts plyr n wn y ollown t sovr onnton strty. I H-sr ns wnnn SC or t plyr wo ust mov, tn t plyr to mov n only onsr movs ns t rrr o ts SC, sn ll otr movs r provly losn; tt s, ll movs outs t wnnn SC s rrr lv t opponnt wnnn SC on tr nxt turn. T mustply or t plyr to mov s t ntrston o tr opponnt s wnnn SC rrrs. By t sm rsonn, ll movs outs t mustply r provly losn. S Fur. F.. Two Wt wnnn SC rrrs n t orrsponn mustply or Bl. F.. Lol nror ll pttrns. Empty lls n prun rom onsrton y Bl s on tr rvrsl/omnt rlton to t orrsponn ott lls. C. Computn Connton Strts H-sr s n lortm tt ns ll-to-ll onnton strts n vn Hx poston [], []. Strtn wt t s s o trvlly onnt nt ll prs, H- sr nutvly uls lrr onntons y omnn smllr ons n srs n prlll untl no urtr onnton strts r oun. H-sr s nown to nomplt; n prtulr, t msss som rltvly smpl onntons wt ovrlppn sustrts tt umns n sly tt. Tus mnor H-sr xtnsons v n vlop [0], []. H-sr pruns mny losn movs n prous prt nm ply, ut s tm-ostly: nt mplmnttons n omput t onnton strts or out postons pr son on tournmnt-sz ors. III. ALPHA-BETA HEX PLAYERS Totr wt E.F. Moor, Clu Snnon vlop t rst utomt Hx plyr n t 0s, n ltron rut ntwor w st postv rs or on plyr s stons, ntv rs or t otr plyr s stons, n tn ply t rtn sl pont. T omputr ply stron opnn movs ut somtms rr n ttl stutons []. Snnon lso vlop omputr to ply Br C, now nown s Br-t, m smlr to Hx. Ts rut
3 ntwor st t rsstn o on plyr s stons to zro, t rsstn o t otr plyr s stons to nnty, n tn ply t ll wt rtst volt rop []. In 000 Hxy won t rst Computr Olymp Hx omptton []. Hxy s vluton unton uss n umntton o Snnon s Br C rut n w xtr wrs r w orrspon to onntons oun y H-sr []. Hxy uss ts vluton unton n strtorwr lp-t sr. In 00, 00, n 00 Sx won t nxt tr Computr Olymp Hx ompttons [], [], []. Sx snntly rns t Hxy rmwor y mprovn H-sr ny v urst lmts, rstrtn t lp-t rnn tor, tunn t vluton unton, n usn smll mount o nror ll nlyss []. Sx uss trunt -ply lp-t sr. Sx s opn sour n s n ply y mny umns. It s nrlly onsr to vry stron, ltou not qut xprt, plyr on or szs up to, wt nr-prt ply on or szs up to []. In 00 Wolv won t t Computr Olymp Hx omptton, tn t prvous mpon our ms to zro []. Wolv ws vlop n onunton wt MoHx n srs t sm os. Wolv mprovs on Sx s sn y moyn t vluton unton, mprovn H- sr ny, usn mu mor nror ll nlyss, usn n opnn oo nrt y sl-ply, n usn solvr n prlll wt t plyr. L Sx, t 00 vrson uss -ply sr. T 00 vrson o Wolv srs to -ply wn tm llows. In summry, omputr plyrs usn lp-t sr v provn to stron ompttors nst sll umn plyrs, n untl 00 omnt omputr Hx ompttons. IV. BASIC MOHEX FRAMEWORK A. Mont Crlo Tr Sr Mont Crlo Tr Sr (MCTS) s st-rst sr lortm tt s u y t outom o rnom m smultons. T lortm s ompos o tr s pss: ) tr trvrsl rom t root to som l no, ) rnom m smulton rom t l no s orrsponn m poston, n ) tr upt usn normton rom t smult m. T s lortm s nytm, rptn ts stps untl no mor tm rmns. Atr t tr trvrsl ps, t sr tr s xpn y n t lrn o t slt l no. Wn MCTS trmnts, t l wt t lrst sutr (.. w prou t most smultons) s slt s t st mov. MoHx s MCTS s ult on t os o Fuo, t Go prorm vlop y Mrtn Müllr t l. t t Unvrsty o Alrt [0]. B. Tr Trvrsl n Upt MoHx uss t UCT rmwor omn wt t llmovs-s-rst (AMAF) urst to slt t st l urn tr trvrsl [], []. L Fuo, MoHx plys stronst wn t uss n xplorton onstnt o zro, tvly turnn o UCT xplorton n rlyn solly on t AMAF urst to n stron nt movs. T tr upt ps upts t wn/loss sttsts or mov n t smult m. Upts our t no lon t pt rom t l to t root, try nlunn l no slton n utur tr trvrsls. C. Rnom Gm Smulton Plyn Hx untl t or s ompltly ll rsults n t sm wn/loss outom s stoppn on on plyr s wnnn n, so t rnom m smulton n not or m trmnton tr mov. Hn Hx m smultons n ntly mplmnt: ll mpty lls to n rry, sul tm rnomly, n ply t rmnn movs n orr. A onsqun o ts prtulr mplmntton s tt ll mov s AMAF sttsts r upt tr m smulton. F.. Br rollout pttrn: Wt plys on mpty ll, Bl plys t otr. As wt Go, t s nl to pply som nowl urn smult ms []. MoHx uss snl pttrn urn rnom m smulton: plyr pros ny opponnt r, tn t opponnt lwys rpls so s to mntn t onnton. S Fur. I multpl rs r pro smultnously, tn on su r s rnomly slt n tn mntn. F.. -- VC rollout pttrn: Wt plys s ll, Bl mntns t onnton y plyn t orrsponn ott ll. Yopt, notr MCTS Hx prorm, uss tonl pttrns s on notr ommonly ourrn VC []. S Fur. Howvr, n our tsts MoHx sow no strnt n rom su pttrns. A. Tr Knowl V. MOHEX ENHANCEMENTS L mny otr MCTS plyrs, MoHx uss nowlntnsv lortms n mportnt prts o t tr, s wll s ls to nt solv stts [].
4 Usn x nowl trsol, ny no s vst otn nou urn tr trvrsl, tn ot nror ll nlyss n t H-sr lortm r run on tt poston. Tr r two possl outoms: tr lln or H-sr solvs t poston, or t poston vlu s stll unnown. In t ormr s, ll l sutrs r lt, n t tr no s mr su tt ny tr trvrsl tt nountrs ts no omts t rnom m smulton, n smply upts ts nstor nos usn t orrt outom. In t lttr s, sutrs orrsponn to movs tt n lmnt v nror ll nlyss or mustply rsults r prun rom t tr. Furtrmor, or vry tr no tt surpsss t nowl trsol, ts lln omputton s stor prmnntly n ppl to vry tr trvrsl. Sn lln s omput nw or tr no, tr n srmnt twn t lln o no n ts l, so t snnt no s lln ts prn, n ny pror lln nowl s nor. w lln mov s ply. To vo ts, unprun l s sutr s lso lt xptn tr roots n ll rlvnt sttsts (.. UCT n AMAF t). Not tt ny susqunt tr xpnsons low t prnt no wll not onlt wt t lln. S Fur. T nowl trsol s typlly smll (.. t 00 Olymp vrson trsol 0), so t sz o ny trunt sutr s smll, n t susqunt loss o normton s pprntly mor tn ompnst y t n n prormn. ) No rs nowl trsol; prorm nror ll nlyss n H-sr omputtons. ) Prun lrn tt r nror or outs o mustply. F.. A Hx poston n ts lln ru poston. ) Rmov sutrs o ll rmnn lrn. F.. Applyn nowl to t Mont-Crlo tr. Ts lln prous two nts. Frstly, t rnom m smultons r sortr (sn t numr o mpty lls s rs), n tus llows mor m smultons pr son. Sonly, t ury o t m smultons soul mprov, sn ny rsolv rons o t or r urnt to v t orrt outom. Altou l no orrsponn to lln mov s lt, lln mov mt stll vll n som l s sutr, possly yln n lll m squn n For nstn, onsr t top Hx poston n Fur. I MoHx vluts ts poston or Bl wtout nowl, t onsstntly ts sor o -% or 0s srs. By pplyn nowl n t Mont Crlo tr, lln prous t ottom Hx poston n Fur, n t vluton sors plummt to % or 0s srs. Ts vour lso ols or otr MCTS Hx plyrs: ts poston ws tn rom n olymp m twn Yopt n Wolv, wr Yopt s vluton sor surpss 0% n t nm spt ts losn poston. B. Lo-Fr Prlllzton MoHx uss t Fuo os, n so nts rom Fuo s lo-r prlll MCTS [0]. MoHx s nowl omputtons r nl wtn ts lo-r rmwor. Tus t s possl or rnt trs to prorm uplt nowl omputtons onurrntly, ut ts s xtrmly rr n prt.
5 C. Prlll Solvr MoHx runs Hx solvr onurrntly wt t sr. Ts prous prt ply wnvr t m poston n solv quly. Ts solvr uss nror ll nlyss n H-sr n t sm wy s MoHx: to nty stt vlus, n prun nror movs rom onsrton. T solvr s s on pt-rst proo numr (DFPN) sr, n so lso rls on ts nowl omputtons to rstrt ts rnn tor n u t to t stronst movs [], [0]. It lso uss mov orrn urst to tmporrly prun t wr movs o no n t DFPN sr tr, rully rvln tm s tr sln movs r provn to losn. Ts tnqu rus t omntorl xploson o lr rnn tors, n lso prvnts t wnnn plyr rom spnn tm xplorn (urstlly) wr movs. Our solvr n solv ll opnns n rouly 0 ours. It s lso solv ovr l o t opnns []. D. Tm Mnmnt Ovr tousns o ms twn stron plyrs (.. ny o MoHx, Wolv, Sx), t vr m lnt s out 0 movs. Wtn on-mnut sr, MoHx s solvr usully omputs t m s vlu y mov, n lmost lwys y mov. Ts ouns t numr o movs MoHx ns to nrt n m. Hx tournmnts llow 0 mnuts pr plyr, so MoHx n sly llot on mnut pr mov. In t 00 Computr Olymp, MoHx llott sons pr mov. VI. EXPERIMENTAL RESULTS Bus o t swp rul, n utomt Hx plyr ns to l to rspon omptntly to vry opnn mov t opponnt mt slt. Tus our tstn trts ovr ll opnns wt t swp rul o, wt prorm plyn on s Bl n on s Wt or vry opnn. Tus on roun onssts o = ms. In orr to ru t stnr rror to smll prnt, w typlly run svrl rouns. Wl ts tstn ormt s lpul n ntyn wnsss n our lortm s prormn (.. opnns wr w prorm poorly s ot Bl n Wt), t snntly mpns ny strnt ns otn, s polrz opnns (.. opnns sly won y Bl or Wt) r ply tw, n ssntlly urnt t lst som wns or t wr plyr. Tus t Elo ns rport n our xprmnts unrrprsnt t xpt tournmnt ply Elo mprovmnt. A. Sln MCTS s prlllzl nytm lortm, so t sln o ts prormn wt rspt to tm n numr o trs s mportnt. As wt mny otr MCTS prorms, MoHx s strnt sls lortmlly vrsus tm, wt ouln o t m smultons proun rouly n tonl Elo o strnt: s/mov MoHx ts s/mov MoHx.% o t tm. S Fur 0. Prnt wns Numr trs / tm multplr snl tr lo-r lo F. 0. Prormn o lo, lo-r, n tm-sl snl tr MoHx nst snl tr s/mov MoHx. As wt Fuo, t lo-r vrson o multtr MoHx sls r ttr tn t lo ltrntv. In, t n r s vn mor rmt tn tt n Go sln o t lo vrson s wors wt two trs, n prormn tully rs wt only our trs prsumly us t m smultons n Hx r so mu str tn n Go, n t trs spn most o tr tm n t tr. S Fur 0. B. Hurst Tnqus Bot t r pttrn n AMAF urst v mor strnt ns or MoHx. T r pttrn prous 0 Elo strnt n nst nv UCT mplmntton (wt n optmz xplorton onstnt o 0.), n ts mprov vrson s surpss y notr Elo y n t AMAF urst (n turnn o UCT xplorton). Bs on t sln normton ov, ts totl strnt n s rouly quvlnt to 0-ol nrs n omputn tm. S Fur. Inrmntlly A Ftur Wn % Elo n Br pttrn.% ±.% 0 AMAF urst.% ±.% F.. T r pttrn n AMAF urst mprov plyn strnt y Elo. W tst mny nror ll nlyss pttrns s m smulton pttrns. Unortuntly, n ll ss ts pttrns v MoHx no strnt n. Ts provs vn tt provly orrt normton n m smultons n wn MCTS plyrs. C. Tr Knowl An tr nowl to MoHx s rouly quvlnt to ouln t numr o m smultons. T optml nowl trsol sms to 00 or t snl tr vrson, n to rs proportonlly wt t numr o
6 t nvrtlss w los ms, tr o w w rly xmn r. S Fur. 0 Prnt wns Knowl trsol multpl y numr o trs tr tr tr tr F.. Tr s/mov MoHx wt nowl nst -ply Wolv. A nowl trsol o zro mns tt no nowl s omput. trs. A vry low nowl trsol n worsn prormn, u to t orrsponn rs n t numr o rollouts. S Fur. D. Opnn Boo Prnt wns no oo 0 oo 0 oo 0 oo MoHx wt * ^(x-) Smultons / Mov F.. MoHx wt oos o nrsn sz nst 00 / mov MoHx wt no oo. MoHx s opnn ply n nonsstnt, prps us tr s so lttl xstn strutur to u t rnom m smultons. W r nvsttn t onstruton o n opnn oo usn Ln s mto []. Our ntl rsults r promsn, wt n opnn oo or Hx tt ws onstrut n y proun ns o Elo, w s wort mor tn ouln o smultons. As t oo sz nrss, t strnt ns row lortmlly. S Fur. VII. TOURNAMENT PERFORMANCE As mnton rlr, MoHx won ol n t 00 Computr Olymp. Its opponnts wr Yopt, notr MCTSs plyr, n t prvous lp-t s mpons Wolv n Sx. S III. Altou MoHx ws unt, F.. MoHx 00 Computr Olymp ms. From top to ottom, opponnts r Wolv, Sx, n Yopt. MoHx s Wt n ll ms. Anst Wolv, MoHx s vluton ws los to 0% trouout t m, untl ts prlll solvr nt tt t m ws won. Wolv s mov loos w sms prrl to ut Wolv lry lost t m t tt pont. Anst Sx, t m s prorsson s surprsn, sn MoHx s opnn ply s w n Sx vlops stron wll o nlun wtn movs. Nvrtlss, MoHx mns to ply tvly, wt n splly rllnt mov. MoHx s prlll solvr nrts movs 0 n yon. Post-m nlyss rvl t m ws los: MoHx lunr on mov n Sx lunr on mov. Anst Yopt, t m s strtorwr, wt prlll
7 r struturs n mny pros o su onntons. Postm nlyss nts tt Yopt ply wnnn movs n, ut tt mov s lunr: loss wl wns. MoHx mntns ts wnnn poston rom ts pont on. S t Olymp rport or urtr ommntry []. A. Exprmntl Tournmnts Bus Olymp rsults r s on vry smll smpl sz, w lso rn our own tournmnts twn MoHx n t two prvous mpons, Sx n Wolv. Ts tournmnts omtt opnn oos n prlll solvrs, so tt t rsults msur only t rltv strnt o plyr s sr nn. W not run tsts nst Yopt sn ts prorm s not pullly vll. T rsults r summrz n Fur. Bslly, MoHx omnts Sx n s vnly mt wt Wolv. F.. Opponnt MoHx Wn % Sx. ±. Wolv. ±. MoHx: prormn nst Sx n Wolv. VIII. APPLICATION TO OTHER GAMES T pplton o tr nowl to MCTS, n prtulr t us o lln, n nrlz to otr lssl or ms. Most o Hx s nror ll nlyss ppls to Y (wt only mnor motons rltn to t t tt or ss r not own y plyrs n Y), n so MCTS Y plyr oul lws us lln n nror ll nowl n ts Mont Crlo tr. Flln n nror ll prunn n lso omput or Hvnn, ut t xstn o rns s wnnn onton (nst o ust onntn pts, su s rs n ors) ms ts pross r lss strtorwr. Our roup s prorm som ntl rsr on ts top, ut no mplmntton o ts s s yt n prorm. Du to t rltv rrty o su pttrns n Hvnn, w xpt ts nowl to yl only mnor n, ltou t my mor usul n trms o mprovn plyr s prlll solvr. For ms l Go wr su tortl prunn s xtrmly ult n/or rr, prps urst lln su s prvntn smult ms rom plyn n t trrtory o lv roups n s rons oul prov usul n mprovn ury n spn up smult ms. IX. LIMITATIONS AND FUTURE WORK Dspt MoHx s rnt suss, tr r stll mny spts tt n mprov. MoHx os not ully xplot t onnton strts t omputs. Altou H-sr ns onnton strts, MoHx uss ts normton only or prunn n nm ply. In prtulr, smult ms o not us nown onnton proprts. As xpt, t nr-rnom ms usully o not mntn mportnt onntons, splly wn t rrr s lr. W v tst mny lortms tt norport onnton strts nto t smultons, nlun t ollown two: urstlly slt onntons to mntn urn t smulton; t t root no, llow t o o mntnn som sust o vn lst o onnton strts. Unortuntly, ll o ts tnqus rtly worsn prormn. T st vrton to t nvolvs usn ommon rsponss n t Mont Crlo tr to u t rsponss n t smult m; nvsttons r stll onon. Gvn t strnt o our prlll solvr, stronr ntrton o t solvr n MCTS oul prou vn rtr nts. Currntly t solvr nts only wtr t root poston s n solv; t solvr oul nst norm MoHx o ny tr nos tt t s solv. Furtrmor, t mt srl or t sr tr to nt w movs t urrntly prrs, try nourn t solvr to xplor t most prtnnt lns. Otr posslts nlu usn t AMAF urst to nlun t rnom m smultons, nsurn tt movs prv to stron r ply rly n t smulton. Also, urst ntlzton o movs n t Mont Crlo tr my nsur tt stron movs r lwys xplor or spup t UCT onvrn, ltou su mtos v not yt n sussul n Mont Crlo Hx. X. CONCLUSION W v ppl Mont Crlo Tr Sr tnqus to prou n utomt Hx plyr tt s on pr wt t st plyrs prou v t lssl lp-t ppro. T potntl or urtr mprovmnt rmns, splly n trms o ttr norportn onnton strts n t tr n m smultons, mprov nowl omputton n t tr (.. prlll solvr upts), vlopmnt o n opnn oo, n mprovn smulton qulty. Anowlmnts W r nt to Sylvn Glly o t MoGo tm n Mrtn Müllr, Mrus Enznrr, Dv Slvr o t Fuo tm or mny usul sussons rrn Mont Crlo Tr Sr. W lso tn Gáor Mls n Trstn Cznv or usul sussons rrn tr rsptv utomt Hx plyrs. REFERENCES [] L. Vtor Alls. Srn or Solutons n Gms n Artl Intlln. PD tss, Unvrsty o Lmur, Mstrt, Ntrlns,. [] Vm V. Anslv. T m o Hx: An utomt torm provn ppro to m prormmn. In AAAI/IAAI, ps, Mnlo Pr, 000. AAAI Prss / T MIT Prss. [] Vm V. Anslv. Hxy wns Hx tournmnt. ICGA Journl, ():, 000. [] Vm V. Anslv. A rrl ppro to omputr Hx. Artl Intlln, ( ):0 0, 00. [] Bror Arnson, Ryn B. Hywr, n Plp Hnrson. MoHx wns Hx tournmnt. ICGA Journl, ():, 00. [] Bror Arnson, Ryn B. Hywr, n Plp Hnrson. Wolv 00 wns Hx tournmnt. ICGA Journl, ():, Mr 00.
8 [] Bror Arnson, Ryn B. Hywr, n Plp Hnrson. Solvn Hx: Byon umns. Apt to Computrs n Gms, 00. [] Elwyn Brlmp, Jon H. Conwy, n Rr K. Guy. Wnnn Wys or Your Mtmtl Plys, volum. A.K. Ptrs, n ton, 000. [] Rém Coulom. Computn Elo rtns o mov pttrns n t m o Go. ICGA Journl, 0(): 0, Dmr 00. [0] M. Enznrr n M. Müllr. Fuo omp, 00. Dt o pulton: My, 00. Dt rtrv: Mr, 00. [] D. Fotln. T Mny Fs o Go, vrson, 00. Dt rtrv: Aprl, 00. [] Mrtn Grnr. T n Snt Amrn Boo o Mtmtl Puzzls n Dvrsons, ptr, ps. Smon n Sustr, Nw Yor,. [] S. Glly, Y. Wn, R. Munos, n O. Tytu. Moton o UCT wt pttrns n Mont-Crlo Go, 00. Tnl Rport RR-0. [] Sylvn Glly n Dv Slvr. Comnn onln n oln nowl n UCT. In Zoun Grmn, tor, ICML, volum o ACM Intrntonl Conrn Pron Srs, ps 0. ACM, 00. [] Ryn Hywr, Ynv Börnsson, Ml Jonson, Morn Kn, Ntn Po, n J vn Rsw. Solvn Hx wt omnton, ll-n, n vrtul onntons. Tortl Computr Sn, ():, 00. [] Ryn B. Hywr. A not on omnton n Hx. Tnl rport, Unvrsty o Alrt, 00. [] Ryn B. Hywr. Sx wns Hx tournmnt. ICGA Journl, ():, 00. [] Ryn B. Hywr n J vn Rsw. Hx n omntors. Dsrt Mtmts, 0( 0):, 00. [] Pt Hn. Vl lr Polyon? Poltn, Dmr. [0] Plp Hnrson, Bror Arnson, n Ryn Hywr. Hx, rs, t rossn rul, n XH-sr. In J. vn n Hr n P. Spron, tors, ACG, volum 0 o Ltur Nots n Computr Sn, ps. Sprnr, 00. [] Plp Hnrson, Bror Arnson, n Ryn B. Hywr. Solvn x Hx. In Cr Boutlr, tor, IJCAI, ps 0 0, 00. [] Plp Hnrson n Ryn B. Hywr. Pron t -- tmplt n Hx. In H. Jp vn n Hr, Xn Xu, Zonmn M, n Mr H.M. Wnns, tors, Computrs n Gms, volum o Ltur Nots n Computr Sn, ps 0. Sprnr, 00. [] Plp Hnrson n Ryn B. Hywr. Cptur-rvrsl movs n str omposton omnton n Hx. Sumtt to Intrs, 00. [] GmCntr [] Lvnt Koss n Cs Szpsvár. Bnt s mont-rlo plnnn. In Jonns Fürnrnz, Tos Sr, n Myr Splopoulou, tors, ECML, volum o Ltur Nots n Computr Sn, ps. Sprnr, 00. [] Toms R. Ln. Strts or t utomt onstruton o opnn oos. In T. Antony Mrsln n In Frn, tors, Computrs n Gms, volum 0 o Ltur Nots n Computr Sn, ps. Sprnr, 000. [] Lttl Golm [] Gáor Mls. Sx. sx.rts.u/, 00. [] Gáor Mls n Ryn Hywr. Sx wns Hx tournmnt. ICGA Journl, (): 0, 00. [0] A. N. D-pn Alortm or Srn AND/OR Trs n Its Appltons. PD tss, Dpt. o Inormton Sn, Unvrsty o Toyo, Toyo, Jpn, 00. [] Jon Ns. Som ms n mns or plyn tm. Tnl Rport D-, RAND, Frury. [] Run Rsmussn n Fréér Mr. An xtnson o t H-sr lortm or rtl Hx plyrs. In Gory I. W n Xnuo Yu, tors, Austrln Conrn on Artl Intlln, volum o Ltur Nots n Computr Sn, ps. Sprnr, 00. [] All Sn. Utlzton UCT u Hx. Tnl rport, Eol Norml Suprur Lyon, 00. [] Clu E. Snnon. Computrs n utomt. Prons o t Insttut o Ro Ennrs, :,. [] Jn Wllmson n Ynv Börnsson. Sx wns Hx tournmnt. ICGA Journl, ():0, 00. [] Mr H. M. Wnns, Ynv Börnsson, n Jn-Ts Sto. Mont-Crlo tr sr solvr. In H. Jp vn n Hr, Xn Xu, Zonmn M, n Mr H.M. Wnns, tors, Computrs n Gms, volum o Ltur Nots n Computr Sn, ps. Sprnr, 00.
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