IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 156 Stronr Vrtul Conntons n Hx Ju Pwlwz n Ryn Hywr n Plp Hnrson n Bror Arnson Astrt For onnton ms su s Hx or Y or Hvnn, nn urnt ll-to-ll onnton strts n omputtonl ottln. In utomt plyrs n solvrs, sts o su vrtul onntons r otn oun wt Anslv s H-sr lortm: ntlz trvl onntons, n tn rptly pply n AND-rul (or omnn onntons n srs) n n OR-rul (or omnn onntons n prlll). W prsnt FstVC Sr, nw lortm or nn su onntons. FstVC Sr s mor tv tn H-sr wn nn rprsnttv st o onntons quly s mor mportnt tn nn lrr st o onntons slowly. W tst FstVC Sr n n lp-t plyr Wolv, Mont Crlo tr sr plyr MoHx, n proo numr sr mplmntton ll Solvr. It os not strntn Wolv, ut t snntly strntns MoHx n Solvr. Inx Trms Hx, onnton ms, vrtul onnton, H- Sr I. INTRODUCTION HEX s two-plyr prt normton onnton m nvnt npnntly y Pt Hn n 1942 [1] n Jon Ns n 1948 [2] [4]. Hx s n n tv omn o rtl ntlln rsr sn Clu Snnon s smnl wor n t 1950s [5]. It s lly to rmn n tv omn, s t m s sy to mplmnt yt llnn to mstr, n solvn rtrry Hx postons s PSPACE-omplt [6]. Automt Hx plyrs rly on t omputton o onnton strts [7]. Ts omputton s ostly n so rus t numr o postons tt n xplor n tr sr, wtr lp-t, mont-rlo, or proo numr sr, ut usully pys o y nn wns rly. For 11 11 ms, Anslv rports tt H-sr onnton omputtons routnly n wn 20 or mor ply or t n o t m n yl snnt mov prunn [8]. Ts ns n loo n prunn r otn wort t omputtonl ost. But t numr o onnton strts o Hx poston n row xponntlly wt t numr o lls. Evn on mortly-sz ors su s 9 9, nn ll o poston s onnton strts s omputtonlly nsl. Fnn rtl sust o ts onntons, n on so mor ntly tn s n on or, woul snntly nrs t strnt o urrnt onnton-s plyrs n solvrs [9]. In ts ppr, w v nw mto or omputn onntons. Rtr tn nn mny onntons, w n rprsnttv sust o rtl onntons. An rtr tn t usul sr, w us mor nt sr. W tst our J. Pwlwz s wt t Insttut o Inormts, Unvrsty o Wrsw. E-ml: pn@mmuw.u.pl R. Hywr n B. Arnson r wt t Dprtmnt o Computr Sn, Unvrsty o Alrt. E-ml: ywr@ulrt., ror@s.ulrt. Mnusrpt prpr Frury, 2014. mto n stt-o-t-rt Hx plyrs. T strnt ns r snnt. In II w rvw t ruls o Hx n t lr o omputn onnton strts. In III w rvw prvous mplmnttons, n mprovmnts, or ts lr. In IV w prsnt our moton o H-sr. In V w sr sms-omnr, nw wy o prormn t OR-rul tt ypsss omputtonl ottln. Ts nw lortm ls to n xplosv rowt n t numr o onntons tt n oun n sort tm. In VI w tl t prolm o ln wt su lr sts o onntons. In VII w v xprmntl rsults n n VIII w onlu. A. Ruls o Hx II. CONNECTION STRATEGY ALGEBRA Hx s ply on n n n or wt xonl lls. Two plyrs, Bl n Wt, ltrnt turns olorn ny unolor ll wt tr olor. T wnnr s t plyr wo orms pt o tr olor onn tr opposn two ss. S Fur 1. F. 1. Wt s pt o lls onn t two Wt ss, so Wt wns. Hx nnot n n rw, n or n n ors tr xsts wnnn strty or t rst plyr. Ts rst-plyr vnt s notl n prt, splly on rltvly smll ors. To mtt t, t swp rul s otn opt: t rst plyr olors ll l; tn t son plyr ooss wtr to Bl; tn Wt olors ll, n ply ontnus n ltrntn son. Wn ply wt t swp rul, t son plyr s wnnn strty, ut to ply prtly must now t wn/loss vlu o vry opnn mov. To t, utomt solvrs v oun ll su vlus or ll or szs up to 9 9. Computr tournmnts otn us 11 11 ors; 13 13 n 19 19 ors r lso populr. On ll ts ors t vr rnn tor n typl m s mor tn 100. B. Connton strts n tr pplton In Hx poston, n s mxml onnt roup o sm-olor lls. A P -n s n wt P s olor. T poston n Fur 1 s 3 l ns n 1 wt n. For Hx poston n plyr P, onnton strty sps t two nponts n onnt n t rrr, nmly t
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 157 st o mpty lls P rqurs to rry out t ltrntn-turn strty. An npont s tr n mpty ll or P -n or P -olor s. Follown Anslv [10], vrtul onnton (VC) s son-plyr onnton strty, n vrtul sm-onnton (VSC) s rst-plyr onnton strty. T rst mov o VSC w yls VC wt smllr rrr twn t sm nponts s ts y. A VC (VSC) twn x n y s n x-y VC (VSC). An x,y V(S)C rrr s mnml t s not strt suprst o som otr V(S)C rrr. Ovously, ronzn VCs n VSCs s usul. I plyr s s-to-s VC on wos nponts r t plyr s two ss tn t plyr s wnnn strty. W ll su VC wnnn VC: t plyr n wn vn t s t opponnt s turn to ply. Smlrly, t plyr-to-mov s s-to-s VSC, tn t plyr-to-mov s wnnn strty. W ll su VSC wnnn VSC. F. 2. From lt: T rst two urs sow t rrr n y o wnnn Wt VSC. So, Bl movs nxt, Bl must ply n ll tt ntrsts ot rrrs, s sown n t tr ur. I t plyr-to-mov P ns wnnn opponnt-vsc, P must mov wtn ts rrr to vo rn losn poston (ssumn tt t opponnt wll lso n ts VSC). Smlrly, P ns svrl wnnn opponnt-vscs, P must mov n t ntrston o t ssot rrrs to vo rn losn poston. Hywr t l. ll ts ntrston t mustply, n us t to prun t lst o possl movs (n so ru t rnn tor) wn plyn n solvn Hx postons [11]. S Fur 2. As n otr ll-olorn ms su s Go or Hvnn, Hx postons n somtms ompos nto npnnt sums. A our-s sum s ssntlly smllr sum on possly rrulrly-sp or. For su sum, nn VC or t wnnr P (wo n onnt tr two opposn P -ss) llows on to ll t sum s mpty lls (..P -olor tm) wtout nn t poston s vlu, tus prunn ll movs n t sum rom utur onsrton [12], [13]. S Fur 3. F. 3. Lt: our-s sum o lrr poston. T ounry o t sum s n y t two Wt ns, t two Bl ns, n t tr ott ll prs ( su pr orms r onnton twn oppost-olor ns). Bl s VC wos nponts r t two Bl ounn ns n wos rrr ls wtn t sum ntror, so s sown t rt on n Bl-ll t sum ntror wtout nn t poston s vlu. C. Connton strty lr Anslv v rrl lr tt omputs som ut not ll VCs n VSCs [10]. A onnton strty s trpl S = (x,c,y) wr x,y r t nponts, C s t rrr, n x,y / C,.. ntr npont s n t rrr. I S s VSC or plyr P wt y, tn (x,c \{},y) s VC n t poston otn y P -olorn ll. A s VC s onnton strty (x,,y), nmly wt x,y nt. For xmpl, or plyr P, nt mpty lls orm s VC, s os n mpty ll nt to P -n or P -s. Strtn wt s onnton strts, on n trtvly onstrut mor onstruton strts usn t AND- n OR-ruls. T ormr omns onntons n srl; t lttr omns tm n prlll. 1) AND-rul. I S 1 = (x,c 1,u) n S 2 = (u,c 2,y) r P -VCs, n C 1 C 2 =, x / C 2, y / C 1, tn ) u s P -olor, (x,c 1 C 2,y) s P -VC, ) u s unolor, (x,c 1 {u} C 2,y) s P -VSC. 2) OR-rul. I S z = (x,c z,y) r P -VSCs, n C z =, tn (x, C z,y) s P -VC. E trton o ts onstruton lortm ppls t AND-/OR-ruls to ll possl nown onnton strts, usn t urrnt strts to prou nw nrton o strts. Itrton ontnus untl no nw strts r prou. Ts rrl (y nrton) AND-/OR- losur lortm s ll H-sr [8]. For poston n plyr, t st o V(S)Cs oun y n trton-lmt H-sr pns on t orr n w t AND- n OR-ruls r ppl to prtulr npont prs. But H-sr s omput to omplton.. untl no nw V(S)C n rt tn t st o mnml V(S)C rrrs tt r oun s x. For ts poston n plyr, w ll ts t mnml VC n VSC rrr sts. III. PREVIOUS APPROACHES Tr r mny wys to mplmnt H-sr. Hr r som lortm tps rom vrous ppros. Dsr ny V(S)C β tt s suprst 1 o notr V(S)C α wt t sm nponts. Bot uton ruls rqur t rrrs n omn to v mpty ntrston, so t st o strts nrt rom st S ontnn α s qul to t st nrt rom S {β} [14], [15]. Bor pplyn t OR-rul to ll urrnt x-y VSCs, onrm tt t ntrston o ll su rrrs s mpty. I t s not mpty tn t ntrston o vry non-mpty sust o rrrs s not mpty, so no nw x-y VCs n rt [15]. I pplyn t OR-rul rursvly, tn tr wnvr t most rnt x-y VSC os not ru t ntrston o ll su VSCs. Ts VSC nnot lp onstrut nw VCs, n wll only nrs t rrr sz o ny nw VC [16]. Lmt OR-rul pplton y onsrn VSC susts o sz t most 3 or 4. Cn ll 2 susts o st o 1 W otn nty V(S)C y ts rrr.
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 158 VSCs usully ts too lon, n yls mnsn strnt rturns or 5 [15]. Wn pplyn t AND-rul, llow or ss to mponts. Ts mt sm ountr-ntutv, sn ss r nl stntons, ut t llows t sovry o onnton strts not otrws ul y H-sr [14], [15]. Du to mnsn prormn rturns, lmt tr t numr o V(S)Cs stor or npont pr x, y (r lmt), or t numr o su V(S)Cs onsr n onstrutn lrr V(S)Cs (sot lmt). I su lmts r us, sort V(S)Cs y rrr sz, so tt smllr (mor usul) onnton strts wll us n onstruton [14], [15]. Svrl vrtons n/or nnmnts o H-sr v n propos, nlun nrlz H-sr, t rossn rul, ptur st ntrston, n ommon m rrr ntrston [12], [17] [19]. A. Our nw mto IV. COMPUTING VCS To n st o VCs tt strntn our plyr n solvr, w n mny spts o t nrl n/or-rul losur lortm. On n s to stor onntons mor ntly: s IV-B. Ts llows mny sr optmztons: s IV-C. Anotr n s to pply t OR-ruls so tt ll VCs twn two x nponts r rt y n oprton ll sms-omnr. Rtr tn onsrn sprtly ll VSC susts tt mt v rs to nw VCs, sms-omnr ts on ll nput VSCs t on: s V. B. Stor Bor outlnn our nw lortm, w sr our t struturs. To stor VCs n VSCs, w us mps M n M s nx y npont prs. M (x,y) (rsptvly M s (x,y)) stors x-y VCs (VSCs). For pr x,y, w stor only t mnml rrrs C or w (x,c,y) s VC (VSC). Ts st o rrrs s stor n vtor. Prvous mplmnttons us ln lst, w llow lsts to sly mntn n sort orr y rrr sz. W prr to us vtor, w llows or str upts vn lmnts r osonlly nsrt t rtrry lotons. Rtr tn mntn ts lsts n urnt sort orr, w us mov-to-ront tnqu w s ust s tv ut os not urnt t lsts r sort: s IV-C7. E rrr s l ntn wtr t s n pross. In our psuoo, or t =, s (or VCs,VSCs) w not y M p t (x,y) (rsp. M u t (x,y)) sust o M t (x,y) wt only pross (unpross) VCs/VSCs. A VC s pross on t s n us n t AND-rul wt ll otr pross VCs to rt nw VCs n VSCs. A VSC s pross on t s n us y sms-omnr to rt nw VCs. S IV-C. Our ppro s mnmlst wt rspt to VC stor. W o not stor VSC s y: n, t s romput on mn y AND-rul lls. W prr ts ppro, s or our purposs t s str to oprt on vtors rtr tn lsts. For ny rsons w n som xtr struturs. For npont x w mntn t st o ll nponts y wt w x s onnt v VC. W not ts noroo y N(x). For npont pr (x, y) w mntn t ntrston o pross VCs n ll VSCs, not y I(x,y) p n I s (x,y) rsptvly. As w wll s, w wll upt I(x,y) p wn VC oms pross n I s (x,y) wn nw VSC s rt. W rt our lortm usn quus Q n Q s. Q mntns trpls (x,c,y) o ll unpross VCs tt r ANDrul nts. Q s ps nts or sms-omnr, ut stors only tos npont prs (x,y) or w tr s som unpross VSC n M s (x,y) n I s (x,y) =. Tus tr s n tt sms-omnr ppl to ll x-y VSCs prous nw VCs. Q s stors only npont prs us sms-omnr wll pross ll su urrnt VSCs t on. S V. TABLE I SUMMARY OF DATA STRUCTURES symol mnn M t(x,y) st o rrrs C su tt (x,c,y) s VC (t = ) or M p t (x,y) VSC (t = s); suprsrpt p (u) nots sust o M u pross (unpross) VCs/VSCs t (x,y) N(x) VC noroo o x: {y : M (x,y) } I p (x,y) ntrston o pross x-y VCs: M p (x,y) I s(x,y) ntrston o x-y VSCs: Ms(x,y) Q quu o unpross VCs Q s quu o npont prs (x,y) wt n unpross x-y VSC, n wt I s(x,y) = C. Sr W now sr t psuoo o Alortm 1, our nw lortm. 1) Funton VCSEARCH: Crt ll s VCs,.. tos twn nt lls. Mr ts s unpross n pus onto Q. Loop untl ot quus r mpty (ln 3). T loop nvrnt s tt pross VCs/VSCs v n us n ll possl AND-rul/OR-ruls wt ll otr pross VCs/VSCs, n unpross VCs/VSCs v nvr n us n tr rul. At trton, try t AND-rul, n v smsomnr try t OR-rul only t AND-rul ls. W postpon pplyn t OR-rul s lon s possl, us sms-omnr uss ll VSCs twn vn npont pr n ns ll VCs t on. S V. I som VC rmns unpross (so NONEMPTY(Q ) s tru), pop VC (x,c,y) rom Q (ln 5) n n t rst o ts trton try t AND-rul only on ts VC totr wt ll pross VCs. Try ot ns s t AND-rul mpont (ln 6). Atr t two DOAND lls ns, mr t VC s pross (ln 7) n upt I p (x,y) (ln 8). Atr ll VCs r pross t AND-rul s n tr on ll prs o urrnt VCs try t OR-rul. Pop t npont pr (x,y) rom Q s (ln 10) n ll DOOR, w ppls sms-omnr (ln 11) to t sts o ll VSCs. Mr ts VSCs s pross (ln 12).
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 159 Alortm 1 FstVC Sr 1: unton VCSEARCH 2: ntlz struturs wt s VCs 3: wl NONEMPTY(Q ) or NONEMPTY(Q s ) o 4: NONEMPTY(Q ) tn 5: (x,c,y) POP(Q ) 6: DOAND(x, C, y), DOAND(y, C, x) 7: mr VC (x,c,y) pross 8: I(x,y) p I(x,y) C p 9: ls 10: (x,y) POP(Q s ) 11: DOOR(x, y) 12: mr ll x-y VSCs s pross 13: unton DOAND(x,C 1,u) 14: or ll y N(u) (C 1 {x}) o 15: (C 1 {x}) I(u,y) p = tn 16: or ll C 2 M p (u,y) o 17: ANDRULE(x,C 1,u,C 2,y) 18: unton ANDRULE(x,C 1,u,C 2,y) 19: (C 1 {x}) C 2 tn rturn 20: u s olor tn 21: TRYADDVC(x,C 1 C 2,y) 22: ls 23: TRYADDVSC(M s (x,y),c 1 {u} C 2 ) 24: unton DOOR(x, y) 25: or ll C SEMISCOMBINER(M s (x,y),m (x,y)) o 26: TRYADDVC(x, C, y) 27: unton TRYADDVC(x, C, y) 28: TRYADD(M (x,y),c) tn PUSH(Q,(x,C,y)) 29: unton TRYADDVSC(x, C, y) 30: TRYADD(M s (x,y),c) tn 31: I s (x,y) I s (x,y) C 32: I s (x,y) = n (x,y) / Q s tn 33: PUSH(Q s,(x,y)) 34: unton TRYADD(C,C nw ) 35: or susqunt C C o 36: C C nw tn 37: mov C to ront o C 38: rturn ls 39: C {C C : C nw C} {C nw } 40: rturn tru 2) Funton DOAND(x,C 1,u): For t AND-rul, vn VC (x,c 1,u), n ompnon VC wt nponts u,y. Itrt ovr ll sl y (ln 14). Itrt ovr t VCnoroo N(u), rtr tn ll lls on t or; ts svs tm. Fx y n trt ovr ll u,y VCs, pplyn t ANDrul y lln ANDRULE. Frst wtr t ntrston I p (u,y) o ts VCs s smll nou (ln 15) to llow sutl VC. Ts otn vos utl loop. 3) Funton ANDRULE(x,C 1,u,C 2,y): C t rmnn AND-rul onton n try to nw VC or VSC. Dpnn on wtr u s olor, ll TRYADDVC (ys) or TRYADDVSC (no). 4) Funton DOOR(x, y): SEMISCOMBINER rturns ll nwly rt VCs. Ts mt not mnml, so onsr tm on y on wt TRYADDVC. 5) Funton TRYADDVC(x,C,y): I C s nw mnml x-y VC rrr, s y TRYADD, pus t ontoq. Mr nwly rt VC s unpross. 6) Funton TRYADDVSC(x, C, y): C wtr C s mnml y lln TRYADD. I ys, upt t ntrston o ll x-y VSCs; ts st s mpty tn pus x y onto Q s. Mr nwly rt VCS s unpross. 7) Funton TRYADD(C,C nw ): Ts s t unton n w t most tm s spnt. C wtr nwly rt rrr C nw s mnml,.. s not t suprst o n xstn rrr. Ts must on ntly, so w us mov-toront tnqu. Itrt ovr ll rrrs n t vtor C. T rst rrr C oun tt s sust o C nw (ln 36) s rtn rrr. It rts t nw onnton, n mt rt utur onntons, so mov t to t ront o C (ln 37). Ts ts mortz onstnt tm, sn w trt ovr ll rrrs prn C. Usn mov-to-ront, t st rtn rrrs quly ollt t t ront o C, snntly run t rton tm o non-mnml rrrs. Ts mto s rul, sn nw rrrs r usully non-mnml, wt nw mnml VCs/VSCs sovr rrly. A nw onnton tt s not rt s s nw rrr. Also, ll rrrs w om non-mnml r rmov (ln 39). An, ts s mortz-tm nt, s w lry trt ovr ll rrrs. Howvr, w n to upt VC normton rully: ltrn rmovs pross VC, rlult I(x,y); p t rmovs n unpross VC, rmov t VC rom Q. A. T prvous ppro V. SEMIS-COMBINER Consr x-y onntons. Lt C (rsp. S) t st o ll x-y VC (VSC) rrrs. A strtorwr pplton o t OR-rul s to trt ovr ll susts S o S. I t ntrston o t susts o S s mpty tn omnn ts strts yls nw VCC wos rrr s t unon o ts susts. On must wtrc ovrs (s suprst o) t rrr o ny urrnt VC; not, tn C s mnml n so n to C. Tr r 2 #S su susts, so trtn ovr ll o tm s n nrl nsl. A usul prvous ppro s to trt only ovr susts S S o oun sz, sy 3 or 4. But ts rus t st o nw onntons tt n oun, w n turn rus t strnt o t prorm tt uss t VC nn, wl stll lvn trton ovr VSCs s omputtonl ottln. B. Our nw ppro W now sr our sms-omnr lortm. T mn o our ppro s to ous on wt w ll lo lls, s w sll xpln. To strt, suppos C s mpty (so tr s no x-y VC). Tn w rt nw VC C wos rrr s t unon o ll
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 160 strts o S only t ntrston o ll orrsponn rrrs s mpty. Nxt, suppos C s non-mpty. Assum C ontns xtly on rrr, sy C. T rrr o nw VC C nnot ovr C, so tr must som ll tt s n C ut not n C, n C n rt only rom VSCs not ontnn. W ll lo ll or onstrutn C. Lt S = {S S / S} t st o ll su VSCs not ontnn som lo ll. Crt nw VC only t omn ntrston o S s mpty. So C ontns on rrr C, t sus to trt ovr ll n C. Consr n xmpl. Aov, twn t l s n t ott ston, w rt VCs rom our VSCs A,B,C,D (lt to rt, top to ottom): T ntrston o A,B,C,D s mpty, s s t ntrston o A,B, so A B s t rrr o VC, sy I: Now lo ll (ott) rom I: Ts ll ntrsts B, D ut msss A, C, wos ntrston s mpty, so A C s t rrr o nw VC, sy II: Now lo ll rom II. A oo opton s ll tt lso lons to I: Ts ll msss A,D, wos ntrston s mpty, so A D s t rrr o nw VC: In nrl, nn snl lo ll s not nou to onstrut ll nw VCs. Inst, w n to n st o lo lls B = { 1,..., n } tt stss t ollown: C n C ontns t lst on lo ll (so C B ) n t ntrston o ll VSCs w o not ontn ny lo ll s mpty (so S =, wr S = {S S S B = }). Gvn st B o lo lls, t unon o t strts o S s t rrr o nw VC. Follown ts ppro, w mplmnt Alortm 2 n trn mnnr. T mn unton s BACKTRACK, Alortm 2 SEMISCOMBINER(S, C) Rqur: VSCs S n xstn VCs C rrr sts Ensur: Rturn st o rrrs o nwly rt VCs 1: unton SEMISCOMBINER(S, C) 2: rturn BACKTRACK(, S, C) C 3: unton BACKTRACK(F, S, C) 4: S tn rturn C 5: C = tn C { S} 6: loop 7: A smllst st rom {C F : C C} 8: A = tn rturn C 9: oos lmnt rom A 10: F F {} 11: S FILTER(S,), C FILTER(C,) 12: C C BACKTRACK(F,S,C ) 13: unton FILTER(A,) rturn {A A : A} ll ntlly t pt 0 n SEMISCOMBINER. Wn ll t pt, t st o lo lls B = { 1,..., } s sz. From r, ll possl suprsts o B r sr. B s not stor xpltly. Inst, ltr VSC n VC rrr sts r pss s rumnts S n C, wr ltr rrr st s on n w rrr ontns no lo ll (so s sont wt B). Atonlly, w pss st F o orn os o +1. F ontns ll lls su tt t st o lo lls B {} s n lry sr. BACKTRACK rturns ll o t VC s rrrs, ot ornl n nwly rt rom VSCs, tt r sont wt B. W now xpln BACKTRACK stp y stp. To strt, tst wtr VC n rt (ln 4). I ltrn rmovs ll VCs, rt nw VC (ln 5). Ts VC wll sont wt ny VC rt so r, us r S ontns only VSCs tt r sont rom B, wrs ll prvous VCs ontn t lst on ll rom B. Nxt, loop ovr ll possl os o +1 (vrl ). In orr to rt nw VC, w must ltr out ll VCs rom C. So or C C w must t som pont lo ll o C. But w nnot lo ny ll rom F. In orr to mnmz t rnn tor, w wnt t smllst possl st rn C F mon ll possl C; ts s A (ln 7). T nxt lo ll must n A. I A s mpty, tr s t lst on VC tt nnot ltr, so n t sr (ln 8). I A s not mpty, p n rtrry rom A (ln 9) s +1. Now rursvly ll BACKTRACK on t rrr sts S,C otn rom S,C y rmovn onntons ontnn (lns 11 12). W or s nt or utur slton o, nmly or >, ot n pr rursv lls n n lol utur os o +1 (ln 10), us nlun t woul
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 161 prou no nw lo lls, n so no nw VCs. Tus sr st B o lo lls only on. C. Eny T totl numr o rursv lls s lmt n svrl wys. W nvr uplt st B o lo lls, so on uppr oun s t numr o su sts, nmly 2 t, wr t s t sz o smllst st tt ts ll VCs. T numr o lo ll sts tt orrspon to rurson lls s smllr tn ts, s t rurson s otn ron y t onton t ln 4: B s t lo st orrsponn to t urrnt rursv ll n t ntrston o VSCs s not mpty, tn w ort t sr or suprsts o B. Wl t nput VSCs tus lmt t sr, t mn ounn tor s t numr o VCs, ot urrnt n nwly rt. For xmpl, rurson pt s oun y t numr o VCs, so SEMISCOMBINER runs quly wn rtn ntl onntons. Ts ounn t s ult to msur prsly. T numr o rursv lls pns not on t numr o VSCs, ut rtr on t numr o nwly rt VCs. W prorm n xprmnt to msur typl prormn, usn t poston n Fur 4, w s mny V(S)Cs. 1 1 2 2 3 3 4 4 5 5 6 6 7 x u 7 8 y 8 9 9 10 10 11 11 F. 4. A poston wt mny VCs n VSCs. For ts poston, or npont pr wt mpty VSC ntrston, w ll SEMISCOMBINER n tr t on t 2 780 rsultn lls. E t pont nlus t numr o nput VSCs, t numr o prou nl VCs, n t numr o rursv BACKTRACK lls m. For t pont, t numr o rursv lls ws mor tn 1000. W us Gnuplot [20] to nlyz t 1500 t ponts wt t most rursv lls, nn unton tt orrlts t numr o rursv lls wt t numr o VSCs n t numr o VCs. Atr rwn vrous 3D plots, w onsr svrl untons wt vryn numrs o prmtrs. For su unton w us Gnuplot s t ommn to stls t st vlu or prmtrs. W oun oo pproxmton or t numr o rursv lls to mx(20 (#VCs) 5/4,2 #VSCs). To sow ow wll ts ts, or t pont w lult ttn tor: proporton o tul numr o rursv lls n ts pproxmton. T mxmum vlu o t ttn tor s 2.98. Tr r 8 t ponts wt ttn tor ovr 2, 53 wt ttn tor ovr 1.5, n 294 wt ttn tor ovr 1. Not tt ts unton s polynoml n t numr o VCs n VSCs, n s xpt rom our susson ov rlts tt t numr o ntl VCs v rtr mpt on t nl numr o onntons tn t numr o ntl VSCs. D. Rmrs n mnor optmztons 1) VC mnmlty: Durn ntrmt prossn, SEMIS- COMBINER n onstrut non-mnml VCs. So s to vo usn ts n t onstruton o urtr strts, w sr tm or t nl st o VCs s rturn. 2) Susqunt us o sms-omnr: Durn t onnton onstruton pross, SEMISCOMBINER n ll wt t sm npont pr mor tn on. In su ss, susqunt SEMISCOMBINER ll n rumnt S prttons t VSCs s tr pross (lry prsnt n t prvous ll) or not. Ts llows t sr to stopp wnvr BACKTRACK s ll wt S ontnn only pross VSCs, s ts nnot prou ny nw VC. 3) Gry sum: Wnvr nw onnton s rt y summn VSCs n ln 5 o Alortm 2, rtr tn sum ll VSCs, w rly sussv rrr only t rus t urrnt ntrston o ll VSCs. S Alortm 3. Ts optmzton rss runtm. Alortm 3 Gry sum 1: unton GREEDYSUM(S) 2: X, I = U 3: or ll S S o 4: I S I tn 5: X X S 6: I I S 7: rturn X VI. LIMITING CONNECTION GROWTH Sms-omnr runs quly on lr VSC sts n ns ll possl H-Sr VCs. But ts n too mny onntons to usul to m plyr. Consr t poston rom Fur 4. Hr omputn l VCs ts 8 mnuts, too lon or ny plyn or solvn pplton. Ts runtm s u smply to tr n 696 901 VCs n 4550587 VSCs. Btwn lls x (7) n y (8) l s 65698 VSCs. Applyn Alortm 2 to ts yls 6925 VCs. T numr o rursv BACKTRACK lls s 860 264. Tr r 14 238 x y VSCs wt y u (7); ts s t most ommon y or x y VSCs. Ts r rt y pplyn t AND-rul to t 60 x-u n 266 u-y VCs; most su VC prs yl nw VSC. Tus t numr o x-y VSCs wt y u n lmost s lr s t prout o t numr o x-u n u-y VCs, w xplns t xponntl rowt o t numr o VSCs. Bus o ts rowt, t s nssry to lmt t rton o nw VCs.
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 162 A. Nw VC ptn urst Prvous ppros to lmt t numr o nw VCs us sot n r lmts. S III. W propos notr ppro. Assum tt C s t st o VC rrrs twn two x lls, n tt nw VC wt rrr C s rt. Wn s t wort n C to C? Alwys, C s strt sust o rrr n C. Nvr, C s suprst o su rrr. But wt C s ntr su strt sust nor suprst? Tn C s usul only t ls to t rton o ny nw VSCs y t AND-rul. A st o VSCs omn to orm nw VC only tr omn ntrston s mpty; n t s usul to ollt VCs wos ntrston (wt otrs wt t sm nponts) s s smll s possl. So our ptn urst s s ollows: nw VC only (1) t s strt sust o n xstn VC, or (2) t rus t omn ntrston o ll urrnt VCs. Wt ts urst, t numr o VCs stor twn two vn lls os not x t numr o lls on t or, w s onstnt. So, t lmts to polynoml n t numr o lls t totl numr o onntons rt, tus von xponntl rowt. Not surprsnly, vn unlmt omputton tm, ts urst wns t rsultn VC nn. Howvr, or x tm omputtons, ts urst nrlly strntns t VC nn. S VII. Ts urst s not unvrslly ttr tn t stnr onnton-nn lortm. Fur 5 sows poston rom t 2011 Computr Olymp Hx omptton, wt Wt to mov. Hr our urst ls to n wt s-to-s VSC. Su VSCs r usul n prunn losn movs, n solvr usn our nw urst s slowr r tn solvr usn t stnr VC omputton. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 F. 5. A VC nn usn t ptn urst ls to n wt sto-s VSC r. Wt wns,.. 10 9 10 10 9 9 8 8 6 7 6 7 5 5 4 6 4 5 4 4 5 9 8 5 6. B. Fst sms-omnr In our ptn urst, (2) otn mpls (1). Ts susts smplr urst: us only (2). Ts llows str mplmntton o sms-omnr. Lt S n C SEMISCOMBINER rumnts,.. xstn VSCs/VCs wt x nponts. Lt A = C t ntrston o ts VCs. W wnt to rt VC C tt s not suprst o A. For ts w n to n lo ll A su tt t ntrston o VSCs not ontnn s mpty. W o ts y trtn ovr A. Atr trton w tr rmov rom A or, w nw VC C, upt A to A C. In t lttr s, sn C, t sz o A s ru y t lst on. W stop t loop wnvr w r unl to rt nw VC ullln (2). Ts ppns A = or S. W sp up ts pross y ltrn out o S ll suprsts o A, sn rom now on ty r uslss. S Alortm 4. Alortm 4 FASTSEMISCOMBINER(S, C) Rqur: VSCs S n xstn VCs C Ensur: Rturn nw VCs run ntrston o C 1: unton FASTSEMISCOMBINER(S, C) 2: C nw 3: A C 4: wl A o 5: S {S S : A S} 6: S tn rturn C nw 7: oos lmnt rom A 8: S FILTER(S,) 9: S = tn 10: C GREEDYSUM(S ) 11: C nw C nw {C} 12: A A C 13: ls 14: A A {} 15: rturn C nw FASTSEMISCOMBINER s t most s mny trtons s t sz o t VCs ntrston, ut usully mu wr. C. Comprson A VC nn usn tr FASTSEMISCOMBINER or SEMISCOMBINER totr wt t ptn urst s usully wr tn VC nn usn only SEMISCOM- BINER. W ll ts VCEs rsptvly st, lmt, n unlmt. For t poston n Fur 4, Fur 6 sows lls VC-onnt to t ottom or ts VCEs. Not tt unlmt VCE ns mor omplt VCs. A wt mustply st or ts poston s sown n Fur 7. Hr st VCE ns no VSCs twn t l ss n so ns no mustply st, wl t mustply o lmt VCE s tw s lr s tt o unlmt VCE. Altou unlmt VCE s t stronst o ts tr VCEs, or x tm omputtons t sp ns o lmt VCE n st VCE mor tn ompnst or t loss n oun onntons. As w sow n VII, st VCE s t stronst o ts VCEs. VII. EXPERIMENTAL RESULTS W prorm xprmnts to sow t strnt o our mtos. W us ts VC nns: s, t stnr nn rom III; v1, n w s s mprov y usn mov-toront n y mn otr mnor mprovmnts; v2, totlly
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 163 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 F. 6. Clls l-vc-onnt to t ottom, s omput y unlmt VCE (s), lmt VCE (l ot), n st VCE (wt ot). 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 F. 7. Wt mustply, s oun y unlmt VCE (s), lmt VCE (ott), n st VCE (vo: no mustply). rwrttn nn, wt our nw t struturs rom IV. W onsr t tr vrnts o v2 rom VI-C. W mplmnt lortms n t opn-sour Hx rpostory Bnzn [19], w n turn s ult on t opnsour m-npnnt rmwor Fuo [21]. Tsts wr prorm on tr Bnzn prorms tt rly on tr VC nns: Solvr, MoHx n Wolv. A. Solvr Solvr ns t tortl vlu o poston. It uss Fous DFPN [9], [12], vrson o DFPN sr [22] tt s nn y t 1 + ε tr [23], VC nn, n nror ll nn, n ltr rut rsstn or mov orrn. VCs ply ntrl rol: omputtonl ny n t strnt o t omput VC sts r t mn tors n trmnn runtm. Bus Solvr pns so rtlly on VC omputton, t s oo nmr or our mprovmnts. W us postons o vrous ulty, wt solvn tms o v2-unlmt vryn twn 400 n 20 000 sons. W us only rltvly llnn postons, sn on smpl postons tr s otn lr vrn n runtm or rnt runs. T rst st o postons onssts o t tn rst 8 8 1- mov opnns. S Fur 8. T son st o postons on 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 F. 8. 1-mov 8 8 Hx opnns (ott). Cll olor sows wnnr l opns tr. ssts o sx 11 11 postons rom t 2011 ICGA Olymp Hx omptton [24]. Ts wr oun y strtn wt t nl poston n pron wrs to mortly ult poston. S Tl II. TABLE II POSITIONS FROM THE 2011 OLYMPIAD. THE GAME 6 POSITION HAS BEEN MODIFIED. GAME 3 WAS NOT USED IN THIS BENCHMARK, BUT WAS USED IN VI-B. m. mov squn l movs rst 3 3 6 7 8 5 7 8 9 6 8 7 9 8 7 7 8 6 8 8 10 10 11 10 9 9 5 2 5 7 8 7 8 7 8 7 8 8 9 8 7 10 11 10 11 10 11 7 6 6 2 5 6 7 8 7 7 9 9 11 8 9 8 9 8 9 10 4 5 6 5 11 10 11 10 11 10 11 11 10 9 10 10 11 10 9 8 8 7 2 7 7 6 5 4 5 6 6 4 5 4 3 2 3 2 3 2 3 4 4 5 5 6 6 8 7 2 3 8 7 2 3 8 9 1 6 6 5 4 3 4 3 4 3 4 3 4 4 6 5 5 7 6 7 6 10 1 7 6 7 7 9 8 5 5 4 6 5 6 5 6 4 5 3 3 4 11 2 6 6 5 7 8 5 7 8 10 9 10 9 10 9 10 Tl III sows our rsults. W rn our xprmnts on n Intl Xon 2.4 GHz. T trnsposton tl sz ws 2 25, w s mor tn sunt. All vrs r omtr mns, sn ts r mor sutl tn rtmt mns wn msurn spup rto. TABLE III AVERAGE SOLVING TIMES AND VC BUILDS FOR DIFFERENT VERSIONS OF SOLVER. 8 8 opnns Olymp Gms vrson tm VC uls tm VC uls s 4819 355322 10834 203820 + sms-omnr 5442 294169 10783 177722 + AND-rul prorty 5901 289586 12828 171459 + stor VC-nours 4946 290862 10411 166276 + mov-to-ront (v1) 3926 289366 7733 171593 v2-unlmt 2015 183144 5099 140475 v2-lmt 2398 294490 4808 244548 v2-st 1695 295163 3185 244791 T rst prt o t tl sows t mprovmnts otn y n vrous turs to t s vrson VC nn. Sms-omnr yls only smll ruton n runtm, n only or t Olymp Gms st, ut or ot sts t rus t numr o sr stts (w, wt sun optmztons, rsult n snnt runtm ruton).
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 164 A VC nn n postpon runnn sms-omnr untl suntly lr st o VSCs s umult. W mplmnt su n AND-rul prorty sm. Wl ts sm sltly rs t numr o VC uls, tr ws no runtm ruton, prsumly us t s vrson ls som optmztons tt r mplmnt n v2. T otr two s optmztons v snnt spup, splly mov-to-ront. Storn VC-nours llows or trton only ovr lls tt r VC-onnt wt vn npont. Ts s usul mnly wtn pplton o t AND-rul. As mnton rlr, mov-to-ront rqurs t us o vtor nst o ln lst; owvr, t lmnts t sortn o V(S)Cs y rrr sz, w s no lonr nssry us o t nturl orr oun y mov-to-ront. Its prormn s s xpt. As xpln n IV, w sn nw t struturs to xplot our numrous optmztons. T son prt o t tl sows t mprovmnt n solvn tm n t numr o vst stts. Vrson v2-st vsts omprl numr o stts to v2-lmt, sustn tt t ormr mt supror to t lttr. Wl v2-lmt vs no mprovmnt ovr v2-unlmt, v2-st s str, splly on lrr ors. Tus v2-st s ruly t st o ts vrnts or us n Solvr. Howvr, t suprorty o vs-st ovr t otr vrnts s not unvrsl: tr r postons.. n Fur 5 wr t ls u to mssn VCs. B. Plyn prorms In orr to msur t omprtv strnts o ts vrnts n -to- omptton, w ply lr tournmnt nvolvn mny vrsons o t two Bnzn plyrs, nmly MoHx n Wolv. For o ts two plyrs, w rt vrous vrsons y sltn VC nn, sltn VC omputton vrnt, n n wtr to us prlll Solvr. T VC nn s on o s, v1 n v2. For v2, w us on o t vrnts: unlmt, lmt, or st. Tus, or VC nn, w onsr 5 rnt vrsons o t VC nn, w totr wt t prlll Solvr o yls 10 rnt plyrs. So n totl 20 rnt plyrs (10 MoHx, 10 Wolv) ompt n t tournmnt. T tournmnt ws ply wt 10 sons pr mov on n 11 11 or. Bus w r ntrst n plyn strnt pr unt tm, n us t rnt VC vrnts t rn mounts o tm n n rnt sts o onntons, usn x tm lmt s ttr msur tn tr xn t numr o plyouts n MoHx or xn t sr pt n Wolv. E two plyrs ply 72 ms wt otr. For opnns, w us 36 rltvly ln snl ston opnns: 2 to 2, 10 to 10, 1 to 1, n 11 to 11. For opnn n pr o plyrs, two ms wr ply: plyr on s l, n on s wt. So, plyr ply 1368 ms, n t totl numr o tournmnt ms ws 13 680. Tl IV sows t rsults. T Elo sor s omput y BysElo [25] wt rror ±11 n onn 80%. A s Elo sor o zro ws ssn to Mox wt t s VC nn n no solvr. T rsults r suss sprtly or MoHx n Wolv. TABLE IV RESULT OF MOHEX AND WOLVE TOURNAMENT FOR DIFFERENT SETTINGS. Rn Prorm VC nn & vrnt solvr Elo sor 1 MoHx v2-unlmt ys 111 2 MoHx v2-lmt ys 93 3 MoHx v2-st ys 88 4 Wolv s ys 85 5 Wolv v1 ys 70 6 Wolv s no 69 7 MoHx v2-unlmt no 48 8 MoHx v1 ys 47 9 Wolv v1 no 46 10 MoHx v2-st no 44 11 MoHx v2-lmt no 40 12 MoHx v1 no 37 13 MoHx s ys 37 14 Wolv v2-unlmt no 33 15 Wolv v2-unlmt ys 28 16 Wolv v2-lmt ys 27 17 Wolv v2-st ys 27 18 Wolv v2-st no 11 19 Wolv v2-lmt no 1 20 MoHx s no 0 1) MoHx: MoHx [7] uss Mont Crlo tr sr [26], [27]. T VC nn s us s ollows. As soon s no s vst 400 tms, VCs r ult or t no, n mustply ron s omput. Ts llows t prunn o nror movs, n tts wn or loss lon or t or s ull. T prlll Solvr us xuts on sprt tr. I Solvr tts wn or loss, t ssot mov s us y t plyr; otrws, t plyr uss MCTS to slt ts mov. It s no surprs tt rrsptv o wtr Solvr s us s s t wst VC nn. It s lso no surprs tt Solvr nrss plyn strnt: s s 37 Elo stronr (wt Solvr on tn wt Solvr o), v1 s 10 Elo stronr, n t tr vrnts o v2 r 40 48 Elo stronr. Wt Solvr, v2 s mu stronr tn otr VC nn vrsons. Amon v2 vrnts, v2-unlmt s st, wl v2-st n v2-lmt r omprl. Our xprmnt sows tt MoHx nts rtly y nrsn t strnt n/or sp o ts VC nn, n susts tt v2-unlmt s t prrr vrnt. Howvr, unr rl tournmnt sttns tos us n ompttons, typlly 16 MCTS trs or MCTS n out 1 mnut pr mov t numr o VCs osonlly xplos to t xtnt tt MCTS sr s rppl. Tus, n su tournmnts, v-st s prrr: ts vs t stst VC omputton, llown t st numr o plyouts pr son, yt mntns rltvly urt mov slton. 2) Wolv: Wolv uss ltr rut rsstn vluton nn y n s or VC-onnt lls [10]. Mov orrn n prunn s s on ts vluton. VCs r lso us or mov prunn, y nn t mustply ron. T qulty o t omput VCs nluns mov orrn n prunn, wl t pt o sr pns on sp. Tus t o o st VCE vrnt s not ovous. As wt MoHx, Solvr n run n prlll, ut r ts nt s qustonl. Ts s prsumly us o t lp-t sr us y Wolv otn yls smlr
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 165 rsults to t DFPN sr us y Solvr, yln lttl xtr normton. By ontrst, MCTS otn yls rnt rsults rom DFPN sr, so usn Solvr s mu xtr normton. Hr, t rsults r ountr-ntutv: t stronr or str t VC nn, t wr t vrnt o Wolv. In prtulr, t vrnts v2-lmt n v2-st, w ul sprs n so rltvly low qulty sts o VCS, prorm vry poorly. Tus on mt onlu tt t ury o t VC nn s mportnt n Wolv. Howvr v1 n v2-unlmt, w v t st possl ury n uln VCs, r wr tn s. Morovr, v1 s stronr tn v2-unlmt; t mn rn twn ts vrnts s tt v2-unlmt s str tn v1, w mns tt lp-t n sr mor ply. Ts ountr-ntutv vour s n osrv n plyrs smlr to Wolv su s Sx [15], n w pr sr n yl wr prorm. On possl xplnton s t pulrts o t vluton unton, w s s on n ltr rsstn mol o t poston. Nos tt r r prt n t tr, splly ty r t rnt pts, r otn norrtly rn y ts unton, n so p srs my vt snntly rom orrt ply. W suspt tt ts vluton s s towrs postons wt mor stons, n tt t rltv rror mon postons wt mny stons s mu rtr tn t rltv rror mon postons wt w stons. E o ts tors oul l to nstlty n p srs. A. Our Rsults VIII. CONCLUSIONS W v prsnt n mprov lortm or nn n usn vrtul onntons n onnton-m utomt plyr. Our lortm rornzs H-sr n stors onntons mor ntly. It uss mov-to-ront, w sps t vrton o nw mnml onntons. Our most mportnt ontruton s ruly our nw ORrul (sms-omnr), w ns rstrt st o nw mnml onntons, ut mor quly tn ornry H- sr. Ts llows us to pply our OR-rul or rtrry lr st o VSCs; prvous tnqus to lmt t numr o VSCs us n n OR-rul to our. Sms-omnr n prou n ntrtly lr numr o onntons, omn too slow or prtl us. Tus w sn VC ptn urst tt llows t pplton o nw OR-rul (st sms-omnr). Ts nl optmzton yls DFPN-s solvr tt s out tr tms str tn wt t prvous ppro. B. Futur Wor Our lortm s not unvrslly ttr tn prvous mtos or som postons, us t lmts t st o VCS t ns, t ls to solv t poston n rsonl tm. Otr ppros to lmtn t st o sovr VCs mt sussul. Furtr mnor optmztons mt possl. For xmpl, nn t orr n w onntons r pross mt mprov runtm, or t strnt o nw onnton sts. FstVC Sr oul ppl to otr prorms w, l MoHx n Wolv, rly on onnton omputtons. MoHx spns most o t tm n onnton omputton, so our nw lortms ru ts runnn tm snntly. By ontrst, our nw lortms not strntn Wolv, prsumly us or Wolv t rtl tor s not ow lon t ts to omput onntons, ut rtr t rnss o t sovr st tn osonlly nrs sr pt. Explorn vrous spt o onnton strts n ow ty ontrut to t strnts o vrous plyrs n solvrs s notr r or stuy. ACKNOWLEDGEMENTS W tn Mrtn Mullr or t us o s mns or som o our xprmnts. REFERENCES [1] P. Hn, Vl lr Polyon? Poltn, Dmr 1942. [2] J. Ns, Som ms n mns or plyn tm, RAND, T. Rp. D-1164, Frury 1952. [3] H. W. Kun n S. Nsr, Es., T Essntl Jon Ns. Prnton Unvrsty Prss, 2002. [4] S. Nsr, A Butul Mn: A Borpy o Jon Fors Ns, Jr. Smon n Sustr, 1998. [5] C. E. Snnon, Computrs n utomt, Prons o t Insttut o Ro Ennrs, vol. 41, pp. 1234 1241, 1953. [6] S. Rs, Hx st PSPACE-vollstän, At Inormt, vol. 15, pp. 167 191, 1981. [7] B. Arnson, R. B. Hywr, n P. Hnrson, Mont Crlo Tr Sr n Hx, IEEE Trnstons on Computtonl Intlln n AI n Gms, vol. 2, no. 4, pp. 251 258, 2010. [8] V. V. Anslv, T m o Hx: An utomt torm provn ppro to m prormmn, n AAAI/IAAI. Mnlo Pr: AAAI Prss / T MIT Prss, 2000, pp. 189 194. [9] B. Arnson, R. B. Hywr, n P. Hnrson, Solvn Hx: Byon umns, n Computrs n Gms 2010, sr. LNCS, H. J. vn n Hr, H. I, n A. Plt, Es. Sprnr, 2011, vol. 6515, pp. 1 10. [10] V. V. Anslv, A rrl ppro to omputr Hx, Artl Intlln, vol. 134, no. 1 2, pp. 101 120, 2002. [11] R. Hywr, Y. Börnsson, M. Jonson, M. Kn, N. Po, n J. vn Rsw, Solvn 7 7 Hx wt omnton, ll-n, n vrtul onntons, Tortl Computr Sn, vol. 349, no. 2, pp. 123 139, 2005. [12] P. Hnrson, Plyn n solvn Hx, P.D. ssrtton, Unvrsty o Alrt, 2010, ttp://wos.s.ulrt./ ywr/tss/p.p. [13] P. Hnrson n R. B. Hywr, Cptur-rvrsl movs n str omposton omnton n Hx, Intrs, vol. 13, p. #G1, 2013. [14] G. Mls n R. Hywr, Sx wns Hx tournmnt, ICGA Journl, vol. 26, no. 4, pp. 277 280, 2003. [15] G. Mls, Sx, sx.rts.u/, 2006. [16] R. Rsmussn, Alortm ppros or plyn n solvn Snnon ms, P.D. ssrtton, Qunsln Unvrsty o Tnoloy, Brsn, Qunsln, Austrl, 2007. [17] V. V. Anslv, T m o Hx: T rrl ppro, July 2000, omntorl Gm Tory Worsop. MSRI, Brly. ttp://www.msr.or/pultons/ln/msr/2000/mtory/nslv/1/. [18] P. Hnrson, B. Arnson, n R. Hywr, Hx, rs, t rossn rul, n XH-sr, n ACG, sr. Ltur Nots n Computr Sn, J. vn n Hr n P. Spron, Es., vol. 6048. Sprnr, 2010, pp. 88 98. [19] B. Arnson, P. Hnrson, n R. B. Hywr, Bnzn, 2009 2012, ttp://nzn.souror.nt/. [20] T. Wllms, C. Klly, n mny otrs, Gnuplot 4.4: n ntrtv plottn prorm, ttp://nuplot.souror.nt/, Mr 2011. [21] M. Enznrr, M. Müllr, B. Arnson, R. Sl, F. X, n A. Hun, Fuo, 2007 2012, ttp://uo.souror.nt/.
IEEE TRANSACTIONS ON COMPUTATIONAL INTELLIGENCE AND AI IN GAMES, VOL 7, NO 2, JUNE 2015, PAGES 156-166 166 [22] A. N, D-pn lortm or srn n/or trs n ts ppltons, P.D. Tss, Dpt. o Inormton Sn, Unvrsty o Toyo, Toyo, Jpn, 2002. [23] J. Pwlwz n L. Lw, Improvn pt-rst pn-sr: 1+ε tr, n Computrs n Gms 2006, sr. LNCS, H. J. vn n Hr, P. Cnrn, n H. Donrs, Es. Sprnr, 2007, vol. 4630, pp. 160 170. [24] R. B. Hywr, Mox wns Hx tournmnt, ICGA Journl, vol. 35, no. 2, pp. 124 127, Jun 2012. [25] R. Coulom, Bysn lo rtn, 2010, ttp://rm.oulom.r.r/bysn-elo. [26], Ent Sltvty n Bup Oprtors n Mont-Crlo Tr Sr, n Pro. 5t Int. Con. Comput. n Gms, LNCS 4630, Turn, Itly, 2007, pp. 72 83. [Onln]. Avll: ttp://portl.m.or/tton.m?=1777826.1777833 [27] L. Koss n C. Szpsvár, Bnt s mont-rlo plnnn, n ECML, sr. Ltur Nots n Computr Sn, J. Fürnrnz, T. Sr, n M. Splopoulou, Es., vol. 4212. Sprnr, 2006, pp. 282 293. Bror Arnson ols n MS n Computn Sn (Alrt 2007, suprvsors Potr Run n Lorn Stwrt). From 2007 untl 2014 ws rsr ssstnt n t Hx n Go roups n Computn Sn t t Unvrsty o Alrt. In prtulr, mplmnt most o t o or Bnzn, t prmry o s or t Hx roup. Sn 2014 s sotwr nnr t Gool. Ju Pwlwz rv t P.D. r n Computr Sn rom Unvrsty o Wrsw, Poln, n 2009. Hs mn l o ntrst s Artl Intlln n Gms. In 2011-2012 l post-otorl poston n Computn Sn t t Unvrsty o Alrt n Cn, urn w tm m o-utor o t Bnzn o prot. Currntly s n unt prossor t Unvrsty o Wrsw. Hs mn rsr ntrst s t sn o nw tr sr lortms. Ryn B. Hywr s prossor n t Dprtmnt o Computn Sn t t Unvrsty o Alrt. H ols PD n omputr sn (MGll 1987, suprvsor Vs Cvtl) n n MS (Qun s/knston 1982, suprvsors Ptr Tylor n Slm Al) n BS (onours, Qun s/knston 1981) n mt. Bor onn t prtmnt n 1999, ws n ssstnt n tn ssot prossor t Ltr (92-99), sssstnt prossor t Qun s/knston (90-92) n Rutrs (86-89), n n Alxnr von Humolt Fllow n Bonn (89/90). Hs rsr ntrsts nlu srt lortms n m-tr sr. Totr wt Bror Arnson, Plp Hnrson, S-C Hun, n Ju Pwlwz, s o-utor o MoHx, t ol-ml wnnn Hx plyr, n Solvr, t Hx solvr tt s solv ll 9x9 n two 10x10 opnns. Plp Hnrson s sotwr nnr t Gool, mprovn t mn lrnn systms tt prt t ltrou rt or sr s. H s ntrst n rp tory n omntorl m tory, n n 2010 omplt s PD on t m o Hx t t Unvrsty o Alrt.