A 4-stt solution to th Firing Squ Synhroniztion Prolm s on hyri rul 60 n 102 llulr utomt LI Ning 1, LIANG Shi-li 1*, CUI Shung 1, XU Mi-ling 1, ZHANG Ling 2 (1. Dprtmnt o Physis, Northst Norml Univrsity, Chnghun, 130024, Chin; 2. Collg o sin, ChngChun Univrsity o Sin n Thnology, ChngChun, 130022,Chin) 1 Astrt: In this ppr, w prsnt 4-stt solution to th Firing Squ Synhroniztion Prolm (FSSP) s on hyri rul 60/102 Cllulr Automt(CA),This solution solvs th prolm on th lin o lngth 2 n with two gnrls. Prvious work on FSSP or 4-stt systms ous mostly on linr llulr utomt, whr synhronizs n ininit numr o lins ut not ll possil lins. W giv tim-optiml solutions to synhroniz n ininit numr o lins y rul 60 n rul 102 rsptivly, n onstrut hyri rul 60 n 102 stts trnsition tl. Compr to th known solutions o llulr utomt, th hyri CA wy is simplr n str, th miniml tim is (n-1) stp. Kywors: Prlll lgorithms, llulr utomt, Firing Squ Synhroniztion, Rul 60, Rul 102 1. Introution Th Firing Squ Synhroniztion Prolm (FSSP), is on o th st stui prolms or llulr utomt. This prolm in whih ntwork o intil lls(init utomt) work synhronously t isrt tim stps. Figur 1 shows init on-imnsionl llulr rry onsisting o n lls, All lls (solirs) xpt th lt n ll (gnrl), r initilly in th quisnt stt t tim t=0. Th nxt stt is trmin y oth its own stt n th prsnt stts o its right n lt nighors. At som utur tim, ll o th lls will simultnously n, or th irst tim, ntr spil iring stt. * Corrsponing uthor. E-mil: lsl@nnu.u.n, Tl.: +86-431-85099665
Fig. 1. A on-imnsionl llulr utomton. Th FSSP prolm hs n stui or long tim ([1]). Minsky n MCrthy ([2] )onstrut solution synhronizing n lls in 3n stps using ivi-n-onqur mtho. Atr tht Wksmn ([3]) n Blzr ([4]) sign nw lgorithm whih n synhroniz in miniml tim in T (n) =(2n stps, n hv stt omplxity o 16 n 8 stts rsptivly. Mzoyr ([5]) show 6-stt miniml-tim solution,ut still unknown whthr iv-stt ull solution xists,yunès ([6, 7]), Sttl n Simon ( [8]) n Umo ( [9]) sign non miniml-tim solutions with w stts. Umo ([10,11]) n in lss thn 6-stt solutions to synhroniz n ininit numr. Whn it oms to our-stt spil solution, som popl try to in solutions to synhroniz n ininit numr ut not ll lins. Blzr prov tht th miniml tim 4-stt solution os not xist, n tht Blzr s rsult shows tht thr is no 4-stt solution whih synhronizs vry lin in miniml-tim. Now th 4-stt solutions r ll uilt using som lmntry lgr, n mostly s on linr llulr utomt with rul 60 n 150, Jn-Bptist ([12]) prsnt on o ths solutions in only 4-stt whih synhronizs vry lin whos lngth is powr o 2. In this ppr, w qust 4-stt solution to lin o 2 n FSSP with two gnrls s on hyri rul 60/102 Cllulr Automt. 2. Prliminry For th our-stt FSSP solution, Yunès sign n lgorithm y Wolrm Rul 60 or 2 n ll rry to mt n ininit numr o lins. Our onstrution is lso s on Wolrm s linr llulr utomt using rul 60 n rul 102.Hr r som si FSSP initions n lmntry onpts. 2.1 Dinitions Thr is llulr utomt A(Q,δ) whr Q is init st, ll th stts st o A, nδis trnsition untion rom Q 3 ->Q. ) A llulr utomt S is oupl A whih is n pplition rom Z in Q. A onigurtion C volvs to nothr onigurtion C* so tht C*(Z)=δ(C(z-1),C(z),C(z+1)) W n in C*=Δ(C)s glol trnsition untion. So th initil onigurtion o llulr utomt is C0 (t th tim o t=0), th onigurtion o tim t is C t =Δ t (C). ) At lst our istinguish stts long to Q. ) Stt Q is th quisnt stt. It stisisδ(q,q,q)=q,δ(q,q,!)=q, δ(!,q,q)=q ) Stt * is th ounry stt. It stisis: q q Q, (q,*,q ) * 1, 2 1 2 ) Stt G is th gnrl stt n th stt F is th Fir stt, suh tht, strting rom th
initil onigurtion in y: ( ) z 0, C[ n]( z)! () z n 1, C[ n]( z)! ()C[n](1) G () z {2,3...,n},C[n](z) Q ) Th volution o th onigurtion C[n] is suh tht, or synhroniztion tim t(n): () z N, t {1,...,t(n) 1},C[n] (z) F () z {1,...,n},C[n] t(n) (z) F 2.2 Som FSSP lmntry onpts ) Synhroniztion Tim[3] [4] [9] [10] [13]. Th solution to th iring squ synhroniztion prolm n shown: Synhroniztion o n lls in lss thn 2n-2 stps is impossil, n Synhroniztion o n lls in xtly 2n-2stps is possil. So th miniml synhroniztion tim is 2n-2 stps. ) Th numr o stts[4] [9] [10]. To sign trnsition tls, thr is t lst thr stts: th stt o quisnt, th stt o gnrl n th stt o iring. Bsis, w lso sign ounry stt. It is shown tht thr xists no thr-stt solution n no our-stt symmtri solution on rings. Blzr prov tht thr is no our-stt ull solution or this prolm. A miniml tim solution o six stts ws introu y Jqus Mzoyr in 1987[13]. ) Th numr o trnsition ruls[9] [10]. Any k-stt trnsition tl or th synhroniztion hs t most (k-1)k 2 ntris in (k-1) mtris o siz k*k. Th numr o trnsition ruls rlts th omplxity o synhroniztion lgorithms. t 3. Yunès s Rul 60 lgorithm Yunès[12] sign our stts solution to hiv this synhroniztion, th stt gnrl (g) is 1 0 0 1, th quisnt stt is, th stt x 1 is, n th stt o iring is, th 0 0 1 1 symol $ rprsnts th ounry stts. Th rul 60 volution igur is ig 2:
Fig. 2. Th Rul 60 Cllulr Auotmt I w ol th psl s tringl moulo2, w n gt ollow igur 3: Fig. 3. Th Rul 60 s solution. Th gry olor stts rprsnt th stt o gnrl, th whit olor stts rprsnt th stt o quisnt, th lk olor stts rprsnt th stt o 1 x, n th rk r olor stts rprsnt th stt o iring. Mnwhil, Yunès rwrit th trnsition untion into lgri orm: (mo $ (mo $ (mo (mo
4. Rul 102 lgorithm Bs on th work o Yunès, w sign our-stt solution y rul 102, in this wy th gnrl is lot in th right n o ll rry. Th rul 102 volution pross n sn in ig. 4: Fig.4. th Rul 102 Cllulr Automt n=16 I w ol th psl s tringl o rul 102, w will gt ig. 5: Fig.5. th Rul 102 s solution n=8 Similrly, w n lso rwrit th trnsition untion into this lgri orm: (mo $ (mo $ (mo (mo
An th rul 102 trnsition tl n got s Tl 1.: Tl 1. Th Rul 102 trnsition tl 5. Th hyri solution o Rul 60 n Rul 102 Bs on th work o ormr txt, w omin th Rul 60 n 102 to onstrut solution or 2 gnrls tht r t lt n right o th lin o 2 n FSSP rsptivly, w lso hv our stts: th stt o gnrl (th gry olor) ; th stt o quisnt (th whit olor), th stt o x 1 (th lk olor stts), th stt o iring(th rk r olor), Hr, w giv snpshot or synhroniztion oprtion or this lgorithm on ight lls: Fig.6. th Rul 102 n Rul 60 s solution n=8 W n rw th onlusion sly tht this lgorithm n short th synhroniztion tim or 2-gnrl, w in th stt o gnrl 1 is G, th stt o gnrl 2 is H, th stt o quisnt is Q, th stt o iring is F, th ounry stt is *, th trnsition tls r s ollows:
Tl 2. Th Rul 102 n Rul 60 trnsition tls In trms o th trnsition tls w n gt th stts hng tls s ollows: Tl 2. Th Rul 102 n Rul 60 stts trnsition tls 6. Conlusion An xistn or non-xistn o our-stt iring squ synhroniztion protool hs n long-stning, mous opn prolm or long tim. In this ppr, w hv prsnt 4-stt solution to th iring squ whih synhronizs th lin o lngth 2 n in (n-1) stp. Bs on th rsult o th Rul 60 n rul 102 to FSSP, w omin with th Rul 60 n Rul 102 to giv solution or 2 gnrls tht r t lt n right o th lin o 2 n FSSP rsptivly, in th n w lso gt stts hng tls. W think this is vry promising pproh or th srh o solutions sign or non miniml-tim solutions with w stts. Rrns [1] Y. Nishitni n N. Hon, Th Firing Squ Synhroniztion Prolm or Grphs, Thortil Computr Sin 14 (1981), 39 61. [2]Minsky, M.: Computtion: Finit n Ininit Mhins. Prnti-Hll, Englwoo Clis (1967). [3] A. Wksmn, An optimum solution to th iring squ synhroniztion prolm, Inormtion n Control 9 (1966) 66 78.
[4]Blzr, R.: An 8-Stt Miniml Tim Solution to th Firing Squ Synhroniztion Prolm. Inormtion n Control 10 (1967) 22 42. [5]Mzoyr, J.: A Six-Stt Miniml-Tim Solution to th Firing Squ Synhroniztion Prolm. Thortil Computr Sin 50(1987) 183 238. [6]Yun`s, J.-B.: Svn Stts Solutions to th Firing Squ Synhroniztion Prolm. Thortil Computr Sin 127( (1994) 313 332. [7]Yun`s, J.-B.: An Intrinsilly non Miniml-Tim Minsky-lik 6-Stts Solution to th Firing Squ Synhroniztion Prolm. RAIRO ITA/TIA 42(1) (2008) 55 68. [8] A. Sttl, J. Simon, Non-miniml tim solutions or th iring squ synhroniztion prolm, Thnil Rport 97-08, Univrsity o Chigo, 1997. [9] H. Umo, M. M, K. Hongyo, A sign o symmtril six stt 3n-stps iring squ synhroniztion lgorithms n thir implmnttions, in: Proings o ACRI 2006, in: Ltur Nots in Computr Sin, 4173(2006) 157 168. [10] H. Umo, T. Yngihr, A smllst iv-stt solution to th iring squ synhroniztion prolm, in: J. Durn-Los,M.Mrgnstrn (Es.), Proings, 5th Intrntionl Conrn on Mhins, Computtions n Univrslity, MCU 2007 Orléns, Frn, in: Ltur Nots in Computr Sin, 4664(2007) 291 302. [11] H. Umo, N. Kmikw, A 4-stts solution to th iring squ s on Wolrm s rul 150, Privt ommunition. [12]Jn-Bptist,Yunès, A 4-stts lgri solution to linr llulr utomt Synhroniztion, Inormtion Prossing Lttrs 107 (2008) 71 75. [13] J. Mzoyr, A six-stt miniml-tim solution to th iring squ synhroniztion prolm, Thortil Computr Sin 50(1987) 183 238.