Decomposability of Global Tasks for Multi-agent Systems

Transcription

1 49th IEEE Cofrc o Dcisio Cotrol Dcmbr 15-17, 2010 Hilto Atlt Hotl, Atlt, GA, USA Dcomposbility of Globl Tsks for Multi-gt Systms Mohmm Krimii Hi Li Abstrct Multi-gt systm is rpily vlopig rsrch r with strog support from both civili militry pplictios. O of th sstil problms i multi-gt systm rsrch is how to sig locl itrctio ruls cooritio pricipls mog gts such tht th whol systm chivs sir globl bhviors. To tckl this problm, ivi--coqur pproch ws propos i [1], th bsic i is to compos th rqust globl spcifictio ito subtsks for iiviul gts i such wy tht th fulfillmt of ths subtsks by ch iiviul gt shoul l to th stisfctio of th globl spcifictio. Th, th sig rucs to chivig th ssig subtsks for corrspoig iiviul gts. I [1], it ws show tht ot ll globl tsks c b compos, cssry sufficit coitio o th composbility of tsk utomto btw two gts ws prst. For mor th two gts, w th propos hirrchicl lgorithm s sufficit coitio for composbility. This ppr ims to xt th cssry sufficit composbility coitios for y rbitrry fiit umbr of cooprtiv gts. A w cssry sufficit coitio o composbility of tsk utomto is propos, hr. Svrl xmpls r provi to illustrt th compositio schm coitios. I. INTRODUCTION Multi-gt systm is rpily vlopig crossiscipliry rsrch r tht hs b obtiig strog support from both civili militry pplictios such s coorit survillc, trgt cquisitio, rcoissc, urwtr or spc xplortio, ssmblig trsporttio rpi mrgcy rspos [2]. It is kow tht sophistict collctiv bhviors c mrg through th cooprtio of ruimtry gts with simpl locl itrctio ruls. But, most stuis i th multi-gt systm litrtur hv b focus o bottom-up pprochs [3], mily through simultio, mpiricl huristic pprochs [4]- [7]. Th pst svrl yrs hv s sigifict rsrch fforts i th thorticl lysis o multi-gt systms usig grph thory [8] symbolic cotrol of swrmig systms [9], [10]. Howvr, th cooprtiv cotrol of multigt systm is still i its ifcy with sigifict prcticl thorticl chllgs tht r ifficult to b formult tckl by th tritiol mthos [9], [11]. O of th sstil problms i cooprtiv cotrol of multi-gt systms is how to sig th locl itrctio ruls cooritio pricipls mog gts such tht th whol systm chivs sir globl spcifictio. To tckl this chllg, i [1], w propos ivi-coqur pproch by composig th rqust globl M. Krimii H. Li r both from th Dprtmt of Elctricl Computr Egirig, Ntiol Uivrsity of Sigpor, Sigpor. Corrspoig uthor, H. Li llh@us.u.sg spcifictio ito subtsks for iiviul gts such tht th fulfillmt of ths subtsks by ch iiviul gt will l to th stisfctio of th globl spcifictio. Th, th sig rucs to chivig th ssig subtsks for corrspoig iiviul gts. I orr to pursu th i, svrl qustios to b swr, such s how to scrib th globl spcifictio subtsks i succict forml wy, how to compos th globl spcifictio, whthr it is lwys possibl to compos, if ot wht is th coitio for composbility. This ppr cotius th fforts i our prvious work [1] ims to swr ths qustios pc th wy towrs forml sig mtho for multi-gt systms. Hr, th tsk for group of itlligt gts r rprst s utomto u to its xprssibility similrity to our hum logicl comms [12], [13]. Th, th compositio of globl tsk ctully bcoms th utomto compositio problm tht hs b stui i th computr scic litrtur. Roughly spkig, two iffrt clsss of problms hv b stui so fr. Th first problm is to sig th vt istributio so s to mk th utomto composbl, which is typiclly stui i th cotxt of cocurrt systms. For xmpl, [14] chrctriz th coitios for compositio of sychroous utomt i th ss of isomorphism bs o th mximl cliqus of th pc grph. Thir chrctriztio of ipcy rlis o forwr imo (FD) ipt imo (ID) ruls, rprstig th ituitiv otio of ipt orr ipt choic of ipt vts (privt vts from iffrt locl vt sts). Our composbility coitios lso iclu two coitios rssig th otio o ipc (w will cll thm DC1 DC2). Howvr, i DC1 DC2, lthough ipt vts strtig from o stt r llow to occur i y orr, iffrt occurrig orrs my l to iffrt stts. This rlxtio llows us to obti th compositio i th ss of bisimultio rthr th isomorphism i [14]. Grlly, som utomt my stisfy DC1 DC2, but ot cssrily FD ID. O th othr h, th sco problm ssums tht th istributio of th globl vt st is giv th gol is to fi coitios o th utomto such tht it is composbl. This is usully cll sythsis moulo problm [13], bisimultio sythsis moulo for globl utomto ws rss i [15] by itroucig cssry sufficit coitio for utomto compositio bs o lgug prouct of th utomto trmiism of its bisimultio quotit. Obtiig th bisimultio quotit, howvr, is grlly ifficult tsk. This motivts us to vlop w /10/$ IEEE 4192

2 cssry sufficit coitio cosir mor grl css, which my occur i multi-gt systms. Followig [1], w ssum tht th globl sir bhvior c b rprst s trmiistic utomto, whos vt st is th uio of th collctio of locl vt sts for ch gt. Ur this ssumptio, th locl tsk utomto for iiviul gt c b obti through turl projctio of th globl tsk utomto ito its corrspoig locl vt st. Ufortutly, it ws show i [1] tht ot ll globl tsk utomto c b compos i this wy, which ms tht th compositio of th obti sub-tsk utomt is ot quivlt to th globl tsk utomto. Formlly, th compositio is through clssicl prlll compositio, whil th quivlc is i th ss of bisimultio. Furthrmor, i [1] cssry sufficit coitio o th composbility of tsk utomto btw two gts ws propos. For mor th two gts hirrchicl lgorithm ws prst tht ws show to b oly sufficit coitio. I this ppr, w im to furthr xt th composbility rsult to y rbitrry fiit umbr of cooprtiv gts, s cssry sufficit coitio. Th rst of th ppr is orgiz s follows. Prlimiry rsults, ottios, fiitios problm formultio r rprst i Sctio II. Sctio III itroucs th cssry sufficit coitio for compositio of utomto with rspct to prlll compositio rbitrry fiit umbr of vt sts. Filly, th ppr coclus i Sctio IV with rmrks iscussios. II. PROBLEM FORMULATION W first rcll th fiitio of utomto [16]. Dfiitio 1: (Automto) A utomto is tupl A = (Q, q 0, E, δ) cosistig of st of stts Q; th iitil stt q 0 Q; st of vts E tht cuss trsitios btw th stts, trsitio rltio δ Q E Q such tht (q,, q ) δ if oly if δ(q, ) = q (or q q ). Th trsitio rltio c b xt to fiit strig of vts, S E, whr E sts for Kl Closur of E (th st of ll fiit strigs ovr lmts of E), s δ(q, ε) = q, δ(q, S) = δ(δ(q, S), ) for S E E. W focus o trmiistic tsk utomt tht r simplr to b chrctriz, covr wi clss of spcifictios. Th qulittiv bhvior of trmiistic systm is scrib by th st of ll possibl squcs of vts strtig from th iitil stt. Ech such squc is cll strig, collctio of strigs rprsts th lgug grt by th utomto, ot by L(A). Th xistc of trsitio ovr strig S E from stt q Q is ot by δ(q, S)!. Cosirig lgug L, by δ(q, L)! w m tht ω L : δ(q, ω)!. Nxt, th succssiv vt pir jct vt pir for utomto r fi s follows. Dfiitio 2: (Succssiv vt pir) Two vts r cll succssiv vts if q Q : δ(q, )! δ(δ(q, ), )! or δ(q, )! δ(δ(q, ), )!. Dfiitio 3: (Ajct vt pir) Two vts r cll jct vts if q Q : δ(q, )! δ(q, )!. To compr th tsk utomto th compositio of its compos utomt, w us th simultio rltio [12], fi s follows. ( Dfiitio 4: (Simultio) Giv two utomt A i = Qi, qi 0, E, δ ) i, i = 1, 2, rltio R Q1 Q 2 is si to b simultio rltio from A 1 to A 2 (ot s A 1 A 2 ) if 1) ( q1, 0 q2) 0 R 2) (q 1, q 2 ) R, δ 1 (q 1, ) = q 1, th q 2 Q 2 such tht δ 2 (q 2, ) = q 2, (q 1, q 2 ) R. A mutul symmtric similrity btw A 1 A 2 is cll bisimilrity ot s A 1 = A2. Two utomt r (bi)similr wh th (bi)simultio rltio is fi ovr ll (Q 1 Q 2 ) Q 1, for ll E. I this ppr, w ssum tht th tsk utomto A S th sts of locl vts E i r ll giv. It is furthr ssum tht A is trmiistic utomto whil its vt st E is giv s th uio of locl vt sts, i.., E = i E i. Th problm is to chck whthr th tsk utomto A S c b compos ito sub-utomt A Si o th locl vt sts E i, rspctivly, such tht th collctio of ths subutomt A Si is quivlt to A S wh put thm togthr. Th quivlc is i th ss of bisimilrity s fi bov, whil th clustrig procss for ths sub-utomt A Si coul b i th usul ss of prlll compositio s fi blow. Prlll compositio [16] is commo wy to mol th itrctios btw utomt, is mploy hr to rprst th globl bhvior of tm of cooprtiv multi-gts. ( Dfiitio 5: (Prlll Compositio) [16] Lt A i = Qi, qi 0, E ) i, δ i, i = 1, 2 b utomt. Th prlll compositio (sychroous compositio) of A 1 A 2 is th utomto A 1 A 2 = (Q, q 0, E, δ), fi s Q = Q 1 Q 2 ; q 0 = (q1 0, q0 2 ); E = E 1 E 2 ; (q 1, q 2 ) Q, E { : δ((q 1, q 2 ), ) = δ1 (q (δ 1 (q 1, ), δ 2 (q 2, )), if 1, )!, δ 2 (q 2, )! ; E 1 E 2 (δ 1 (q 1, ), q 2 ), if δ 1 (q 1, )!, E 1 \E 2 ; (q 1, δ 2 (q 2, )), if δ 2 (q 2, )!, E 2 \E 1 ; ufi, othrwis Th prlll compositio of A i, i = 1, 2,..., is cll prlll istribut systm, is fi bs o th ssocitivity proprty of prlll compositio [12] s A i = A 1... A = A 1 (A 2 ( (A 1 A ))). Nxt, lt us rcll th oprtio of turl projctio tht will b us ltr to obti th compos subtsk utomt. Dfiitio 6: (Nturl Projctio o Strig) Cosir globl vt st E its locl vt sts E i, i = 1, 2,...,, 4193

3 with E = E i. Th, th turl projctio p i : E Ei is iuctivly fi s p i (ε) = ε; { S E pi (S) if i loc();, E : p i (S) = p i (S) othrwis. Hr loc() = {i E i }. Th turl projctio is lso fi o utomt s P i (A S ) : A S A Si, whr, A Si r obti from A S by rplcig its vts tht r blog to E\E i by ε-movs, th, mrgig th ε-rlt stts. Th turl projctio is formlly fi o utomto s follows. Dfiitio 7: (Nturl Projctio o Automto) Cosir utomto A S = (Q, q 0, E, δ) locl vt sts E i, i = 1, 2,...,, with E = E i. Th, P i (A S ) = (Q i = Q / Ei, [q 0 ] Ei, E i, δ i ), with δ i ([q] Ei, ) = [q ] Ei if thr r stts q 1 q 1 such tht q 1 Ei q, q 1 Ei q, δ(q 1, ) = q 1. Hr, [q] E i = [q] i ots th quivlc clss of q fi o E i, whr, th rltio E i is th lst quivlc rltio o th st Q of stts such tht δ(q, ) = q i / loc() q Ei q. Th followig xmpl lborts th cocpt of turl projctio o giv utomto. Exmpl: Cosir utomto A S : 4 b with th vt st E = E 1 E 2 locl vt sts E 1 = {, b, }, E 2 = {, b,, 4 }. Th turl projctios of A S ito E 1 is obti s P 1 (A S ): b ˇ by rplcig, 4 E\E 1 with ε mrgig th ε-rlt stts. Similrly, th projctio P 2 (A S ) is obti s P 2 (A S ): 2 4 b. To ivstigt th itrctios of trsitios i two utomt, prticulrly i P 1 (A S ) P 2 (A S ), th itrlvig of strigs is fi s follows. Dfiitio 8: Cosir two squcs q 1 q q q 1 q 2... q, th itrlvig of thir corrspoig strigs, S =... S = m, is ot by S S, fi s S S = L{PA(q 1, S)PA (q 1, S )}, whr, PA(q 1, S) = ({q 1,..., q }, {q 1 }, {,..., },δ PA ) with δ PA (q i, i ) = q i+1, i = 1,..., 1, δ PA is fi i similr wy. Bs o ths fiitios, w my ow formlly fi th composbility of utomto with rspct to prlll compositio turl projctios s follows. Dfiitio 9: (Automto composbility) A tsk utomto A S with th vt st E locl vt sts E i, i = 1,...,, E = E i, is si to b composbl with rspct to prlll compositio turl projctios if P i (A S ) = A S. m It is sy to show by simpl coutr xmpls (s Exmpls 2-5) tht ot ll utomt r composbl with rspct to prlll compositio turl projctios. Th, turl follow-up qustio is wht mks utomto composbl. It c b formlly stt s follows. Problm 1: Giv trmiistic tsk utomto A S locl vt sts E i, i = 1,,, wht is th cssry sufficit coitio tht A S is composbl with rspct to prlll compositio turl projctios P i : A S P i (A S ), i = 1,,, such tht P i (A S ) = A S? III. DECOMPOSABILITY OF TASK AUTOMATON A. Dcomposbility Coitio for 2 gts First, lt us rcll th mi rsult i [1] for th composbility of utomto A S for two cooprtiv gts, to b us s bsis for th mor grl cs of rbitrry fiit umbr of gts. Lmm 1: (Thorm 1 i [1])) A trmiistic utomto A S = (Q, q 0, E = E 1 E 2, δ) is composbl with rspct to prlll compositio turl projctios P i : A S P i (A S ), i = 1, 2, such tht A S = P1 (A S )P 2 (A S ) = (Z, z 0, E, δ ) if oly if it stisfis th followig composbility coitios (DC): E 1 \E 2, E 2 \E 1, q Q, S E, DC1: [δ(q, )! δ(q, )!] [δ(q, )! δ(q, )!]; DC2: δ(q, S)! δ(q, S)!, DC3: S, S E, shrig th sm first pprig commo vt E 1 E 2, S S, q Q: δ(q, S)! δ(q, S )! δ(q, p 1 (S) p 2 (S ))! δ(q, p 1 (S ) p 2 (S))!; DC4: i {1, 2}, x, x 1, x 2 Q i, x 1 x 2, E i, t E i, δ i(x, ) = x 1, δ i (x, ) = x 2 : δ i (x 1, t)! δ i (x 2, t)!. Rmrk 1: Ituitivly, th composbility coitio DC1 ms tht for y jct pir of privt vts (, ) {(E 1 \E 2, E 2 \E 1 ), (E 2 \E 1, E 1 \E 2 )} (from iffrt privt vt sts), both orrs shoul b lgl from th sm stt, ulss thy r mit by commo strig. Whil DC1 rsss th cisio o jct vts, DC2 ls with th cisio o th orr of succssiv vts stts tht if y orr of pir of jct privt vts (, ) {(E 1 \E 2, E 2 \E 1 ), (E 2 \E 1, E 1 \E 2 )} (from iffrt privt vt sts) is cssry coitio of occurrc of y strig S E, th th othr orr lso shoul b lgl for such occurrc (s Exmpl 3). Not tht, s spcil cs, S coul b ε. Th coitio DC3 ms tht if two strigs S S shr th sm first pprig commo vt, th y itrlvig of ths two strigs shoul b lgl i A S. This rquirmt is u to sychroiztio of projctios of ths strigs i P 1 (A S ) P 2 (A S ). Th lst coitio DC4, surs th symmtry of mutul simultio rltios btw A S P 1 (A S )P 2 (A S ). Giv th trmiism of A S, this symmtry is gurt wh ch locl tsk utomto bisimults trmiistic 4194

4 utomto, lig to th xistc trmiistic utomto tht is bisimilr to P 1 (A S )P 2 (A S ). If th simultio rltios r ot symmtric, th som of th squcs tht r llow i A S will b isbl i P 1 (A S )P 2 (A S ). Th followig svrl xmpls r rviw from [1] to illustrt th composbl ucomposbl utomt bs o th composbility coitios i Lmm 1. Exmpl: This xmpl shows utomto tht stisfis DC2, DC3 DC4, but ot DC1, lig to ucomposbility. Lt utomto A S to b with locl vt sts E 1 = { } E 2 = { }. Th prlll compositio of P 1 (A S ) : 1 P 2 (A S ) : 2 is P 1 (A S ) P 2 (A S ) : 1. Thrfor, A S is ot composbl, sic two jct vts E 1 \E 2 E 2 \E 1 o ot rspct DC1. O c obsrv tht, if i this xmpl E 1 \E 2 E 2 \E 1 wr sprt by commo vt E 1 E 2, th th utomto with locl vt sts E 1 = {, } E 2 = {, }, ws composbl. Exmpl 3: This xmpl shows utomto which rspcts DC1, DC3 DC4, but is ucomposbl u to violtio of DC2. Cosir utomt A S : 1 2 with E 1 = {, }, E 2 = {, }, lig to P 1 (A S )P 2 (A s ):. Th trsitio δ (z 0, )! i P 1 (A S )P 2 (A S ), but δ(q 0, )! i A S. If E 1 \E 2 E 2 \E 1 wr sprt by commo vt E 1 E 2, th th utomto 1 2 with locl vt sts E 1 = {, } E 2 = {, }, ws composbl. Also if both orrs of E 1 \E 2 E 2 \E 1 wr lgl for occurrc of, i.., if A S ws, th gi it ws composbl. Exmpl 4: This xmpl illustrts utomto tht stisfis DC1, DC2 DC4, but it is ucomposbl s it os ot fulfil DC3, sic w strigs ppr i P 1 (A S )P 2 (A S ) from th itrlvig of two strigs i P 1 (A S ) P 2 (A S ), but thy r ot lgl i A S. Cosir th tsk utomto A S : with E 1 = {, }, E 2 = {, }, lig to P 1 (A S ) =, P 2 (A S ) = P 1 (A S )P 2 (A S ) = 2 tht is ot bisimilr to A S sic its two lft brchs r wly grt, whil thy o ot ppr i A S, lthough both P 1 (A S ) P 2 (A S ) r trmiistic. Exmpl 5: This xmpl illustrts utomto tht stisfis DC1 DC2, DC3, but is ucomposbl s it os ot fulfil DC4. Cosir th tsk utomto A S : q 1 b q 2 q 3 q 0 q 4 with E 1 = {, b, }, E 2 = {, b}, lig to P 1 (A S ): x 1 b x 2 x 3, x 0 P 2 (A S ): x 4 b y 1 y 2, y 0 y 3 P 1 (A S )P 2 (A S ): z 1 b z 2 z 3 z 0 z 5 z 4 which is ot bisimilr to A S. This tsk utomto A S stisfis DC1 DC2 s cotis o succssiv/jct trsitios fi o iffrt locl vt sts. It os stisfis DC3 sic y strig i T = {p 1 (S) p 2 (S ), p 1 (S ) p 2 (S)} (S S r th top bottom strigs i A S shr th first pprig commo vt E 1 E 2 ), pprs i A S. But A S os ot fulfil DC4 sic thr os ot xist trmiistic utomto tht bisimults P 2 (A S ). This rsults i th xistc of trsitio o strig from z 0 to z 5 tht stops i P 1 (A S )P 2 (A S ), whrs, lthough trsits from q 0 i A S, it os ot stop ftrwrs. This illustrt issymmtry i simultio rltios btw A S P 1 (A S )P 2 (A S ). Not tht A S P 1 (A S )P 2 (A S ) with th simultio rltio R 1 ovr ll vts i E, from ll stts i Q ito som stts i Z, s R 1 = {(q 0, z 0 ), (q 1, z 1 ), (q 2, z 2 ), (q 3, z 3 ), (q 4, z 4 )}. Morovr, P 1 (A S )P 2 (A S ) A S with th simultio rltio R 2 ovr ll vts i E, from ll stts i Z ito som stts i Q, s R 2 = {(z 0, q 0 ), (z 1, q 1 ), (z 2, q 2 ), (z 3, q 3 ), (z 4, q 4 ), (z 5, q 2 )}. Thrfor, lthough A S P 1 (A S )P 2 (A S ) P 1 (A S )P 2 (A S ) A S, P 1 (A S )P 2 (A S ) A S, sic (z 5, q 2 ) R 2, whrs (q 2, z 5 ) / R 1. If similr to P 1 (A S ), 4195

5 P 2 (A S ) lso h trmiistic bisimilr utomto, th for stoppig of strig i P 1 (A S )P 2 (A S ), thr ws stt i Q rchbl from q 0 by stoppig thr, th w h q Q, z Z : (q, z) R 1 (z, q) R 2 P 1 (A S )P 2 (A S ) = A S. B. Dcomposbility Coitio for gts I prctic, multi-gt systms r typiclly compris of my iiviul gts tht work s tm. Th propos procur of compositio c b grliz for mor th two gts, cosirig y -tupl of succssiv/jct privt vts from iffrt privt vt sts. Howvr, this pproch bcoms rpily complx s th umbr of gts icrss, s it shoul chck! possibilitis for th orr of ths vts. To tckl this problm o wy is to us hirrchicl compositio mtho to hv oly two iiviul vt sts t tim for prtitioig: locl vt st; th st of th rst of locl vt sts. I ch stp, th lgorithm sks ths two sts such tht thy stisfy th composbility coitios. This lgorithm, howvr, ps strogly o th orr of th vt sts tht w choos for compositio, s it is lbort i th followig xmpl. Exmpl 6: Th utomt A S : with E = E 1 E 2 E 3, E 1 = {, }, E 2 = {, b, }, E 3 = {b, 3 }, P 1 (A S ): 1, P 2 (A S ) = 2 b P 3 (A S ) = b 3, is composbl s A s = P1 (A S )P 2 (A S )P 3 (A S ) = P 1 (A S )(P 2 (A S )P 3 (A S )) = P 3 (A S )(P 1 (A S )P 2 (A S )) = P 2 (A S )(P 1 (A S )P 3 (A S )). I this xmpl, P E2 E 3 (A S ) = P2 (A S )P 3 (A S ) P E1 E 2 (A S ) = P1 (A S )P 2 (A S ), but P E1 E 3 (A S ) P 1 (A S )P 3 (A S ). This ms tht whil choosig P 1 (A S ) or P 3 (A S ) s th first st llows th hirrchicl lgorithm to procs up to A s = P1 (A S )P 2 (A S )P 3 (A S ), choosig P 2 (A S ) will stuck th lgorithm i th sco stp s A s = P2 (A S )P E1 E 3 (A S ), but P E1 E 3 (A S ) P 1 (A S )P 3 (A S ). Thrfor, th lgorithm ps o th orr of slctios. Nxt, s illustrt i th followig xmpl, th bov hirrchicl compositio mtho is oly sufficit. Exmpl 7: Th utomt A S : b 3 c 5 2 b 3 c 5 with E = E 1 E 2 E 3, E 1 = {, c,,, 5 }, E 2 = {, b,, }, E 3 = {b, c, 3 }, P 1 (A S ): b 3 b 3 c 5 c P 2 (A S ) = 2 b P 3 (A S ) = b 3 c, is composbl s A s = P1 (A S )P 2 (A S )P 3 (A S ). Howvr, P E2 E 3 (A S ) P 2 (A S )P 3 (A S ), P E1 E 2 (A S ) P 1 (A S )P 2 (A S ) P E1 E 3 (A S ) P 1 (A S )P 3 (A S ). This ms tht lthough A S is composbl with rspct to P 1 (A S ), P 2 (A S ) P 3 (A S ), choosig y of locl vt sts E 1, E 2 E 3 psss th first stg of hirrchicl compositio, but th lgorithm will stuck t th sco stp. Thrfor, it woul b vry vtgous if w c fi cssry sufficit coitio for composbility of trmiistic utomto with rspct to rbitrry fiit umbr of locl vt sts. Th mtho woul b ipt of th orr of th locl vt sts shoul b bl to chck th composbility coitio by irct ivstigtio. I th followig, s th mi rsult, w propos such cssry sufficit coitio for tsk utomto compositio for rbitrry fiit umbr of gts. ( Thorm 1: A trmiistic ) utomto A S = Q, q 0, E = E i, δ is composbl with rspct to prlll compositio turl projctios P i : A S P i (A S ), i = 1,..., such tht A S = 5, P i (A S ) if oly if A S stisfis th followig composbility coitios (DC):, E, q Q, S E, E i {E 1,..., E },{, } E i : DC1: [δ(q, )! δ(q, )!] [δ(q, )! δ(q, )!]; DC2: δ (q, S)! δ (q, S)!; DC3: δ(q 0, p i (S i ))! for S i L(A S ), whr, L(A S ) L (A S ) is th lrgst subst of L (A S ) such tht S L(A S ) S L (A S ), E i, E j {E 1,..., E }, i j, p Ei E j (S) p Ei E j (S ) strt with th sm vt, DC4: i {1, 2}, x, x 1, x 2 Q i, x 1 x 2, E i, t Ei, δ i(x, ) = x 1, δ i (x, ) = x 2 : δ i (x 1, t)! δ i (x 2, t)!. Proof: I orr for A S = P i (A S ), from th fiitio of bisimultio, it is rquir to hv A S P i (A S ); P i (A S ) A S, th simultio rltios r symmtric. Ths rquirmts r provi by th followig thr lmms. Du to limittio i spc, th proofs for ths lmms r omitt from hr, will b provi i th xt vrsio of this rsult. Firstly, i grl, P i (A S ) lwys simults A S. ( Lmm 2: For ) trmiistic utomto A S = Q, q 0, E = E i, δ turl projctios 4196 P i : A S A S P i (A S ). P i (A S ), i = 1,...,, it lwys hols tht

6 This lmm shows tht, i grl, A S. Th similrity of P i (A S ) simults P i (A S ) to A S, howvr, is ot lwys tru (s Exmpls 2 3), s som coitios s stt i th followig lmm. ( Lmm 3: Cosir ) trmiistic utomto A S = Q, q 0, E = E i, δ turl projctios P i : A S P i (A S ), i = 1,...,. Th, DC1, DC2 DC3 hol tru for A S. P i (A S ) A S if oly if Nxt, w to show tht th two simultio rltios R 1 (for A S P i (A S )) R 2 (for P i (A S ) A S ) fi by th bov two lmms r symmtric. Lmm 4: Cosir utomto A S = (Q, q 0, E = E i, δ) with turl projctios P i : A S P i (A S ), i = 1,...,. If A S is trmiistic, A S P i (A S ) with th simultio rltio R 1 P i (A S ) A S with th simultio rltio R 2, th R1 1 = R 2 (i.., q Q, z Z: (z, q) R 2 (q, z) R 1 ) if oly if DC4: i {1,..., }, x, x 1, x 2 Q i, x 1 x 2, E i, t Ei, δ i (x, ) = x 1, δ i (x, ) = x 2 : δ i (x 1, t)! δ i (x 2, t)!. Now, ccorig to Dfiitio 4, A S = P i (A S ) if oly if A S 2), P i (A S ) (tht is lwys tru u to Lmm P i (A S ) A S (tht it is tru if oly if DC1, DC2 DC3 r tru, ccorig to Lmm 3) th simultio rltios r symmtric, i.., R1 1 = R 2 (tht for trmiistic utomto A S, wh A S P i (A S ) with simultio rltio R 1 P i (A S ) A S with simultio rltio R 2, u to Lmm 4, R1 1 = R 2 hols tru if oly if DC4 is stisfi). Thrfor, A S = P i (A S ) if oly if DC1, DC2, DC3 DC4 r stisfi. Th composbility coitios i Thorm 1 r illustrt i th followig xmpl. Exmpl 8: Cosir th utomto i Exmpl 6. W ot th strigs o th bottom top brchs of A S s S 1 S 2, rspctivly. Sic {, } E 1, {, } E 2, {, b} E 2, {b, 3 } E 3, th DC1 DC2 r stisfi. Morovr, S 1, S 2 coti E 1 E 2, b E 2 E 3, whr pprs s th first vt i both p E1 E 2 (S 1 ) p E1 E 2 (S 2 ). Th, ccorig to DC3, followig ight itrlvig trsitios r chck: δ (q 0, p 1 (S i ) p 2 (S j ) p 3 (S k ))!, i {1, 2}, j {1, 2}, k {1, 2}, i.., DC3 is stisfi. Morovr, DC4 is lso stisfi s P 1 (A S ), P 2 (A S ) P 3 (A S ) r trmiistic, hc, A S is composbl. IV. CONCLUSION Th ppr propos forml mtho for utomto compositio, pplicbl i top-ow cooprtiv cotrol sig for multi-gt systms. For th clss of prlll istribut systms whos globl spcifictio is rprst s trmiistic utomto with giv vt istributio, w provi cssry sufficit coitio for composbility of utomto with rspct to prlll compositio turl projctios ito rbitrry fiit umbr of locl vt sts. Aothr rsrch qustio is tht wh tsk utomto is ot composbl, th how o c moify th vt istributio to mk th globl tsk utomto composbl. Aothr futur work c b ivstigtio of composbility ur vt filur, mly, ur wht coitios composbl tsk utomto prsrvs its composbility i spit of filur of som vts. REFERENCES [1] M. Krimii H. Li, Gurt Globl Prformc Through Locl Cooritios, Submitt to Automtic, [2] P. U. Lim L. M. Custio, Multi-Robot Systms, Book Sris Stuis i Computtiol Itlligc, Book Iovtios i Robot, Mobility Cotrol, Chptr 1, vol. 8, Sprigr Brli / Hilbrg, [3] V. Crspi, A. Glsty K. Lrm, Top-ow vs bottom-up mthoologis i multi-gt systm sig, Jourl: Autoomous Robots, Publishr: Sprigr Nthrls, vol. 24, o. 3, pp , April [4] C. Ryols, Flocks, hrs schools: A istribut bhviorl mol, Proc. of th4th ul cofrc o Computr Grphics, vol. 21, o. 4, July [5] M. G. Hichy, J. L. Rsh, W. F. Truszkowski, C. A. Rouff R. Strrit, Autoomous utoomic swrms, I Proc. Th005 Itrtiol Cof. o Softwr Egirig Rsrch Prctic (SERP 05), CSREA Prss, Ls Vgs, Nv, USA, pp , 27 Ju005. [6] W. F. Truszkowski, M. G. Hichy, J. L. Rsh C. A. Rouff, Autoomous utoomic systms: A prigm for futur spc xplortio missios, IEEE Trs. o Systms, M Cybrtics, Prt C, [7] C. A. Rouff, W. F. Truszkowski, J. L. Rsh M. G. Hichy, A survy of forml mthos for itlligt swrms, Tchicl Rport TM , NASA Gor Spc Flight Ctr, Grblt, Mryl, [8] A. Jbbi, J. Li A. S. Mors, Cooritio of groups of mobil utoomous gts usig rst ighbor ruls, IEEE Trsctio o Automttic Cotrol, vol. 48, o. 6, pp. 988C-1001, [9] C. Blt, A. Bicchi, M. Egrstt, E. Frzzoli, E. Klvis G. J. Ppps, Symbolic plig cotrol of robot motio, IEEE Robotics Automtio Mg., spcil issu o Gr Chllgs of Robotics, vol. 14, o. 1, pp , [10] M. Klotzr C. Blt, Tmporl logic plig cotrol of robotic swrms by hirrchicl bstrctios, IEEE Trs. o Robotics, vol. 23, o. 2, pp , [11] P. Tbu G. J. Ppps, Lir tim logic cotrol of iscrttim lir systms, IEEE Trs. Automt. Cotr., vol. 51, o. 12, pp , [12] C. G. Cssrs S. Lfortu, Itrouctio to Discrt Evt Systms, Sprigr, U.S.A, [13] M. Muku, From globl spcifictios to istribut implmttios, i B. Cillu, P. Drou, L. Lvgo (Es.), Sythsis Cotrol of Discrt Evt Systms, Kluwr, pp , [14] R. Mori, Dcompositios of sychroous systms, I CONCUR 98, LNCS 1466, pp , Sprigr, [15] I. Cstlli, M. Muku, P.S. Thigrj, Sythsizig istribut trsitio systms from globl spcifictio, I: Pu Rg, C., Rm, V., Rmujm, R. (s.) FSTTCS LNCS, vol. 1738, pp , Sprigr, Hilbrg, [16] R. Kumr V. K. Grg, Molig Cotrol of Logicl Discrt Evt Systms. Kluwr Acmic Publishrs, Bosto,