Cooperative Triangulation in MSBNs. without Revealing Subnet Structures. Y. Xiang. Department of Computer Science, University of Regina

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1 Cooprtiv Tringultion in MSBNs without Rvling Sunt Struturs Y. Xing Dprtmnt o Computr Sin, Univrsity o Rgin Rgin, Ssthwn, Cn S4S 0A, yxing@s.urgin. To ppr in Ntwors Astrt Multiply stion Bysin ntwors (MSBNs) provis ohrnt rmwor or proilisti inrn in ooprtiv multignt istriut intrprttion systm. Inrn in MSBNs n prorm tivly using ompil rprsnttion. Th ompiltion involvs th tringultion o th olltiv pnny strutur ( grph) n in trms o th union o st o lol pnny struturs ( st o grphs). Privy o gnts limints th option to ssml ths grphs t ntrl lotion n to tringult thir union. Erlir wor solv istriut tringultion in rstrit s. Th mtho is onptully omplx n th orrtnss o its xtnsion to th gnrl s is iult to justiy. In this ppr, w prsnt nw mtho tht is onptully simplr n is int. W prov its orrtnss in th gnrl s n monstrt its prormn xprimntlly. 1 Introution Multiply stion Bysin ntwors (MSBNs) provis ohrnt rmwor or proilisti inrn in lrg omin [9]. It n ppli unr singl gnt prigm [8] or ooprtiv multi-gnt prigm [6]. It supports ojt-orint inrn [3]. Inrn in MSBN n prorm tivly using ompil rprsnttion ll lin juntion orst (LJF) [9]. Th ompiltion o MSBN into LJF ts svrl stps tht r similr in prinipl to th ompiltion o Bysin ntwor (BN) into its juntion tr o li univrss []. This wor ouss on th tringultion stp. In prtiulr, th input o th pross is st o unirt grphs with ovrlpping nos whih olltivly ns grph union (to n ormlly), n th output is orrsponing st o horl suprgrphs whih olltivly ns horl suprgrph o. Unr th multignt prigm, h gnt my onstrut y n inpnnt vnor who ms th now-hows out suomin into th gnt. For xmpl, th vnor o omponnt in omplx systm n uil n gnt with th now-hows o th omponnt m. Th vnor my not willing to rvl th now-hows whn th gnt is 1

2 intgrt into multignt MSBN. To prott th now-hows o suh vnors, it is sirl not to or h gnt to rvl its intrnl strutur (th grph lol to th gnt). This privy rquirmnt limints th option to ssml ths grphs t ntrl lotion n to tringult thir union s in th singl-gnt prigm. An lgorithm to solv th prolm unr rstritiv onition ws prsnt in [9]. Th mtho is onptully omplx n th orrtnss o its xtnsion to th gnrl s is iult to justiy ormlly. In this wor, w prsnt onptully simpl mtho to solv th prolm n w prov its orrtnss in th gnrl s. In Stion, w motivt this rsrh with n ovrviw o MSBNs n its pplitions. W riy introu th grph-thortil trminologis to us in th rst o th ppr in Stion 3. W n th prolm o multignt tringultion in Stion 4. In Stion 5 w prsnt our mtho in th simplst s n prov its orrtnss. Th mtho is xtn to mor gnrl s in Stion 6. Builing on th rsults o Stions 5 n 6, w prsnt th most gnrl s in Stions 7 n 8. Th xprimntl stuy is prsnt in Stion 9. W nlyz th omplxity o th propos lgorithms in Stion 10. Ovrviw o MSBNs In this stion, w riy introu th rmwor o MSBNs. A BN [4] S is triplt (N;D;P) whr N is st o omin vrils, D is DA whos nos r ll y lmnts o N, n P is joint proility istriution (jp) ovr N spi in trms o proility istriutions o h no in D onition on its prnts. 1 g g () 4 3 () Figur 1: () A igitl iruit. () Th strutur o BN rprsnting th iruit. Figur 1 () shows simpl igitl iruit. A BN () n us to mol th iruit or troul-shooting purpos. Th BN rprsnts h vi n h input/output s no/vril, n rprsnts irt pnn twn h pir o vrils y n r. Th pnn is proilisti in gnrl, spilly or th ulty hvior o vis. Eh no is thus ssoit with proility istriution onition on its prnts. For xmpl, th no g (output o n OR gt) is ssoit with th istriution p(gj; ; ). Th istriution ns how th vlu o g pns on th vlus o its prnt vrils. For instn, p(g =0j =0; =0; = norml) =1:0 rprsnts how input = 0 n =0 to trmin g whn th gt is norml. On th othr hn, p(g =0j =0;=0; = norml) =0:3 rprsnts tht g ts th orrt vlu or th sm input 30% o tim

3 whn th gt is norml (n ts th wrong vlu 70% o th tim). Th BN n thn us to nswr quris suh s \p( = normljg =1;=0;= 1) =?". A MSBN [9] M is olltion o Bysin sunts tht togthr ns BN. Ths sunts shoul stisy rtin onitions to prmit ohrnt istriut inrn. On onition rquirs tht nos shr y two sunts orm -spst, s n low. Lt i =(N i ;E i )(i =0; 1) two grphs. Th grph =(N0 [ N1;E0 [ E1) is rrr to s th union o 0 n 1, not y = 0 t 1. Dnition 1 Lt D i =(N i ;E i )(i =0; 1) two DAs suh tht D = D0 t D1 is DA. Th intrstion I = N0 \ N1 is -spst twn D0 n D1 i or vry x I with its prnts in D, ithr N0 or N1. Eh x I is ll -spno. Just s th strutur o BN is DA, th strutur o MSBN is multiply stion DA (MSDA) with hyprtr orgniztion: Dnition A hyprtr MSDA D = F i D i, whr h D i is onnt DA, is onnt DA onstrutil y th ollowing prour: Strt with n mpty grph (no no). Rursivly DA D, ll hyprno, to th xisting MSDA F 1 i=0 D i sujt to th onstrints: [-spst] For h D j (j<), I j = N j \ N is -spst whn th two DAs r isolt. [lol ovring] Thr xists D i (i<) suh tht, or h D j (j < ; j 6= i), w hv I j N i : For n ritrrily hosn suh D i, I i is th hyprlin twn D i n D whih r si to jnt. W us P T (N) to not proility istriution ovr st N o vrils tht is ssoit with n ojt T. A MSBN is thn n s ollows: Dnition 3 A MSBN M is triplt M =(N ; D; P). N = S i N i is th totl univrs whr h N i is st o vrils. D = F i D i ( hyprtr MSDA) is th strutur whr nos o h DA D i r ll y lmnts o N i. P = Q i P Di (N i )= Q P I (I ) is th jp. Eh P Di (N i ) is istriution ovr N i suh tht whnvr D i n D j r jnt in D, th mrginliztions o P Di (N i ) n P Dj (N j ) onto thir -spst r intil. Eh P I (I ) is suh mrginl istriution ovr hyprlin I o D. Eh triplt S i =(N i ;D i ;P i ) is ll sunt o M. S i n S j r jnt i D i n D j r jnt l q m n o j h i g Figur : An xmpl MSDA o thr DAs. Figur illustrts MSBN with trivil MSDA. Th hyprtr is pit in th lt n th thr DAs r shown in th right. Th -spnos r highlight with oul ovls. 3

4 Snsors Bysin Sunt Dision Mr Atutor Snsitivity Anlyzr Rsonr Communitor Distriut lol Strutur Vriir Figur 3: Min omponnts o n gnt in MSBN-s multi-gnt systm. MSBNs provi rmwor or proilisti rsoning in ooprtiv multi-gnt istriut intrprttion systms. Eh gnt hols its prtil prsptiv o lrg prolm omin (Sunt in Figur 3), sss lol vin sour (Snsors in Figur 3), ommunits with othr gnts inrquntly (Communitor), rsons with th lol vin n limit glol vin (Rsonr), n nswrs quris (Rsonr) or ts tions (Dision Mr/Atutor). I gnts r ooprtiv, n h pir o jnt gnts r onitionlly inpnnt givn thir shr vrils n hv ommon initil li on th shr vrils, thn joint systm li is wll n whih is intil to h gnt's li within its suomin n supplmntl to th gnt's li outsi th suomin [6]. Evn though multipl gnts my quir vin synhronously in prlll, th ommunition oprtions o MSBNs nsur tht th nswrs to quris rom h gnt r onsistnt with vin quir in th ntir systm tr h ommunition. Sin ommunition is inrqunt, th oprtions lso nsur tht twn two sussiv ommunitions, th nswrs to quris or h gnt r onsistnt with ll lol vin gthr so r n r onsistnt with ll vin gthr in th ntir systm up to th lst ommunition. Potntil pplitions o th rmwor inlu ision support to ooprtiv humn usrs in unrtin omins n troulshooting omplx systm y multipl nowlg s susystms [6]. 3 rph-thortil trminologis In this stion, w riy introu th grph-thorti trminologis tht th rmining o th ppr pns on. Aitionl trminologis tht r us ut r not ssntil to unrstning r inlu in th Appnix. W shll ll grph =(N;E) sovr N. Th jny o no x is th st o nos jnt to x, n is not y j(x). A st X o nos in is omplt i h pir o nos in X is jnt. A hor is lin onnting two nonjnt nos. is tringult or horl i vry yl o lngth > 3 hs hor. A no x in grph =(N;E) islimint i j(x) is m omplt y ing lins (i nssry) or x n lins inint to x r rmov. Eh lin thus is ll ll-in. Lt F th st o ll-ins in liminting ll nos in som orr. Thn th grph 0 =(N;E [ F ) is horl. A grph is horl i ll nos n limint on y on in som orr without ll-ins []. Our mtho is s on no limintion in rtin orr. W stt this onpt 4

5 prisly: Dnition 4 Lt grph ovr N1 [ N suh tht N1 \ N =. is liminl in th orr (N1;N) i it is possil to limint ll nos in N1 on y on rst n thn limint ll nos in N on y on without ny ll-ins. Th onpt n xtn suh tht th orr is spi s n-tupl. For simpliity, w shll writ (g; g; g) s(; ; ). W n th onpt o grph-onsistny to rr to pir o grphs whos sugrphs ovr shr nos r intil. Dnition 5 Lt 1 ovr N1 n ovr N two grphs suh tht N1\N 6=. Thn 1 n r si to grph-onsistnt i th sugrphs o 1 n spnn y N1 \ N r intil. 4 Th prolm o ooprtiv tringultion Prorming inrn irtly in MSBN is iult sin its MSDA is otn multiply onnt. As on o th most int wys o inrn in multiply onnt BN uss juntion tr (JT) (Appnix) rprsnttion, w ompil MSBN into st o intr-rlt JTs ll lin juntion orst (LJF) [9] or inrn. In th LJF, th JTs r orgniz into th intil hyprtr strutur s th MSDA. Th ompiltion o MSBN into LJF ts svrl stps, som o whih prlll th ompiltion o BN. W isuss only th rst our stps tht r rlvnt to th purpos o this ppr: Th rst stp, ll morliztion, is to onvrt th MSDA into its morl grph y omplting prnts or h no n ropping th irtions o lins. Sin th MSDA is th union o st o DAs, its morl grph is n in trms o th union o morl grphs o ths DAs. Th son stp is to tringult th morl grph o th MSDA into horl grph. This is n or th sm rson in ompiling BN: JT o grph xists i th grph is horl. Agin, this horl grph is n s th union o st o horl grphs h o whih is otin rom th morl grph o DA. Eh omponnt grph shoul horl sin w wnt to orgniz h sunt into JT or lol inrn. Th thir stp is to onvrt h horl grph in th union into juntion tr (JT) o li univrss [] to us in th inrn omputtion. Th ourth stp is to ompil h hyprlin (-spst) o MSDA into ling tr [6]. This stp llows n gnt's li on -spst ( proility istriution) to toriz omptly so tht ommunition twn gnts n prorm mor intly whn th -spst is lrg. Mor tils on this n oun in [5]. W giv hr nition o ling trs quivlnt to tht in [6] (ut omputtionlly lss int) to ilitt th isussion. Th proo o quivln is trivil. Dnition 6 Lt I th -spst twn JTs T n T in LJF. Rpt th ollowing until no vril n rmov: (1) Rmov vril x 6 I i x is ontin in singl liqu C. () I C oms sust o n jnt liqu D tr (1), union C into D. 5

6 Lt L th grph rsultnt rom th prour. Thn L is ling tr o T with rspt to I i [ ll l = I, whr h liqu l in L is ling. Dn liqu in T tht ontins l s its ling host n r tis ritrrily. It n shown [7] tht th ling tr L is JT, n tht i T is n I-mp (Appnix), thn L is n I-mp ovr I. Thror, to propgt li on I rom T to T uring inrn, it is suint to propgt li on h ling [5] sin B(I) =[ Y l B(l)]=[ Y q B(q)]; whr h l is ling in L with its li tl B(l) n h q is sprtor in L with its li tl B(q). On th othr hn, i [ ll l I whn th prour in Dnition 6 hlts, in whih s L is not ling tr, thn B(I) nnot omptly rprsnt y th ov qution n n only otin y mor xpnsiv omputtion [9]. Th ollowing thorm intis th onition unr whih th ling tr xists 1. Th onition is tht th grph rom whih T is onstrut must tringult in rtin wy. Thorm 7 Lt grph ovr N rom whih JT T is onstrut. Lt I sust o nos in. Thn ling tr o T xists with rspt to I i is liminl in th orr (N ni;i). Proo: [Suiny] Suppos is liminl in (N n I;I). Consir no x N n I tht n limint rst without ll-ins. Thn x must ppr in singl liqu C in T sin othrwis j(x) is inomplt. Hn x n rmov rom C. Rpting this rgumnt or h no in N n I, w will l to limint N n I rom T n th rsultnt grph is th ling tr. [Nssity] Suppos is not liminl in (N n I;I). Tht is, N n I nnot limint without ll-ins in ny prtiulr orr tht is onsistnt with (N n I;I). This mns tht no mttr wht orr w us, thr xists non-mpty sust o N n I (th sust my ir or irnt orrs) suh tht h no in th sust pprs in t lst two liqus o T. Hn th ling tr os not xist. For h JT T in th hyprtr, li propgtion ns to prorm rltiv to h jnt JT. To nsur th xistn o ling tr or h jnt JT, w hv th ollowing rquirmnt or tringultion: Rquirmnt 1 For h hyprno in th hyprtr o MSBN, th tringult morl grph ovr N or th hyprno must suh tht or vry hyprlin I inint to th hyprno, is liminl in th orr (N n I;I). For h hyprlin in th hyprtr, th two npoints (hyprnos) my potntilly onnt th -spst irntly uring tringultion. Th ollowing xmpl shows tht suh tringultion shoul prvnt. 1 This onition n shown to quivlnt to th host omposition onition in [9]. Howvr, th host omposition onition is sriptiv whil th onition prsnt hr is prourl n hn provis irt guilin to tringultion. 6

7 Q,,,,,,, Figur 4: Illustrtion o Rquirmnt. Th -spst is shown y oul ovls. Consir th two morl grphs n Q (ignor th sh lin) in Figur 4 (lt). Using th orr in Thorm 7, is limint without ny ll-ins n Q is limint with th ll-in ; g (sh). Th two orrsponing JTs r shown in th right. Not howvr, tht th JT rom xprsss ls onitionl inpnn rltion, i.., tht ; g is inpnnt o ; g givn. To voi this prolm, th -spst I = ; ; g shoul intilly onnt in n Q (y ing th ll-in ; g to ). Hn, w hv th ollowing rquirmnt or tringultion: Rquirmnt For h pir o jnt hyprnos in th hyprtr o MSBN, th orrsponing -spst must onnt intilly in th two tringult morl grphs. Tht is, th two tringult grphs must grph-onsistnt. Th smntis o Rquirmnt is tht ssumptions on th onitionl inpnn within -spst xprss in irnt JTs must onsistnt. Unr th multignt prigm, MSBN my intgrt rom st o gnts, sy, or troul-shooting omplx systm. Th MSBN signr my now only th intr (spst) twn suomins (sunts intrnl to gnts) ut not th tils o h sunt. Eh gnt my onstrut y n inpnnt vnor who ms th now-hows out suomin into th gnt. For xmpl, th vnor o omponnt in omplx systm n uil n gnt with th now-hows o th omponnt m. Th vnor my not willing to rvl th now-hows whn th gnt is intgrt into multignt MSBN. To prott th now-hows o suh vnors, it is sirl not to or h gnt to rvl its intrnl strutur uring ompiltion. An gnt will givn, y th MSBN signr, th inormtion whih othr gnts it intrs to n wht r th -spsts, ut not muh mor thn tht rgring th intrnl o othr gnts. This privy rquirmnt limints th option to ssml DAs in th MSDA t ntrl lotion or ompiltion. In othr wors, it is sirl to istriut ll stps o ompiltion. Th rst stp (morliztion) n istriut sily. First, h gnt prorms morliztion lolly. Thn jnt gnts in th hyprtr xhng lins mong -spnos so tht h pir o -spnos r onnt intilly t irnt lol grphs. Hn privy o gnts is not ompromis in this stp. On th tringultion (son stp) is omplt, th thir n ourth stps (onstrution o JT rom th horl grph n onstrution o th ling tr rom JT) r lol n thy o not thrtn privy o gnts. W stt th privy rquirmnt or th son stp s ollows: Rquirmnt 3 For h hyprno H in th hyprtr o MSBN n h o its jnt hyprno H 0, th intrnl pnny strutur o H yon th sugrph spnn y thir -spst shoul not rvl to H 0. 7

8 Th ous o this wor is th istriution o th son stp: tringultion. Th prolm n strt s ollows: A st o gnts is orgniz into hyprtr whr h hyprno is ll y n gnt. Eh gnt hs n unirt grph m in it. Eh pir o grphs m in jnt gnts on th hyprtr shr st o ommon nos n r grph-onsistnt. W shll rr to gnts n grphs intrhngly. Th grphs r so lot on th hyprtr tht whnvr two grphs hv st S o shr nos with thir grph on th hyprpth twn thm, S is lso shr y th thir grph 3. Th prolm is to n n tiv wy in whih h gnt tringults its lol grph sujt to Rquirmnts 1 through 3 suh tht th union o ll lol horl grphs is tringult. W shll rr to this prolm s th prolm o ooprtiv tringultion. An lgorithm to solv th prolm in hyprstr ( gnrt hyprtr) ws prsnt in [9]. Th mtho is onptully omplx (.g., six typs o lins n onpt -pth r n, n th srh o suh -pths is nssry) n its orrtnss in th gnrl hyprtr s is iult to justiy ormlly. In this ppr, w prsnt onptully simpl mtho s on no limintion [1] n w prov its orrtnss in th gnrl s. W prsnt th mtho stpwis n rss only Rquirmnts n 3 in Stions 5 through 7. Rquirmnt 1 is rss in Stion 8. 5 Cooprtiv tringultion o two gnts In this stion, w onsir th simplst s o th prolm o ooprtiv tringultion, whr only two gnts/grphs r involv. Algorithm 1 os th ooprtiv tringultion. ivn st F o lins ovr st N o nos, w shll ll sust E F rstrition o F to S N i E = (x; y)jx S; y S; (x; y) F g: Algorithm 1 Dsription: Lt N, V n S isjoint nonmpty sts o nos. Lt grph ovr N [ S n Q grph ovr V [ S suh tht n Q r grph-onsistnt. Lt A n B two gnts suh tht is m in A n Q is m in B. gin 1 A limints nos in in th orr (N;S); not th ll-ins y F ; A s F to ; not th rsultnt grph y 0 ; 3 A sns B th rstrition o F to S; 4 B s to Q th rstrition o F to S riv; not th rsultnt grph y Q 0 ; 5 B limints nos in Q 0 in th orr (V;S); not th ll-ins y F 0 ; 6 B s F 0 to Q 0 ; not th rsultnt grph y Q 00 ; 7 A sns A th rstrition o F 0 to S; 8 B s to 0 th rstrition o F 0 to S riv; not th rsultnt grph y 00 ; n This onition n nsur y istriut morliztion. 3 This is quivlnt to th lol ovring onition. 8

9 Algorithm 1 lrly stiss th privy rquirmnt sin h xhng twn th two gnts rvls only th onntion ovr S. Figur 5 illustrts th lgorithm, whr grphs n Q r shown in th lt. Th shr nos S = ; g r shown s oul ovls. Elimintion in is prorm in th orr (;;;g; ; g). No n limint without ll-in. No nos in ;;g n now limint without ll-ins. I is limint nxt, thn ll-in ; g is rquir. No n thn limint without ll-in. To limint nxt, nothr ll-in ; g is n. Hn limintion in in th orr (;;;;;) prous 0 in Figur 5 (right), whr th ll-ins r shown s sh lins. Q l q Q" j m n o l n m q o j Figur 5: Illustrtion o Algorithm 1. Th ll-in ; g is thn to Q to otin Q 0 (not shown). Elimintion in Q 0 in th orr (;o;j;l;q;m;n;;) prous ll-in ; qg n th rsultnt grph Q 00 is shown in Figur 5 (right). For this xmpl, no ll-ins rom Q 00 n to to 0, n hn 00 = 0. Howvr, this is not lwys th s. Q Q Figur 6: Illustrtion o lst two stps o Algorithm 1. To s th n o stps 7 n 8 in Algorithm 1, onsir th xmpl in Figur 6. Th grphs n Q r shown in th lt, whr S = ; ; ; g. Atr limintion in in th orr (;;;;;), th rsultnt 0 is shown in Figur 6 (right). Atr th rstrition o ll-ins to S is to Q, w otin Q 0 in Figur 6 (right). Elimintion in Q 0 shoul strt with, whih rquirs ll-in ; g. This is th only ll-in to otin Q 00 (not shown). Sin ; g is twn nos in S, it will to 0 to otin 00 (not shown) in stps 7 n 8 o Algorithm 1. 9

10 Th ollowing proposition shows tht th rsultnt grphs rom Algorithm 1 r grphonsistnt n thir union is horl: Proposition 8 Whn Algorithm 1 hlts, th ollowing hol: (1) 00 n Q 00 r grph-onsistnt. () Th grph union 00 t Q 00 is horl. Proo: (1) It is tru rom stps 4 n 8. () It sus to show tht nos in 00 t Q 00 is liminl in th orr (V;N;S). V n limint rst without ll-ins sin th sugrph o 00 t Q 00 spnn y V [ S is Q 00 y stps 7 n 8, n Q 00 is liminl in th orr (V;S) rom stps 5 n 6. Th rmining grph is 00 y stp 8. N n limint rst rom 00 without ll-ins sin sugrphs o 00 n 0 spnn y N r intil y stp 8, n 0 is liminl in th orr (N;S) y stps 1 n. Th rmining grph is spnn y S. It is liminl sin Q 00 is liminl in th orr (V;S). Proposition 8 not only srvs to illustrt th si i o our mtho, it is lso n in proving th mor gnrl s low. 6 Cooprtion with hyprstr orgniztion Nxt, w onsir ooprtiv tringultion whr n> gnts r orgniz into hyprstr. W not th gnt t th ntr o th hyprstr y A0, n th gnt t h l y A i (i =1; :::; n 1). W not th grph m in gnt A i (i =0; :::; n 1) y i ovr nos N i. W ssum tht ths grphs stisy th lol ovring onition: N i \ N j N0 or vry i n j (i 6= j). W not N0 \ N i y S i. Algorithm st LINK = ; or h l gnt A i,o limint N0 in th orr (N0 n S i ;S i ) n not th rsultnt ll-ins y F ; F to 0 n LINK; sn A i th rstrition o F to S i ; riv st F 0 o ll-ins ovr S i rom A i ; F 0 to 0 n LINK; not th rsultnt grph y 0 0; or h l gnt A i,o sn A i th rstrition o LINK to S i ; Algorithm is xut y A0. It is orgniz into two stgs init y lin lin in-twn. Algorithm 3 is xut y h A i (i =1; :::; n 1). It is lso orgniz into two stgs. W ssum tht i n xisting lin is to to grph, it hs no t. 10

11 Algorithm 3 riv st F o ll-ins ovr S i rom A0; F to i ; limint N i in th orr (N i n S i ;S i ) n not th rsultnt ll-ins y F 0 ; F 0 to i ; sn A0 th rstrition o F 0 to S i ; riv st LINK 0 o ll-ins ovr S i rom A0; LINK 0 to i n not th rsultnt grph y 0 i; Th pross strts y A0. In h itrtion o th rst or loop (Algorithm ), it prorms lol limintion rltiv to S i, s ll-ins lolly n sns thm to A i. Whn A i rivs th ll-ins (Algorithm 3), it upts its grph, prorms lol limintion rltiv to S i, s ll-ins lolly n sns thm to A0. Atr th ov hs n on with h A i, A0 nlizs 0 0 n strts th son or loop (Algorithm ). It sns ll rlvnt ll-ins otin in th rst loop to h A i. In rspons, h A i rivs th ll-ins n nlizs 0 i (Algorithm 3). A A 0 A 1 () 1 0 l q m n o j h i g () l q 1 0 m g n o j h i () 1 0 l n m q o j h g i () Figur 7: Illustrtion o Algorithm n 3. Figur 7 illustrts Algorithms n 3. Th hyprstr with thr gnts is shown in (), n th thr grphs r shown in (). Atr A0 itrt on in th rst or loop (rltiv to A1) n A1 omplt its rst stg, th rsultnt grphs r shown in (). Th orrsponing limintion orrs r (o;;j;l;q;m;n;;) n (;;;;;), rsptivly. Atr A0 itrt th son tim in th rst or loop (rltiv to A) n A omplt its rst stg, th rsultnt grphs r shown in (). Th orrsponing limintion orrs r (o;l;;;q;m;n;j;) n (i;h;g;j;), rsptivly. For this xmpl, no ll-ins r to istriut in th son stg o Algorithms or A0. This is not th s in th nxt xmpl. Suppos th thr gnts on th hyprstr o Figur 7() r ssoit with th grphs in Figur 8(). Atr th rst itrtion o A0 in th rst or loop n th rst stg o A1, th rsultnt grphs r shown in (). Th limintion orrs r (; ; ; ; ) n (i; ; ; ), 11

12 0 0 i 1 0 () i 1 0 () i 1 () i 1 () Figur 8: Illustrtion o Algorithm n 3. rsptivly. Atr th son itrtion o A0 in th rst or loop n th rst stg o A, th rsultnt grphs r shown in (). Th limintion orrs r (;;;;)n(;;;;;), rsptivly. Atr th son stg o A0, th rsultnt grphs r shown in (). Not tht without this stg, th ll-in ; g nnot to 1. Also not tht in th son stg o A0, it is unnssry to itrt through th l gnt involv in th lst itrtion o th rst or loop (i.., A in this xmpl). W i not xlu this itrtion in Algorithm in orr to p it simpl. Clrly, Algorithms n 3 stisy th privy rquirmnt. W show tht th rsultnt grphs is grph-onsistnt n thir union is horl: Proposition 9 Whn Algorithms n 3 hlt t ll orrsponing gnts, th ollowing hol: (1) For i =1; :::; n 1, 0 0 n 0 i r grph-onsistnt. () Th grph union t n 1 i=0 0 i is horl. Proo: (1) For h i, 0 n i r grph-onsistnt y ssumption (Stion 4). All ll-ins ovr S i uring th ntir pross r to 0 0, umult in LINK, n snt to A i y A0. Thy r to 0 i y A i. Hn, 0 0 n 0 i r grph-onsistnt. () W prov th rsult through moi vrsion o Algorithms n 3: Without losing gnrlity, w shll ssum tht th rst or loop in Algorithm pros in th orr i =1; ; :::; n 1. For Algorithm, w to th moi vrsion th ollowing s th lst stp o th rst or loop: sn A j (j =1; :::; i 1) th rstrition o LINK to S j ; For Algorithm 3, w m its son stg (th lst two lins) into loop tht itrts i tims, whr th rst i 1 itrtions orrspons to th itionl sning y A0. 1

13 Th t o th moition is tht s soon s A0 n A i hv h omplt lol limintion rltiv to h othr, th nw ll-ins r immitly ommunit to A1 through A i 1 n r inlu in thir lol grphs. Thror, th moi vrsion will prou xtly th sm n rsult s th originl lgorithms. Lt 1 0 not th moi 0 tr A0 hs omplt th rst itrtion o th rst or loop (i = 1). W shll us th vlu o i s th suprsript to inx th urrnt grphs in irnt gnts. Lt 1 not th moi 1 tr A1 hs omplt its rst stg. Aoring to Proposition 8, th grph union Q 1 = 1 0 t 1 is horl. Atr A0 hs omplt th son itrtion o th rst or loop (i = ), it upts 1 0 into 0. Aoring to th moi lgorithm, A0 will ommunit th nw ll-ins ovr S1 to A1 so tht A1 n upt 1 into 1. This upting y A 0 n A1 n olltivly onsir s quivlnt to upting Q 1 into Q 1 = 0 t 1. Not tht 0 n 1 r grph-onsistnt. Th upting n lso onsir s n limintion y A1 ollow y n limintion y A0 s sri in Algorithm 1. Hn Q 1 is horl oring to Proposition 8. On th A si, tr A hs omplt its rst stg, it upts into. Th upting y A0, A1 n A n olltivly onsir s quivlnt to n limintion on Q 1 ollow y n limintion on s sri in Algorithm 1 us th lol ovring onition hols. Applying Proposition 8, w onlu tht th grph union Q = Q 1t = 0t 1t is horl. Using th ov rgumnt rursivly on i (i =3; :::; n 1), th n rsult Q n 1 n 1 = n 1 0 t ::: t n 1 n 1 = t n 1 i=0 0 i is horl. Not th us o th lol ovring onition in th ov proo. Th horlity rsult woul not hol without th lol ovring onition. Not lso tht just s th itionl ommunition in th moi vrsion o Algorithms n 3 us in th proo os not t th n rsult, th sttmnt \ F 0 to i " in Algorithm 3 os not t th n rsult n n rmov. 7 Cooprtion with hyprtr orgniztion W now onsir th most gnrl s o th ooprtiv tringultion prolm, whr n>3 gnts r orgniz into hyprtr. W prsnt rursiv lgorithms or h gnt. Th xution o h lgorithm y n gnt is tivt y ll rom n ntity nown s llr. W not th gnt ll to xut th lgorithm y A0. Th llr is ithr n jnt gnt o A0 in th hyprtr or th systm. I llr is n gnt, w not it y A (with grph ovr N m). I A0 hs jnt gnts othr thn A, w not thm y A1; :::; A n. W not N \ N0 y S n N0 \ N i y S i (i =1; :::; n). In lgorithm DpthFirstElimint, n gnt A0 prorms limintion n upting with rspt to ll jnt gnts. In lgorithm DistriutDlin, n gnt A0 rivs ll-ins rom llr n thn istriuts ll ll-ins mong -spnos to 0 sin th strt o th ooprtiv tringultion to h othr jnt gnt. 13

14 Algorithm CoTringult is xut y th systm to tivt th ooprtiv tringultion y multipl gnts. Algorithm 4 (DpthFirstElimint) i llr is n gnt A,o riv st F o ll-ins ovr S rom A ; F to 0; st LINK = ; or h gnt A i (i =1; :::; n), o limint N0 in th orr (N0 n S i ;S i ) n not th rsultnt ll-ins y F ; F to 0 n LINK; sn A i th rstrition o F to S i ; ll A i to run DpthFirstElimint n riv ll-ins F 0 ovr S i rom A i whn nish; F 0 to 0 n LINK; i llr is n gnt A,o limint N0 in th orr (N0 n S ;S ) n not th rsultnt ll-ins y F 0 ; F 0 to 0 n LINK; sn A th rstrition o LINK to S ; Algorithm 5 (DistriutDlin) i llr is n gnt A,o riv st F o ll-ins ovr S rom A ; F to 0; st LINK to th st o ll ll-ins to 0 so r; or h gnt A i (i =1; :::; n), o sn A i th rstrition o LINK to S i ; Algorithm 6 (CoTringult) hoos n gnt A ritrrily; ll A to run DpthFirstElimint; tr A hs nish, ll A to run DistriutDlin; Figur 9 illustrts CoTringult with systm o 11 gnts. Th hyprtr is pit in th gur with h no ll y n gnt. Suppos th gnt A hosn in th lgorithm is A5. For h rrow, n limintion (s DpthFirstElimint) y th gnt t th til on its lol grph is prorm n th rlvnt ll-ins gnrt r thn snt to th gnt t th h. For xmpl, th rrow rom A5 to A3 rprsnts th limintion o A5 on 5 in th orr (N5 n N3;N5 \ N3), ollow y sning A5 th rstrition o ll-ins to N5 \ N3. Th ll o h rrow shows th orr o th oprtion. It's sy to s tht th orr is similr to pth-rst trvrsl n hn th nm o th oprtion. 14

15 Atr A5 hs nish DpthFirstElimint, th ow o ll-ins uring xution o DistriutDlin is shown y only thos rrows pointing wy rom A5. A 0 A 1 A A A A 5 A A A A6 A8 Figur 9: Illustrtion o CoTringult. In Thorm 11, w show th proprtis o CoTringult. W n th ltitu o no in hyprtr to us in th proo. ivn no A0 on th hyprtr, n th longst pth rom A through A0 to l, n not th l y A. Th ltitu o A0 is thn th lngth o th pth twn A0 n A. In Figur 9, or xmpl, i A is A5, thn th ltitus o A, A4 n A5 r 0, 1 n, rsptivly. Lmm 10 stlishs th pth-rst proprty o DpthFirstElimint to us in proving Thorm 11. W shll rgr A s th root o th hyprtr. Lmm 10 All limintions on grphs lot t th su(hypr)tr root t A0 r prorm tr A0 is ll to run DpthFirstElimint n or A0 rturns rom th ll. Proo: W prov y inution on th ltitu o A0. Whn =0,A0 is l n xtly on limintion is prorm on 0 (s Figur 9, whr th totl numr o limintions on grph lot t (hypr)no is shown y th numr o outgoing rrows o th no). Only th two i sttmnts in DpthFirstElimint r xut in this s, whr th son on ontins th limintion. Hn th lmm is tru. Assum tht th lmm is tru whn = m 0. Now onsir th s = m + 1. Th limintions prorm on grphs lot t th sutr root t A0 r thos prorm on 0 n thos prorm on grphs lot t th sutr root t h A i (i =1; :::; n). Extly n + 1 limintions r to prorm on 0 (on rltiv to h A i (i =1; :::; n) n on rltiv to A ). Th n limintions rltiv to A i (i =1; :::; n) r ontin in th or loop. All limintions on grphs lot t th sutr root t h A i (i =1; :::; n) r lso prorm in th or loop uring th ll to A i y th inutiv ssumption. Th lst limintion on 0 is prorm in th i sttmnt ollowing th or loop. W now prov th proprtis o CoTringult. Thorm 11 Whn CoTringult hlts, th ollowing hol: (1) Eh pir o jnt grphs on th hyprtr r grph-onsistnt. () Th grph union o ll grphs on th hyprtr is horl. 15

16 Proo: (1) This is tru u to th xution o DistriutDlin y gnt A, whih rursivly propgts ll-ins outwrs rom A to th ntir hyprtr. () W prov using moi DpthFirstElimint whr th ollowing sttmnt is t th n: or h gnt A j (j =1; :::; n), o sn A j th rstrition o LINK to S j n ll A j to run DistriutDlin; Its t is tht s soon s ll limintions on grphs lot in th sutr root t A0 hs omplt (Lmm 10), jnt grphs on th sutr r m grph-onsistnt. Hn, th n rsult o CoTringult using th moi DpthFirstElimint is invrint. Using th moi lgorithm, w prov y inution on th ltitu o gnt A0. Whn =1,A1 through A n r lvs o th hyprtr. By Lmm 10, ll limintions on grphs lot on th sutr root t A0 r prorm whn A0 is ll to run DpthFirstElimint. Th prossing is intil to tht o Algorithms n 3, xpt th itionl limintion in 0 rltiv to A. Hn y Proposition 9, th grph union Q o 0 through n is horl or th itionl limintion. Sin th itionl limintion n th upting tht ollows hng nithr th horlity o 0 nor tht o th rmining sugrph o Q, th grph union o 0 through n is horl tr th itionl limintion. Now ssum tht whn th ltitu o A0 is = m 1, th grph union o 0 through n is horl tr A0 is ll to run th moi DpthFirstElimint. Consir th s = m +1. Whn DpthFirstElimint is ll y A0 in h A i (i =1; :::; n), it is quivlnt to rgr A i s (hypr)l with Q i m in it, whr Q i is th union o ll grphs on th sutr root t A i. It is quivlnt sin ll limintions on grphs in th sutr root t A i r prorm uring th ll o DpthFirstElimint on A i y Lmm 10, n sin th lol ovring onition hols. By th inutiv ssumption, whn th ll is omplt, Q i is horl. Du to th quivln with th prossing o Algorithms n 3, w onlu tht th grph union o 0, Q1; :::; Q n is horl y Proposition 9. Thorm 11 shows tht whn CoTringult hlts, th union o ll grphs on th hyprtr is horl. Rll tht h lol grph ns to horl s wll (Stion 4). Thorm 1 shows tht this is utomtilly stis u to th wy in whih MSDA is n. Thorm 1 Lt horl grph n g grph otin y lting som nos (n lins inint to ths nos) rom. Thn g is horl. Proo: It sus to show tht tr th ltion o singl no x, th rmining grph g is horl. W prov y ontrition: Suppos tht g is nonhorl. Thn thr must horlss yl o lngth > 3in g. Sin is horl, shoul prou y th ltion o x n lins inint to x. Howvr, no mttr wht th jny o x is in, w nnot m th horlss yl to go wy. Hn, is in or x is lt. This implis tht is nonhorl: ontrition. 16

17 8 Ensur limintion orrs or multipl -spsts In Stion 7, w hv prsnt CoTringult tht solvs th ooprtiv tringultion prolm sujt to Rquirmnts n 3. Woul Rquirmnt 1 stis s wll? Th ollowing xmpl shows tht this is not th s in gnrl. 1 0 g h Figur 10: Illustrtion o violtion o Rquirmnt 1. Consir th thr lol grphs (ignor th sh lins) in Figur 10 with th hyprstr 1 0. Following CoTringult, gnt A0 rst limints 0 in th orr (;;;;;) rltiv to A1. Th limintion prous no ll-ins. Thn gnt A1 limints 1 in th orr (g;;;;) rltiv to A0 lso without ll-ins. Nxt, gnt A0 limints 0 rltiv to A in th orr (;;;;;), whih prous ll-in ; g (th sh lin in 0). It will snt to A n to (th sh lin in ). A thn limints in th orr (h; ; ; ) rltiv to A0 without itionl ll-ins. CoTringult now hlts. Howvr, i w limint th nw 0 gin rltiv to A1 in th orr (;;;;;), nothr ll-in ; g is now n. Hn, CoTringult os not gurnt stistion o Rquirmnt 1 in gnrl. On th othr hn, situtions li th ov xmpl o not sm to ris otn. In our xprimntl stuy (s Stion 9) with thr MSBNs, ll o thm stis Rquirmnt 1 whn CoTringult hlt. Hn w suggst th ollowing lgorithm to rss Rquirmnt 1: Algorithm 7 (SCoTringult) prorm CoTringult; h gnt prorms n limintion rltiv to th -spst with h jnt gnt; i no gnt ny ll-ins, hlt; ls rstrt this lgorithm; Not tht th limintion prorm in SCoTringult is stritly lol to h gnt. S Stion 10 or th omplxity nlysis n possil msurs or improving th iny. W hv th ollowing thorm on th proprty o SCoTringult: Thorm 13 SCoTringult will hlt, n whn it hlts, Rquirmnt 1 is stis. Proo: Eh roun o SCoTringult will som ll-ins to som lol grphs, sin othrwis it will hlt. Sin only nit numr o ll-ins n to h lol grph 17

18 or it oms omplt, n omplt grph is liminl in ny orr, w onlu tht SCoTringult will hlt. Whn SCoTringult hlts, Rquirmnt 1 hs n xpliitly vri. By Thorms 11, 1 n 13, w onlu tht th prolm o ooprtiv tringultion with Rquirmnts 1 through 3 is solv y SCoTringult. 9 Exprimntl Stuy So r, w hv shown tht SCoTringult orrtly solvs th prolm o ooprtiv tringultion. Howvr, our nlysis sys nothing out how sprs th rsultnt tringultion is. In t, DpthFirstElimint spiys only th (prtil) orr o limintion in trms o susts o nos in grph. Th orr o limintion within h sust is lt unspi. Dpning on th (totl) orr us, irnt ll-ins n prou. In gnrl, w prr limintion orrs tht prou th minimum numr o ll-ins. As it is NP hr [10] to n suh orrs, huristis shoul us. A simpl n tiv huristis or ntrliz tringultion is liminting th no with th smllst ll-ins []. W hv opt th sm huristis to supplmnt th prtil orr spi in Dpth- FirstElimint. To vlut how wll SCoTringult (supplmnt y th huristis) prorms, w onut th ollowing xprimntl stuy: W hv implmnt our lgorithms in WEBWEAVR-III (th sussor o WEBWEAVR- II writtn in JAVA). To vlut th sprsnss o rsults rom ooprtiv tringultion, w ompr with th rsults rom th ntrliz tringultion using th sm huristis. A 0 A A A 1 4 A 3 gt1 gt4 gt gt3 0 h gt10 g gt9 g gt5 gt6 n m o gt8 q gt7 g gt1 i j 1 l gt11 p gt14 gt17 w q gt18 gt13 s l x r gt15 t gt16 p z y gt19 u v 4 3 Figur 11: CIRCS: n xprimntl prolm or ooprtiv tringultion. Tsts wr run using thr gnrt MSBNs: 5PARTS (not shown), CIRCS (Figur 11) n BIB (Figur 1). Th hyprtr o CIRCS with v gnts is shown in Figur 11 (lt). Lol grphs tr morliztion r shown in th right (without th sh lins). Fill-ins uring ooprtiv tringultion r rwn s sh lins. Th rsult or BIB is shown in Figur 1. For ll thr MSBNs, SCoTringult trmint with just on xution o CoTringult. 18

19 A 3 A 0 A 1 A A 4 T1 S1 R1 O1 N1 y M1 U1 Q1 w P1 t L1 v u A V1 s g 3 y x 0 v w t u A B z F1 m n i p l h s 1 o E1 j H1 D1 q x V I1 B J1 R W H z K1 I S D E J U T F X1 Z1 g W1 m n l r H C I D E J F Y1 B1 h o C K N 1 1 C1 i p j q P Q L M O 1 A1 Z X r 4 Y Figur 1: BIB: n xprimntl prolm or ooprtiv tringultion. Th xprimntl rsults r summriz in Tl 1. Th son olumn lists th totl vrils (shr vrils r ount on) in h MSBN. Th numr o ll-ins prou y h ntrliz tringultion is list in th thir olumn, n th orrsponing rsult or ooprtiv tringultion is list in th lst olumn. For 5PARTS, smll MSBN, th sm ll-in ws prou y oth mthos. For CIRCS, h mtho prou ight ll-ins ut two ll-ins wr irnt. For BIB, rsults rom th two mthos shr only two ll-ins, n th ooprtiv tringultion prou two mor. Totl numr Fill-ins Fill-ins MSBN o vrils (ntrliz) (ooprtiv) 5PARTS CIRCS BIB Tl 1: Summry o xprimntl rsults Th rsults monstrt tht SCoTringult prous rsonly sprs tringultions ompr with th ntrliz tringultion. 19

20 10 Complxity nlysis First, w nlyz th tim omplxity o CoTringult. W onntrt on DpthFirstElimint n ous on its limintion prossing only, s th mount o omputtion in ommunition o ll-ins to jnt gnts uring DpthFirstElimint n DistriutDlin is minor. ivn hyprtr o n gnts, rom Figur 9, it's sy to s tht n limintions r prorm uring DpthFirstElimint. Lt th mximum numr o nos in lol grph, n i th mximum gr o no. To limint no, th ompltnss o its jny is h. Th omplxity o th hing is O(i ). Using th huristis, O() nos r h or on is limint. Hn th tim omplxity o liminting ll nos in grph is O( i ). Th omplxity o CoTringult is thn O(n i ). Nxt, w onsir th omplxity o SCoTringult. Clrly, th worst s omplxity o SCoTringult is muh highr thn CoTringult. Howvr, our xprimntl stuy provis vin tht th vrg s omplxity o SCoTringult will los to tht o CoTringult. Furthrmor, iny o SCoTringult n improv in svrl wys: For instn, i -spst oms omplt uring tringultion, thn th hyprlin it rprsnts is \lo" n n ntir su(hypr)tr in on si o th hyprlin ns not pross in ltr itrtions o CoTringult i no ll-ins r prou in th sutr. Anothr onition or rly prtil trmintion is th hng o lol topology. I n gnt nithr prous ny ll-ins lolly nor rivs ny rom jnt gnts uring on roun o CoTringult, it os not hv to prtiipt in th nxt roun (lthough it my hv to prtiipt in ltr rouns). Bs on ths possil improvmnts n our mpiril rsults, w liv tht lthough th worst s omplxity o SCoTringult is quit xpnsiv, only vry smll numr o (otn on) itrtions o CoTringult n to prorm in mny MSBNs. Most gnts n not prtiipt in most o th itrtions o CoTringult xpt th rst on. Thror, th vrg s omplxity o SCoTringult (with ths improvmnts) will similr with tht o CoTringult. 11 Conlusion Inrn with multi-gnt MSBN n prorm tivly in ompil rprsnttion. Th ompiltion without ompromising gnts' privy rquirs ooprtiv tringultion o grph union without rvling h gnt's lol grph yon th sugrph ovr shr nos. W hv propos onptully simpl n int lgorithm (SCoTringult) or ooprtiv tringultion o grph n s th union o st o grphs orgniz into hyprtr. Th lgorithm hs n implmnt n smll sl tst monstrt sprs tringultions ompr with th ntrliz prossing. 0

21 Anowlgmnts This wor is support y th Rsrh rnt OP rom th Nturl Sins n Enginring Rsrh Counil (NSERC) o Cn. Hlpul wr riv on n rly vrsion rom F. Jnsn, U. Kjrul n K.. Olsn. Appnix: Othr rph-thortil trminologis A juntion tr (JT) T ovr st N is tr whr h no is ll y sust (ll lustr) on n h lin is ll y th intrstion (ll spst) o its inint lustrs, suh tht th intrstion o ny two lustrs is ontin in vry spst on th pth twn thm. Two simpl JTs r shown in Figur 4 (right). A mximl omplt st o nos in grph is ll liqu. ivn horl grph ovr st N o nos, JT T o is rt y lling h no o T with liqu o. Suh JT xists i is horl. Th two JTs in Figur 4 (right) r JTs o n Q in Figur 4 (lt), rsptivly. For isjoint susts X, Y n Z o nos in grph, wus<xjzjy > to not tht nos in Z grphilly sprt nos in X n nos in Y. rphil sprtion my n irntly in irnt typs o grphs. I is n unirt grph, thn th norml grphil sprtion pplis. In Figur 4 (lt), ; g n ; g r sprt y g in. In JT T ovr N, grphil sprtion is n s ollows: Lt X, Y n Z isjoint susts o N. Suppos tht or h x X n h lustr C x suh tht x C x, n or h y Y n h lustr C y suh tht y C y, thr xists spst S on th pth twn C x n C y suh tht S Z. Thn X n Y r si to grphilly sprt y Z in T. In Figur 4 (right), ; g n ; g r sprt y g in th lt JT. Lt N th st o vrils in omin n P (N) th proility istriution ovr N. For thr isjoint susts X, Y n Z o vrils, X n Y r onitionlly inpnnt givn Z i P (XjY;Z) =P (XjZ) whnvr P (Y;Z) > 0: Dnot th rltion y I(X; Z; Y ). A grph is n I-mp [4] o PDM ovr N i thr is n on-to-on orrsponn twn nos o n vrils in N suh tht or ll isjoint susts X, Y n Z o N, <XjZjY > =) I(X; Z; Y ). Rrns [1] M.C. olumi. Algorithmi rph Thory n Prt rphs. Ami Prss, [] F.V. Jnsn. An introution to Bysin ntwors. UCL Prss, [3] D. Kollr n A. Pr. Ojt-orint Bysin ntwors. In D. igr n P.P. Shnoy, itors, Pro. 13th Con. on Unrtinty in Artiil Intllign, pgs 30{ 313, Provin, Rho Isln, [4] J. Prl. Proilisti Rsoning in Intllignt Systms: Ntwors o Plusil Inrn. Morgn Kumnn,

22 [5] Y. Xing. Optimiztion o intr-sunt li upting in multiply stion Bysin ntwors. In Pro. 11th Con. on Unrtinty in Artiil Intllign, pgs 565{573, Montrl, [6] Y. Xing. A proilisti rmwor or ooprtiv multi-gnt istriut intrprttion n optimiztion o ommunition. Artiil Intllign, 87(1-):95{34, [7] Y. Xing. Bli upting in MSBNs without rpt lol propgtions. Thnil Rport CS-98-01, Univrsity o Rgin, [8] Y. Xing, B. Pnt, A. Eisn, M. P. Bos, n D. Pool. Multiply stion Bysin ntwors or nuromusulr ignosis. Artiil Intllign in Miin, 5:93{314, [9] Y. Xing, D. Pool, n M. P. Bos. Multiply stion Bysin ntwors n juntion orsts or lrg nowlg s systms. Computtionl Intllign, 9():171{ 0, [10] M. Ynnis. Computing th minimum ll-in is NP-omplt. SIAM J. o Algri n Disrt Mthos, (1), 1981.