Fluent Merging: A Generl Tehnique to Improve Rehility Heuristis n Ftore Plnning Menkes vn en Briel Deprtment of Inustril Engineering Arizon Stte University Tempe AZ, 85287-8809 menkes@su.eu Suro Kmhmpti Deprtment of Computer Siene Arizon Stte University Tempe AZ, 85287-8809 ro@su.eu Thoms Vossen Lees Shool of Business University of Coloro t Bouler Bouler CO, 80309-0419 vossen@oloro.eu Astrt Fluent merging is the proess of omining two or more fluents (stte vriles) into single super fluent. By ompiling in some of the inter-fluent intertions, fluent merging n (1) help improve informeness of relxe rehility heuristis, n (2) improve the effiieny of ftore plnning s it removes some of the intervrile epenenies. Although some speil ses of fluent merging hve een roun (uner other nmes), the tehnique in its full generlity hs not een exploite or nlyze. In this pper, we isuss the generl motivtions for n treoffs in fluent merging. We will rgue tht existing tehniques re too onservtive in ientifying mergele fluents. We will then provie some novel tehniques se on usl grph nlysis for ientifying mergele fluents. Introution In this pper we esrie fluent merging, the proess of omining two or more stte vriles into single super stte vrile. Fluent merging, when one juiiously, n le to etter heuristi estimtes n more effetive ftore plnning. Let us strt with n informl exmple to illustrte the ie of fluent merging (we shll formlize this lter). In simple Logistis prolem instne with one truk, one pkge, n two lotions, we my omine the fluents t(truk1, lo1) n t(truk1, lo2), whih tke vlues from {T, F }, into super-fluent. This new fluent tkes vlues from {T, F } {T, F }. On the fe of it, fluent merging seems like rther quixoti ie s it runs ounter to the onventionl wisom tht it is vntgeous to represent n reson with omins in terms of iniviul fluents (the so lle ftore representtions ). After ll, the strtegy of merging fluents n, in the extreme, le us to non-ftore representtion where single super-fluent hs n exponentil omin size, with eh omin vlue orresponing to eh of the sttes in the omin. The reson merging wins up eing useful in some ses is tht if we merge fluents tht hve strong epenenies, then their effetive omin n e muh Copyright 2007, Assoition for the Avnement of Artifiil Intelligene (www.i.org). All rights reserve. smller thn the Crtesin prout of the iniviul vrile omins. In the logistis exmple ove, the merging proess n explite the ft tht only the vlues (T, F) n (F, T) re rehle for the merge fluent. Thus it removes the vlue omintions (T, T) n (F, F) from onsiertion. The ensuing omin reution s well s the ompiltion of negtive intertions turns out to e quite useful, s we shll see elow. Notie tht when we merge fluents, we nturlly win up with super-fluents tht re multi-vlue (even if we strte with oolen fluents). The populr ie of onverting omin from oolen to multi-vlue representtion, s esrie y (Eelkmp & Helmert 1999; Fox & Long 1998; Gerevini & Shuert 2000; Helmert 2006) n thus e seen s just speil se of fluent merging. Speifilly these methos fous on merging fluents tht hve strit mutul exlusion reltionships. Fluent merging oes not however hve to e onfine to fluents with strit mutul exlusions. We shll see tht merging fluents with strong inter-usl epenenies n lso e vntgeous. The most importnt spet of fluent merging is tht it ompiles-in some of the negtive intertions etween the fluents. This is illustrte well in the logistis exmple ove. The negtive intertion etween t(truk1, lo1) = T n t(truk1, lo2) = T is effetively remove y the ft tht the merge fluent oesn t hve {T, T } in its omin. This hs signifint impt on two moern ies for speeing up plnning: 1 Relxe rehility heuristis whih o rehility nlysis y ignoring negtive intertions n o more informe jo of istne estimtion fter fluent merging (sine some of the negtive intertions hve lrey een ompile-in). Ftore plnning tehniques tht ttempt to fin plns for iniviul fluents n omine them n enefit if fluents with epenenies re merge up front (sine this mens tht the merging phse will 1 While we fous on the ompiltion of negtive intertions, it is worth noting tht the erlier work y Eelkmp et. l. (Eelkmp & Helmert 1999) motivtes fluent merging from the perspetive of minimizing stte enoing length.
likely hve fewer ktrks). Both these vntges hve een mply, if iniretly, estlishe in the plnning literture. Prt of the reson for the improve performne of Fst-Downwr plnner (Helmert 2006) n e ttriute to the ft tht it oes fluent merging (in onverting oolen fluent omin esription into multi-vlue fluent esription). Our own reent work (vn en Briel et l. 2007) shows tht this type of informeness vntge lso hols for more generl forms of fluent merging. The vntge of fluent merging for ftore plnning is estlishe y our work (vn en Briel, Vossen, & Kmhmpti 2005) whih shows tht multi-vlue representtions n le to signifint performne improvements. While the foregoing pints mostly positive piture of fluent merging, s exhorte t the outset, fluent merging n only e goo in senrios where the originl omin esription ontins signifint numer of strongly epenent fluents. Speifilly, while merging reues the numer of fluents, it inreses their omin sizes. The ltter inrese n e exponentil in the numer of vriles merge. This worst se ours when the merge vriles re ompletely inepenent. However, the inrese n e offset with omin reution if the vriles re strongly epenent. 2 The prolem with existing fluent merging methos however is not so muh tht they on t le to omputtionl vntges, ut rther tht they re too onservtive. In prtiulr, the mutul-exlusion se multivlue moels foun y (Eelkmp & Helmert 1999; Helmert 2006) onsier merging fluents only when the effetive omin size goes from exponentil to liner. Speifilly, they will merge m oolen fluents tht form mutex lique into single multi-vlue vrile with m vlues (whih is reution of omin size from 2 m to m). While these merging strtegies will give onsierle omputtionl vntges when they re pplile, they re too onservtive n re often not pplile. Speifilly, we my hve sets of oolen fluents tht hve strong epenenies n yet o not quite form mutex lique. Fining n merging suh sets of fluents oul still e quite useful. The omin reution in suh ses my only e from 2 m to m k (for some smll k) inste of m n yet it is impressive nonetheless. For exmple, our reent work (vn en Briel et l. 2007) shows tht more ggressive merging n improve heuristi informeness. The hllenge of ourse is to ome up with pprohes tht n ientify suh fluent sets utomtilly. 2 It is even possile to hve omins where merging ll the fluents n still e goo ie. Consier the extreme exmple with n toms, two legl sttes (T,..., T) n (F,..., F), one tion tht toggles ll vriles from T to F, n one tion tht toggles ll vriles from F to T. In this se, we re etter off merging ll fluents into single superfluent tht esries the omplete rehle stte spe of the prolem. In the reminer of the pper, we provie some first steps towrs formlly efining the fluent merging prolem n eveloping methos tht re more ggressive in ientifying mergele fluents. Towrs the ltter, we esrie some tehniques se on novel nlysis of the usl grph. In wy, our work n e seen s n pplie pproh to the work on ftore plnning y Brfmn n Domshlk 2006. This pper is orgnize s follows. First, we provie some kgroun n efine the proess of fluent merging more formlly. Seon, we isuss the potentil use of fluent merging in improving heuristi estimtes n ftore plnning. Some onlusions re given t the en. Fluent Merging We ssume tht we re given SAS+ plnning tsk Π = C, A, s 0, s, whih llows oth oolen n multi-vlue stte esriptions, where: C = { 1,..., n } is finite set of stte vriles, where eh stte vrile C hs n ssoite omin V n n impliitly efine extene omin V + = V {u}, where u enotes the unefine vlue. For eh stte vrile C, s[] enotes the vlue of in stte s. The vlue of is si to e efine in stte s if n only if s[] u. The totl stte spe S = V 1... V n n the prtil stte spe S + = V + 1... V + n re impliitly efine. A is finite set of tions of the form pre, post, prev, where pre enotes the pre-onitions, post enotes the post-onitions, n prev enotes the previlonitions. For eh tion A, pre[], post[] n prev[] enotes the respetive onitions on stte vrile. The following two restritions re impose on ll tions: (1) One the vlue of stte vrile is efine, it n never eome unefine. Hene, for ll C, if pre[] u then pre[] post[] u; (2) A previl- n post-onition of n tion n never efine vlue on the sme stte vrile. Hene, for ll C, either post[] = u or prev[] = u or oth. We use A E to enote the tions tht hve n effet in stte vrile, n A V to enote the tions tht hve previl onition in. s 0 S enotes the initil stte n s S + enotes the gol stte. We sy tht stte s is stisfie y stte t if n only if for ll C we hve s[] = u or s[] = t[]. This implies tht if s [] = u for stte vrile, then ny efine vlue f V stisfies the gol for. Two importnt onstruts tht we use re the solle omin trnsition grph n usl grph. The omin trnsition grph DTG = (V, E ) of stte vrile is lele irete grph with noes for eh vlue f V. DTG ontins lele r (f, g) E if n only if there exists n tion with pre[] = f n post[] = g or pre[] = u n post[] = g. Eh r is lele y the set of tions with orresponing pre-
n post-onitions. For eh r (f 1, f 2 ) with lel in DTG we sy tht there is trnsition from f 1 to f 2 n tht tion hs n effet in. The usl grph CG Π = (V, E) of plnning tsk Π is irete grph with noes for eh stte vrile C. CG ontins n r ( 1, 2 ) E if n only if there exists n tion tht hs previl onition or preonition in 1 n n effet in 2. We efine fluent merging s the omposition of two or more stte vriles s follows. The term omposition is lso use in moel heking to efine the prllel omposition of utomt (Cssnrs & Lfortune 1999). Definition (Composition) Given the omin trnsition grph of two stte vriles 1, 2, the omposition of DTG 1 n DTG 2 is the omin trnsition grph DTG 1 2 = (V 1 2, E 1 2 ) where V 1 2 = V 1 V 2 ((f 1, f 2 ), (g 1, g 2 )) E 1 2 if f 1, g 1 V 1, f 2, g 2 V 2 n there exists n tion A suh tht one of the following onitions hol. pre[ 1 ] = f 1, post[ 1 ] = g 1, n pre[ 2 ] = f 2, post[ 2 ] = g 2 pre[ 1 ] = f 1, post[ 1 ] = g 1, n prev[ 2 ] = f 2, f 2 = g 2 pre[ 1 ] = f 1, post[ 1 ] = g 1, n f 2 = g 2 We sy tht DTG 1 2 is the ompose omin trnsition grph of DTG 1 n DTG 2. Exmple Consier the set of tions A = {,,, } n the set of stte vriles C = { 1, 2 } whose omin trnsition grphs hve V 1 = {f 1, f 2, f 3 }, V 2 = {g 1, g 2 } s the possile vlues, n E 1 = {(f 1, f 3 ), (f 3, f 2 ), (f 2, f 1 )}, E 2 = {(g 1, g 2 ), (g 2, g 1 )} s the possile trnsitions s shown in Figure 1. Merging stte vriles 1 n 2 retes new stte vrile whose omin is efine y the Crtesin prout V 1 V 2 s shown in Figure 1. Note tht some vlue omintions eome isonnete omponents, suh s (f 3, g 2 ). These isonnete omponents re unrehle from the initil stte n thus n sfely e ignore. Also, note tht some tions generte multiple instnes in the omposition, suh s tions n. These multiple instnes re generte if n tion hs n effet in one fluent, ut no effet or previl onition in the other fluent 3. The omposition of more thn two stte vriles n e otine y reting omposition over one or more ompose omin trnsition grphs. For exmple, DTG 1 2 3 n e otine y reting the omposition etween DTG 1 2 n DTG 3. 3 As we shll see lter, one onsiertion in piking effetive merging strtegies is to ensure tht they on t inrese the numer of tions too muh. Ientifying Mergele Fluents Previously, mergele fluents hve een ientifie y looking t the oolen fluents tht form mutex lique. These type of fluent mergings re goo sine they eliminte mny unrehle vlue omintions. We introue two other wys to ientify mergele fluents se on usl grph nlysis. First, in orer to ientify mergele fluents we look for yles in the usl grph. Cusl yles re unesirle s they esrie two-wy epenenies etween stte vriles. Tht is, hnges in stte vrile 1 will epen on onitions in stte vrile 2, n vie vers. While it is possile tht usl yles involve more thn two stte vriles, we only onsier 2-yles (yles of length two). In prtiulr, we merge two fluents 1 n 2 if they form 2-yle in the usl grph n if the following onition hol. For ll A E 1 we hve (A E 2 A V 2 ) For ll A E 2 we hve (A E 1 A V 1 ) In other wors, for every tion tht hs n effet in stte vrile 1 ( 2 ) we hve tht tion hs n effet or previl onition in stte vrile 2 ( 1 ). The min reson for requiring this itionl onition is to ensure tht the tions o not generte multiple instnes in the omposition. This onition is quite restritive, ut s shown y the next exmple effetive nevertheless. Moreover, vn en Briel et l. 2007 show tht this type of fluent merging les to improve network flow se rehility heuristis. Exmple Figure 2 shows n exmple of how fluent merging n remove usl 2-yles from the usl grph. The figure on the left shows the usl grph for typil stte esription of Zenotrvel prolem with two irplnes, two pssengers, n ny numer of ities. The stte esription is etermine y six stte vriles: one for eh pssenger Lo(person1) n Lo(person2) with vlues tht enote the lotion of the pssengers, one for eh irplne Lo(irplne1) n Lo(irplne2) with vlues tht enote the lotion of the irplnes, n one for the fuel tnk of eh irplne F uellevel(irplne1) n F uellevel(irplne2). The figure on the right shows the usl grph of the sme prolem, ut is se on stte esription in whih the stte vriles Lo(irplne1) n F uellevel(irplne1), n Lo(irplne2) n F uellevel(irplne2) hve een merge into super stte vriles. The vntgeous of the resulting stte esription shoul e ler. Fewer yles in the usl grph will le to etter hierrhil eompositions, whih oul le to improve plnning performne. Seon, in orer to ientify mergele fluents we look t pirs of toms (f 1, f 2 ) suh tht there exists n tion tht hs f 1 s previl onition n f 2 s elete effet. Speifilly, we look for usl links in the usl grph tht re introue y the tions with previl onition in one fluent n n effet in nother flu-
ent. Some hierrhil se plnners n hnle suh uslities quite well n simply inorporte them iretly into the hierrhil struture. For exmple, in the Logitis omin hierrhil plnner my first fin pln for eh pkge, use these plns to impose orere onitions on the truks, n then fin pln for eh truk. However, rehility heuristis tht o not exploit hierrhies my sometimes give poor estimtes even in some very simple plnning tsks. Exmple Figure 3 shows n exmple of how fluent merging n improve heuristi estimtes. The figure onsiers simple Logistis prolem with one truk, one pkge, n two lotions. In the initil stte we hve the truk t 2 (= t(truk1, lo2)) n the pkge t 1 (= t(pkge1, lo1)). Severl known rehility heuristis, inluing FF s relxe pln heuristi (Hoffmnn & Neel 2001), fil to reognize tht the truk nees to rive k to lotion 2 in orer to unlo the pkge. The figure shows the merge tom pirs n their orresponing trnsitions. If FF s relxe pln heuristi onsiers the tom pirs s single toms, it woul hve etete tht it nees to rive to lotion 1 to lo the pkge n then rive k to unlo the pkge. In Proeeings of the 17th Ntionl Conferene on Artifiil Intelligene (AAAI-2000), 761 767. Helmert, M. 2006. The Fst Downwr plnning system. Journl of Artifil Intelligene Reserh 26:191 246. Hoffmnn, J., n Neel, B. 2001. The FF plnning system: Fst pln genertion through heuristi serh. Journl of Artifiil Intelligene Reserh 14:253 302. vn en Briel, M.; Benton, J.; Kmhmpti, S.; n Vossen, T. 2007. An LP-se heuristi for optiml plnning. In Proeeings of the Interntionl Conferene of Priniples n Prtie of Constrint Progrmming (CP-2007). (To pper). vn en Briel, M.; Vossen, T.; n Kmhmpti, S. 2005. Reviving integer progrmming pprohes for AI plnning: A rnh-n-ut frmework. In Proeeings of the Interntionl Conferene on Automte Plnning n Sheuling (ICAPS-2005), 161 170. Conlusions We esrie the proess of fluent merging n showe how it n help improve rehility heuristis n ftore plnning. While fluent merging hs een roun uner the ie of onverting oolen to multi-vlue representtions, we introue methos tht re more generl in ientifying mergele fluents. Our reent work (vn en Briel et l. 2007) shows tht we n erive more informe heuristis y merging fluents without experiening too muh omputtionl overhe. We elieve, however, tht there my e other wys to ientify mergele fluents, whih either exten or generlize the wys tht we esrie. Referenes Brfmn, R., n Domshlk, C. 2006. Ftore plnning: How, when, n when not. In Proeeings of the 21st Ntionl Conferene on Artifiil Intelligene (AAAI-2006), 809 814. Cssnrs, C., n Lfortune, S. 1999. Introution to Disrete Event Systems. Kluwer Aemi Pulishers. Eelkmp, S., n Helmert, M. 1999. Exhiiting knowlege in plnning prolems to minimize stte enoing length. In Proeeings of the Europen Conferene on Plnning (ECP-1999), 135 147. Fox, M., n Long, D. 1998. The utomti inferene of stte invrints in TIM. Journl of Artifiil Intelligene Reserh 9:367 421. Gerevini, A., n Shuert, L. K. 2000. Disovering stte onstrints in DISCOPLAN: Some new results.
f 1,,g 1 f 3,g 2 f 1,g 2 f 1 f 2 g 1 f 3,g 1 f 2,g 1 f 3 g 2 f 2,g 2 DTG 1 DTG 2 DTG 1 2 Figure 1: Two omin trnsition grphs n their omposition. Smll in-rs enote the initil stte of eh stte vrile. Lo(person1) Lo(person2) Lo(person1) Lo(person2) Lo(irplne1) Lo(irplne2) Lo(irplne1) Fuellevel(irplne1) Lo(irplne2) Fuellevel(irplne2) Fuellevel(irplne1) Fuellevel(irplne2) Figure 2: Fluent merging removes usl 2-yles from the usl grph for typil stte esription of the Zenotrvel omin. DTG(Pkge1,Truk1) 1,1 1,2 Unlo(p1,t1,l1) 2,1 2,2 Unlo(p1,t1,l2) Lo(p1,t1,l1) Lo(p1,t1,l2) T,1 T,2 Figure 3: Fluent merging improves heuristi estimtes in the Logistis omin.