Deciding the Consistency of Branching Time Interval Networks

Transcription

1 Deiing the Consisteny of Brnhing Time Intervl Networks Mro Gvnelli Deprtment of Engineering University of Ferrr, Itly Alessnro Pssntino Deprtment of Mthemtis n Computer Siene University of Ferrr, Itly lessnr.pssntino@stuent.unife.it Guio Sivio Deprtment of Mthemtis n Computer Siene University of Ferrr, Itly guio.sivio@unife.it Astrt Allen s Intervl Alger (IA) is one of the most prominent formlisms in the re of qulittive temporl resoning; however, its pplitions re nturlly restrite to liner flows of time. When eling with nonliner time, Allen s lger n e extene in severl wys, n, s suggeste y Rgni n Wölfl [20], possile solution onsists in efining the Brnhing Alger (BA) s set of 19 si reltions (13 si liner reltions plus 6 new si nonliner ones) in suh wy tht eh si reltion etween two intervls is ompletely efine y the reltive position of the enpoints on tree-like prtil orer. While the prolem of eiing the onsisteny of network of IA-onstrints is well-stuie, n every suset of the IA hs een lssifie with respet to the trtility of its onsisteny prolem, the frgments of the BA hve reeive less ttention. In this pper, we first efine the notion of onvex BA-reltion, n, then, we prove tht the onsisteny of network of onvex BA-reltions n e eie vi pth onsisteny, n is therefore polynomil prolem. This is the first non-trivil trtle frgment of the BA; given the ler prllel with the liner se, our ontriution poses the ses for eeper stuy of frgments of BA towrs their omplete lssifition ACM Sujet Clssifition Theory of omputtion Constrint n logi progrmming Keywors n phrses Constrint progrmming, Consisteny, Brnhing time Digitl Ojet Ientifier /LIPIs.TIME Aknowlegements G. Sivio knowleges prtil support y the Itlin INDAM GNCS projet Forml methos for verifition n synthesis of isrete n hyri systems. 1 Introution Allen s Intervl Alger [1] (IA) is one of the most prominent formlisms in the re of qulittive temporl, n lso sptil, resoning. However, its pplitions re nturlly restrite to liner flows of time. Allen s lger is onsiere one of the most influentil formlisms in qulittive resoning, n it hs foun pplition in wie rnge of ontexts, Mro Gvnelli, Alessnro Pssntino, n Guio Sivio; liense uner Cretive Commons Liense CC-BY 25th Interntionl Symposium on Temporl Representtion n Resoning (TIME 2018). Eitors: Ntsh Alehin, Kjetil Nørvåg, n Wojieh Penzek; Artile No. 12; pp. 12:1 12:15 Leiniz Interntionl Proeeings in Informtis Shloss Dgstuhl Leiniz-Zentrum für Informtik, Dgstuhl Pulishing, Germny

2 12:2 Deiing the Consisteny of Brnhing Time Intervl Networks suh s sheuling, plnning, tse theory, nturl lnguge proessing, mong others. In Allen s IA we onsier the omin of ll intervls on liner orer, n efine thirteen si reltions etween pirs of intervls (suh s, for exmple, meets or efore). A onstrint etween two intervls is ny isjuntion of si reltions, n network of onstrints is efine s set of vriles plus set of onstrints etween them, interprete s logil onjuntion. The most relevnt prolem in IA is eiing whether network n e stisfie, tht is, eiing if every vrile of network n e instntite to n intervl without violting ny onstrint. The onsisteny of network of onstrints is rhetypil of the lss of onstrints stisftion prolems (CSP), euse network is onjuntion of onstrints; other onsisteny prolems, even in temporl lgers, re more generl, n llow some form of isjuntion. A very wie rnge of progrmming tehniques, strtegies, n heuristis hve een n re eing stuie to evise effiient implementtions; in the prtiulr se of the IA, whose onsisteny prolem is NP-omplete, two min strtegies hve een minly opte, tht re se either on lever rute-fore enumerting lgorithms (see, e.g. [13, 23]), or on trtle frgments of the lger, whih re interesting on their own [14] s well s heuristis, ime to reue the rnhing ftor in rnh-n-oun pprohes [15, 19]. Among the severl trtle frgments of the IA, prtiulrly interesting one is known s IA onvex, introue in the erly yers of this line of reserh, n enompssing 44 si n non-si reltions [25]. To otin IA onvex, vn Beek n Cohen [25] stuy, first, the simpler Point Alger (P A), whih hs only three si reltions, n efine the notion of onvexity for P A-reltion; hving ientifie the set of P A onvex -reltions, they efine the set IA onvex s the mximl suset of IA-reltions with the following property: every network of IA onvex -onstrints n e trnslte into n equi-stisfile network of P A onvex -onstrints. Then, using the properties of suh trnsltion, they prove tht the pth onsisteny lgorithm is omplete for eiing the onsisteny of IA onvex -network, otining the sme result for P A onvex -networks s orollry. Sine then, mny other trtle frgments of the IA hve een isovere, n, in ft, we know the sttus of every frgment of the IA with respet to the trtility/intrtility of the orresponing onsisteny prolem [14]. Brnhing time hs een less stuie from the lgeri point of view, in shrp ontrst with the huge mount of reserh on point-se n intervl-se temporl logis, suh s CTL, CTL, ATL, or rnhing PNL [3, 4, 11]. The Brnhing Point Alger (BP A) hs een stuie in [9, 12], n the omputtionl ehviour of the onsisteny prolem of the BP A n its frgments hs een nlyze in [5], where, in prtiulr, polynomil lgorithm to eie the onsisteny of network of BP A-onstrints is given. In [20], the uthors efine rnhing version of Allen s IA, whih we refer to s BA (Brnhing Alger) n introue two possile sets of si reltions tht my hol etween two intervls on tree-like prtil orer. One of these sets, ompose of 24 si reltions, n lso stuie from the (first-orer) expressive power point of view in [10], is hrterize y si reltions whose semntis nnot e lwys written in the lnguge of enpoints, therefore requiring quntifition; on the other hn, the si reltions of the seon set re orser, still jointly exhustive n mutully exlusive, n their point-se semntis epens only on the reltive position of the enpoints. The ltter set (BA si ), ompose of 19 si reltions, is therefore preferle in mny spets. As expete, the onsisteny prolem for the full BA is NP-omplete, n we know only one trtle frgment of it tht inlues t lest one nonliner reltion, tht is, the set BA si itself, sine network of BA si -onstrints n e fully trnslte into network of BP A-onstrints. In this pper:

3 M. Gvnelli, A. Pssntino, n G. Sivio 12:3 e x () () Figure 1 A pitoril representtion of the four si rnhing point reltions, in whih =, <, >, n e (left-hn sie), n two rnhing reltions tht nee quntifition (right-hn sie). (i) in the the spirit of [25], we efine the notion of onvex rnhing point reltion n the notion of onvex rnhing intervl reltion, n prove tht, s in the liner se, the onsisteny of network of onvex rnhing intervl (n therefore point) reltions n e eie enforing its pth onsisteny, n (ii) following [22], we implement simple rnh-n-oun lgorithm for BA-networks to empirilly stuy the expete improvement in omputtion time when the splitting is riven y onvex reltions inste of si reltions. This pper is orgnize s follows. First we give some neessry preliminries n nottion. In Setion 3 we give the min result of this pper, tht is, we efine onvex rnhing reltions n show tht the onsisteny prolem n e eie y enforing pth onsisteny in onvex networks. Then, in Setion 4, we give some experimentl eviene tht onvex rnhing reltions n e use to spee up the proess of eiing rnhing reltion networks, efore onluing. 2 Preliminries Nottion. Let (T, <) e prtil orer, whose elements re generlly enote y,,..., n where enotes tht n re inomprle with respet to the orering reltion <. We use x, y,... to enote vriles in the omin of points. A prtil orer (T, <), often enote y T, is future rnhing moel of time (or, simply, rnhing moel) if for ll, T there is gretest lower oun of n in T, n, if then there exists no T suh tht > n > (tht is, it is tree). In rnhing moel (T, <), ny mximl linerly orere suset B of T will e lle rnh. There re four si reltions tht my hol etween two points on rnhing moel: equls (=), inomprle ( ), less thn (<), n greter thn (>); the first two re symmetri, while the lst two re inverse of eh other. These reltions re epite in Fig. 1, n re lle si rnhing point reltions. The set of si rnhing point reltions is enote y BP A si. In the liner setting, the set of si reltions hs only three elements, <, =, n >, n it is lle P A si (si point reltions). An intervl in T is pir [, ] where <, n [, ] = {x T : x }. Intervls re generilly enote y I, J,..., n we use X, Y,... to inite vriles in the omin of intervls. Following Allen [1], we opt the so-lle strit interprettion y sking tht intervls with oinient enpoints re exlue. In the se of liner time, theory tht enompsses oth intervls n points hs een presente in [7]. There re severl wys to efine si reltions etween intervls on rnhing orer. Following [10], one n esrie 24 si rnhing reltions se on the possile reltive position of two pirs of orere points on rnhing moel, tht is, y iretly generlizing the universlly known set of 13 si intervl reltions [1] (IA si ). While towrs preise stuy of the T I M E

4 12:4 Deiing the Consisteny of Brnhing Time Intervl Networks (i) I efore J < m (mi) I meets J = o (oi) I overlps J < < < (i) I uring J < < < s (si) I strts J = < < f (fi) I finishes J < < = e I equls J = < = i (ii) I init.efore J < im (imi) I init.meets J < < ie I init.equls J = < u I unrelte J Figure 2 A pitoril representtion of the nineteen si rnhing intervl reltions. In this piture, we ssume < n <. Soli lines re tul intervls, she lines omplete the unerlying tree struture. expressive power of rnhing reltions in first-orer ontext this is n optiml hoie, this is no longer true when stuying the omputtionl properties of the onsisteny prolem. In prtiulr, some of these reltions require first-orer quntifition to e efine: for exmple, in Fig. 1 we see tht, in orer to istinguish the two situtions, we nee to quntify of the existene, or non-existene, of points, omprle with, etween n. To overome this prolem, tht eomes relevnt when we stuy the ehviour of rnhing reltions in ssoition with the ehviour of rnhing point reltions (tht is, y stuying the properties of their point-se trnsltions), Rgni n Wölfl [20] introue set of orser reltions, hrterize y eing trnsltle to point-se reltions using only the lnguge of enpoints, tht is, without quntifition. These 19 reltions re epite in Fig. 2, n form the set of si rnhing intervl reltions BA si ; for eh reltion, the symol in rkets orrespons to its inverse one, if the reltion is not symmetri. A reltion in the set BA si is either liner reltion, or the reltion u (unrelte), or it orrespons to the isjuntion etween pir of fine reltions. For exmple, the reltion im is the isjuntion of the two reltions in Fig. 1. Opertions n lgers. The set BP A of rnhing point reltions is the set of ll possile non-empty isjuntions of si rnhing point reltions, n it enompsses = 15 reltions. Similrly, the set BA of rnhing intervl reltions is the set of ll possile isjuntions of si rnhing intervl reltions, n it enompsses

5 M. Gvnelli, A. Pssntino, n G. Sivio 12:5 Tle 1 Composition of si BP A-reltions (left-hn sie), n of si P A-reltions (righthn sie). < > = < {<} lin {<} {, <} >? {>} {>} { } = {<} {>} {=} { } { } {>, } { }? < > = < {<} lin {<} >? {>} {>} = {<} {>} {=} reltions. In generl, given the si reltions r 1,..., r l, we enote y R = {r 1,..., r l } the isjuntive reltion r 1... r l ; thus, reltion is seen s set, n si reltion s singleton. A BP A-onstrint is n ojet of the type xry, where x, y re point vriles n R BP A; nlogously, BA-onstrint is n ojet of the type XRY, where X, Y re intervl vriles n R BA. There re three si opertions with reltions: (Boolen) intersetion, inverse, n omposition. The inverse of reltion R = {r 1,..., r l } is the reltion R 1 = {r1 1,..., r 1 l }, where, for eh i, r 1 i is the inverse si reltion of the si reltion r i. In our nottion, for exmple, i (lter) is the inverse of the si reltion (efore). The omposition of two si reltions r 1, r 2 is efine s follows: for vriles s, t, z, we sy tht s is in the ompose reltion r 1 r 2 with t, enote s(r 1 r 2 )t, if there exists z suh tht sr 1 z n zr 2 t. The omposition of two reltions R 1, R 2 is efine omponent-wise: R 1 R 2 = {r r 1 R 1 r 2 R 2 (r = r 1 r 2 )}. Clerly, to ompute the omposition of two non-si reltions we se ourselves on the omposition etween si reltions. As for the set BP A si (resp., the set P A si ), the omposition tle n e esily ompute y hn, s in T. 1, left-hn sie (resp., right-hn sie), where we use the revitions lin = {<, =, >} n? = lin { }. The result of omposing two reltions in the set BA si (resp., IA si ) n e ompute utomtilly from T. 1, n it is fully reporte in [21] (resp., [2]). Given set A of reltions, n A-network is irete grph N = (V, E), where V is set of vriles n E V V is set of A-onstrints etween pirs of vriles. To enote onstrint etween the vriles s n t in network, we use inistintly the nottion (s, t) or the infix nottion srt (when we wnt to speify the reltion). Given network N = (V, E), we sy tht N is su-network of N if N = (V, E ), V V, n E is the projetion of E on the vriles in V. Given network, we sy tht it is onsistent if there exists moel suh tht eh vrile n e mppe (relize) to onrete element so tht every onstrint is respete; estlishing if n A-network is onsistent is the A-onsisteny prolem, n two networks N n M re si to e equi-stisfile if it hppens tht N is onsistent if n only if M is onsistent. Given onstrint (s, t) in network N, we sy tht r (s, t) is fesile if there exists moel of N suh tht s n t re relize respeting r, n onstrint (s, t) is si to e miniml if every r (s, t) is fesile n (s, t) nnot e extene with other fesile reltions; estlishing the miniml onstrints for every onstrint in network is lle the miniml lels prolem. Enforing the miniml lel in network implies eiing its onsisteny, ut, in generl, we my hve onsistent network with non-miniml lels. The opertions of inverse, intersetion, n omposition n e use to esign onstrint stisftion prolem (CSP) tehnique to eie the onsisteny of A-network. The sets BP A n BA re lle, respetively, the rnhing point lger n the rnhing intervl lger, n they exten, respetively, the intervl lger IA n the point lger P A. Sine the onsisteny prolem for the IA is NP-omplete [26], the prolem of fining trtle frgments of it is interesting, n it hs een lrgely stuie T I M E

6 12:6 Deiing the Consisteny of Brnhing Time Intervl Networks in the reent literture [1, 25, 14]. The rnhing setting presents similr sitution, s the onsisteny prolem for the BA is NP-omplete s well, ut, in ontrst with the liner se, only one trtle frgment is known, tht is, BA si [20]; the onsisteny of network of si rnhing intervl onstrints n e eie y trnslting it into n equi-stisfile network of BP A-onstrints, for whih eterministi polynomil onsisteny lgorithm exists [5]. Lol onsisteny. On the one hn, the BA-onsisteny prolem is in NP euse there exists simple non-eterministi lgorithm tht solves it, whih, given BA-network N = (V, E), guesses the reltive position of 2 V points n heks if every onstrint is respete. On the other hn, these prolems re often solve vi populr heuristis suh s onstrint propgtion n lol onsisteny. A network N is si to e k-onsistent if, given ny onsistent reliztion of k 1 vriles, there exists n instntition of ny k-th vrile suh tht the onstrints etween the suset of k vriles n e stisfie together; it is si to e strongly k-onsistent if it is k -onsistent for every k k (see [17]); if network is strongly k-onsistent, then it must lso hve miniml lels. Beuse of the prtiulr nture of networks of onstrints in temporl lgers, they re lwys 1-onsistent (lso lle noe onsistent) n 2-onsistent (lso lle r onsistent), y efinition. Enforing pth onsisteny, tht is, 3-onsisteny, in network N, orrespons to pply the following simple lgorithm: for every triple (s, t, z) of vriles in N = (V, E) suh tht srt, sr 1 z, tr 2 z E, reple srt y s(r (R 1 R 2 ))t. Clerly, if enforing pth onsisteny results in t lest one empty onstrint, the entire network N is not onsistent. But, in generl, enforing pth onsisteny (in ft, k-onsisteny for ny onstnt k) oes not imply onsisteny; this is true for BA-networks s well s for IA-networks. In [25], however, it is proven tht enforing pth onsisteny is equivlent to omputing the miniml lels of IA si -network, whih, in turn, llows one to hek the existene of moel. This property of pth onsisteny is shown for more generl set of reltions, lle onvex intervl reltions, whih re efine strting from the set P A onvex of onvex point reltions, n, in prtiulr, it is prove tht the set IA onvex (the onvex intervl lger) inlues IA si, n tht its onsisteny prolem (n, s orollry, the onsisteny of P A onvex -onstrints) n e eie y pth onsisteny. This result, prtiulrly interesting for us, hs the following onsequenes. First, IA onvex is frgment of the IA with trtle (in ft, ui time) onsisteny prolem. Seon, one n implement simple rnh-n-oun lgorithm to eie the onsisteny of ny IA-network, se on IA si : t eh step, the lgorithm tries one si reltion for eh reltion, n then fores the pth onsisteny of the resulting network; if t ny step the network is pth onsistent, it returns true, n if every omintion hs een trie n enforing pth onsisteny hs lwys resulte in n empty reltion, it returns flse. Thir, the set IA onvex n e use to rive the splitting in suh n lgorithm, s heuristis to spee up the rnh-n-oun proess: if, t ny step, one ens up with IA onvex -network, tht prtiulr rnh n e eie y simply enforing pth onsisteny. In the following, we shll efine the lger BA onvex of onvex rnhing reltions. We shll prove tht, sine BA onvex extens BA si, n sine enforing pth onsisteny n e use to eie the onsisteny of BA onvex -network, one n pply the sme shem, effetively lifting ll ove results to the setting of rnhing time. A motivting senrio. Sheuling is the prolem of istriuting omputing resoures (suh s proessor time, nwith, or memory) to vrious proesses, thres, t flows, n pplitions tht nee them. In rootis, sheuling is use to orgnize tsks to e ssigne

7 M. Gvnelli, A. Pssntino, n G. Sivio 12:7 to roots of vrious kins, in suh wy tht ll physil n sujetive onstrints re met. A reent pplition of Allen s Intervl Alger to the sheuling of tsks for root hs een propose y Murová n Hwes [18]. The uthors propose sheuling tehnique tht tkes into ount series of onstrints, inluing elines n proessing time for eh one of series of tsks tht root is ske to omplete. They propose to pply onsisteny heking lgorithm to the network of qulittive onstrints tht unerlies the sheuling prolem, in orer to prune ny orering of the tsks tht oes not meet the qulittive onstrints, n to e le to selet, systemtilly, possile moels of the prolem. To eh of the moels, then, suessive phse of quntittive onstrint heking is pplie. Murová n Hwes solve the sheuling prolem of single root, whih enompsses ssuming the time to e liner n introuing the itionl onstrint tht no two tsks n overlp. Using Brnhing Alger inste of Intervl Alger s prt of the sheuling lgorithm, n using the reltion to relx, when possile, the non-overlpping onstrint, one my otin, inste, rnhing moels mong the solutions. A rnhing moel of the sheuling n e interprete s sheuling in whih more thn one root is involve, tht is, in whih every new rnh impliitly refers to new root eing tivte (t the rnhing point), n, if we ssume tht ootstrpping root hs some fixe ost (higher thn mintining root tive), then it mkes sense to look for rnhing sheuling s we hve efine it, tht is, tree-like (in whih rnhes never join gin). Thus, Brnhing Alger llows one to generlize this sheuling prolem to more omplex senrio, whih nnot e esily hnle in the originl formultion. 3 Convex Brnhing Intervl Reltions We strt y efining the onept of onvex reltion in the rnhing setting. In this setion, we operte with trnsltions from intervl onstrints to point onstrints; when neessry, for n intervl vrile X, we use the symols X, X + to enote the point vriles tht orrespon to its enpoints. Definition 1. The onvex rnhing point lger is the set of reltions: BP A onvex = {{=}, {<}, {<, =}, {<, =, >}, {=, >}, {>}, { }}. Eh reltion of set BP A onvex is lle onvex rnhing point reltion. Oserve tht BP A onvex extens the onvex point lger P A onvex s efine in [25] y ing the reltion { }. While the set P A onvex is lose uner omposition, inverse, n intersetion, the set BP A onvex is lose uner uner inverse n intersetion only; this, however, oes not prevent us from pplying onstrint propgtion lgorithms suh s pth onsisteny enforing. Definition 2. The onvex rnhing intervl lger BA onvex is the set of ll (n only) BA-reltions R suh tht the onstrint XRY n e trnslte to n equi-stisfile onjuntion of onstrints etween the enpoints of X n Y using BP A onvex -reltions only. A rnhing reltion with suh property is lle onvex rnhing intervl reltion. For exmple, in the liner setting, {, m} is onvex euse, if X = [X, X + ] n Y = [Y, Y + ], the onstrint X{, m}y is equivlent to the onjuntion of the onstrints X < X +, Y < Y +, n X 1 + {<, =}Y ; onversely, the reltion {, i} is not onvex, euse trnslting it results in isjuntion of point-se onstrints. Clerly, the set BA onvex extens the set IA onvex of onvex intervl reltions s efine in [25]. The T I M E

8 12:8 Deiing the Consisteny of Brnhing Time Intervl Networks N k 1 (int. vr.) M k 1 (point vr.) x 3 =, =, <, = x 2 >, = =, >, x 4 ple 2 new point vr. show k-ons. T k 1 (enpoints) x 1 () () Figure 3 A generl view of the strtegy for the inutive se of Theorem 5 (left-hn sie), n pth-onsistent network with non-miniml lels (right-hn sie). ruil property of BA onvex -network N is tht it n e trnslte to n equi-stisfile BP A onvex -network M (notie tht this is not true for generi BA-network: non-onvex onstrints my result in isjuntions tht nnot e represente in the lnguge of full BP A-networks), suh tht, if N is k-onsistent for ny k, then M is 2 k-onsistent. As it n e esily heke, there re preisely 91 onvex rnhing intervl reltions. In the liner se, the following results hol [25]. Theorem 3. Enforing pth onsisteny in IA onvex -network is suffiient to ompute its miniml lels. Corollry 4. Enforing pth onsisteny in P A onvex -network is suffiient to ompute its miniml lels. Our purpose in this setion is to generlize the ove theorem to the rnhing se. We wnt to prove tht we n eie the onsisteny of BA onvex -network y enforing its pth onsisteny. To this en, we prove tht enforing pth onsisteny of BA onvex - network tully enfores the miniml lels on eh onstrint y proving tht, in ft, enforing pth onsisteny of network entils enforing its strongly k-onsisteny for every k. As we hve lrey oserve, this llows us to hek the onsisteny of network. Theorem 5. Enforing pth onsisteny in BA onvex -network is suffiient to ompute its miniml lels. Proof. Let N e BA onvex -network, n let M e the BP A onvex -network tht results from trnslting N in the lnguge of enpoints. We ssume tht pth onsisteny hs een fore on N, n we wnt to show tht N is strongly k-onsistent for every k; sine strongly k-onsistent network must hve miniml lels, we hve the result. Let us proee y inution. As se se, we know tht N is k-onsistent for k 3. As for the inutive se, we suppose now tht N is k 1-onsistent n we prove tht it is lso k-onsistent. Consier suset S of k 1 intervl vriles in N. Let us ll N k 1 the su-network, with the k 1 intervl vriles X 1,..., X k 1, orresponing to the projetion of N over set S, n let us ll M k 1 the orresponing BP A onvex -network whose vriles re preisely the 2 (k 1) enpoints of X 1,..., X k 1. Our strtegy, s skethe in Fig. 3, n e summrize s follows: sine N k 1 is onsistent y hypothesis, M k 1 must e onsistent s well, tht is, it must relize in rnhing moel T k 1 ; if we pik the point vriles orresponing to the enpoints of ny k-th intervl vrile n ommote them in T showing tht every onstrint is respete, then we otin rnhing moel for k intervl vriles, proving tht N k is lso onsistent. Let X e ny intervl vrile in N ifferent

9 M. Gvnelli, A. Pssntino, n G. Sivio 12:9 from X 1,..., X k 1, n let XR i X i the BA onvex -reltion etween the vriles X n X i, for eh i. Let M k the BP A onvex -network otine y ing to M k 1 the point vriles X, X +, the onstrint X < X +, n every onstrint etween the enpoints of X n the enpoints of X 1,..., X k 1 tht results from trnslting the onstrints of the type XR i X i. On T k 1 we n ientify the set R = { 1,..., n } with the following hrteristis: for eh i, i is the reliztion of some point vrile y in M k 1 (tht is, i is the reliztion of some enpoint of the intervl vriles X 1,..., X k 1 ) n tht, for every point vrile y M k 1, relize in some point T k 1, it is not the se tht < i. Inee, onsier the rnhing moel T k 1 : sine it must e tree, it my e the se tht, in orer to relize two vriles tht re onstrine to e inomprle to eh other, gretest ommon preeessor must e e; therefore, if projete to the points tht relize some point vrile in N k 1, T k 1 is forest of trees, rther then tree. Every point in R is the root of one of the trees in T k 1 ; let us ll their gretest ommon preeessor. Now, let x 1,..., x m e the point vriles tht hve een relize in 1,... n (oserve tht n m k 1: two vriles my hve een relize in the sme root, n m nnot exee k 1 euse, t most, every intervl vrile hs its left enpoint relize in root). We wnt to show, first, tht the point vrile X n e suessfully relize on T k 1, n we proee se y se. Suppose, first, tht (x l, X ) = { } for every point vrile x l relize in some root. In this se, we relize X with new point suh tht i for eh root i, n tht <. To prove tht this is onsistent hoie, onsier ny point vrile y of M k 1 relize t some point i for some root i. Suppose tht i is the reliztion of some point vrile x l, whih mens tht (y, x l ) lin. If < (y, x l ), then (y, x l ) (x l, X ) = {<, }. By intersetion with BA onvex, then either (y, X ) = {<} or (y, X ) = { }; in the first se, however, we otin, y pth onsisteny, tht (X, x l ) lin, whih is ontrition. Therefore, (y, X ) = { }. If, on the other hn, </ (y, x l ), then, (y, x l ) {>, =}, n, sine {>, =} { } = { }, it must e the se tht (y, X ) = { }. Suppose, now, tht (x l, X ) lin for some point vrile x l relize in some root i. Oserve, first, tht if (X, x l ) = {<}, then we n selet the suset R R suh tht, for eh x l R, we hve tht (X, x l ) = {<}; in this se, y the rgument in the ove se, for eh x l R \ R, we hve tht (X, x l ) = { }. Consier eh j tht is the reliztion of some vrile in R : we relize X in point >, suh tht is less thn every suh j, n inomprle with every other root in R \ R. If, otherwise, (X, x l ) {>, =}, then, for eh x l relize in some root j i, we must hve (X, x l ) = { }. In this se, we n sy tht i is the root of the tree in whih we hve to relize X ; let us ll it T i. Oserve tht, y the sme rgument s in the ove se, wherever we relize X in T i, this reliztion is onsistent with ny point tht elongs to some T j with j i. Now, we onsier the point T i whih is the lest point (greter thn or equl to i ) with t lest two immeite suessors 1, 2 suh tht 1 2, if it exists. We hve the following ses. Suppose tht oes not exists. This mens tht T i is linerly orere. Let e the lest point (greter thn i ), suh tht is the reliztion of some vrile y suh tht (y, X ) = { }. If there is no suh, then, y Theorem 3, we n fin reliztion for X onsistent with T i ; sine we lrey know tht suh reliztion is onsistent with every other tree, we onlue tht it is onsistent. If exists, then we relize X in point suh tht n tht > where is the immeite preeessor of. By the rgument use in the first se, this hoie must e onsistent with T i, n therefore it must e onsistent. T I M E

10 12:10 Deiing the Consisteny of Brnhing Time Intervl Networks Suppose, now, tht exists. If y is relize in, n {<, =, } (X, y), then we proee s in the previous se. If, on the other hn, (X, y) = {>}, we hve the following two ses. First, if < (X, y) for every y j relize in some point j suh tht j is immeite suessor of, then relize X in point suh tht < n tht < j for every immeite suessor j of, whih must e onsistent hoie, given tht, y pth onsisteny, the reltion etween X n every vrile relize in point greter thn must ontin <. If, for some immeite suessor j of, whih is the reliztion of some vrile y, it is the se tht </ (X, y), then we n tret every immeite suessor j of s the root of some su-tree of T i, n therefore we n pply the sme entire rgument, reursively. Hving relize the vrile X, the network M k 1 enrihe with X (n ll reltive onstrints) must e onsistent. By repplying the entire rgument, we n show tht ny other point vrile n e onsistently relize in the resulting network; if we hoose X + mong these, we prove tht the originl network N is, in ft k-onsistent, ompleting the inution. Corollry 6. Enforing pth onsisteny in BP A onvex -network is suffiient to ompute its miniml lels. It woul e nturl, t this point, to sk whether the set BP A onvex n e enrihe while retining the ove nie properties. A nturl nite in this perspetive woul e the set of ll BP A-reltions suh tht, for eh R, it is the se tht R lin (whih is liner reltion) is onvex. This set is lose uner omposition, inverse, n intersetion, llowing one to pproh its onsisteny prolem in the sme wy s we i in this work; in prtiulr, it woul e possile to efine the set of BA-reltions tht n e trnslte in this lnguge, otining, in ft, greter set of rnhing reltion whose miniml lels oul e enfore y pth onsisteny. Unfortuntely, there is n esy ounter-exmple to this lim, shown in Fig. 3, in whih we hve pth onsistent network ut with non-miniml lels. Therefore, if there exists ny extension of BP A onvex whose miniml lels n e enfore y pth-onsisteny, it nnot inlue t lest one of the reltions in Fig. 3. Oserve tht enforing the miniml lels vi pth onsisteny is not the only wy to prove tht the onsisteny of network n e eie vi pth onsisteny, n it is ertinly not the only wy to prove tht frgment of reltions is trtle. For exmple, the trtility of the onsisteny prolem for network of IA-reltions in the ORD-Horn frgment is proven vi emeing into the Horn frgment of propositionl logi [14]; s n nother exmple, the trtility of the onsisteny prolem for network of full BP A-reltions is proven with speilize lgorithm in [5], n it is the sis for Rgni n Wölfl s result out the trtility of the onsisteny prolem for network of BA si -reltions. 4 Experiments In orer to evlute the usefulness of the onvex frgment s heuristis for the tsk of heking the onsisteny of BA-network, we evise set of experiments, following the lssil methoologies in the literture. To generte rnom set of instnes, we use (moifition of) tehnique suggeste y Renz n Neel [22] tht onsists of the following steps. Given numer n of noes, n verge ensity n proility p, we generte rnom instne s follows:

11 M. Gvnelli, A. Pssntino, n G. Sivio 12:11 Algorithm 1 Bktrking lgorithm. funtion Consistent(P, Split) enfore generlize r onsisteny on P if there is vrile ν XY suh tht D XY = then return flse else hoose n unproesse vrile ν XY suh tht D XY / Split if there is no suh vrile then return true {D 1,..., D p }=Prtition(D XY, Split) for ll D i {D 1,..., D p } o P = P DXY /D i if Consistent(P, Split) then return true return flse (i) we generte grph with n noes, n selet n (n 1) 2 eges t rnom; (ii) for eh selete ege (s, t), we generte BA-reltion R y seleting, with proility p, eh BA si -reltion to e inserte in R, n (iii) to eh non-selete ege (s, t), we ssign the universl reltion. As muh s the solver is onerne, we se ourselves on the generl strtegy of onstrint logi progrmming on finite omins, y mens of whih we re le to hek the stisfiility of temporl network. The solver itself is se on the ul CSP, enoe with trnsltion into ternry onstrints, s propose y Conott et l. [6]. In prtiulr, given n A-network N = (V, E), we efine CSP P s triple V, D, C, suh tht: the set V ontins CSP vrile ν XY for eh pir of vriles X, Y in V ; the omin D XY of eh vrile ν XY is preisely the onstrint (X, Y ), n for eh triple of vriles, C ontins ternry onstrint, so tht eh ternry onstrint will e stisfie y triple (r, r 1, r 2 ) BA 3 si if n only if r r 1 r 2. As note y Conott et l. [6], enforing the (generlize) r onsisteny on the prolem P is equivlent to enforing pth onsisteny on the originl A-network. Sine, s we know, oth pth onsisteny n (generlize) r onsisteny re inomplete lgorithms for BAnetworks, ktrking serh is pplie, n to eh noe of the serh tree, (generlize) r onsisteny is enfore. The generi shem of ktrking lgorithm n e esrie s in Alg. 1 [19]. In Alg. 1, the fmily of sets Split plys key role. When solving the generl CSP, without exploiting ny trtle frgment of the BA, Split n e thought s ontining ll singletons, eh one of them orresponing to single BA si -reltion. Therefore, the tehnique to solve prolem P onsists of simply hoosing vrile, sustitute its omin with one of its omponents reting new prolem P, n reursively solve P. This lgorithm is orret euse the serh termintes with noe with si reltions only, for whih pth onsisteny is omplete metho. When igger frgment for whih enforing pth onsisteny is known to e omplete for onsisteny, we n tke vntge from it y setting Split to e the fmily of its reltions; in suh se Alg. 1 stops the serh even if the omin of some of the CSP vriles is not singleton, otining, e fto, smller rnhing ftor of the serh tree. Unfortuntely, estlishing if set n e prtitione into smller sets tken from some fmily of sets orrespons to the set prtitioning T I M E

12 12:12 Deiing the Consisteny of Brnhing Time Intervl Networks 100 Time (s) Density (%) Figure 4 Geometri men of the omputtion time to solve the instnes vrying the ensity of the network. Numer of noes n = 15, 20 instnes solve per point. The lk squres show the running time of the ktrking lgorithm with Split = BA si, while the re tringles represent the urve with Split = BA onvex. 300 Time (s) Numer of noes Figure 5 Geometri men of the omputtion time to solve the instnes vrying the numer of noes n of the network. Density = 70%, 20 instnes solve per point. The lk squres show the running time of the ktrking lgorithm with Split = BA si, while the re tringles represent the urve with Split = BA onvex. prolem, whih is NP-omplete in generl (in Alg. 1 this step is enoe into funtion Prtition(D, Split)). In our se, the omin n e prtitione into si reltions (so polynomil solution exists), ut suh solution is inonvenient, s it oes not exploit the trtility of the frgment. Sine this prolem shoul e solve in every noe of the serh tree, quik, lthough non-optiml, metho is mntory to otin resonle effiieny. A first solution woul e to pre-ompute the (possily, optiml) solutions of the set-prtitioning for eh possile suset of the omins; this woul generte tle of size 2 BA si, whih my fit into the min memory of moern omputers, ut poses the prolem of effiiently essing to the t struture tht ontins it. Another option is to use greey metho to quikly provie possily non-optiml prtitioning (note, lso, tht it is not neessry for Prtition(D, Split) to return the omplete prtitioning, s eh of the sets D 1,..., D p n e generte on emn). We eie to follow the ltter pproh n to store the set of onvex reltions into trie, whih is t struture whose ess time epens on the orer in whih the elements of set re store, ut tht resulte effiient in prtie. The lgorithm ws implemente in the onstrint logi progrmming system ECL i PS e [24], using the CLP(FD) lirry. To implement the ternry onstrints we use the propi

13 M. Gvnelli, A. Pssntino, n G. Sivio 12:13 80 Time (s) Numer of noes Figure 6 Geometri men of the omputtion time to solve the instnes vrying the numer of noes n of the network. Eh point is the geometri verge of 140 instnes, otine with ensity vrying from 70% to 100%. The lk squres show the running time of the ktrking lgorithm with Split = BA si, while the re tringles represent the urve with Split = BA onvex. lirry [16] tht provies generl n very elrtive wy to implement new onstrints, lthough we re wre tht more effiient implementtions oul e possile. The ojetive of the experimenttion ws isussing the reltive improvement given y the exploittion of the onvex frgment, rther then evluting the solute performnes of our implementtion. All experiments were run on Intel Core i7-3720qm 2.60GHz running ECL i PS e Version 6.1 #224 on Linux Mint 18.1 Seren 64 its, n using only one ore. Timeout ws fixe to 10 minutes. In Fig. 4, we fixe the numer of verties of the A-network to n = 15, the proility p = 1/2 n vrie the onstrint ensity from 5% to 100%. Eh point in the urve represents the geometri men otine running 20 instnes. Sttistilly speking, smll numer of very iffiult networks my e proue in set of 20 rnom instnes; the net effet on the omputtion time of suh instnes n e softene using the geometri men rther then the, more ommon, rithmeti men [8]. The shpe of the urve shows the expete phse trnsition: when the ensity is low, most of the instnes re esily stisfile, while to high ensity orrespon networks for whih proving unstisfiility is esy. The phse trnsition ours t ensity roun = 80%, in whih oth stisfiility n unstisfiility re hr to prove. The urves in Fig. 4 (in whih the re urve represents the performne of the lgorithm when the onvex frgment is tken into ount) show tht exploiting the onvex frgment is prtiulrly onvenient for hr prolems ner the phse trnsition, while the overhe tht, impliitly, is introue in suh solution mkes it not worth for esily stisfile prolems. In Fig. 5 we fixe the ensity to point lose to phse trnsition ( = 70%), n vrie the numer of noes in the grph, from 3 to 20. Eh point is the geometri men of 20 runs. At 70% ensity, most prolems uner 20 noes re iffiult, n, gin, exploiting the onvex frgment is onvenient with respet to the expete performne. Finlly, we investigte the omputtion time of the two solutions vrying the numer of noes (from 5 to 20) inepenently of the ensity. To this en, we generte, for eh numer of noes, 140 instnes with ensities vrying from 70% to 100%, n onsiere the geometri men of the time neee to solve them. In Fig. 6 we show the result of suh n nlysis, tht proves tht ertin improvement in omputtion time exists when the onvex frgment is tken into ount. T I M E

14 12:14 Deiing the Consisteny of Brnhing Time Intervl Networks 5 Conlusions Allen s Intervl Alger is one of the most prominent formlisms in the re of qulittive temporl resoning. However, its pplitions re nturlly restrite to liner flows of time, rising the question of whether one n reson out rnhing (tree-like) flows of time in similr mnner. We onsiere, in this pper, the set of 19 rnhing reltions suggeste y Rgni n Wölfl, whih enjoy the esirle hrteristis of eing expressile in the lnguge of enpoints without quntifition. Rgni n Wölfl hve shown tht the onsisteny prolem for network of rnhing reltions is intrtle (s expete), while the onsisteny prolem for network of si rnhing reltions is polynomil. In ler prllelism with the liner se, we efine the set of onvex rnhing reltions, whih extens the set of si rnhing reltions, n we prove tht enforing pth onsisteny of network of onvex reltions is suffiient to eie its onsisteny, effetively proviing the first non-trivil trtle (vi pth onsisteny) frgment of the rnhing lger. As nother onsequene of this work, we me it possile to tret the onsisteny prolem of network of onstrints s onstrint propgtion prolem, llowing not only the possiility of quik implementtions using well-known lirries, ut, lso, the possiility of implementing lever rnh-n-oun lgorithm for generi network tht exploits the trtility of onvex reltions s n heuristis. Finlly, we teste suh solution, giving experimentl eviene of the expete improvement. The most interesting open prolems t the moment inlue, mong other, the question of whether the onvex rnhing lger is mximl with respet to trtility of the network onsisteny prolem (whih seems unlikely) n/or with respet to the possiility of enforing the miniml lels of network vi pth onsisteny, the question of whether other populr n well-ehve frgments of the intervl lger in the liner se n e generlize to the rnhing setting preserving their omputtionl ehviour, n the question of whether effiient enumerting lgorithms n e evise for the rnhing se s it hs een one in the liner se. We lrey know tht there is set of rnhing point reltions whih oul e onsiere nturl generliztion of onvex rnhing point reltions n whose miniml lels prolem nnot e solve y pth onsisteny; however, frgments of the rnhing point lger tht stritly inlue BA onvex n for whih the miniml lels of network n e enfore y pth onsisteny re still possile. Referenes 1 J.F. Allen. Mintining knowlege out temporl intervls. Communitions of the ACM, 26(11): , J.F. Allen n P. J. Hyes. Short time perios. In Pro. of IJCAI 1987: 10th Interntionl Joint Conferene on Artifiil Intelligene, pges , R. Alur, T.A. Henzinger, n O. Kupfermn. Alternting-time temporl logi. Journl of the ACM, 49(5): , D. Bresolin, A. Montnri, n P. Sl. An optiml tleu for right propositionl neighorhoo logi over trees. In Pro. of TIME 2008: 15th Interntionl Symposium on Temporl Representtion n Resoning, pges IEEE, M. Broxvll. The point lger for rnhing time revisite. In Pro. of KI2001: Avnes in Artifiil Intelligene, volume 2174 of Leture Notes in Artifiil Intelligene, pges Springer, J.F. Conott, D. D Almei, C. Leoutre, n L. Sïs. From qulittive to isrete onstrint networks. In Pro. of the Workshop on Qulittive Constrint Cluli hel with KI 2006, pges 54 64, 2006.

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