When does a Composition Table Provide a Complete and Tractable. Proof Procedure for a Relational Constraint Language?

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1 When does Composition Tle Provide Complete nd Trtle Proof Proedure for Reltionl Constrint Lnguge? Brndon Bennett, Amr Isli nd Anthony Cohn Division of Artiil Intelligene Shool of Computer Studies University of Leeds, Leeds LS2 9JT, Englnd Astrt Originting in Allen's nlysis of temporl reltions, the notion of omposition tle (CT) hs eome key tehnique in providing n ef- ient inferene mehnism for wide lss of theories. In this pper we hllenge reserhers working with CTs to give generl hrteristion of the lss of theories nd reltionl onstrint lnguges for whih omplete proof proedure n e speied y CT. Severl relted prolems nd onjetures will e disussed. A seondry im is to lrify the terminology used to desrie CTs nd to estlish generl oneptul frmework pplile to ny CT whtever reltions re involved. One of the min dvntges of using CTs is tht they n led to trtle omputtion of signint lsses of inferene. An importnt spet of our proposed reserh progrmme is to seprte omputtionl from purely logil issues nd then to systemtilly investigte reltionships etween logil nd omputtionl omplexity. 1 Resoning out Reltions In representing nd resoning out mny domins useful informtion n often e stted in terms of limited voulry of inry reltions holding mong ojets. Typilly suh fmilies of reltions will e logilly onstrined in tht ertin omintions of reltions re possile whilst others re impossile. The logil dependenies etween reltions my e stted in mny wys. If suh representtions re to e useful for some omputtionl pplition, we must hve prtil method of determining onsequenes of sets of reltionl fts. The trditionl pproh to this prolem is to formulte dependenies etween reltions s forml theory This work ws supported y the EPSRC under grnt GR/K stted in some generl-purpose logil lnguge (e.g. 1storder logi). If this theory is onjoined with set of reltionl fts, onsequenes of these fts n e determined using ny proof proedure whih is omplete for tht lnguge. From omputtionl point of view, this pproh is unstisftory for ll ut the simplest sets of reltions. Resoning with suiently expressive generl-purpose logi is in most ses undeidle nd t est n NP-omplete prolem. Another method widely employed is to formulte the tsk s onstrint stisftion prolem (CSP) [Tsng, 1993]: eh reltion is interpreted s onstrint restriting possile vlues of its rguments. Although testing stisility of CSPs is in generl intrtle, for mny restrited lsses of CSP eetive lgorithms exist. We re onerned with investigting reltionl onstrint lnguges onsisting of nite voulry of si reltionl expressions ( sis) nd n enumerle set of onstnts. In the urrent work we restrit ttention to dydi reltions. 1 We lso will wnt to tlk out omplex reltions whih re logil funtions of the si reltions. A reltionl onstrint lnguge my e prt of some more omprehensive theory. Alterntively we my know tht for prtiulr prolem domin ertin voulry of reltions is useful so tht the hoie of reltions preedes ny theoretil nlysis of the domin. 1.1 Composition Tles Given set Rels of si reltion symols, omposition tle is mpping CT : Rels Rels! 2 Rels i.e. CT speies for eh ordered pir hr; Si, where R nd S re elements of Rels, suset S Rels, lled the omposition of R nd S nd written CT (R; S). It is usul to ssume tht the elements of Rels refer to reltions tht form jointly exhustive nd pirwise disjoint (JEPD) prtition of the possile reltions whih n hold etween pirs of ojets in the domin under 1 Mondi properties n strightforwrdly e represented s speil kind of reexive reltion. In ft ny reltion n e represented in terms of inry reltions y reifying the reltions ut the resulting theory my e muh less nturl.

2 onsidertion. In other words, ny (ordered) pir of ojets in the domin re relted y extly one of the memers of Rels. Under these onditions ny Boolen omintion of reltions is equivlent to disjuntion of memers of Rels. The preise mening of CT depends to some extent on the ontext in whih it is employed. Sometimes it is reord of ertin kinds of onsequene of some underlying theory whih my lredy e fully or prtilly formlised. Alterntively, the speition of CT my preede the development of forml theory of the reltions involved nd is n initil step in speifying the theory of some set of intuitively understood reltions. In either se, the fundmntl mode of resoning enoded in CT is to test onsisteny of trids of reltions of the forms R(; ), S(; ), T (; ), where R; S; T 2 Rels: suh trid is onsistent i T 2 CT (R; S). The notion of the ompleteness of CT ppels to some underlying theory or intuition of the menings of the reltions involved. We sy tht CT of some reltions Rels is (refuttion) omplete w.r.t. some (possily unformlised) theory, if whenever set of (ground) fts involving only reltions in Rels nd onstnts is inonsistent this n e deteted y referene to the tle: i.e. we n nd reltions R(; ), S(; ), T (; ), s.t. T 62 CT (R; S). 1.2 Comptness Sets of ground reltions re nturlly represented y networks. The size of network is the numer of its nodes nd in this ontext this is the numer of distint onstnts whih our in the reltionl fts. Completeness of CT is intimtely onneted with the reltionship etween lol onsisteny of trids of reltions nd overll onsisteny of network. Beuse of this the notion of the mximum network size whih must e heked for onsisteny is very useful. We sy tht sis set Rels is k-ompt w.r.t. theory i for ny network of reltions drwn from Rels the network is inonsistent i it inludes n inonsistent su-network of size k or less. For some sets of reltions we my nd tht there n e ritrrily lrge inonsistent networks ll of whose su-networks re onsistent. We sy tht these re not nitely ompt. 2 We oserve tht CT is omplete w.r.t. i its sis set is 3-ompt w.r.t.. Clerly not ll theories re 3-ompt: e.g., if theory denes its ojets to hve the properties of diss of equl size nd we hve set of reltions inluding the reltion of externl onnetion then ny given irle n e externlly onneted to mximum of six other irles. Hene if we speify sitution in whih seven regions 2 If there n e n innite inonsistent network with no nite inonsistent su-network the reltion set is not ompt t ll. re ll mutully externlly onneted this is inonsistent; ut this nnot e deteted y heking only the reltions etween sets of three regions [Cui et l., 1993]. 1.3 Forml Theories nd CTs To give preise mening to the lim tht theory entils or is equivlent to CT ( j= CT or CT ) we need to speify the mening of CT in terms of theory. A wek speition, 3 whih we ll the onsistenysed denition, is s follows: Given theory in whih set Rels of se reltions is dened, the omposition, CT (R; S), where R; S 2 Rels is dened to e: the set of ll reltions Ti 2 Rels, s.t. there exists individul onstnts,,, for whih the formul R(; ) ^ S(; ) ^ Ti(; ) is onsistent with. This mens tht if CT (R; S) = ft 1 ; : : :; T n g then j= 8x8y8z[R(x; y) ^ S(y; z)! (T 1(x; z) _: : : _ Tn(x; z))] nd, furthermore, ft 1 ; : : :; T n g is the smllest suset of Rels for whih suh formul is provle. 4 Thus CT (R; S) is the disjuntion of ll possile se reltions whih ould hold etween nd. So, if one hs onsisteny heking lgorithm for sets of instnes of the reltions in Rels, the omposition of ny pir of reltions in Rels n e omputed. The onsisteny-sed denition of omposition is equivlent to tht given in [Rndell et l., 1992] nd lso seems to orrespond with wht [Egenhofer nd Frnzos, 1991] men y the omposition of two reltions. However, there is nother denition of the omposition of two reltions whih is stndrd in set theory: Let R 1 e reltion from A to B nd R 2 e reltion from B to C (i.e. A, B nd C re sets, R 1 AB nd R 2 BC), Then the omposition of R 1 with R 2, R 1; R 2 is the set of ll ordered pirs, h; i 2 AC suh tht, for some 2 B, h; i 2 R 1 nd h; i 2 R 2. This purely extensionl denition is stritly stronger tht the onsisteny-sed denition euse not only does it ensure tht whenever we hve R(; ) nd S(; ) we must lso hve R; S(; ) ut it lso requires tht whenever R; S(; ) (i.e. nd re relted y ny one of the se reltions mking up the generlly disjuntive reltion R; S(; )) there must exist some region, sy, s.t. R(; ) nd S(; ). 8x8y[R; S(x; y) $ 9z[R(x; z) ^ S(z; y)]] () We oserve tht the extensionl omposition R; S lwys yields reltion whose extension is suset of the unions of the extensions of the reltions given y 3 There re even weker speitions. 4 This suset is unique if the reltions T 1; : : : ; Tn re disjoint.

3 CT (R; S), where CT stises the onsisteny-sed definition given ove. Also, if CT n e interpreted extensionlly then, s well s providing mens for onsisteny heking of ground reltion sets, it n e employed to justify ertin kind of existentil inferene: from reltion holding etween two ojets we dedue the existene of third ojet relted to the originl two in spei wy. We suggest tht CT is omplete w.r.t. theory i implies ll formule orresponding to the extensionl interprettion of omposition. If omposition is not extensionl then CT (R; S) my e more generl reltion thn R; S. This mens tht informtion is lost when (onsisteny-sed) ompositions re omputed vi CT. Thus if onsisteny of network is tested y propgtion of onstrints imposed y CT we my nd tht it seems to e onsistent when it is tully inonsistent. One might further onjeture tht y rening reltions in set Rels one n lwys rrive t set Rels 0 whih is more expressive thn Rels nd whose CT n e interpreted extensionlly. However, if, s we suggest, n extensionl omposition tle yields omplete onsisteny heking proedure, this is unlikely to e true, sine then resoning with ny reltionl onstrint lnguge would e deidle. 1.4 Algeri Anlysis of Reltions A formlism, whih hs een vlule in the nlysis of temporl reltions nd resoning lgorithms [Ldkin nd Mddux, 1994], is reltion lger, in whih reltions re onsidered s elements of Boolen lger ugmented with omposition nd onverse opertors oeying xioms rst speied y [Trski, 1941]. A reltion lger (RA) is Boolen lger whih hs in ddition to the usul sum (+), produt (:) nd omplement (?), two dditionl opertors: inry omposition opertor, `;', nd unry onverse opertor, `^'. It lso hs onstnt 1 0, denoting the identity reltion. The ojets of reltion lger re intended to e inry reltions oneived of s sets of pirs. 5 Under the intended interprettion, ; ehves s if dened y the 1st-order formule () given ove nd ^ nd 1 0 s if dened y R^(x; y) def R(y; x) nd 1 0 (x; y) def (x = y). But in RA reltions re si entities nd the opertors re given n lgeri hrteristion. The menings of ;,^ nd 1 0 re thus xed y mens of set of equtions whih re tken s xioms [Trski, 1941]. It is known tht, in gererl, resoning in RA is undeidle nd this ounts ginst the potentil usefulness of these lgers in utomted resoning. However 5 However, it turns out tht this stndrd interprettion is not possile for every RA. for mny spei lgers, onsisteny heking is deidle nd indeed my e polynomil. 6 Indeed if n extensionlly interpreted omposition tle n e given for voulry of si reltions in reltion lger, this n e used to eliminte the omposition opertion from omplex lgeri terms nd this will led diretly to deision proedure. We elieve tht the viiliy of reltion lger s formlism for utomted resoning deserves further explortion. The si reltions in set Rels my themselves e menle to nlysis in terms of equtions on some lgeri struture. If so, then eh reltion R(; ) 2 Rels is equivlent to n eqution () = (), where nd re opertors in some lger (typilly this will e Boolen lger with dditionl opertors). We elieve tht this kind of nlysis n give gret insight into the logil struture of set of reltions nd would like to know how generl is this pproh. The omintion of RA with further lgeri representtion of se reltions results in two-tier lgeri lnguge whih seems to e oth elegnt nd powerful. We enourge reserhers to look for uses for this kind of lnguge. 1.5 CTs nd CSPs In CSP [Tsng, 1993] 7 we hve set of vriles nd set of onstrints on possile vlues of these vriles. These onstrints n e regrded s set of tuples of possile ssignments (perhps not expliitly given ut heked on demnd y some proedure) or s speied y some theory. There re two wys in whih the notion of CT n e ssimilted into the frmework of CSPs. One is to tret the CT s set of ternry onstrints on vriles rnging over reltion nmes (see e.g. This pproh is pplile to ny CT nd does not tell us nything out the reltions involved. A more illuminting pproh is to regrd the reltions in sis set Rels s themselves onstituting the onstrints of CSP. This requires further nlysis of the logil struture of the reltions involved, suh s will e illustrted in setion Complexity nd Disjuntion Testing whether network of si reltions mong n ojets is onsistent with CT is trtle: it mounts to heking ll sunetworks of size three nd this n e omputed in n 3 time. There hs een onsiderle interest in generlising the use of CTs to del with resoning out reltions whih re disjuntions of those ppering expliitly in the tle. A prtiulr gol hs een to 6 Some omplexity results for resoning with reltion lgers re given y [Ldkin nd Mddux, 1994]. 7 Note tht the term `k-onsisteny' employed in these works is not diretly relted to our term k-omptess. The ondition of so-lled k-onsisteny of network is in ft independent from its eing onsistent in the usul logil sense.

4 identify sets of disjuntions whose onsisteny n lso e omputed in polynomil time. From the point of view of the ompleteness of CT the se of disjuntive ompositionl resoning is not signifintly dierent from the se of simple non-disjuntive resoning. As long s resoning for the se reltions is omplete then heking onsisteny of network ontining disjuntive reltions is just mtter of showing tht y piking one disjunt from eh r of the network one n otin non-disjuntive network whih is onsistent with the CT; nd if suh network exists it n lwys e found y ktrking lgorithm. Whether onsisteny of disjuntive networks over given set of se reltions n e heked in polynomil time is n importnt issue ut is one of trtility rther thn ompleteness. Nevertheless, like ompleteness, trtlity is losely onneted with the reltionship of lol onsisteny to overll network onsisteny. Speilly if we know tht set of reltions, some of whih my e disjuntive, is k-ompt (w.r.t. some theory) then one we hve reord of ll onsistent k-ry networks involving these reltions (suh reord my e regrded s k? 1 dimensionl nlogue of CT) we n hek onsisteny of ny network in n k time. 2 Temporl nd Sptil Reltions We now illustrte the nlyti methods desried ove y exmining their pplition to the domins of sptil nd temporl reltions. Our im is to motivte the exmintion of underlying priniples nd the pplition of these methods to other domins. 2.1 Temporl Reltions Allen's set of thirteen (JEPD) qulittive reltions etween temporl intervls [Allen, 1983] nd the lger generted from these reltions hve een quite extensively explored. The NP-hrdness of resoning out ritrry disjuntions of the temporl reltions ws demonstrted y [Vilin nd Kutz, 1986]. Ldkin hs investigted the model theory of the reltions nd their representtion in the frmework of RA [Ldkin nd Mddux, 1994]. A mximl trtle su-lger over the Allen reltions, lled the `ORD-Horn' sulss, hs een identied y [Neel, 1995]. This sulss inludes suset known s the onvex reltions. The rguments of Allen's temporl reltions re intervls nd these re esily interpreted s pirs of (rel or rtionl) numers hs; fi, where s is the strting point nd f the nishing point of the intervl. Under this interprettion the reltions re esily desried y order onstrints involving the end-points of intervls. It is then esy to demonstrte tht tht the CT for the Allen reltions is extensionl. [Ligozt, 1994] showed tht if we tke one intervl s xed nd represent the positions of the end points of nother intervl y oordintes in the plne, the onvex reltions tully orrespond to onvex regions on this 2D representtion of the spe of possile Allen reltions (see gure 1. m f d o s o eq f d s m Figure 1: Ligozt's nlysis of the Allen reltions 2.2 Sptil Reltions [Rndell et l., 1992] presented n xiomti theory of sptil reltions, known s the RCC (Region Connetion Clulus), whih is intended to provide logil frmework for the inorportion of sptil resoning into AI systems. This 1st-order theory is sed on primitive `onnetedness' reltion, C(x; y). The set of eight JEPD topologil reltions (known s RCC-8) shown in gure 3 re identied s eing of prtiulr importne (see lso [Egenhofer nd Frnzos, 1991]). A deision proedure for the RCC-8 reltions sed on n enoding in 0-order intuitionisti logi ws given in [Bennett, 1994]. This hs een shown to e trtle in [Neel, 1995]. The omposition tle in [Rndell et l., 1992] gives, the omposition of EC nd TPP s fec; PO; TPP; NTPPg. The extensionl interprettion of this is speied y 8x8y[ 9z[EC(x; z) ^ TPP(z; y)] $ (EC(x; y) _ PO(x; y) _ TPP(x; y) _ NTPP(x; y))] whih mens tht whenever regions, re relted y either of EC,PO,TPP or NTPP, there must e third region suh tht EC(; ) ^ TPP(; ). This is illustrted in Figure 2. As long s is n ordinry ounded region, region stisfying the pproprite onditions n e found. However the domin of RCC inludes universl region u

5 EC(,) PO(,) TPP(,) NTPP(,) Figure 2: (Almost) extensionl omposition of EC, TPP suh tht every other region is n NTPP of u. If is n ordinry region nd = u then NTPP(; ) ut no region n e found whih is TPP of. This mens tht the CT for RCC-8 nnot e given ompletely existentil reding. There re two ovious wys round this: one is to get rid of the universl region nd the other is to rene the set of reltions so tht reltions whih re true when one or more rguments is the universe re dierent from those reltions whih re true for ordinry non-universl regions. A Sptil Reltion Alger The RA formlism n e used to speify sptil RA whih desries the sme domin s the RCC theory. As in the 1st-order RCC theory, we strt with onnetedness reltion, whih is xiomtised to e symmetri nd reexive. We now denote reltions y the sme letters s their RCC ounterprts ut in lower se. Being symmetri nd reexive, must oey the xioms ^ = nd = : In terms of we n dene the reltions prt, overlps, proper prt nd tngentil prt y: p = def?(;?) o = def p^; p pp = def p?1 0 tp = def ;?o With these reltions we n go on to dene ll the reltions in the 8 reltion RCC sis firly strightforwrdly: d = def? tpp = def pp (;?o) e = def?o ntpp = def pp?(;?o) po = def o?p?(p^) tppi = def tpp^ eq = def 1 0 ntppi = def ntpp^ Thus, ll the RCC-8 reltions n e dened s RA expressions formed from the single reltion. It is likely tht dditionl xioms will e needed to pture existentil properties of the domin of sptil regions. E.g., if there is universl region whih onnets with every region in the domin then the identity ; = 1 must hold. All the reltions in RCC-8 n themselves e represented y mens of equtions nd the negtions of reltions operting on n interior lger. This is Boolen lger with n dditionl interior opertor oeying pproprite equtions [Trski, 1944]. The prolem of testing whether set of ground equtions of this theory is onsistent is deidle. When is the RCC CT Complete? The results of [Bennett, 1994] strongly suggest tht the RCC-8 CT is omplete if one interprets regions s open sets in ny ritrry topologil spe. 8 However, if one is interested in regions of more restrited nture, the CT my well eome inomplete w.r.t. the pproprite theory of regions. E.g. if we require tht ll the regions e 2D nd ounded y Jordn urves in the plne the onsisteny heking prolem eomes NP-hrd [Grigni et l., 1995] nd so ny CT for RCC-8 restrited to this domin must e inomplete. A plusile onjeture is tht ny network of RCC-8 reltions whih is onsistent ording to the omposition tle i it n e relised in model where the regions re identied with susets of Euliden 2-D spe (C(x; y) holds if the losures of x nd y shre ommon point nd O(x; y) if they shre n (interior) point). To prove this one would hve to show tht the RCC-8 reltions re 3-ompt w.r.t. the theory of this domin. 3 Neighourhoods nd Convexity We hve surveyed nd lried wht we see s the most importnt previous work on omposition tles nd hve tken re to seprte the issue of ompleteness from omputtionl issues. We now undertke more speultive exmintion of the onnetion etween the logi of the se reltions nd the omplexity of ompositionl resoning with these reltions. The ide oneptul neighourhood ws originlly pplied to Allen's temporl intervls y [Freks, 1992] s wy of grouping together reltions whih re intuitively `lose' to eh other. This loseness ws ssoited with the possility of sitution hnging from one reltion to nother y ontinuous trnsition. A oneptul neigourhood is set of reltions whih n e trversed y suh trnsitions. Freks notied tht ll the entries in the Allen CT re oneptul neighourhoods. Coneptul neighourhoods of sptil reltions (e.g. see gure 3) hve een investigted in numer of works (e.g. [Glton, 1993] ). Neighourhoods of the Allen reltions re lso known s onvex reltions. They form trtle suset of 8 In ft this hs now een proved y Bennett ut the result hs not yet een pulished.

6 = DC EC PO TPP TPPI NTPP NTPPI Figure 3: Coneptul neighourhood grph for RCC-8 those reltions whih re dijuntions of Allen's 13 intervl reltions. Moreover eh onvex reltion is determined y mximum nd minimum extreme ses nd these extremes determine their ehviour w.r.t. omposition. We suggest tht the ide of onvexity n e usefully generlised to ritrry sets of reltions. We sy tht reltion is onvex if it n e dened in terms of ertin extreme ses (whih we ll limits) y sying tht ll situtions lying etween these limits re instnes of the reltion. The most esy se to visulise is where we hve liner series of reltions. Here onvex reltion orresponds to disjuntion of reltions whih form ontinuous suseries of the possile reltions. In this se onvex reltions ould lwys e dened y two limiting ses. One ould however imgine more omplex sitution in whih possile reltions were rrnged two dimensionl spe nd three or more limit points might dene onvex reltion. 9 We elieve tht if the ide of onvex reltion is to e useful then the omposition of two reltions R nd S whih re onvex must lwys e determined y the ompositions whih hold mong the their limits. More speilly, if we ompute the ompositions of ll pirs of limit reltions (tking one from eh onvex reltion) then the omposition of R nd S should e the smllest onvex reltion ontining the union of the resulting sets of possiilities. We propose this s the denition of reltionl onvexity in generl setting. In the se of the Allen reltions it hs een found tht the lss of disjuntive reltions for whih our generl denition of onvexity holds is lso menle to polynomil time onsisteny heking. We suggest tht this my e generl phenomenon nd my provide mens of identifying trtle disjuntive lsses of other kinds of reltion. A further interesting possiility is tht, whenever reltion is, in our sense, onvex w.r.t. ertin onep- 9 E.g. this might rise if we wnted to use reltionl voulry inluding onepts suh s reltive weight nd density s well s sptil reltions euse omining these kinds of reltion genertes rih oneptul spe whih is not deomposle into simpler, logilly independent spes. tul domin, there is some n-dimensionl digrmti representtion of the spe of possile reltions in tht domin, suh tht, the reltion is identied with geometrilly onvex region of this digrm (e.g. Ligozt's nlysis of the Allen reltions). Convexity of reltion, R, whih is disjuntion of memers of JEPD set seems to e ssoited with the possiility of using dierent nlysis of the reltions in whih R n e represented without the use of logil disjuntion. e.g. If we nlyse the Allen reltions in terms of their endpoints then the disjuntion proeeds(x; y) _ meets(x; y) _ overlps(x; y) n e re-represented s strt(x) < strt(y). This illustrtes generl methodology in the representtion of systems of reltions. Considertion of some knowledge domin will often led to the lssition of the reltionship etween two ojets into some JEPD set of reltions (suh s RCC-8 or Allen's 13 intervl reltions). To represent prtil knowledge or to represent onstrints in this domin we will wnt to reson with disjuntions of these si JEPD reltions. But nive inferene with suh disjuntions will led to exponentil omputtionl omplexity. However, nlysis of the struture of onstrints in terms of whih the reltions n e dened my led to n lterntive formultion in whih lrge numer of reltions n e dened in wy whih enles omputtionlly trtle inferene. There is further notion of onvexity whih seems to e relevent to the exmintion of reltionl onvexity. Aording to [Oppen, 1980] lss of entilments is onvex i whenever? j= 1 _: : :_ n then? j= i ; for some i 2 f1 : : :ng: This property hs een used in [Bennett, 1994] to onstrut omplete nd trtle resoning system for the RCC-8 reltions. We hypothesise more generl link etween onvexity of reltions in onstrint lnguge nd onvexity of entilment in the underlying lgeri theory of these reltions. 4 Summry We hve proposed s hllenge the prolem of nding generl onditions under whih CT n provide omplete nd trtle inferene system. We hve identied the property of 3-omptness of reltionl theory nd the extensionlity of CT s key onepts in determining the ompleteness of ompositionl resoning. More speilly we hve onjetured tht ompleteness of CT requires n extensionl interprettion w.r.t. some underlying theory whih is 3-ompt w.r.t. the se reltions of the CT. We hve suggested tht the formlism of Reltion Alger is prtiulrly suited to representing ompositionl

7 resoning nd deserves further study nd wider pplition. We hve dened the RCC-8 topologil reltions from primitive onnetion reltion in terms of RA equtions. Finlly we hve presented some tenttive ides out the onepts of logil nd reltionl onvexity. We speulte tht tht the trtility of resoning with reltionl onstrint lnguge might lwys e explinle in terms of these notions of onvexity. Referenes [Allen, 1983] J F Allen. Mintining knowledge out temporl intervls. Communitions of the ACM, 26(11):832{843, [Bennett, 1994] B. Bennett. Sptil resoning with propositionl logis. In J Doyle, E Sndewll, nd P Torsso, editors, Priniples of Knowledge Representtion nd Resoning: Proeedings of the 4th Interntionl Conferene (KR94), Sn Frniso, CA., Morgn Kufmnn. [Cui et l., 1992] Z Cui, A G Cohn, nd D A Rndell. Qulittive simultion sed on logil formlism of spe nd time. In Proeedings AAAI-92, pges 679{ 684, Menlo Prk, Cliforni, AAAI Press. [Cui et l., 1993] Z Cui, A G Cohn, nd D A Rndell. Qulittive nd topologil reltionships in sptil dtses. In D Ael nd B C Ooi, editors, Advnes in Sptil Dtses, volume 692 of Leture Notes in Computer Siene, pges 293{315. Springer Verlg, Berlin, [Egenhofer nd Frnzos, 1991] M Egenhofer nd R Frnzos. Point-set topologil sptil reltions. Interntionl Journl of Geogrphil Informtion Systems, 5(2):161{174, [Freks, 1992] C Freks. Temporl resoning sed on semi-intervls. Artiil Intelligene, 54:199{227, [Glton, 1993] A P Glton. Perturtion nd dominne in the qulittive representtion of ontinuous stte-spes. Sumitted for pulition, Otoer [Grigni et l., 1995] M. Grigni, D. Ppdis, nd C. Ppdimitriou. Topologil inferene. In C.S. Mellish, editor, proeedings of the fourteenth interntionl joint onferene on rtiil intelligene (IJCAI-95), volume I, pges 901{906. Morgn Kufmnn, [Ldkin nd Mddux, 1994] P. Ldkin nd R. Mddux. On inry onstrint prolems. Journl of the ACM, 41(3):435{469, [Ligozt, 1994] G Ligozt. Towrds generl hrteriztion of oneptul neighourhoods in temporl nd sptil resoning. In F D Anger nd R Lognnthrh, editors, Proeedings AAAI-94 Workshop on Sptil nd Temporl Resoning, [Neel, 1995] B. Neel. Computtionl properties of qulittive sptil resoning: First results. In Proedings of the 19th Germn AI Conferene, [Neel, 1995] B Neel. Resoning out temporl reltions: mximl trtle suset of Allen's intervl lger. Journl of the Assoition for Computing Mhinery, 42(1):43{66, Jnury [Oppen, 1980] D.C. Oppen. Complexity, onvexity nd omintions of theories. Theoretil Computer Siene, (12):291{302, [Rndell et l., 1992] D A Rndell, A G Cohn, nd Z Cui. Computing trnsitivity tles: A hllenge for utomted theorem provers. In Proeedings CADE 11, Berlin, Springer Verlg. [Rndell et l., 1992] D A Rndell, Z Cui, nd A G Cohn. A sptil logi sed on regions nd onnetion. In Pro. 3rd Int. Conf. on Knowledge Representtion nd Resoning, pges 165{176, Sn Mteo, Morgn Kufmnn. [Trski, 1941] A Trski. On the lulus of reltions. Journl of Symoli Logi, 6:73{89, [Trski, 1944] A Trski. The lger of topology. Annls of Mthemtis, 45:141{191, [Tsng, 1993] P.K. Tsng. Foundtions of Constrint Stisftion. Ademi Press, [Vilin nd Kutz, 1986] M.B. Vilin nd H. Kutz. Constrint propgtion lgorithms for temporl resoning. In Proeedings of the 5th AAAI onferene, Phildelphi, pges 377{382, 1986.