Modify Bayesian Network Structure with Inconsistent Constraints

Transcription

1 Modfy Bayesan Network Structure wth Inconsstent Constrants Y Sun and Yun Peng Department of Computer Scence and Electrcal Engneerng Unversty of Maryland Baltmore County Baltmore, MD, Abstract Ths paper presents a theoretcal framework and related methods for ntegratng probablstc knowledge represented as low dmensonal dstrbutons (also called constrants) nto an exstng Bayesan network (BN), even when these constrants are nconsstent wth the structure of the BN due to dependences among relevant varables n the constrants beng absent n the BN. Wthn ths framework, a method has been developed to dentfy structural nconsstences. Methods have also been developed to overcome such nconsstences by modfyng the structure of the exstng BN. 1 Introducton Knowledge ntegraton nvolves modfyng the exstng knowledge base accordng to newly dscovered knowledge. Ths process can be dffcult because peces of new knowledge comng from dfferent sources may conflct wth each other or wth the exstng knowledge base. In ths stuaton we say the peces of new knowledge are nconsstent. In ths paper we focus on a specfc knd of knowledge ntegraton,.e., the dscrete probablstc knowledge ntegraton for uncertan knowledge, where the knowledge base s represented as a dscrete ont probablty dstrbuton (JPD), and peces of new knowledge are represented as lower dmensonal dstrbutons, whch are also called constrants. We are especally nterested n representng the knowledge base usng a Bayesan network (BN) for ts compact representaton of nterdependences (Pearl 1988). Each node n a BN represents a varable n the knowledge base, and arcs between the nodes represent the nterdependences among varables. All the nodes and edges form a drected acyclc graph (DAG) of the BN, whch we refer to as the structure of BN. A condtonal probablty table (CPT) s attached to each node to represent the strength of the nterdependences. Usng BN ntroduces an addtonal challenge for ntegraton,.e., a constrant may conflct wth the structure of the exstng BN when the probablstc dependences among some of the varables n the constrant are absent n the BN. We call ths knd of constrant structural nconsstent constrant. When the structural nconsstency occurs, one can choose to 1) keep the structure of the BN and ntegrate as much as possble of the knowledge n the constrant that s consstent and gnore or reect the nconsstent part of the knowledge n the constrant; 2) modfy the structure of the BN so that the knowledge n the constrant can be completely ntegrated nto the BN; or 3) fnd a trade-off between the above two optons. Exstng methods n ths area manly take the frst opton. The ntegraton of peces of new knowledge s carred out by only updatng the parameters of the exstng BN whle keepng the network structure ntact. The ratonale behnd ths s that n many stuatons the structure of the exstng BN represents more stable and relable aspects of the doman knowledge, and therefore s desrable to not change frequently. In ths paper, we explore the second opton to completely ntegrate the structural nconsstent constrant. The motvaton of dong ths s that when new peces of knowledge are more up-to-date or come from relable sources and the new dependences they brng should be accepted, t s necessary and benefcal to modfy the structure of the exstng BN so that the new constrants, ncludng the new dependences they contan, can be ntegrated nto the BN n ts entrety. The rest of ths paper s organzed as follows. In Secton 2 we ntroduce some background and formally defne the problem we set to solve. In Secton 3 we brefly revew the exstng works related to ths problem. In Secton 4 we explore how to dentfy structural nconsstences between the constrants and the BN. In Secton 5 we nvestgate how to overcome the dentfed structural nconsstences durng knowledge ntegraton. Fnally, we show the effectveness of the proposed methods through experments n Secton 6 and conclude the paper n Secton 7 wth drectons of future research.

2 2 Background and the Problem We wll follow the followng conventons n ths paper. To name a set of varables, we wll use captal talc letters such as X, Y, Z and wth superscrpts such as X, Y, Z k for subsets of varables. To name ther nstantatons, we wll use lower case talc letters such as x, y, z correspondngly. To name an ndvdual varable, we use captal talc letters such as A, B, C. To name ther nstantatons, we wll use lower case talc letters such as a, b, c correspondngly. If the ndvdual varable belongs to a set, we wll use subscrpts to ndcate ts poston n the set, such as X for the th varable n set X, and x for ts nstantaton. P, Q, R are specfcally used to ndcate probablty dstrbutons, and bold P, Q, R are used for sets of dstrbutons. Knowledge ntegraton refers to the process of ncorporatng new knowledge nto an exstng knowledge base. The probablstc knowledge base can be represented usng JPD PX ( ), where X represents the set of varables n the doman. The new knowledge that needs to be ntegrated nto PX ( ) usually comes from more up-to-date or more specfc observatons for a certan perspectve of the doman. It can be represented as a lower-dmensonal probablty dstrbuton over a subset of X, whch s called probablstc constrant, or constrant for short. We use the probablty dstrbuton R ( Y ) to denote the th pece of new knowledge or the th constrant whch s a dstrbuton over the set of varables Y, where Y X. A set of m constrants can be represented as R { R ( Y ),..., R ( Y )}. 1 m Defnton 1 (Constrant Satsfacton) Let PX ( ) be a JPD, and RY ( ) be a constrant, where Y X. PX ( ) s sad to satsfy RY ( ) f P( Y) R( Y),.e., the margnal dstrbuton of PX ( ) on varable set Y equals RY ( ). Defnton 2 (Consstent Constrants) If there s at least one JPD PX ( ) that can satsfy all the constrants n R,.e., R ( Y ) R, P( Y ) R( Y ) holds, then we say constrants n R are consstent. Defnton 3 (Inconsstent Constrants) If there does not exst a JPD that can satsfy all the constrants n R, then we say constrants n R are nconsstent. The probablstc knowledge ntegraton problem s to modfy PX ( ) to satsfy all the constrants n R. Meanwhle, t s desrable to mnmze the modfcaton to PX ( ) durng the ntegraton process, whch can be measured usng some metrcs such as I-dvergence (Csszar 1975; Kullback and Lebler 1951; Vomlel 1999) or ts weghted verson I- aggregate (Vomlel 2003) defned afterwards. Defnton 4 (I-dvergence) Gven two probablty dstrbutons PX ( ) and QX ( ), I-dvergence from PX ( ) to QX ( ) s: 1 m Px ( ) P( x)log f P Q I( P Q) xx Qx ( ) (1) otherwse where P Q means Q domnates P,.e., { X P( X ) 0} ' ' { X Q( X ) 0}. Defnton 5 (I-aggregate) Let P be a JPD, Q { Q1,..., Q m } be a set of dstrbutons, and be a nonnegatve weght for Q. I-aggregate s the weghted sum of I-dvergence for all of the dstrbutons n Q,.e., ( P, Q ) I( P Q) (2) where 0< <1 and 1. BNs have been wdely used for representng large uncertant knowledge bases. For a gven BN G, we use G s to refer to the DAG,.e., the network structure of G, and G P to refer to the set of CPTs. A BN can then be represented as G ( GS, GP), where GS {( X, )}. Here s the set of parents for X, and GP { P( X )}. The followng are some graph defntons based on the DAG of the BN (Geger, Verma and Pearl 1990). Defnton 6 (I-Map) Gven a DAG G s, and a JPD P whch uses the same set of varables as G s, f every ndependency mpled by Gs holds n P, then we say Gs s an I-Map of P (Geger, Verma and Pearl 1990). A complete DAG s an I-Map of any dstrbuton. Defnton 7 (Mnmal I-Map) The mnmal I-Map of P s a DAG G s that by removng any arc fromg s ntroduces ndependences that do not hold n P (Geger, Verma and Pearl 1990). From the defnton of a mnmal I-Map we know that, to represent P, the BN structure Gs needs to be an I-Map of P, f t s not a mnmal I-Map of P. That means Gs can have more dependences than P, but not less (Geger, Verma and Pearl 1990). Ths leads to our defnton for structural nconsstent constrant. Defnton 8 (Structural Inconsstent Constrant) Gven DAG G s over a set of varables X, and a constrant R over a set of varables Y X, f every ndependency mpled by G s nvolvng varables n Y holds n R, then we say R s structurally consstent wth Gs. Otherwse, we say R s structurally nconsstent wth G s, or we say R s a structural nconsstent constrant for G. s Structural nconsstent constrant (defned n Defnton 8) and nconsstent constrants (defned n Defnton 3) refer to dfferent types of nconsstences, and we refer them as Type I and Type II nconsstency, respectvely, n the rest of ths paper. Type I nconsstency n Defnton 3 s defned for a set of constrants. It can arse whether or not the knowledge base s represented as a BN. Type II ncon-

3 sstency n Defnton 8 s defned for an ndvdual constrant wth respect to a partcular BN. It can occur only when the knowledge base s represented as a BN, and some dependency relatons n the constrant are absent n the BN. Fgure 2 below shows some examples of nconsstent constrants for the BN n Fgure 1. Constrants n Fgure 2(a) have Type I nconsstency because R1( A) R2( A). Constrant n Fgure 2(b) has Type II nconsstency because t contradcts wth the dependency relaton n the BN structure of Fgure 1, where B and C are ndependent gven A, whle R3( B, C A) R3( B A) R3( C A). Fgure 1: A 3 Node BN (a) Constrants wth Type I Inconsstency (b) Constrant wth Type II Inconsstency Fgure 2: Inconsstent Constrants Wth the defnton of Type I and Type II nconsstences, the problem we set to solve s formally defned as follows: Gven BN G ( GS, GP) wth JPD PX ( ), and a set of 1 m constrants R { R1 ( Y ),..., Rm ( Y )} wth at least one constrant havng Type II nconsstency, construct a new BN ' ' ' G ( GS, GP) wth probabllty dstrbuton P ' that meet the followng condtons: ' C1: Constrant satsfacton: R ( Y ) R, P ( Y ) R( Y ) ; ' C2: Mnmalty: I( P ( X ) P( X )) s as small as possble. 3 Related Works In ths secton we brefly revew the exstng works related to the problem. IPFP (Iteratve Proportonal Fttng Procedure) (Kruthof 1937) and vrtual evdence method (Pearl 1990) are the two bass methods that support the other methods n ths secton. IPFP s a mathematcal procedure that teratvely modfes a JPD to satsfy a set of constrants whle mantanng mnmum I-dvergence to the orgnal dstrbuton. Vomlel appled IPFP to knowledge ntegraton problems wth consstent constrants (Vomlel 1999). The core of IPFP s I- proecton (Valtorta, Km and Vomlel 2002). Specfcally, for a set of constrants R and an ntal JPD, the IPFP procedure s carred out by teratvely modfyng the current JPD Q ( ) k 1 X accordng to the followng formula, usng one constrant R ( Y ) n R at a tme: R ( Y ) Qk( X ) Qk 1( X ), (3) Q ( Y ) k 1 where k mod m, whch determnes the constrant used at teraton k, and m s the number of constrants n R. Here, Qk s sad to be an I-proecton of Q ( ) k 1 X on the set of all JPDs that satsfy R ( Y ), and as such, Qk has the mnmum I-dvergence from Q ( ) k 1 X among all the JPDs that satsfy R ( Y ). One lmtaton of IPFP-based ntegraton methods s that t works on JPDs, not on BNs. To solve ths problem, Peng and Dng proposed E-IPFP (Peng and Dng 2005) based on IPFP by addng a structure constrant: n Q ( X ) Q ( X ), (4) k k 1 1 where ( X, ) Gs. Qk s the JPD of the BN whose CPTs are extracted from Qk 1 accordng to G s (Peng et al. 2012). When constrants n R are consstent, E-IPFP wll converge to a new BN whch 1) has the same structure as the gven BN, 2) satsfes all the constrants n R, and 3) wth changes to the gven BN mnmzed. Another lmtaton of IPFP s that t wll not converge but oscllates when constrants are nconsstent (Csszar 1975; Peng and Dng 2005; Vomlel 2003). To deal wth nconsstent constrants (.e., Type I), Vomlel proposed CC-IPFP (Vomlel 1999; 2003) and GEMA (Vomlel 2003) by extendng IPFP. Later Peng et al. proposed SMOOTH (Zhang and Peng 2008) whch has better performance than CC-IPFP and GEMA. SMOOTH crcumvents the nconsstency by makng b-drectonal modfcatons durng the ntegraton process. When the knowledge base s represented as a BN, SMOOTH can be extended to E-IPFP- SMOOTH (Peng and Zhang 2010), whch can deal wth both Type I and Type II nconsstences by modfyng the constrants. Ths s acceptable when constrants are consdered to contan some nose. But when new dependency relatons are consdered relable, E-IPFP-SMOOTH wll not be able to ntegrate ths nformaton nto the revsed BN snce the structure of BN s not changed.

4 Next we brefly dscuss vrtual evdence method, not only because t plays an mportant role n BN belef update wth uncertan evdences, but also because t serves as a bass for our proposed methods n Secton 5. Researchers have dentfed 3 knds of evdences n BN belef update: hard evdence, vrtual evdence, and soft evdence (Pan, Peng, and Dng 2006). Hard evdence specfes the partcular state the varable s n, soft evdence specfes the probablty dstrbuton for the states the varable s n (Valtorta, Km, and Vomlel 2002), and vrtual evdence specfes the lkelhood rato for the uncertanty of the observatons of the states the varable s n (Pearl 1990). For example, f one beleves that the event represented by varable A occurs wth probablty p, and does not occur wth probablty 1 p, then the lkelhood rato can be represented as L( A) p : (1 p), whch does not necessarly need to be specfc probabltes. For belef update wth vrtual evdence, Pearl proposed the vrtual evdence method (Pearl 1990). For vrtual evdence on varable A wth lkelhood rato L(A), ths method extends the gven BN by creatng a bnary vrtual nodev as the chld of A, wth state v standng for the event that A ahas occurred, and the CPT of V satsfyng the followng equaton: P( v A a) : P( v A a) LA ( ). (5) It then treats v as a hard evdence by nstantatngv to v. Ths wll update the belef n the gven BN, and the updated BN wll satsfy the gven vrtual evdence. For belef update wth soft evdence, Chan and Darwche extended the vrtual evdence method by frst convertng each soft evdence to a vrtual evdence (Chan and Darwche 2005). Let PX ( ) be a dstrbuton and RY ( ) be a soft evdence, where Y X. We use y (1),..., y ( l ) to represent all the possble nstantatons of Y whch form a mutually exclusve and exhaustve set of events. RY ( ) can then be converted to a vrtual evdence wth the followng lkelhood rato: R( y(1) ) R( y(2) ) R( y( l) ) LY ( ) : :...:. (6) P( y ) P( y ) P( y ) (1) (2) ( l) For multple vrtual evdences, the belef update for one vrtual evdence wll not affect the belef update for the other vrtual evdences (Pan, Peng, and Dng 2006). However, ths s not the case for multple soft evdences. For BN belef update wth multple soft evdences, Peng et al. proposed BN-IPFP (Peng, Zhang and Pan 2010) whch combnes Pearl s vrtual evdence method wth IPFP. It s able to preserve margnal dstrbutons specfed n all soft evdences after convertng them to vrtual evdences. BN- IPFP converges teratvely, and the resultng BN can satsfy multple soft evdences at the same tme. 4 Identfy Structural Inconsstences In ths secton we examne how to dentfy such structural nconsstences for a gven constrant and a BN. Ths s done by checkng f every dependency relaton n the constrant also holds n the BN. Frst we dscover dependency relatons from the constrant and from the BN. We can use d-separaton method (Pearl 1988) to dscover dependency relatons n the BN. Ths method tells whether every two varables are condtonally ndependent under another set of varables by lookng at the connecton type between these two varables n the BN. We can use the ndependence test to fnd out dependency relatons n the constrant. The zero-order (uncondtonal) ndependence test s to check whether R( A, B) R( A) R( B) holds for each par of varables A and B n the varable set of R. Smlarly, the -order ndependence test for R s to check whether Equaton 7 holds for each par of varables A and B, where Z s a set that contans any number of varables n the varable set of R other than A and B. R( A, B Z) R( A Z) R( B Z) (7) The followng s the algorthm for dentfyng structural nconsstences between the constrant and the BN. Algorthm INCIDENT Input: BN G ( Gs, GP) wth orderng of nodes n G s, and constrants R { R1, R2,..., R m }. Output: Structural Consstent constrant set R, structural nconsstent constrant set R and Dependency Lst DL. Steps: 1. Create empty sets R and R, and empty lst DL ; 2. Perform the followng steps for each constrant R n R : 2.1 Create an empty lst DL ; 2.2 For each par of varables A and B n R, do from zero-order to ( R 2) -order ndependence test on them, where R s the number of varables n R. If the test fals, add R, A, B, Z to DL ; 2.3 For each entry R, A, B, Z n DL, use d- separaton method to test whether A and B are ndependent gven Z,.e., A B Z. If the test fals, remove ths entry from DL ; 2.4 If DL s empty, add R to R. Otherwse, add R to R, and merge DL nto DL ; 3. Return R, R, and DL.

5 5 Overcome Structural Inconsstences In ths secton we focus on modfyng the structure of the exstng BN to accommodate the dentfed structural nconsstences n a way smlar to addng vrtual evdence node n Pearl s vrtual evdence method. We can add one node n BN for each structural nconsstent constrant and make ths node the chld of all varables of that constrant. Then we set the CPT of the node wth the lkelhood rato calculated from the constrant usng (6), and set the state of the node to true. Ths guarantees the varables covered by the constrant to be dependent, whch wll overcome any structural nconsstences for that constrant. Ths forms the core of our AddNode method, whch can also be used wth E-IPFP to ntegrate a set of constrants whch may contan both structural consstent constrants and structural nconsstent constrants. The followng s the algorthm whch adds nodes for structural nconsstent constrants n R, and updates BN wth constrants n R usng E-IPFP. The nput constrants are appled teratvely durng the process. When the algorthm converges, all constrants are satsfed (C1). Besdes, the IPFP style computng makes the JPD of the fnal BN as close to the JPD of the orgnal BN as possble (C2). We are workng to formally prove these clams. Algorthm AddNode+E-IPFP Input: BN G ( GS, GP) wth orderng of nodes n G S, and constrant set R { R1, R2,..., R m }. Output: BN G ' ( G ', ' S GP) that satsfes R wth JPD as close to that of G as possble. Steps: 1. Run INCIDENT to partton R nto R and R ; 2. For each constrant R n R, add a new node to G wth varables n R as ts parents; S 3. /* ths step s the same as AddNode method */ For each constrant R n R, 3.1 Calculate lkelhood rato for R usng ( (1) ) ( (2)) ( ( )) R R y R y R y l LY ( ) : :...: PY ( ) P( y ) P( y ) P( y ), (1) (2) ( l ) where Y s the varable set of R, and l s the number of dstnct nstantatons for Y ; 3.2 Construct CPT ofv wth lkelhood rato LY ( ); 3.3 Set the state ofv to true; 4. Apply one teraton of E-IPFP wth R and the updated BN as nput,.e., 4.1 Do I-proecton for the current probablty dstrbuton P ( ) k 1 X on each constrant R n R : V R ( Y ) P X P X, P ( Y ) k( ) k1( ) k 1 where Y s the varable set of set of all varables n G ; 4.2 Form and apply the structure constrant: n P ( X ) P ( X ), where ( X, ) G ; k k 1 1 s R, and X s the 5. Repeat step 3 and step 4 untl the JPD of the updated BN does not change; 6. Return ' ( ', ' ' G GS GP), where GS s the updated structure after addng nodes to G S, and G ' P s the CPTs for the nodes n G. ' S 6 Experments and Results Our experments are based on the BN shown n Fgure 3, and constrant set R { R1, R2, R3}, where R1( A, C, E ) = ,0.2232,0.03,0.03,0.1672,0.2728,0. 07,0.07, R 2 ( BC, ) = 0.6, 0.1, 0.2, 0.1 and R3( C, D, E ) = (0.228,0.372, 0.076,0.124,0.04,0.04,0.06,0.06). Fgure 3: A 5 Node BN Frst we run the INCIDENT algorthm to dscover the structural nconsstences. R { R1, R2}, R { R3} and DL= {< R 1, A, C, >, < R 1, A, E, >, < R 1, A, C,{ E } >, < R 2, B, C, >} are returned as the output of INCIDENT. Then we run AddNode+E-IPFP to ntegrate constrants n R. The BN structure has been updated as n Fgure 4 after addng nodes for constrants n R. Fgure 4: Modfed BN Structure

6 Table 1 gves the performance comparson for dfferent methods. From the table we can see that E-IPFP-SMOOTH does not add any extra nodes to the exstng BN. But ts I- aggregate s not 0. AddNode+E-IPFP adds extra nodes to the exstng BN. But t has 0 I-aggregate. Table 1 ncludes experment results for two more effcent varatons of AddNode+E-IPFP. AddNode+Merge frst merges the constrants n R nto a sngle constrant usng IPFP algorthm, and then adds a sngle node to the exstng BN for the merged constrant. AddNode+Fatorzaton frst decomposes each constrant n R nto sub-constrants accordng to the ndependences among ts varables, and then merges the sub-constrants that are structural nconsstent wth the exstng BN, and at the end adds a sngle node to the exstng BN for the merged constrant. Table 1: Experment Result wth 5 Node BN To further compare the performance of the dfferent versons of our methods and get a sense of ther scalablty, we have conducted experments wth an artfcally composed BN of 10 dscrete varables. The nput constrant set conssts of 3 consstent constrants and 3 structural nconsstent constrants. The results are gven n Table 2 below. AddNode+Merge and AddNode+Fatorzaton have better performance than AddNode+E-IPFP on large BN because these two methods add fewer nodes to the exstng BN. Table 2: Experment Result wth 10 Node BN 7 Conclusons In ths paper we propose several methods to ntegrate structural nconsstent constrants nto a BN. The methods frst dentfy the structural nconsstences between the constrant and the exstng BN, and then overcome the dentfed structural nconsstences by addng nodes to the exstng BN n a way smlar to the vrtual evdence method. Experments show that the proposed methods have an advantage n ncorporatng new structural nformaton. We look forward to applyng the framework and related methods to update a BN bult for a demand-supply network wth uncertan demand-reply relatons. The nformaton used to update ths BN may requre addng new demand-supply relatons to the exstng BN. Ths s mportant n supply chan management to better satsfy the needs of the customers. Our future work also ncludes mprovng the performance of our methods so they can be easly appled to ndustry problems modeled wth large BNs. References [1] H. Chan and A. Darwche On the Revson of Probablstc Belefs usng Uncertan Evdence. Artfcal Intellgence, [2] I. Csszar I-dvergence Geometry of Probablty Dstrbutons and Mnmzaton Problems. The Annuals of Probablty, 3(1): [3] D. Geger, T Verma and J. Pearl Identfyng Independence n Bayesan Networks. Networks, 20: [4] R.Kruthof Telefoonverkeersrekenng. De Ingeneur, 52: [5] S. Kullback and R.A. Lebler On Informaton and Suffcency. Ann. Math. Stats, 22: [6] R. Pan, Y. Peng and Z. Dng Belef Update n Bayesan Networks Usng Uncertan Evdence. In Proceedngs of the IEEE Internatonal Conference on Tools wth Artfcal Intellgence, [7] J. Pearl Probablstc Reasonng n Intellgent Systems: Networks of Plausble Inference. San Mateo: Morgan Kaufman. [8] J. Pearl Jeffery s Rule, Passage of Experence, and Neo-Bayesansm. Knowledge Representaton and Defeasble Reasonng. H.E. Kyburg, Jr., R.P. Lou, and G.N. Carlson, Eds. Boston: Kluwer Academc Publshers. [9] Y. Peng and Z. Dng Modfyng Bayesan Networks by Probablty Constrants. In Proc. 21st Conference on Uncertanty n Artfcal Intellgence. Ednburgh. [10] Y. Peng, Z. Dng, S. Zhang and R. Pan Bayesan Network Revson wth Probablstc Constrants. Internatonal Journal of Uncertanty, Fuzzness and Knowledge- Based Systems, [11] Y. Peng and S. Zhang Integratng Probablty Constrants nto Bayesan Nets. In Proceedngs of 9th European Conference on Artfcal Intellgence (ECAI2010), Lsbon, Portugal. [12] Y. Peng, S. Zhang and R. Pan Bayesan Network Reasonng wth Uncertanty Evdences. Internatonal Journal of Uncertanty, Fuzzness and Knowledge-Based Systems, 18(5): [13] M. Valtorta, Y. Km and J. Vomlel Soft Evdental Update for Probablstc Multagent Systems. Internatonal Journal of Approxmate Reasonng, 29(1): [14] J. Vomlel Methods of Probablstc Knowledge Integraton. Ph.D. dss., Department of Cybernetcs, Faculty of Electrcal Engneerng, Czech Techncal Unversty. [15] J. Vomlel Integratng Inconsstent Data n a Probablstc Model. Journal of Appled Non-Classcal Logcs, 14(3): [16] S. Zhang and Y. Peng An Effcent Method for Probablstc Knowledge Integraton. In Proceedngs of The 20th IEEE Internatonal Conference on Tools wth Artfcal Intellgence (ICTAI-2008), Nov Dayton, Oho.