Hierarchical CP-networks

Transcription

1 Hierarchical CP-networks Denis Mindolin Det. of Comuter Science and Engineering, Universit at Buffalo, Buffalo, NY Jan Chomicki Det. of Comuter Science and Engineering, Universit at Buffalo, Buffalo, NY ABSTRACT We resent here a variant of acclic CP-networks. It allows not onl finite but also infinite domain attributes. It also has the roert that a reference over each attribute in the network has higher riorit then all the descendants references. We rovide an algorithm of constructing a reference formula reresenting the order induced b a hierarchical CPnetwork, thus making it ossible to work with hierarchical CP-networks in the database context. We also rovide a coexit analsis of the size of reference formula constructed b the algorithm. 1. INTRODUCTION In making an kind of choice in the everda s life, the notion of reference alwas comes to mind. A number of reference handling models have been develoed. Two ver oular ones are the CP-network model[1] and the binar relation model[4, 9]. Being ver general, the binar relation framework can be used in different contexts like reference construction [9] or reference modification [6]. Moreover, the ower of relational databases can be used to find the otimal outcomes of references reresented as binar relations [4, 9]. At the same time, the CP-network model is ver sie and intuitive. It reresents a coex reference over objects using a set of atomic references each of which is a reference over a single object attribute given that the values of the other attributes are equal (the ceteris aribus rincile). This set of atomic references is reresented as a directed grah (sometimes called the reference grah) whose nodes are atomic references, and edges between nodes corresond to conditional references over attributes - i.e. the values of the arent attributes influence the references over the child attributes. The hierarchical CP-network model, which is introduced in this aer, addresses some semantical and reresentational limitations of CP-networks: 1) it allows continuous attributes b reresenting conditional reference tables as Permission to co without fee all or art of this material is granted rovided that the coies are not made or distributed for direct commercial advantage, the VLDB coright notice and the title of the ublication and its date aear, and notice is given that coing is b ermission of the Ver Large Data Base Endowment. To co otherwise, or to reublish, to ost on servers or to redistribute to lists, requires a fee and/or secial ermission from the ublisher, ACM. VLDB 07, Setember 23-28, 2007, Vienna, Austria. Coright 2007 VLDB Endowment, ACM /07/09. binar reference relations, 2) the attributes in hierarchical CP-networks are rioritized: an attribute in reference grah is more imortant than its descendants; 3) the ceteris aribus rincile can be selectivel relaxed. According to [1], in some CP-network instances edges between attributes in reference grah corresond to attribute riorities. Exae 1. Let a CP-net N 1 over the roblem with two attributes X = {X 1, } be defined as in Figure 1. X 1 x a 1 x b 1 x a 1 x a 2 x b 2 x b 1 x b 2 x a 2 (a) CP-network N 1 (x a 1, x a 2) (x a 1, x b 2) (x b 1, x a 2) (x b 1, x b 2) (b) The reference order iied b N 1. The references obtained from the CPTs are shown as solid arcs. The entailed references are shown as dashed arcs. Figure 1: CP-net N 1 from Exae 1 The CP-net N 1 induces a total ordering of all the outcomes (Figure 1.b) which iies that the most imortant is to satisf the references over the attribute X 1, then over the attribute (the outcomes with values X 1 = x a 1 are alwas referred to those with X 1 = x b 1). However, attribute rioritization does not hold for all CPnetwork instances. In articular, it is not alwas the case that violation of a reference is worse than violation of two or more descendant references. Exae 2. Let CP-net N 2 over the roblem with three attributes X = {X 1,, } be defined as in Figure 2. The entailed reference arcs are skied in Figure 2.b for siicit. However the can be obtained b erforming the transitive closure of the grah. Note that in N 2, (x a 1, x b 2, x a 3) is not referred to (x b 1, x b 2, x b 3) even though the the value x a 1 of X 1 is referred to x b 1.

2 X 1 x a 1 x b 1 x a 1 x a 2 x b 2 x b 1 x b 2 x a 2 x a 2 x a 3 x b 3 x b 2 x b 3 x a 3 (a) CP-network N 2 (x a 1, x a 2, x a 3) (x a 1, x a 2, x b 3) (x a 1, x b 2, x a 3) (x b 1, x a 2, x a 3) (x a 1, x b 2, x b 3) (x b 1, x a 2, x b 3) (x b 1, x b 2, x a 3) (x b 1, x b 2, x b 3) (b) The reference order iied b CPnetwork N 2 Figure 2: CP-net N 2 for Exae 2 Models in which descendant references are less imortant then their ancestors are natural for frameworks in which references are iterativel constructed b a user in a todown manner; namel, when a new reference is added as a leaf node or a node starting a new reference subgrah. In this case, the nodes added as less imortant references (leaf nodes) are guaranteed not to violate more imortant, alread introduced ancestor references. Another drawback of CP-networks [12] is that sometimes the ceteris aribus semantics is too strict, and when one introduces a reference over an attribute, it s not alwas ossible to sa that this reference has to be of the everthing else being equal kind. Exae 3. Assume a erson wants to bu a car. Let a car seller have a database of cars which are described b the attributes {make, ear, rice, mileage, engine-te}. At the same time, the erson onl cares about the attributes {make, ear, rice, mileage}. However, in the CP-network framework there is no wa to secif that engine-te is irrelevant: if a CP-network has a conditional reference table over engine-te, then engine-te is relevant; if it does not, then one outcome will be referred to another according to the CP-network onl if the engine-tes of the outcomes are the same. According to the definition of CP-networks [1], each attribute involved in a CP-network is categorical. At the same time, there are man roblems in which continuous attributes arise and for which it would be useful to have an aroach similar to CP-networks. Due to these limitations, the references from Exae 4 below cannot be reresented as a CP-network. However, the can be reresented using hierarchical CP-networks, as we show in Exae 5. Exae 4. Assume a user wants to bu a car, and her reference over make has the same imortance as the reference over ear, rice, and mileage. At the same time, assume that ear is more imortant than rice and mileage. Let the reference be constructed in a to-down manner from more to less imortant variables. So the first reference introduced b the user is over make: given two cars with the same age, rice and mileage, she refers VW to Kia, and Kia to all the other makes. The second reference is over ear: given two cars of the same make, she would bu the newer one. The third reference is over mileage: if the makes, ages, and rices are the same, but the cars are relativel new (ear 2004), she refers the one with less mileage (less than 60000). However, if the makes, ages, and rices are the same, but the cars are old (ear < 2004), she refers the one with the mileage less than The last reference is over rice: if two cars have the same make, age, and mileage, but are new (ear 2006), she refers the cheaer one. However, given two cars with the same make, age, and mileage, but not new (ear < 2006), she refers to send not more then on it. The aer is organized as follows. Section 2 contains the definition of hierarchical CP-networks and a discussion of some of their roerties. In Section 3, we describe an algorithm constructing a reference formula reresenting the order induced b a hierarchical CP-network. This allows to work with hierarchical CP-networks in the database context. Section 4 contains the coexit analsis of the reference formula constructed b the algorithm. Section 5 concludes the aer with a discussion of related and future work. 2. NOTIONS In this aer we adot the notations from both aroaches: CP-networks and binar reference relations. Assume we have a roblem over outcomes described b n attributes X = {X 1,..., X n} such that each attribute X i is associated with a domain D(X i) (categorical or continuous). Let us denote the set of all ossible outcomes as D D = D(X 1)... D(X n), and the set of all ossible assignments to the set of attributes U X as D(U) = X U D(X). Then given an outcome o D, we denote the value of an attribute X X of o as o.x. A relation instance is a finite set of outcomes. We limit our attention to reference relations that are strict artial orders (SPOs): transitive and irreflexive binar relations. 2.1 Hierarchical CP-network Let a conditional reference table CP T(X) associated with an attribute X X be defined as a trile CPT(X) = (Φ X, W X, U X) in which W X X, U X X such that U X W X =, X U X, X W X, and Φ X = {ϕ 1,..., ϕ k }. Let ϕ be defined as u ϕ : R ϕ where u ϕ is a relation such that u ϕ D(U X), R ϕ is a strict artial order over D(X), and for all airs of different ϕ 1, ϕ 2 Φ X, we have u ϕ1

3 u ϕ2 =. Let P uϕ be a finite formula reresenting the relation u ϕ, and a P Rϕ be a finite formula reresenting the relation R ϕ. In the coexit analsis, we will assume that P uϕ and P Rϕ are such that the can be evaluated in olnomial time for given valuations of the free variables. Define for each ϕ a binar relation over outcomes ϕ = {(uxw, ux w ) :u D(U X), u ϕ(u); w, w D(W X); D(Y X);(x, x ) R ϕ} where Y X = X ({X} W X U X). We define the binar relation induced b conditional reference table CP T(X) as CPT (X) = ϕ ΦX ϕ. Thus from the exression for ϕ it follows that the reference over attribute X is conditionall deendent on the attributes from U X. Moreover, the reference over the attribute X is indeendent of the attributes from W X. In other words, instead of the CP-network rincile everthing else being equal we use the everthing else being equal excet for the attributes W X rincile which was introduced in [12]. Summarizing the notation introduced so far: U X, W X, and Y X are such that 1) U X W X = ; 2) X U X, X W X; 3) Y X = X ({X} W X U X). Proosition 1. ϕ and CPT (X) are strict artial orders. Definition 1. Let Γ be a set of conditional reference tables for the attributes X Γ X Γ = {CPT(X) : X X Γ}. We define the reference order Γ induced b Γ as the transitive closure of CPT(X) Γ CPT (X) Γ TC( CPT(X) Γ CPT (X)). Note that our definition of the order induced b Γ is different from the definition of the order entailed b CP-network [1]. Definition 2. [1] Let Γ be a set of conditional reference tables. Let be a total order over D. Then is said to satisf CPT(X) if for all o, o D : (o, o ) CPT (X) iies o o. A total order is said to satisf Γ if it satisfies ever CP T(X) Γ. Proosition 2. Let the order Γ entailed b the hierarchical CP-network Γ be the intersection of all total orders satisfing Γ. Then the order Γ induced b Γ and the order Γ entailed b Γ are equivalent. Proof: Prove that ever linear extension of Γ is a total order satisfing Γ and vice versa. Let be a linear extension of Γ, i.e. TC( X XΓ CPT (X)). It iies X X Γ CPT (X). Thus satisfies Γ. Let be a total order satisfing Γ, i.e. X XΓ CPT (X). being an SPO iies TC( X XΓ CPT (X)). Therefore is a linear extension of Γ. We define now hierarchical CP-networks. Let H(X) = {(Y, X) : Y U X}, where H(X) can be viewed as a directed grah with incoming edges going from the attributes Y U X to a single attribute X. These edges corresond to the conditional reference of attribute X on attributes Y U X. Then we define the reference grah of Γ as H Γ = X XΓ H(X). Working with reference grah H Γ, let us use the following notation: Anc Γ(X) = { ancestors of X in H Γ }, Anc-self Γ(X) = { ancestors of X in H Γ or X }, Desc Γ(X) = { descendants of X in H Γ }, Sibl Γ(X) = X Γ (Desc Γ(X) Anc Γ(X) {X}), Pa Γ(X) = { arents of X in H Γ}, Ch Γ(X) = { children of X in H Γ }. Definition 3. A set of conditional reference tables Γ is called a hierarchical CP-network if 1. for ever X in X Γ, W X = Z ChΓ (X)({Z} WZ); 2. if an attribute X has no child attributes in H Γ, then W X = X X Γ. From Definition 3 it follows that for each attribute X in the reference grah of a hierarchical CP-network Γ, W X = Desc Γ(X) (X X Γ). As a result, a reference over an attribute is more imortant than the references over this attribute s descendants in H Γ. Definition 3 also iies that a reference grah H Γ of a hierarchical network is acclic, otherwise W X of some attribute X involved in a ccle would contain attribute X leading to a contradiction. The formula reresentation P ϕ of ϕ is defined as P ϕ (o, o ) = [ Z Anc(X) Sibl(X) o.z = o.z] P uϕ (o.u X) P uϕ (o.u X) P Rϕ (o.x, o.x), according to Definition 3. The formula reresentation P CPT (X) of CPT is defined as P CPT (X)(o, o ) = ϕ Φ X P ϕ (o, o ), or using the exression for P ϕ (o, o ), P CPT (X)(o, o ) = [ Z Anc(X) Sibl(X) o.z = o.z] ϕ Φ X P uϕ (o.u X) P uϕ (o.u X) P Rϕ (o.x, o.x). Let Q CPT (X)(o, o ) = ϕ Φ X P uϕ (o.u X) P uϕ (o.u X) P Rϕ (o.x, o.x). Then P CPT (X)(o, o ) = [ Z Anc(X) Sibl(X) o.z = o.z] Q CPT (X)(o, o ). Note that since P uϕ and P Rϕ are finite formulas and can be evaluated in olnomial time, P ϕ, P CPT (X), and Q CPT (X) are finite formulas and can be evaluated in olnomial time as well. We introduce formula Q CPT (X) here since it las an imortant role further in the aer. In our model, a conditional reference table CPT(X) is grahicall reresented as a two-column table in which a row corresonds to a single ϕ Φ X. The first column of each row holds the formula P uϕ, and the second column holds the formula P Rϕ. Comared to conditional reference tables of the traditional CP-network model, hierarchical CPTs differ in the following:

4 the first column of a CP-net CPT is required to store a single assignment to U X, while a hierarchical CPT has there a formula P uϕ. As a result, since P uϕ reresents the relation u ϕ, hierarchical CPTs can be defined for infinite domain attributes U X; the second column of a CP-net CPT(X) is required to store a total order of D(X). In our aroach, the second column holds a formula P Rϕ reresenting an SPO relation over D(X). Therefore hierarchical CPTs can be defined for infinite domain attributes X: Exae 5. Take Exae 4. A hierarchical CP-network Γ which corresonds to the reference from this exae is shown in Figure 3. Note that according to Exae 4 the references over the attributes and m are unconditional, i.e. there is no attribute in X Γ whose value influences the reference over or m. Therefore U = U m = and the first columns of CPT() and CPT(m) are skied (shown filled with dashes). CPT(m) - m - [o.m = vw o.m = kia] [o.m {kia, vw} o.m {kia, vw}] m CPT() CPT() 2006 o. < o. CPT() - - o. > o o. < o < 2004 o. < o < 2006 o o. > 8000 Figure 3: Hierarchical CP-net for Exae 5. Note that hierarchical CP-networks do not subsume CPnets, although it ma aear to be so. In CP-nets, W X = but this ma violate the first condition of Definition Subnet of a hierarchical CP-network Loosel seaking, a subnet of a hierarchical CP-network Γ is just a subset of the conditional reference tables from Γ. However not an subset of Γ is a subnet. Its formal definition is given further. Definition 4. Let and Γ be two hierarchical CP-networks such that 1. X X Γ: all attributes used in the network are also used in the network Γ; 2. if some attribute X from X Γ is in X then all ancestors of X from H Γ are in X ; 3. Given two conditional reference tables CPT (X) and CPT Γ(X) for an attribute X Γ corresondingl, the first and the third comonents (namel, Φ X and U X) of the two conditional reference tables are equal. The W X comonent of CPT (X) is set to Desc (X) (X X ), i.e. it is set according to the reference grah H of. Then is called a subnet of a hierarchical CP-network Γ. The notion of subnet of a hierarchical CP-network will be used further in the aer to construct a reference formula reresenting the order induced b a hierarchical reference network. 3. FROM HIERARCHICAL CP-NETS TO PREFERENCE FORMULAS Dealing with references, the two most common tasks are 1) given two outcomes, find the more referred one, and 2) find the otimal outcomes from the given set of outcomes. The first roblem is called dominance testing. In the CPnetwork aroach, this roblem can be solved in olnomial time in the number of attributes when a reference grah is a directed tree or oltree. If the grah is directed-ath singl-connected, dominance testing is NP-coete, and it is in NP if the number of aths between an air of nodes in the grah is olnomiall bounded [1]. In the hierarchical CP-network framework, this roblem can be solved in olnomial time in the size of the hierarchical CP-network descrition b the following roosition which is analogous to a result in [12]. Proosition 3. Let Γ be a hierarchical CP-net. Let o, o be two outcomes. Let Diff = {X 1,..., X l } X Γ be the attributes in whose values o and o are different, and To be the set of all nodes from Diff which have no ancestors in Diff. Then o Γ o X To : Q CPT (X)(o, o ) As shown in [12], the roblem of finding the otimal outcomes can be also solved b Proosition 3 b taking ever outcome and checking if there is an outcome dominating it. Thus the otimal outcomes can be comuted in time olnomial in the size of the network descrition and the number of outcomes. To find the otimal outcomes in the relational database framework, the winnow oerator [4] can be used. It is an algebraic oerator which icks from a given relation (containing all ossible outcomes) the set of the most referred outcomes, according to a given reference relation. Formall, it is defined as follows. Definition 5. If R is a relation schema and a reference relation over R, then the winnow oerator is written as ω (R), and for ever instance r of R: ω (r) = {t r t r. t t} From Definition 5, it follows that using the winnow oerator requires the order induced b a hierarchical CP-network to be reresented as a binar reference relation. According to Proosition 3, one of the was to construct a reference formula reresenting the order induced b a hierarchical CP-network Γ is b 1) taking ever nonemt subset Diff i of X Γ, 2) for ever Diff i, finding the corresonding To i, 3) for ever Diff i, writing down the formula D i(o, o ) = ( X X Diffi o.x = o.x) ( X Diffi To i o.x o.x) ( X Toi Q CPT (X)(o, o )),

5 and 4) finding the disjunction of all D i(o, o ), which will be the order induced b Γ. However, the formulas constructed b this algorithm will be clearl exonential in the size of the hierarchical CPnetwork descrition, since the algorithm requires enumerating all nonemt subsets of X Γ. At the end of this section, we resent another algorithm, Algorithm 1, for constructing a reference formula reresenting the order induced b a hierarchical CP-network. Algorithm 1 roduces olnomial-size reference formulas for a large class of hierarchical CP-networks. 3.1 Disassembling a hierarchical CP-network into a set of connectives and subnets To construct a reference formula for a hierarchical CPnet Γ, we consider its reference grah H Γ as a set of manto-one connectives and arallel subnets. The reference formula construction algorithm rovided at the end of this section is iterative. It starts from the set of arallel subnets each of which consists of a tomost node of H Γ. Each ste of the algorithm joins two or more arallel subnets of Γ and/or extends the existing subnets with a node from X Γ Parallel subnets P Γ (o, o ) (P 1 (o, o ) o. = o. o. = o. o. = o.) (P 2 (o, o ) o.m = o.m) [o. = o. o. = o. o. = o. o.m = o.m] where P 1 is a reference formula reresenting the order induced b 1, and P 2 is a reference formula reresenting the order induced b 2 which is defined as P 2 o.m = vw o.m = kia o.m {kia, vw} o.m {kia, vw} Man-to-one connectives Let 1,..., k be some subnets of Γ. Let also X e be such an attribute from X Γ (X 1... X k ) that each arent of X e in H Γ is in one of X 1,..., X k ; if an attribute Y X i is a arent of X e in H Γ, then all the other nodes in X i are the ancestors of Y in H i (i.e. Y is the bottom most node of H i ). Then the set of 1,..., k along with X e is called manto-one connective. 1 2 k k... Figure 4: Parallel subnets Proosition 4. Let 1,..., k be some subnets of Γ and 1,..., k be SPO relations reresenting the orders induced b 1,..., k corresondingl. Let P 1,..., P k be formula reresentations of 1,..., k corresondingl. Let also be a subnet of Γ such that X = X 1... X k. Then the formula P defined as P (o, o ) (P 1 (o, o ) Z X 1 o.z = o.z)... (P k (o, o ) Z X k o.z = o.z) Z X o.z = o.z defines an SPO, and P P. B Proosition 4, if we know reference formulas reresenting the orders induced b a set of arallel subnets, we can easil find a reference formula reresenting the order induced b the union of these subnets. Note that according to Proosition 4, the subnets 1,..., k are not required to be disjoint (i.e. if some subnet i contains an attribute, then the other subnets can also contain the same attribute). Exae 6. Take the hierarchical CP-network from Exae 5. It can be considered as a union of two subnets 1 consisting of the attributes {,, }, and 2 consisting of a single attribute {m}. Then according to Proosition 4, the reference formula reresenting the order induced b Γ is X e Figure 5: Man-to-one connective Proosition 5. Let a set of subnets 1,..., k of Γ and an attribute X e X Γ be a man-to-one connective and 1,..., k be SPO relations reresenting the orders induced b 1,..., k corresondingl. Let P 1,..., P k be formula reresentations of 1,..., k corresondingl. Let also be such a subnet of Γ that X = X 1... X k {X e}. Then the formula P defined as P (o, o ) (P 1 (o, o ) [ Z X 1 o.z = o.z])... (P k (o, o ) [ Z X k o.z = o.z]) [ X 1... X k o.z = o.z] [ Z X 1... X k o.z = o.z] Q CPT (X e)(o, o ) defines an SPO, and P P. B Proosition 5, if we know reference formulas reresenting the orders induced b all comonents of a manto-one connective, we can easil find a reference formula reresenting the order induced b the entire connective. Note that if the reference grah of a hierarchical CPnetwork Γ can be reresented as a set of DAGs in which each node has at most one outgoing edge, Proositions 4 and 5 are enough to iterativel construct a reference formula for Γ. Namel, for each DAG in the set, erform the following stes 1) start with the subnets each of which consists of a single tomost attribute of the DAG, 2) iterativel merge and extend the existing subnets b one child node in each iteration (i.e. al Proosition 5), 3) finall, al Proosition 4 to the arallel subnets with no descendant nodes. However this algorithm is not alwas alicable because a hierarchical CP-network can have attributes with more than one outgoing edge (e.g. Γ from Exae 5).

6 3.1.3 One-to-man connectives In this section, we resent a method which allows to transform a hierarchical network whose reference grah contains nodes with more then one outgoing edge (one-to-man connective) to a set of arallel subnets whose reference grahs have no nodes with more than one outgoing edge. Let 1 be a subnet of a hierarchical CP-network of Γ and X s be such attribute in H 1 that all the other attributes in 1 are its ancestors (i.e. X s is the bottom most node in H 1 ). Let X s have k (where k 1) outgoing edges in H Γ. Formall, let there exist such attributes X j,..., X j+k X Γ that X s is their arent (ossibl one of man arents). In order to avoid the situation when a node has more than one outgoing edge, we will make k 1 coies of the subnet 1. As a result, we will make subnets 2,..., k each of which is a co of 1. Note that the order induced b the subnet which is a union of 1,..., k will be the same as the order induced b onl 1 (b Proosition 4). After that, we will make each attribute X s of each of 1,..., k a arent of onl one attribute X j,..., X j+k corresondingl. As a result, each co of X s now has onl one outgoing edge. Note also that this oeration does not violate the semantics of the hierarchical CP-network because the reference over each of X j,..., X j+k is still conditionall deendent on X s. We call this rocess one-to-man connective elimination. Exae 7. Take the subnet 1 from Exae 6. It consists of the attributes {,, } such that has two outgoing edges to and. The one-to-man connective elimination technique slits this subnet into two arallel subnets 3 and 4 as it is shown in Figure Figure 6: Eliminating the one-to-man connective from 1. The hierarchical CP-networks 3 and 4 are defined as in Figure 7. Given a hierarchical CP-network with man-to-one connectives, this technique roduces a set of subnets without one-to-man connectives. Each of the subnets is a hierarchical CP-network and thus one can al Proosition 5 to construct a reference formula for each of them. Finall, Proosition 4 can be used to merge the subnets back and roduce a reference formula for it. 3.2 Constructing reference formulas Below we rovide an algorithm for constructing an SPO reference formula reresenting the order induced b a hierarchical CP-network. It takes two arameters: a hierarchical CP-network Γ and the grah H Γ which is the result of erforming the one-to-man connective elimination on H Γ. Algorithm 1. FormulaCons( Γ, H Γ ) 1. S = toologicall sorted sequence of nodes H Γ; 2. For each node X in S, 3 = {CPT 3 (), CPT 3 ()} CPT 3 () : CPT 3 () : - - o. > o o. < o. < 2006 o o. > = {CPT 4 (), CPT 4 ()} CPT 4 () : CPT 4 () : - - o. > o o. < o < 2004 o. < o Figure 7: The hierarchical CP-networks 3 and if X has no arents in H Γ, PA[X] = {X}; 4. else PA[X] = PA[Y 1]... PA[Y k ] {X}, where 5. Y 1,..., Y k are the arents of X in H Γ; 6. For each node X from S do 7. If X has no arents in H Γ, then 8. P[X] = P CPT(X) 9. If X has arents Y 1,..., Y k, then 10. P[X] =(P[Y 1] [ Z PA[Y1 ] o.z = o.z]) (P[Y k ] [ Z PA[Yk ] o.z = o.z]) 13. ([ Z PA[Y1 ]... PA[Y k ] o.z = o.z] 14. P CPT(X)) 15. Let D be the sequence of all nodes from H Γ 16. with no outgoing edges 17. If D = 1, then 18. OUT = P[D[1]] 19. else ( D = k > 1) 20. OUT = (P[D[1]] [ Z PA[D[1]] o.z = o.z]) (P[D[k]] [ Z PA[D[k]] o.z = o.z]) 23. ( [ Z PA[D[1]]... PA[D[k]] o.z = o.z]) 24. return OUT; Since a reference grah H Γ is acclic, H Γ is also acclic. Therefore the sequence D constructed in line 15 is nonemt. The algorithm oututs relation OUT(o, o ) which is an SPO reference formula reresenting the order induced b Γ. Below we rovide an exae of aling the algorithm to the hierarchical CP-network from Exae 5. Exae 8. Given a hierarchical CP-network Γ and a reference grah from Exae 5, we al the one-to-man elimination technique and decomose the reference grah into two arallel subnets: 1 and 2 as it is in Exae 6. B aling one-to-man elimination technique, 1 was decomosed into 3 and 4 as it is in Exae 7. B roosition 5, the formula P 3 reresenting the order induced b 3 is P 3 (o, o ) o. > o. (o. = o. (o o o. < o. o. < 2006 o. < 2006 o o. > 8000)),

7 and the formula P 4 reresenting the order induced b 4 is P 4 (o, o ) o. > o. (o. = o. ( o o o. < o o. < 2004 o. < 2004 o. < o )). Thus b Proosition 4, the formula P 1 reresenting the order induced b 1 is P 1 (o, o ) (P 3 (o, o ) o. = o. o. = o.) (P 4 (o, o ) o. = o. o. = o.) (o. = o. o. = o. o. = o.). And finall, a reference formula reresenting the order induced b Γ can be comuted b the exression rovided in Exae 6, i.e. P Γ (o, o ) (P 1 (o, o ) o. = o. o. = o. o. = o.) (P 2 (o, o ) o.m = o.m) [o. = o. o. = o. o. = o. o.m = o.m] 4. COMPLEXITY ANALYSIS The urose of this section is to rovide an analsis of the size of the reference formula roduced b Algorithm 1. We divide this analsis into two arts: the analsis of the size of the constructed reference formula as a function of the number of nodes in its reference grah H Γ assuming that H Γ has no one-toman connectives; the analsis of size of a reference grah H Γ roduced b the one-to-man connective elimination technique alied to the original reference grah H Γ; According to Proosition 6 below, the size of the reference formula is olnomial in the number of conditional reference tables in a hierarchical CP-network, given its reference grah has no one-to-man connectives. Proosition 6. The size of the reference formula roduced b Algorithm 1 for a hierarchical CP-network Γ such that H Γ does not have one-to-man connectives is Θ( X Γ 2 + X Γ B max), where B max is the size of the longest formula Q CPT (X) among all X X Γ. However, given a reference grah H Γ, its version H Γ without one-to-man connectives ma have exonential size as shown below. Thus in general the size of the reference formula constructed b Algorithm 1 ma be exonential in the number of conditional reference tables in a hierarchical CP-network. Proosition 7. Given a hierarchical CP-network Γ, the size of the reference grah H Γ roduced from H Γ b the one-to-man elimination technique is Θ(2 X Γ ). The exonential number of nodes in the reference grah after eliminating one-to-man connectives is caused b the fact that the number of coies of each node in the modified grah (the grah without one-to-man connectives) is N (X) = N out(x) 1 + Y Ch Γ (X) N (Y ), where N out(x) is the number of outgoing edges from node X in H Γ. Therefore, in the worst case, eliminating one-toman connectives for each node of H Γ multilies b c (where c 2) the number of nodes in H Γ. Exae 9. Consider a hierarchical CP-network Γ n whose reference grah has n nodes (where n is an even number) connected as follows: the nodes X 1 and have no outgoing edges; and for an i [3, n], 1) X i is a arent of the nodes X i 3 and X i 2, if i is even, and 2) X i is a arent of the nodes X i 2 and X i 1, if i is odd. An exae of such network is shown below. X 1 X 6 Figure 8: Hierarchical CP-net for Exae 9. Since the nodes X 1 and have no outgoing edges, no new coies of them will be roduced b the one-to-man elimination. However, for each i [3, n], the number of coies of X i will be N (X i) = 1 + 2N (X i 2). Thus the total number of nodes in the network obtained from Γ n b eliminating one-to-man connectives is clearl exonential in n. However, b restricting the structure of a reference grah, the exonential blowu can be avoided. Proosition 8. Let Γ be such a hierarchical CP-network that ever node in H Γ has at most one child node with outgoing edges. Then the size of the reference grah H Γ roduced from H Γ b the one-to-man elimination technique is O( X Γ 3 ). X 1 (a) Figure 9: (a) A reference grah of a hierarchical CP-network satisfing Proosition 8, and (b) the network resulting from one-to-man connective elimination As it follows from the analsis, given an hierarchical CPnetwork, erforming dominance testing via reference formula constructed b our algorithm will require at most exonential time. At the same time, as it was discussed above, using Proosition 3 one can do it in olnomial time. The same alies to the roblem of finding the otimal outcomes. However, having constructed a reference formula, one can use hierarchical CP-networks in the relational framework, where using the winnow oerator creates oortunities for efficient evaluation and algebraic quer otimization [3, 4]. (b) X 1

8 5. RELATED AND FUTURE WORK In this aer, we roose a variant of CP-networks[1] that addresses some of the limitations of the original roosal. Namel, reference riorit can be exressed i.e. an edge in a reference grah catures not onl conditional deendence between different attributes but also the relative reference imortance - the higher an attribute is in the reference grah, the higher riorit it has. Moreover, our framework can be used with infinite domain attributes. We also believe that the everthing else being equal semantics is too strict and sometimes might not be alicable. B introducing a new reference with the ceteris aribus semantics, the user is required to consider all ossible values of all other attributes, which is not alwas feasible. Thus in such situations, using our framework will be referred to using CP-networks. TCP-networks [2] is an extension of CP-networks which adds attribute imortance to them. In TCP-networks, two secial kinds of arcs are added to reference grahs: 1) an i-arc from one attribute to another iies that the first attribute is more imortant than the second; 2) a ci-arc between two attributes iies that relative imortance of the connected attributes is deendent on the values of some other attributes. In contrast to that, relative imortance of attributes in hierarchical CP-networks is dictated b the reference grah structure, thus i-arcs would be redundant here. However, to reresent conditional attribute imortance, the hierarchical CP-network model needs to be extended. The idea of extending CP-networks with irrelevant attributes was introduced in [12]. The class of networks considered in [12] is more general than hierarchical CP-networks. In this model, W X is not limited to the set of siblings and descendants of X and can be an subset of X {X} Pa(X). However, [12] does not consider how to deal with infinite domain attributes, and it does not have the notion of conditional reference table, which is essential for CP-networks. [12] roves that the orders induced b some classes of the extended CP-networks are strict artial orders. However, it does not rovide an algorithm of constructing reference formulas for them. The observation that the semantics of CP-nets can be catured using Constraint Datalog rograms was first made in [6]. It was further develoed into a method of bridging the TCP-network framework and the framework of reference relations in [7]. In [7] a binar reference formula is constructed for a given TCP-network using a set of reference constructors. The reference order is essentiall comuted as transitive closure of the union of all relations reresenting the orders induced b each conditional reference table. The transitive closure is comuted using Constraint Datalog. When Constraint Datalog is used to comute transitive closure, one needs to show that the evaluation terminates (which is the case onl for some constraint theories [8]). Moreover, one ticall obtains at best exonential bounds on the resulting formula size. In contrast to that, we describe classes of hierarchical CP-networks for which our algorithm can construct olnomial-size reference formula reresentations. [7] also gives an exae of embedding references over infinite domain attributes into TCP-networks, namel it shows how can one reresent CPT(X) of an infinite domain attribute X using a limited set of reference constructors for which a transitive closure Datalog rogram terminates. In our work we have no limitations on the class of SPO references formulas used in CPT(X) reresentation. In this aer, we define hierarchical CP-networks that are based on a relaxed ceteris aribus rincile (i.e. some other attributes being equal). An interesting direction of the future work is to extend our semantics using the some other attributes being equivalent rincile where instead of equalit attribute values are considered to be equivalent according to some indifference relation [11]. One should also consider aling the existing reference modification and construction techniques [5, 10] to hierarchical CP-networks. 6. ACKNOWLEDGMENT Research suorted b NSF grant IIS REFERENCES [1] C. Boutilier, R. Brafman, C. Domshlak, H. Hoos, and D. Poole. CP-nets: A tool for reresenting and reasoning with conditional ceteris aribus reference statements. Journal of Artificial Intelligence Research (JAIR), 21: , [2] R. I. Brafman, C. Domshlak, and E. Shimon. Introducing variable imortance tradeoffs into CP-nets. Proceedings of the Eighteenth Conference on Uncertaint in Artificial Intelligence (UAI02), ages 69 76, [3] J. Chomicki. Semantic otimization techniques for reference queries. Information Sstems, 32(5), Jul 2007, ages [4] J. 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