Complexity of Reasoning in Entity Relationship Models

Transcription

1 Complexity of Reasoning in Entity Relationship Models A. Artale 1, D. Calvanese 1, R. Kontchakov 2, V. Ryzhikov 1, M. Zakharyaschev 2 1 Faculty of Computer Science Free University of Bozen-Bolzano I Bolzano, Italy 2 School of Comp. Science and Inf. Sys. Birkbeck College London WC1E 7HX, UK {roman,michael}@dcs.bbk.ac.uk March 29, 2007 Abstract In this work we investigate the complexity of reasoning over various fragments of the Extended Entity Relationship (EER) language, which include different combinations of the constructs for isa between concepts and relationships, disjointness, covering, cardinality constraints, including their refinement. Specifically, we show that reasoning over ER diagrams with isa between relationships is ExpTime-hard, even when we drop relationship covering. Surprisingly, when we drop also isa between relations, reasoning becomes NP-complete. If we further remove boolean constructs, reasoning becomes polynomial. Our lower-bound results are established through direct reductions, while the upper-bounds follow from correspondences with expressive variants of the DL DL-Lite. 1 Introduction This work investigates the complexity of reasoning over conceptual modeling languages. We consider various fragments of the Extended Entity Relationship (EER) language [7]. The full fledged EER language, denoted as ER full, has the ability to represent: isa relations between both entities and relationships; disjointness and covering (denoted in the following as boolean constructors) between both entities and relationships; cardinality constraints for participation of entities in relationships; refinement of cardinalities for sub-entities participating into relationships; and multiplicity constraints for attributes. It has been shown in [3] that reasoning with respect to ER full class diagrams is an ExpTimecomplete problem. The proof shows how to reduce the ExpTime-complete problem of reasoning over ALC [2] knowledge bases to reasoning over ER full. The reduction proposed makes use of both isa and boolean constructors between relationships. The first original result presented in this paper shows that the language ER full without booleans between relationships (denoted as ER isar ) is still ExpTime-complete. This result shows that we don t need boolean constructs between relationships 1

2 and that isa between relationships is enough to have an ExpTime reasoning problem. To show the hardness complexity result we provide a polynomial reduction from ALC knowledge bases to ER isar schemas. The membership in ExpTime follows from the fact that ER isar is a sub-language of ER full. On the other hand, we prove that reasoning on the language ER bool (essentially ER isar without isa between relationships) is an NP-complete problem. This quite surprising result shows that isa between relationships is a major source of complexity for automated reasoners. In [1] it has been shown that the logic DL-Lite bool, which is the boolean extension of DL-Lite [6], has an NPcomplete reasoning problem. We show that ER bool schemas can be captured by knowledge bases in DL-Lite bool proving membership in NP. On the other hand, completeness for NP is shown by a polynomial reduction from the satisfiability problem in propositional calculus. Finally, we consider the simplest language, ER ref, i.e., ER bool without the covering constructor, showing that the reasoning problem is NLogSpace-complete. To prove this complexity result we consider the logic DL-Lite krom, extending DL-Lite with cardinality restrictions and with krom-style axioms, and proved NLogSpace-complete in [1]. Membership in NLogSpace of ER ref is shown by reducing reasoning over ER ref to reasoning over DL-Lite krom, while hardness for NLogSpace is showed by a reduction from the graph reachability problem. 2 The DL-Lite language We consider the extension DL-Lite bool [1] of the description logic DL-Lite introduced in [5, 6]. The language of DL-Lite bool contains atomic concept names A 0, A 1,..., and atomic role names P 0, P 1,.... Complex roles R and concepts C of DL-Lite bool are defined as follows: R ::= P i Pi B ::= A i q R, C ::= B C C 1 C 2, where q 1. Concepts of the form B are called basic concepts. A DL-Lite bool knowledge base, K, consists of axioms of the form C 1 C 2. A DL-Lite bool interpretation is a structure of the form I = (, A I 0, A I 1,..., P I 0, P I 1,... ), (1) where is a nonempty set, A I i, Pi I in I as usual:. The role and concept constructors are interpreted (P i )I = {(y, x) (x, y) P I i }, I =, ( q R) I = {x {y (x, y) R I } q} ( C) I = \ C I, (C 1 C 2 ) I = C I 1 C I 2. The standard abbreviations :=, R := ( 1 R) and q R := ( q + 1 R) we need in what follows are self-explanatory and correspond to the intended semantics. We say that an interpretation satisfies an axiom C 1 C 2 iff C1 I CI 2. A knowledge base K is satisfiable if there is a model 2

3 satisfying all the members of K. A concept C is satisfiable w.r.t. a knowledge base K if there is a model I of K such that C I. We also consider a sublanguage of DL-Lite bool : the Krom fragment, DL-Lite krom. (Krom) A DL-Lite krom knowledge base only contains axioms of the following form: B 1 B 2 or B 1 B 2 or B 1 B 2. 3 The Conceptual Modeling Language In this section, we define the notion of a conceptual schema by providing its syntax and semantics for the full fledged conceptual modeling language ER full. First citizens of a conceptual schema are entities, relationships and attributes. Arguments of relationships specifying the role played by an entity when participating to a particular relationship are denoted by means of so called role names. Given a conceptual schema, we make the following assumptions: Relationships and entity names are unique; names for attributes are local to entities, i.e., the same attribute can be re-used by different entities but the typing must be the same; role names are unique w.r.t. a particular relationship but not w.r.t. the entire schema (this freedom will be limited when considering conceptual models without sub-relationships). 3.1 Syntax We make use of the notion of labelled tuple. Given two finite sets X, Y one of which, X, is considered as the labeling set an X-labeled tuple T over Y is a function from X to Y together with the bijective function p : X {1,..., X } representing fixed enumeration of the elements of the tuple. Given x X, we use the notation T [x] to refer to the element y Y labelled by x. Given X = {x 1,..., x n }, the expression x 1 : y 1,..., x n : y n will be used as the full representation of T, and T [x i ] = y i, for 1 1 n. We also write (x i, y i ) T for T [x i ] = y i, while T Y (X) denotes a set of all X-labeled tuples over Y. From the definition of labeled tuple, we have that x i x j if i j, while this is not necessarily the case for elements in the set Y. Definition 1 (ER full Syntax). An ER full conceptual schema is a tuple: where: Σ full = (L, rel, att, card R, card A, ref, isa, disj, cov) L is the disjoint union of alphabets E (entity symbols), A (attribute symbols), R (relationship symbols), U (role symbols), and D (domain symbols). We will call the tuple (E, A, R, U, D) the signature of the schema Σ. rel : R T E (ν) is a total function that assigns to every relation symbol a tuple over ν U the entity symbols labelled with a non-empty collection of role symbols: rel(r) = U 1 : E 1,..., U m : E m, and m is the arity of R. Note that, from the definition of labeled tuple, roles are unique for a given relationship while entities can be repeated. att : E T D (α) is a total function that assigns to every entity symbol a tuple over α A the domain symbols labelled with some collection (possibly empty) of attribute symbols: att(e) = A 1 : D 1,..., A h : D h. 3

4 card R : R U E N (N { }) is a function denoting cardinality constraints. card R (R, U, E) is defined if and only if (U, E) rel(r). If not stated otherwise, the first component of the value of card R is assumed to be zero, and the second component is assumed to be. card A : A E N (N { }) is a function denoting multiplicity for attributes. card A (A, E) is defined only if (A, D) att(e), for some D D. If not stated otherwise, the first component of the value of card A is assumed to be zero, and the second component is assumed to be. ref : R U E N (N { }) is a function denoting refinement of cardinality constraints for sub-entities. ref(r, U, E) is defined only if E isa E and (U, E ) rel(r). If not stated otherwise, the first component of the value of ref is assumed to be zero, and the second component is assumed to be. Note that, ref can capture classical cardinality constraints. isa is a binary relation: isa (E E) (R R). isa between relationships is restricted to relationships with the same arity. 1 We will use the notation isa E to denote isa restricted to entities and isa R for the restriction to relationships. disj and cov are binary relations over (2 E E) (2 R R), describing disjointness and covering partitions, respectively. As for isa, relationships should be of the same arity. We will use the symbols disj E, cov E to denote disj and cov restricted to entities, and disj R, cov R for the restriction to relationships. 3.2 Semantics The following definition specifies a set-theoretic semantics to interpret ER full schemas. Definition 2 (ER full Semantics). Let Σ be an ER full conceptual schema, and {B D } D D a collection of disjoint and countable sets called basic domains. An interpretation for Σ is a pair B = ( B Λ B, B), where: B is a non empty set, called the interpretation domain. Λ B Λ B D, with Λ B D B D for each D D, called the active domain, such that B Λ B =. D D B is a function such that: (i) D B = Λ B D, for each D D. (ii) E B B, for each E E. (iii) R B ( B ) m, with m arity of R, for each R R. (iv) A B B Λ B, for each A A. An interpretation B of a schema Σ is called a legal database state if it satisfies the following conditions: 1 For the isa relation, we use the notation E 1 isa E 2 as a shortcut for (E 1, E 2) isa. Similarly for disj and cov. 4

5 1. For each R R, if rel(r) = U 1 : E 1,..., U m : E m, then, i {1,..., m}, (e 1,..., e i,..., e m ) ( B ) m.(e 1,..., e i,..., e m ) R B e i E B i. 2. For each E E, if att(e) = A 1 : D 1,..., A h : D h, then, i {1,..., h}, (e, a) B Λ B.(e, a) A B i a Di B. 3. For each R R, if rel(r) = U 1 : E 1,..., U m : E m, then, i {1,..., m}.card R (R, U i, E i ) = (α, β) e E B i.α {(e 1,..., e i,..., e m ) R B e i = e} β. 4. For each E E if att(e) = A 1 : D 1,..., A h : D h, then, i {1,..., h}.card A (A i, E) = (α, β) e E B.α {(e, a) A B i } β. 5. For each R R if rel(r) = U 1 : E 1,..., U m : E m, then, i {1,..., m}, E E.E isa E i, then, ref(r, U i, E) = (α, β) e E B.α {(e 1,..., e i,..., e k ) R B e i = e} β. 6. For each E 1, E 2 E, we have that E 1 isa E 2 implies E B 1 EB 2 (similarly for relationships). 7. For each E, E 1,..., E n E, we have that {E 1,..., E n }disje, implies E B i E B and E B i EB j = for i, j {1,..., n} and i j (similarly for relationships). n 8. For each E, E 1,..., E n E, we have that {E 1,..., E n } cov E implies E B = for relationships). i=1 E B i (similarly 3.3 Reasoning Problems Reasoning tasks over a conceptual schema include verifying whether an entity, a relationship, or a schema are consistent, or checking whether an entity (relationship) subsumes another entity (relationship). The model-theoretic semantics associated to a conceptual schema allows us to formally define these reasoning tasks. Definition 3 (Reasoning services). Let Σ be an ER full schema, E E an entity, and R R a relationship. The following are the reasoning tasks over Σ: Schema consistency. Σ is consistent iff there exist a legal database state B for Σ such that for some entity E E, E B. Strong (schema) consistency. Σ is strongly consistent iff there exist a legal database state B for Σ such that for every entity E E, E B. Entity consistency. An entity E E is consistent w.r.t. a schema Σ iff there exists a legal database state B for Σ such that E B. Relationship consistency. A relationship R R is consistent w.r.t. a schema Σ iff there exists a legal database state B for Σ such that R B. Entity (relationships) subsumption. An entity E 1 E subsumes an entity E 2 E w.r.t. a schema Σ iff for every legal database state B for Σ, we have that E1 B EB 2 (similarly for relationships). 5

6 . 1,1 C R 2 R A2 1, 1 1, 1 C RA R A1 A C R cov C RA 1, 1 R A1 disj A O 1,1 R 1 R 2 1,1 1, 1 1, 1 1, n C R A2 C RA R A1 A C R O. 1,1 R 1 Figure 1: Encoding axioms: (a) A R.C; (b) A R.C. The reasoning tasks Schema/Entity/Relationship consistency and Entity subsumption are reducible one to each other. Indeed, that Entity subsumption is equivalent to Entity satisfiability is shown in [3]. In the following, let O denote a fresh entity symbol and R E a fresh relationship symbol. Schema consistency can be reduced to Entity consistency by extending Σ as follows: E = E {O }, cov = cov {(E, O )}. Now, Σ is consistent iff O is consistent w.r.t. Σ. Vice versa, we extend Σ as follows: E = E {O }, cov = cov {(E, O )}, R = R {R E }, rel(r E ) = U 1 : E, U 2 : O, card R (R E, U 2, O ) = (1, ). Now, E is consistent w.r.t. Σ iff Σ is consistent. Relationship consistency can be reduced to Entity consistency by extending Σ as follows: E = E {O }, let E be some entity such that (U, E) rel(r), then isa E = isa E {(O, E)} and ref extends ref so that ref (R, U, O ) = (1, β) where β is such that card R (R, U, E) = (α, β). Relation R is consistent w.r.t. Σ iff entity O is consistent w.r.t. Σ. Vice versa, we introduce a new relationship, R E, such that rel(r E ) = U 1 : E, U 2 : E. Thus, E is consistent iff R E is consistent. Finally, we note that, in the absence of the covering constructor, Schema consistency cannot be reduced to a single instance of Entity consistency, though it can be reduced to several Entity consistency checks. Thanks to the above equivalences, in the following we will consider Schema consistency as the main reasoning service. 4 Reasoning over ER isar Schemas The modeling language ER isar denotes the language without booleans between relationships but with the possibility to express isa between them, i.e., Σ isar = (L, rel, att, card R, card A, ref, isa, disj E, cov E ). We prove that reasoning in ER isar is an ExpTime-complete problem by showing a reduction from concept satisfiability w.r.t. an ALC knowledge base to entity satisfiability w.r.t. an ER isar conceptual schema. The reduction is similar to the one presented in [3] for capturing ALC knowledge 6

7 bases in ER full. In particular, axioms with boolean constructors between concepts are mapped as in [3]. However, the mapping provided in [3] uses both booleans and isa between relationships to model ALC axioms of the form A R.C. Figure 1 introduces a new mapping for the latter kinds of axioms that does not use booleans between relationships. The following proposition proves that isa between relationships and thus conceptual schemas in ER isar are enough to encode ALC axioms. Proposition 4. An ALC knowledge base is satisfiable iff the conceptual schema Σ constructed as stated above is satisfiable. Proof. We show that the mappings in Figure 1 are correct. ( ) Let B = ( B, B) be a legal database for Σ. We show that I = ( I, I) is a model of the ALC knowledge base, where I = B and I coincides with B on concepts/entities while for a role R we set R I = (R 1 R 2 ) B where is the composition operation. A R.C. Let o A I. We show that o ( R.C) I. Let o I such that (o, o ) R I. Thus, by construction, o.(o, o ) (R 1 ) B (o, o ) R2 B. Then, o CR B = CI R and, due to the covering, o CR I A CR I. We show that o C I A R. By contradiction, let assume that o C I A R, A then,!a.(o, a) R B A1 and a AB = A I. From R B A1 RB 1 and the cardinality constraint on the entity C R, we conclude that a = o. By assumption, o A I, which contradicts o = a A I, due to the disjointness constraint between A and A. Thus, o CR I and!b.(o, b) R B A A2 and b C B = C I. From RA2 B RB 2 and the cardinality constraint on the entity C R, we conclude that b = o. Thus, o B I, and this proves that o ( R.C) I. A R.C. Let o A I we show that o ( R.C) I. Since o A I = A B, then, o.(o, o ) R B AB1 o CR B AB. Since RAB1 B RB 1, then (o, o ) R1 B. Furthermore, o.(o, o ) RAB2 B o B B = B I. Thus, by construction, (o, o ) R I o B I. ( ) Let I = ( I, I) be an ALC interpretation and φ R be a bijective mapping, φ R : R I R, with I R =. A R.C. We build a legal database state B = ( B, B) for the schema in Figure 1(a). B = I R A B = A I ; C B = C I ; O B = B ; A B = B \ A I ; C B R = φ R(R I ) R B 1 = {(r, o) R I r = φ R (o, o ), for some o I } R B 2 = {(r, o) R I r = φ R (o, o.), for some o I } C B R A = {r = φ R (o, o ) o A B }; C B R A = {r = φ R (o, o ) o A B } R B A1 = {(r, o) R I r = φ R (o, o ), for some o I r C B R A } R B A2 = {(r, o) R I r = φ R (o, o), for some o I r C B R A } R B A1 = {(r, o) R I r = φ R (o, o ), for some o I r C B R A } It is now easy to show that if I satisfies A R.C then B satisfies the schema in Figure 1(a). A R.C. The proof is similar to the above case. Just notice that C B R AB = {r = φ R (o, o ) o A B o C B }. 7

8 Since reasoning over ALC knowledge bases is an ExpTime-complete problem [2] and ER isar is a sub-language of ER full which is ExpTime-complete we can state the following result. Theorem 5. Reasoning over ER isar conceptual schemas is an ExpTime-complete problem. 5 Reasoning over ER bool Schemas With ER bool we denote the language without isa and booleans between relationships. The syntax of the language is restricted w.r.t. that of ER full in the following way: isa isa E E E, disj disj E 2 E E, cov cov E 2 E E. For rel we pose the additional restriction that we do not allow reusing the same role symbol by different relations: There is no U U such that for some distinct R, R R it is the case that (U, E ) rel(r ) and (U, E ) rel(r ) for some E, E E. This last restriction is acceptable in ER bool since the language does not allow for sub-relationships. We first define the polynomial translation τ of an ER bool schema to a DL-Lite bool knowledge base. We then show that an entity E is consistent in an ER bool schema Σ iff the translation of the entity, E, is satisfiable in the knowledge base K = τ(σ). The latter problem is known to be in NP [1]. Let Σ bool be an ER bool schema, and be a mapping from the set L to the set of concepts and roles in the knowledge base τ(σ). In particular, maps entity, relationship and domain symbols to concept names, and attribute and role symbols to role names. The translation τ(σ) is defined as the set of assertions τ rel τ att τ cardr τ carda τ ref τ isa τ disj τ cov τ dom, where: τ rel = τ R,U,E rel R R,(U,E) rel(r) τ att = τ A,D att E E,(A,D) att(e) τ cardr = R R,(U,E) rel(r), with τ R,U,E rel, with τ A,D att τ R,U,E card R (R, U, E) as follows: τ carda = E E,(A,D) att(e) as follows: = {R U, 2U, U R, U E}. = { A D}. card R, where τ R,U,E card R is defined according to the value of (α, β) = τ R,U,E card R = τ A,E {E αu, E βu }, α 0, β {E αu }, α 0, β = {E βu }, α = 0, β, otherwise card A where τ A,E card A is defined according to the value of (α, β) = card A (A, E) τ A,E card A = {E αa, E βa}, α 0, β {E αa}, α 0, β = {E βa}, α = 0, β, otherwise 8

9 τ ref = τ R,U,E ref (R,U,E) s.t. ref(r,u,e)=(α,β) ref(r, U, E) as follows: τ isa = {E 1 E 2 } E 1 isae 2 τ disj = τ R,U,E ref = ({E 1 E,..., E n E}, where τ R,U,E ref is defined according to the value of (α, β) = {E αu, E βu }, α 0, β {E αu }, α 0, β = {E βu }, α = 0, β, otherwise {E 1,...,E n}disje τ cov = {E 1 E,..., E n E, E E 1 E n } {E 1,...,E n}cove τ dom = D D,X E R D,D X {D X} 1 i<j n {E i E j }) We now formally prove that the above polynomial translation is useful for reasoning over ER bool schemas. Proposition 6. An ER bool schema Σ is consistent iff the DL-Lite bool knowledge base K = τ(σ) is consistent. Proof. ( ) Let the interpretation B = ( B Λ B, B) with the domain sets {B D } D D be a legal database state for Σ. To define the interpretation I = ( I, I) for K we introduce a new set, R, disjoint from B Λ B and a bijective mapping, φ R : R B R. Now we define an interpretation for K as follows. i) I = B Λ B R ii) D I = D B, for every D D iii) E I = E B, for every E E iv) A I = A B, for every A A v) R I = φ R (R B ) vi) U I I I, for every U U, such that, if rel(r) = U 1 : E 1,..., U : E i,..., U m : E m, for some R R and i {1,..., m}, then: U I = {(r, e i ) R B r = φ R (e 1,..., e i,..., e m ) and (e 1,..., e i,..., e m ) R B }. Now we prove that the interpretation I as defined above satisfies K = τ(σ). We guide the proof by considering the translation of the various statements in Σ. R R 9

10 1. rel(r) = U 1 : E 1,..., U m : E m. For every statement of this form we have the following m collection of assertions: {R U i, 2U i, U i R, U i Ei } in K. We prove that I satisfies each assertion. i=1 (a) R U i. Assume r R I, which implies that r = φ R (e 1,..., e i,..., e m ), for some (e 1,..., e i,..., e m ) R B. According to our construction of I we have that I I (φ R (e 1,..., e i,..., e m ), e i ) U i. This proves r Ui. Thus I = R Ui. (b) 2U i. Towards a contradiction, let (r, e) and (r, e ) in U i I, with e e. By construction, r = φ R (e 1,..., e i 1, e, e i+1,..., e m ) with (e 1,..., e i 1, e, e i+1,..., e m ) R B, and r = φ R (e 1,..., e i 1, e, e i+1,..., e m) with (e 1,..., e i 1, e, e i+1..., e m) R B. Since φ R (e 1,..., e i 1, e, e i+1,..., e m ) = φ R (e 1,..., e i 1, e, e i+1,..., e m), and φ R is a bijection, then, e = e which contradicts our assumption. I I (c) U i Ei. Assume e U i, then (r, e) Ui for some r I. Since U i can be involved only in one relation (R in this case), then, by U I construction, we can state that r = φ R (e 1,..., e,..., e m ) for some (e 1,..., e,..., e m ) R B. Now, according to semantics of R, e Ei B I = E i. So we have shown that I = Ui Ei. (d) U i R. Assume r U i I, then (r, e) Ui I for some e I. Since U i can be involved only in one relation (R in this case), then, by U I construction, we can state that r = φ R (e 1,..., e,..., e m ) for some (e 1,..., e,..., e m ) R B. Thus, r R I, by R I construction. Thus I satisfies τ rel. 2. att(e) = A 1 : D 1,..., A h : D h. For every statement of this form we have the follwing h collection of assertions { A i Di }. We show that for every i the corresponding assertion i=1 is satisfied by I. Let a ( A i ) I, then for some e I, (e, a) A i I and, since Ai I = A B i, then e B and a Λ B. According to the semantics of att(e), a Di B I = A i Di and τ att is satisfied by I. = D I i. Therefore, 3. card R (R, U, E) = (α, β). By definition, there is a (unique) statement such that (U, E) rel(r). Consider the following two cases: (a) α 0. In this case τ R,U,E card R contains the assertion E αu. As long as B is a legal database state, then, fixed e E I = E B, {(e 1,..., e i,..., e m ) R B e i = e} α. Now, by construction of U I, {r (r, e) U I } = {φ R (e 1,..., e i,..., e m ) (e 1,..., e i,..., e m ) R B and e i = e} = (since φ R is a bijection) = {(e 1,..., e i,..., e m ) R B e i = e} α. This proves that e ( αu ) I and thus I = E αu. (b) β. In this case τ R,U,E card R contains the assertion E βu, and the proof that this assertion is satisfied by I proceeds in the same way as before. We thus conclude that I = E βu. The two cases above complete the proof that τ cardr is satisfied by I. 10

11 4. card A (A, E) = (α, β). As for case 3 we consider two cases: (a) α 0. We prove that each assertion E αa in τ A,E card A is satisfied by I. Since, B is a legal database state, then fixed e E I = E B, {(e, a) A B } α, i.e., {a (e, a) A I } α. This proves that I = E αa. (b) β. The proof that the assertion E βa is satisfied by I is similar to the previous case. The two cases above complete the proof that τ carda is satisfied by I. 5. ref(r, U, E) = (α, β). The proof for this case is similar to the case 3. Thus, τ ref is satisfied by I. 6. E 1 isa E 2. The translation τ isa is satisfied since E I 1 = E B 1 EB 2 = EI {E 1,..., E n } disj E. Since B is a legal database state, E B i E B and E B i E B j = for i, j {1,..., n} and i j. Hence, every assertion in τ disj is satisfied by I. 8. {E 1,... E n } cov E. As for the previous case, it is immediate to check that τ cov is satisfied by I. 9. We finally prove that τ dom is satisfied by I. For any two different D 1, D 2 D, D B 1 DB 2 =. By construction of D I, we have that D 1 D 2 is satisfied by I. For every pair (D, E) D E since we know that E I = E B B and D I = D B Λ B and B Λ B =, we get that I = D E. For every pair (D, R) D R we have again D I = D B Λ B and R I R with R Λ B =, so I = D R. This concludes the proof that I = τ dom. Thus, I satisfies τ(σ). ( ) Let the interpretation T = ( T, T ) be a model for K = τ(σ). Without loss of generality, we can assume that T is a tree model (see Chapter 2 in [4]). From this interpretation we construct the domain sets {B D } D D and the legal database state B = ( B Λ B, B) for the ER bool schema Σ as follows. i) B D = Λ B D = DB = D T, for every D D ii) Λ B = D D Λ B D iii) B = T \ Λ B D iv) E B = E T, for every E E v) A B = A T ( B Λ B D ), for every A A vi) R B = {(e 1,..., e m ) ( T ) m r R T T T s.t. (r, e 1 ) U 1,..., (r, em ) U m }, for every R R with rel(r) = U 1 : E 1,..., U m : E m Observe that the function B satisfies properties (i)-(iv) of Definition 2. Indeed, (i) is true by construction; (ii) holds since T satisfies axioms τ dom while B is disjoint from Λ B by construction; (iii) is true since T satisfies τ rel ; (iv) holds by construction. We now show that B satisfies every assertion of the original ER bool schema Σ, thus proving that B is a legal database state for Σ. 11

12 1. rel(r) = U 1 : E 1,..., U m : E m. Let (e 1,..., e m ) R B, then, by construction, there is an r R T T R,U such that (r, e j ) U j for j {1,..., m}. Since T satisfies τ j,e j rel and U j Ej τ R,U j,e j rel we have that e j (E j ) T = Ej B, for every j {1... m}. 2. att(e) = A 1 : D 1,..., A h : D h. Let (e, a) B Λ B such that (e, a) A B i. Then, by T construction, (e, a) A i. Since T satisfies τ A i,d i att and A i Di τ A i,d i att, we have that T a D i = D B i Λ B, for each i {1,..., h}. 3. card R (R, U, E) = (α, β). Then Σ contains a declaration for R of the form rel(r) = U 1 : E 1,..., U i 1 : E i 1, U : E,..., U m : E m. Let e E B, then we have to show that α {(e 1,..., e i,..., e m ) R B e i = e} β. Consider the following two cases. (a) α 0. Since T satisfies τ R,U,E card R, E αu τ R,U,E card R and E B = E T, there exist at least α different r 1,..., r α T such that (r j, e) U T with j {1,..., α}. Furthermore, since T satisfies τ R,U,E rel, and in particular U R, then r 1,..., r α R T. Since T satisfies τ rel and in particular {R U i, 2U i }, for i {1,..., m}, then, j {1,..., α}, k {1,..., m},!e j k T.(r j, e j k ) U T k, where e j i = e j {1,..., α}. Furthermore, since T is a tree-like model, then, j, j {1,..., α}, k, k {1,..., m} \ {i}.e j k ej k, when either k k or j j. Thus, we proved that to each object in R T corresponds exactly one tuple, and vice versa. Now, by R B construction, j {1,..., α}.(e j 1,..., ej i 1, e,..., ej m) R B. This finally prove that {(e 1,..., e i,... e m ) R B e i = e} α. (b) β. Similarly to the previous point, we can show that there exist at most β different r 1,..., r β R T such that (r j, e) U T with j {1,..., β}. Since as we showed before, to each object in R T corresponds exactly one tuple, and vice versa, then we can conclude that {(e 1,..., e i,..., e m ) R B e i = e} β. To conclude the proof it only remains to notice that for the case when α = 0 or β = the corresponding number restriction for the set {(e 1,..., e i 1, e,..., e m ) R B } holds trivially. 4. card A (A, E) = (α, β). Let e E B = E T. Consider the following two cases. (a) α 0. Since T satisfies τ carda and E αa τ carda, and T satisfies τ A,D att, for some D such that (A, D) att(e), and A D τ A,D att, then, {a (e, a) A T and a D B } α. Since A B = A T ( B Λ B ), then {a (e, a) A B } α. (b) β. The proof is similar to the previous case. For the cases α = 0 or β = the corresponding cardinality restrictions are trivially satisfied. 5. ref(r, U, E) = (α, β). The proof essentially repeats that one for case E 1 isa E 2. This holds in B since T satisfies τ isa, E 1 E 2 τ isa and E i B = E i T, for i {1, 2}. 7. {E 1,..., E n } disj E. This holds in B since T satisfies τ disj, {E 1 E,..., E n E} τ disj, E i E j τ disj, for i, j {1,..., n}, i j, and E B i 8. {E 1,... E n } cov E. Similar to the previous case. = E i T, for i {1,..., n}. 12

13 As a direct consequence of the above proposition and the fact that checking consistency of a knowledge base in DL-Lite bool is NP-complete [1] we can prove the following corollary. Corollary 7. Checking consistency of an ER bool conceptual schema can be solved in NP. To show that reasoning in ER bool is an NP-complete problem we provide a polynomial reduction from the NP-complete problem 3SAT to the problem of entity consistency in an ER bool schema. Let the instance of 3SAT be given by a set φ = {c 1,..., c n } of 3-clauses over the literals lit(φ) = {a 1,..., a m }. We define the ER bool schema Σ φ as follows. The signature L is given by: E = a lit(φ){a} {c} {φ, }, A =, R =, U =, D =. c φ isa = c φ{(φ, c)}. cov = c φ disj = {( {a}, c)} {(E \ { }, )} a lit(c) {({a, b}, )}. (a,b) s.t. a,b lit(φ) and b= a {({a, b}, )}. (a,b) s.t. a,b lit(φ) and b= a att, rel, card R, card A, ref are empty functions. Now we prove the following Lemma. Lemma 8. φ is satisfiable iff the entity φ is consistent w.r.t. the schema Σ φ. Proof. ( ) Let J be a model for φ, i.e., J = φ, and construct from it the following interpretation I = ( I, I) for Σ φ. I = {o} I = {o} For every E E \ { }: E I = { {o}, J = E, J = E We show that I = Σ φ. Since J = φ, it is also the case that J = c for all c φ, and, by construction, c I = {o}. This implies that every isa assertion is satisfied by I. Now, fix c, then since c is satisfied by J, we have that for at least one a lit(c), J = a, which means that a I = {o}, and for the remaining a lit(c), a I {o}. This proves that every assertion in {( {a}, c)} cov c φ a lit(c) holds. The assertion (E \ { }, ) cov holds, since for every E E \ { } it holds that E I {o}, φ I = {o} and I = {o}. For every pair a, a lit(φ) it is the case that one of a, a is satisfied by J, hence this one will be interpreted with {o} in I. Since I = {o}, every assertion in {({a, b}, )} cov holds w.r.t. I. Since only one of a, a is satisfied by J, the (a,b) s.t. a,b lit(φ) and b= a other one will be interpreted in I as the empty set, so every assertion in disj holds, too. Thus, I = Σ φ, with φ I. 13

14 ( ) Let I = ( I, I) be a model of Σ φ such that o φ I, for some o I. We construct the evaluation J for φ in the following way. For every propositional letter prop φ, we set: J = prop iff o prop I We show that J = φ. Indeed, since o φ I, and φ isa c i, then o c I i, for {i,..., n}. To prove that J = φ it is enough to show that J = c i, for {i,..., n}. Since ( {a}) cov c i then a lit(c i ) a lit(c i ) such that o a I. Now, if a is a propositional letter then, by construction of J, J = a, and thus J = c i. Otherwise, a = prop, and, since {a, prop} disj, o prop I. Thus, by construction of J, J = prop, i.e., J = a, and thus J = c i. From the above lemma, Corollary 7 and the mutual reducibility between Schema and Entity consistency, we can finally prove the following completeness result. Theorem 9. Checking consistency of ER bool conceptual schemas is an NP-complete problem. 6 Reasoning problem for ER ref The modeling language ER ref denotes the language without booleans and isa between relationships and with the possibility to express isa and disjointness between entities, i.e., Σ ref = (L, rel, att, card R, card A, ref, isa E, disj E ). We prove that reasoning in ER isar is an NLogSpace-complete problem. Consider the reduction τ from Section 5. It is not difficult to check that τ is logspace bounded. At the same time, for any ER ref schema Σ the knowledge base K = τ(σ) is a DL-Lite krom knowledge base, due to emptiness of the τ cov translation while τ disj can be easily transformed noticing that an axiom of the form C C 1 C 2 is equivalent to the pair of axioms C C 1 and C C 2. Thus, as a consequence of Proposition 6, the problem of entity consistency in ER ref can be reduced in logspace to the problem of concept satisfiability w.r.t. a DL-Lite krom knowledge base. As we know that the concept satisfiability problem w.r.t. a DL-Lite krom knowledge base is in NLogSpace [1], we conclude that the entity consistency problem w.r.t. ER ref schemas is in NLogSpace as well. To show the hardness result we consider the reachability problem in oriented graphs, shown to be NLogSpace-hard in [8] and referred to as the maze problem. Let the instance of maze be given as G = (V, E, s, t), where s, t denote initial and terminal vertices of (V, E), respectively. The encoding of a maze instance into ER ref gives the following schema Σ G : isa = {(u, v)}, (u,v) E disj = {({s, t}, O)}, with O a newly introduced entity. Since the translation is obviously logspace bounded we can prove the following result. Proposition 10. The terminal node t is reachable from s in G = (V, E, s, t) iff the entity s is not consistent w.r.t. Σ G Since NLogSpace=coNLogSpace (see Immerman-Szelepcsényi theorem in [8]) it follows that the problem of entity consistency in ER ref is NLogSpace-hard. This result coupled with the membership in NLogSpace showed before give us the following completeness result. Theorem 11. The entity consistency problem for ER ref is NLogSpace-complete. 14

15 References [1] Anonymous. DL-Lite in the light of first-order logic. Technical report, [2] F. Baader, D. Calvanese, D. McGuinness, D. Nardi, and P. F. Patel-Schneider, editors. Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, [3] Daniela Berardi, Diego Calvanese, and Giuseppe De Giacomo. Reasoning on UML class diagrams. Artificial Intelligence, 168(1 2):70 118, [4] Patrick Blackburn, Maarten de Rijke, and Yde Venema. Modal Logic. Cambridge University Press, [5] Diego Calvanese, Giuseppe De Giacomo, Domenico Lembo, Maurizio Lenzerini, and Riccardo Rosati. DL-Lite: Tractable description logics for ontologies. In Proc. of AAAI-05, pages , [6] Diego Calvanese, Giuseppe De Giacomo, Domenico Lembo, Maurizio Lenzerini, and Riccardo Rosati. Data complexity of query answering in description logics. In Proc. of the 10th Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR 2006), pages , [7] R. Elmasri and S. B. Navathe. Fundamentals of Database Systems. Addison-Wesley, 5th edition, [8] Dexter Kozen. Theory of Computation. Springer,