Roth, M. A., Korth, H. F., Silberschatz, A.: Null Values in Nested Relational Databases. Acta Inf. 26, 615-642 (1989).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and iteration facilities. KEY WORDS: nested relations, null values, null extended algebra, faithful extended algebra operators, precise extended algebra operators ############### Appeared in Fundamenta Informaticae, Vol. 19, pp. 303 343, 1993. 2 1. Introduction The nested relational model [1, 2, 8, 13, 15, 16, 19, 20, 21, 22, 23, 26, 28] was developed in order to extend the applicability of the relational model [5, 18, 27] to more complex, nonbusiness applications such as CAD, image processing and text retrieval [2] Nested relations can model complex data directly by recursively defining values of attribute domains to be either ....

....and text retrieval [2] Nested relations can model complex data directly by recursively defining values of attribute domains to be either flat relations in first normal form [5, 18, 27] or nested relations. Recently many extended algebras have been suggested for the nested relational model [1, 3, 6, 8, 10, 15, 16, 17, 21, 22, 23, 26, 28]. From these extended algebras only those of Roth et al. 22] Levene [15] and Levene and Loizou [16] have comprehensively incorporated null values into the nested relational model; Roth et al. 22] deal with a subclass of nested relations, whilst Levene and Loizou [16] deal with an extension of ....

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M.A. Roth, H.F. Korth and A. Silberschatz, Null values in nested relational databases, Acta Informatica 26 (1989) 615-642.

....in certain posets, cf. 6, 19] The idea of using orderings to represent partiality has been quite fruitful. A general theory of partial information and languages to handle it based on orderings was developed in [21] As another example of its applicability, it was shown in [18] that a mistake in [30] discovered in [17] goes away if equivalence with respect to v [ is used instead of equality in the definition of representation systems. Data oriented programming In this subsection we give an overview of the data orientation as a programming language paradigm (cf. Cardelli [7] and ....

M.A. Roth, H.F. Korth and A. Silberschatz. Null values in nested relational databases. Acta Informatica, 26 (1989), 615--642.

....to the property P2 for extended union, extended intersection and extended di erence. 13 Lemma 8 Z= B;X) r 1 [ r 2 ) Z= B;X) r 1 ) Z= B;X) r 2 ) if r 1 [A] r 2 [A] Proof: By assumption, 8t 1 2 r 1 ; t 2 2 r 2 ; t 1 [AY ] 6= t 2 [AY ] This condition implies LHS = RHS. See [20]) 2 Based on the de nition and assuming the scheme ER1 = ER2 = A; B; X;Y ) S 1 r 1 r 2 can be divided into two disjoint subsets T 1 1 and T 1 2 which are de ned in the following notations. Notations T 1 1 = ft j t 2 r 1 (9s 2 r 2 , such that t[A] s[A] t[B] 62 2 A=s[A] B) g; T 2 ....

M. Roth, H. Korth and A. Silberschatz. Null Values in Nested Relational Databases. Acta Informatica 26, 1989, pp. 615-642.

....for the nested relational model. Several variations of the nested relational model have been proposed in the literature, depending on whether null values are permitted [9] whether empty sets are permitted [1] whether atomic attributes form a key and what data manipulation operators are required [15, 14]. The model we use in this paper is the one proposed by [1] and called the Verso model which is based on partitioned normal form (PNF) relations [13] The reason for adopting this model is because of its exibility in supporting empty sets, the assumption that relations are in partitioned normal ....

....of how to extend the de nitions of containment and disjointedness from at relations to PNF relations is not as straightforward as might rst appear. This is discussed in more detail in [10] but we brie y summarise our approach here for the sake of completeness. In [10] we adopted the approach of [9, 14]. In this approach we require that the de nitions for containment for and disjointedness must be faithful and precise. By faithful, we mean that the de nitions for containment and disjointedness for PNF relations should coincide with the de nitions for containment and disjointedness for falt ....

M A Roth, H F Korth, and A Silberschatz. Null values in nested relational databases. Acta Informatica, 26(7):615-642, 1989.

....that might be a part of a university or a corporation database, some values are missing and the symbol ni (no information) is used. Note that there could be several different reasons for using ni. This is reflected in the second relation in figure 1 where three kinds of nulls are used (cf. LL86, RKS89, Zan84] ne means nonexistent; that is, John does not have a phone. un means existing unknown; Mary is on payroll but the precise figure of her salary is unknown. And ni still means no information. For other kinds of nulls see [GZ88, LL93] One of the most important achievements of the early ....

....is impossible. In fact, even milder definition of q(R) leads to similar negative results. Very little is known about null values in complex objects or nested relations, that is, relations whose attributes can be relation valued themselves. An attempt to extend the results of [IL84] was made in [RKS89] only for a restricted subclass of complex objects, those in partitioned normal form, cf. AB86] but later an error was found [LL91] It was then shown [LL93] that some of the results can be recovered if equality of representations of incomplete complex objects is replaced by the Hoare ....

M. A. Roth, H. F. Korth, and A. Silberschatz. Null values in nested relational databases. Acta Informatica, 26(7):615--642, 1989.

....proved. 2. 11 4 Containment and Disjointedness for PNF Nested Relations In this section, we de ne the properties of containment and disjointedness for PNF nested relations. In order to determine the correctness of our de nitions, we extend the notions of faithfulness and preciseness, de ned in [6, 11] for algebraic operators on nested relations, to boolean operators de ned on nested relations. Firstly let us de ne containment and disjointedness for at relations De nition 4.1 (Containment) The at relation r is contained in the at relation r, denoted by Cont(r; r) i r 6= and r r. De ....

M A Roth, H F Korth, and A Silberschatz. Null values in nested relational databases. Acta Informatica, 26(7):615-642, 1989.

....IEs and develop view maintenance algorithms for the nested relational model in this paper. Several different nested relational models have been proposed, depending on whether null values are permitted [8] whether empty sets are permitted [1] and what data manipulation operators are required [11, 12]. The model we use in this paper is the one proposed by [1] and called the Verso Model. The reason for adopting this model is because of its flexibility in supporting empty sets, the assumption that relations are in partitioned normal form (which has clearer semantics than general nested ....

M.A.Roth, H.F.Korth and A.F.Silberschatz, Null values in nested relational databases. Acta Informatica, 26(7):615-642, 1989.

.... [22, 51] in knowledge based systems [3] and in coupled systems [33] As separate research attempts, there have also been advances dealing with fuzzy information in database systems [9, 23, 33, 54, 52] and in knowledgebased systems [31, 58] and with other uncertain information in both sytems [47, 34]. Although there have been some theoretical studies for representing and implementing the complex objects along with their fuzzy attributes in a database model [54] there has not been any serious attempt for handling such fuzzy complex objects in a knowledge based systems. One of the most ....

....representations of information may provide a more intuitive user view of data. 3. Non first normal form reduces the number of tuples and eliminates redundancy. 4. Related to the stated three reasons, querying is also simplified [41] The nested relational data model (Non first normal form) [2, 24, 35, 41, 46, 47, 48, 18, 28] was developed in order to extend the support for the complex objects in the relational data model. Nested relations can naturally represent the hierarchy of complex objects by allowing recursively defined relation valued attributes in addition to atomic attributes. We will give a more formal ....

M.A. Roth, H.F. Korth, and A. Silberschatz. Null Values in Nested Relational Databases. 13(4):615--642, 1989.

....establish the connection between data models and types, i.e. to represent database objects (not necessarily relational databases) as typed objects in programming languages. There have been made a number of attempts to generalize relational databases giving up the firstnormal form assumption, see [1, 4, 6, 9, 10, 13, 16, 17, 18, 20]. They can be divided into two categories. The first one consists of models that do not contain sets. Usually it means that they admit null values and or record structures, the latter including also case, or discriminated union. In this case [5] provides us with the idea how to develop the ....

....domains multivalued dependencies are in one to one correspondence with the decompositions of relations. The model proposed in [5] does not admit constructions containing sets. However, they are necessary in order to describe some models which are being widely studied now, namely nested relations [6, 13, 16, 17, 18, 20] and complex objects [1, 13] which play an essential role in the theory of object oriented databases [2] By complex objects we mean objects constructed from the basis ones by using the operations of forming records (including discriminated union) and sets, i.e. record, variant and set ....

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M.A. Roth, H.F. Korth, A. Silberschatz. Null values in nested relational databases. Acta Informatica, 26(7):615--642, 1989.

....data across versions. This facility is difficult to implement in some of the proposals presented (most notably by Palisser (1989) due to the searching necessary to compile the completed schema. 5.3. Problems presented by null values The three valued null logic proposed by Zaniolo (1984) and Roth, Korth and Silberschatz (1989) presupposes a static schema. The introduction of attributes (or relations) undefined at a given point in time introduces new semantics to null values in the database. Indeed, Zaniolo s 3 valued null logic can be extended by a dimension as follows: Attribute is Defined Attribute is Not Defined ....

Roth, M.A., Korth, H.F. and Silberschatz, A. 1989. `Null values in nested relational databases'. Acta Inf. 26(7):615-642.

....models, normal forms; H.2.3 [Database Management] Languages query languages General Terms: Algorithms, Languages, Theory Additional Key Words and Phrases: Nested relations, nulls, null extended algebra, null extended data dependencies, extended chase 1. INTRODUCTION The nested relational model [ABIT86, ABIT89, GYSS89, LEVE90b, MAKI77, OZSO87, ROTH88, ROTH89, SCHE86, THOM86, VANG88] was developed in order to extend the applicability of the (flat) relational model [CODD79, MAIE83, ULLM88] to more complex, nonbusiness applications such as CAD, image processing and text retrieval [ABIT89] Nested relations can model hierarchical complex data directly by recursively defining ....

.... of the model (the extended algebra) 3) The integrity constraints of the model (extended data dependencies) So far most of the research on the nested relational model has concentrated on (1) and (2) while (3) has been mainly investigated in order to characterise subclasses of nested relations [ABIT86, BIDO87, FISC85, JAES82, MAKI77, MIUR86, ROTH88, ROTH89, TAKE89, 3 THOM86, VANG88] and in order to define normal forms for nested relations [LEVE89b, MAKI77, OZSO87, OZSO89, ROTH88] Furthermore, there have been only few extensions of the nested relational model which comprehensively incorporate nulls into the model [LEVE89a, LEVE90b, ROTH89] The treatment of incomplete ....

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ROTH, M. A., KORTH, H. F., AND SILBERSCHATZ, A. Null values in nested relational databases. Acta Informatica 26, (1989), 615-642.

....) 1 p q(L 0 q ) L rq ) 1 p s(L 0 s ) 2) where L 0 r = X; Y; Z) L 0 q = X 00 ; Y 00 ; Z 00 ) L rq = XX 00 ; Y Y 00 ; U 00 ) L 0 s = X 0 ; Y 0 ; U 0 ) 5. Correctness of decomposition P Join In this section we adapt the criteria, defined by Roth et al. [9], to establish the correctness of our P join for nested relations. The criteria for correctness of an extended operator is that it is faithful and precise. We state the formal definition of faithfulness from [9] Definition 5.1 Let P and P 0 be classes of relations and and 0 binary ....

....of decomposition P Join In this section we adapt the criteria, defined by Roth et al. 9] to establish the correctness of our P join for nested relations. The criteria for correctness of an extended operator is that it is faithful and precise. We state the formal definition of faithfulness from [9]. Definition 5.1 Let P and P 0 be classes of relations and and 0 binary operators on P and P [ P 0 respectively. We say that 0 is faithful to if r 0 q = r q for every r; q 2 P for which r q is defined. 2 Proposition 1 P join is faithful to standard natural join. Proof: By ....

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M.A. Roth, H.F. Korth and A. Silberschatz. Null Values in Nested Relational Databases. Acta Informatica 26, pages 615-642, 1989.

....in Partitioned Normal Form (PNF) is not a precise generalization of standard projection with respect to unnesting and PNF possibility function POSS . The extended projection of a nested relation, r, over relation scheme, R, on attributes, X , denoted by P e X (r) is given in Definition 22 of [1] by P e X (r) tP X (r) e (t) 1) where e is the extended union operator of Definition 17 therein. It was claimed in [1] Proposition 5 thereof, that the extended projection operator, P e , is a precise generalization of the standard projection operator, P, with respect to unnesting, ....

....function POSS . The extended projection of a nested relation, r, over relation scheme, R, on attributes, X , denoted by P e X (r) is given in Definition 22 of [1] by P e X (r) tP X (r) e (t) 1) where e is the extended union operator of Definition 17 therein. It was claimed in [1], Proposition 5 thereof, that the extended projection operator, P e , is a precise generalization of the standard projection operator, P, with respect to unnesting, i.e. P e X (r) P X ( r) 2) where X is the set of zero order attributes corresponding to X of equation (1) ....

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Roth, M. A., Korth, H. F., Silberschatz, A.: Null Values in Nested Relational Databases. Acta Inf. 26, 615-642 (1989).

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