L. S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567--582, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....not been derived to the best of our knowledge. This motivates our study IEs for nested relations. The aim of this research is to derive incremental equations (IEs) for the operators of nested relational algebra. There are two types of nested relational algebra, non recursive [7] and recursive [1], available for deriving IEs. The non recursive algebra is extended from the algebra for at relations by treating values regardless of whether they are atomic or nested. It applies to the outmost level of nested relations. When access to attributes on a deeper level is needed, a relation is ....

....the algebra for at relations by treating values regardless of whether they are atomic or nested. It applies to the outmost level of nested relations. When access to attributes on a deeper level is needed, a relation is unnested until the attributes come to the top level. The recursive algebra [1], on the other hand, has the ability to navigate into deeper levels of a relation. Therefore accessing an attribute at a deeper level does not need to unnest a relation. These two types of algebra are all employed in this paper. The IEs we derive in this paper take the property of exact change. ....

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Latha S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567-82, 1990.

....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 ....

....of projected NRSs and the null extended projection operator. In Subsection 3.4 we define the concept of joinable NDSs and the null extended join and null extended outer join operators. We note that our null extended join is more general than other extended joins defined for nested relations [1, 6, 8, 13, 21, 22, 26], because in our null extended join attributes are joined at all heights of nodes in the scheme trees of a joinable NDS. In Subsection 3.5 we define the null extended selection and the null extended renaming operators. In Subsection 3.6 we define the null extended powerset operator, and we ....

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L.S. Colby, A recursive algebra for nested relations, Inf. Syst. 15 (1990) 567-582.

....operator is to offer a solution to the problem of how to efficiently process queries which include join operation in the complex value model. The P join operator does not require as many restructuring operators and combines the advantages of the standard join and other extended join operators [3, 8]. Detailed discussion of this can be found in [4] The idea behind the P join operation is that we form the join by combining two nested relational operands with common join attributes not only at the top level but also at the subschema levels. First, we describe some terminologies which are used ....

L.S. Colby. A Recursive Algebra for Nested Relations. Information Systems, 15(5):567-582, 1990.

....possibly containing other sets, the databases are no longer in the first normal form. Therefore, we have to deal with nested relations, or complex objects. The language we describe contains the nested relational algebra as a sublanguage. The standard presentations of the nested relational algebra [7, 24, 25] have a cumbersome syntax. Therefore, we have decided to follow the approach of [3] which gives a very clean and simple language that has precisely the expressive power of the nested relational algebra. The relational language introduced in [3] was based on earlier languages for lists [27, 28] ....

L. Colby, A recursive algebra for nested relations, Inf. Syst. 15:567--582, 1990.

....and type system is given in Section 4. We present some equivalence and optimization laws of the algebra in Section 5. Finally, we give the static typing rules for the algebra in Section 6. Section 7 discusses open issues and problems. Cited literature includes: SQL [16] OQL [4, 5, 13] NRA [8, 15, 21, 22], Quilt [11] UnQL [3] XDuce [19] XML Query [33, 34] XML Schema [35, 36] XML QL [17] XPath [32] XQL [25] and YaTL [14] 2 The Algebra by Example This section introduces the main features of the algebra, using familiar examples based on accessing a database of books. 2.1 Data and Types ....

L. S. Colby. A recursive algebra for nested relations. Information Systems 15(5):567--582, 1990.

....the algebra are presented in Section 6. Finally, the static typing rules for the algebra are described in Section 7. Section 8 discusses open issues and problems. Cited literature includes: alternative algebras for XML [3, 11] comprehensions [32, 6] monads [26, 27, 32 34] NRA and equivalents [14, 7, 22, 23, 29, 30], OQL [2, 1, 13, 9] Quilt [10] UnQL [5] SQL [16] tree Automata [15, 31] type systems [24, 25] XDuce [20, 21] XML Query [38, 39] XML Schema [40, 41] XMLQL [17] XPath [37, 35] XQL [28] XSLT [42] and YaTL [12] 2 The Algebra by Example This section introduces the main features of the ....

L. S. Colby. A recursive algebra for nested relations. Information Systems 15(5):567--582, 1990.

....other sets, the databases are no longer in the first normal form. Therefore, we have to deal with nested relations, or complex objects. The language we describe in this paper contains the nested relational algebra as a sublanguage. The standard presentations of the nested relational algebra [8, 25, 26] have a cumbersome syntax. Therefore, we have decided to follow the approach of [3] which gives a very clean and simple language that has 1 precisely the expressive power of the nested relational algebra. The relational language introduced in [3] was based on earlier languages for lists [28, 29] ....

L. Colby, A recursive algebra for nested relations, Inform. Systems 15 (1990), 567--582.

....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 equivalence, which will be defined later. However, LL93] used the standard presentation of languages for complex objects, like in [TF86, SS86, Col90] and consequently inherited all of its problems and drawbacks. In particular, the description of the notion of null extended join operator is almost one page long, and many other operations are rather hard to grasp. The algebra for complex objects proposed in [Lib91] does not have adequate power ....

L. S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567--582, 1990.

....use of stored ancestor theories. The language we describe in this paper contains the nested relational algebra as a sublanguage. The nested relational algebra is a standard query language for database objects that freely combine values of base types, records and sets. Its standard presentations [4, 12, 13] have cumbersome syntax, so we have decided to follow the approach of [2] which gives a clean and simple language that has precisely the same expressive power. In order to represent disjunctive information in our query language, we added a new type constructor for or sets to the nested relational ....

....cartesian product and their counterparts for or sets, see [2, 8] These functions are included in OR SML in the form of a structure called Set. val x1 = mksetint [1,2] val x1 = 1, 2 : int : co smap (pair(id,id) x1; val it = 1, 1) 2, 2) int int) co val x2 = mksetint [3,4]; val x2 = 3, 4 : int : co union(x1,x2) val it = 1, 2, 3, 4 : int : co Set.cartprod(x1,x2) val it = 1, 3) 1, 4) 2, 3) 2, 4) int int) co 3 Normalization As we discussed before, while an object h1; 2; 3i is structurally just a set, conceptually it is a ....

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L. Colby, A recursive algebra for nested relations, Inform. Systems 15 (1990), 567--582.

....our study IEs for nested relations. Nested relations generalize at relations by permitting non atomic and relation values. They allow users to view the database in a way that is closer to their concept of the real world since complex objects can be represented as a whole in a single relation [4]. Because of the advantages of representing complex objects in applications such as CAD CAM, nested relations are widely accepted by the database world [9, 15] Furthermore, materialized views of nested relations are used by database researchers in data warehouses [5] and in maintaining 2 XML ....

....are used by database researchers in data warehouses [5] and in maintaining 2 XML data [1] The aim of this research is to derive incremental equations (IEs) for the operators of nested relational algebra. There are two types of nested relational algebra, non recursive [14, 15] and recursive [4], available for deriving IEs. The non recursive algebra is extended from the algebra for at relations by treating values regardless of whether they are atomic or nested [3] It applies to the outmost level of nested relations. When access to attributes on a deeper level is needed, a relation is ....

[Article contains additional citation context not shown here]

Latha S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567-82, 1990.

....necessary for understanding or using the algebra, 1 therefore the reader may skip this section. In Section 6, we discuss open issues and problems. In Appendix A, we give a formal mapping relating the algebra to the XML Query Data Model. Cited literature includes: SQL [16] OQL [4, 5, 13] NRA [8, 15, 20, 22], Quilt [11] UnQL [3] XDuce [21] XML QL[17] XPath [33] XQL [25] and YaTL [14] 2 The Algebra by Example This section introduces the main features of the algebra, using familiar examples based on accessing a database of books. 2.1 Data and Types Consider the following sample data: bib ....

L. S. Colby. A recursive algebra for nested relations. Information Systems 15(5):567--582, 1990.

....PNF relations. The main aim of these operators is to provide a powerful set of data manipulation operators under which PNF relations are closed; i.e. the result of applying any of the PNF operators is another relation that is in PNF. We first define extended union and difference operators [AbB86, Col90]. These two operators are used to add and delete tuples from relations. Definition 4.1 (Union Operator) Let r and s be two PNF relations defined over nested scheme R. The union of r and s is a relation over R, denoted by r Phi s, and recursively defined by: i) r Phi s = fxjx 2 r or x 2 sg, ....

Colby,S.L. (1990). A recursive algebra for nested relations. In Information Systems, 15(5):567-82.

....explicitly typed calculus in this section. This calculus is based on the formalism of [2, 12, 26] which is recently being used as a query language for complex heterogenous genomic data sources [9] The fragment dealing with sets is known [26] to be equivalent to several nested relational algebras [20, 17, 5, 15]. The types used in the explicitly typed calculus are either eq types t or noneq types T . The eq types are those types where equality tests can be performed on their objects; the noneq types are those types where equality tests are not available. This distinction is necessary because we require ....

L. S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567--582, 1990.

....in Omega and the aggregate functions in Gamma . 3. 4 Expressive Power and Complexity It is well known that the complex values algebra CVA is equivalent to a lot of other algebraic or calculus based languages (without powerset) over complex values and nested collections proposed in the literature [5, 6, 19, 20]. It turns out that CVA expresses only functions (over uninterpreted databases) that have ptime data complexity. When considering the complexity of functions over numeric interpreted domains, a cost model has to be specified, since it can be defined in several different ways. For instance, it is ....

L. S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567--582, 1990.

....The nested model [14, 18] is an extension of the traditional, flat relational database model in which relations can have both flat, atomic entries and structured, relationvalued entries. Since the late 1980s, various query languages have been considered in the context of the nested model [1, 5, 7, 9, 11, 16, 18]. These languages can be classified according to their expressive power [2] The nested algebra [18] which extends the traditional, flat relational algebra with two restructuring operators, called nest and unnest , can only express a fragment of the polynomial time queries over nested ....

L.S. Colby, "A Recursive Algebra for Nested Relations," Information Systems, 15, 1990, pp. 567--582.

....programming syntax. Applying it to complex objects, we obtain a nested relational language called NRL. The language resulting from adding equality test to NRL, denoted by NRL(eq) has the same expressive power as the nested relational algebra, originally developed in Thomas and Fischer [54] Colby [13], and Schek and Scholl [49] NRL(eq) is a better language for the purpose of this paper than these older languages because it has simpler semantics and it is easily extensible with operations such as aggregate functions. A very important property of NRL(eq) is its conservativity [46, 59] the ....

.... relational operations of the well known algebra of Thomas and Fischer [54] In fact, this result can be strengthened [60] because the converse is also true if a few constant relations are added to the algebra of Thomas and Fischer [54] which is known to be equivalent to the language of Colby [13] and to the language of Schek and Scholl [49] Also [60] real booleans can be added to NRL as a base type together with equality tests = s : s Theta s bool and the conditional construct to yield a language that has the same strength as NRL(eq) we list the additional primitives explicitly ....

L. S. Colby, A recursive algebra for nested relations, Information Systems 15, No. 5 (1990), 567--582.

....more pressing. Various implementation oriented modelling techniques exist. A classical distinction can be made between relational, network and hierarchical models. Recently a lot of research has been done on nested relational models, also called non first normal form or NF 2 models (see [1] [6] and [19] These models are interesting for special database applications, involving e.g. textual data, computer aided design or image processing. Current approaches to the transformation of conceptual models into internal models focus mainly on the relational model. The result of the ....

....easily recognised. They form the basis of the generation algorithm described in section 4. 3. 1 The underlying data structure An NF 2 model consists of relation types, where a relation type consists of atomic attribute types and other relation types, called relation valued attribute types (see [6] and [19] NF 2 relation types may be in one of the classical relational normal forms (see e.g. 26] but they do not necessarily have to. In this way the relational theory can be exploited in a wider area. Recent research involves the definition of a recursive relational algebra (see [6] ....

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L.S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567--582, 1990.

....property holds: Lemma 3.2 Edge e = hm; ni with (e) f has the following property: jn f j = 1. For fact type f this unique predicator is as denoted Hook(f) Our concept of nested representations of an information structure is very close to the concept of nested relations, as discussed in [Col90], RKS88] SS86] and [TKSD91] In order to demonstrate this, we show how tree representations are translated into nested tables. Suppose node m has descendants as depicted in figure 3. We then construct a table with a column for m values, and for each fact type hooking to node m. As a ....

....In this section we give the necessary definitions for the construction and destruction of tree representations. Construction will be performed by the nest operator, while destruction will be performed by the unnest operator. The concept of nesting and unnesting has been widely studied (see e.g. [Col90], RKS88] SS86] and [TKSD91] Usually the effect of these operators is expressed in a rather complex way (see e.g. SS86] In order to simplify our definitions, we express these operators in terms of encoded tree representations. First, we introduce the nest operator. Let p be an unused ....

L.S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567-- 582, 1990.

....: f(xn ) that is, it extends f to sets. This function is well defined. Using ext together with j, projections and record formation, conditional and the equality test gives us precisely the nested relational algebra [2] but the presentation is nicer than the standard ones, such as in [8]. This approach to the language design has proved extremely fruitful and allowed to solve some open problems and develop languages for other collections. The reader who is interested in other applications of this approach and in mathematical constructions behind it is referred to [23] In order to ....

L. Colby. A recursive algebra for nested relations. Inf. Syst. 15 (1990), 567--582.

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L. S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567--582, 1990.

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L. S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567--582, 1990.

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L. S. Colby, A recursive algebra for nested relations, Information Systems 15, No. 5 (1990), 567--582.

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L.S. Colby. A recursive algebra for nested relations. Information Systems, 15(5):567--582, 1990.

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