Relational Algebra Expressions Research Articles

Automated test case generation for programs specified by relational algebra queries

Black-box software testing requires test cases to be generated from specifications alone. However, it is impossible to automate the process completely for arbitrary specifications. Specifications are thus restricted to being written entirely in terms of relational algebra expressions. An automated test case generation method is developed for such specifications. >

Query processing techniques in the summary-table-by-example database query language

Summary-Table-by-Example (STBE) is a graphical language suitable for statistical database applications. STBE queries have a hierarchical subquery structure and manipulate summary tables and relations with set-valued attributes. The hierarchical arrangement of STBE queries naturally implies a tuple-by-tuple subquery evaluation strategy (similar to the nested loops join implementation technique) which may not be the best query processing strategy. In this paper we discuss the query processing techniques used in STBE. We first convert an STBE query into an “extended” relational algebra (ERA) expression. Two transformations are introduced to remove the hierarchical arrangement of subqueries so that query optimization is possible. To solve the “empty partition” problem of aggregate function evaluation, directional join (one-sided outer-join) is utilized. We give the algebraic properties of the ERA operators to obtain an “improved” ERA expression. Finally we briefly discuss the generation of alternative implementations of a given ERA expression. STBE is implemented in a prototype statistical database management system. We discuss the STBE-related features of the implemented system.

Updating derived relations: detecting irrelevant and autonomously computable updates

Consider a database containing not only base relations but also stored derived relations (also called materialized or concrete views). When a base relation is updated, it may also be necessary to update some of the derived relations. This paper gives sufficient and necessary conditions for detecting when an update of a base relation cannot affect a derived relation (an irrelevant update), and for detecting when a derived relation can be correctly updated using no data other than the derived relation itself and the given update operation (an autonomously computable update). The class of derived relations considered is restricted to those defined by PSJ -expressions, that is, any relational algebra expressions constructed from an arbitrary number of project, select and join operations (but containing no self-joins). The class of update operations consists of insertions, deletions, and modifications, where the set of tuples to be deleted or modified is specified by a selection condition on attributes of the relation being updated.

Efficient optimization of simple chase join expressions

Simple chase join expressions are relational algebra expressions, involving only projection and join operators, defined on the basis of the functional dependencies associated with the database scheme. They are meaningful in the weak instance model, because for certain classes of schemes, including independent schemes, the total projections of the representative instance can be computed by means of unions of simple chase join expressions. We show how unions of simple chase join expressions can be optimized efficiently, without constructing and chasing the corresponding tableaux. We also present efficient algorithms for testing containment and equivalence, and for optimizing individual simple chase join expressions.

Algebraic Optimization Of Recursive Database Queries

AbstractThe topic is algebraic optimization of recursive database queries. Queries are expressed by relational algebra expressions including a fixpoint operation. A condition is presented under which a natural join commutes with a fixpoint operation. This condition is a simple check of attribute sets of sub-expressions of the query. The work may be considered a generalization of Aho and Ullman, (1979).The result is interpreted in function free logic database terms as a transformation of the recursively defined predicate involving: (a) elimination of an argument, and (b) propagation of selections (instantiations) to the extensionally defined predicates. A collection of examples shows that this transformation abstracts some optimizations which otherwise are done by more complex graph algorithms (e.g. Bancilhon et al., (1986); Chang, (1985); Gardarin and DeMaindreville, (1986); Henschen and Naqvi, (1984); Kifer and Lozinskii, (1986)). Thus, this optimization is expressed in a form which is not biased towards...

Intelligent query answering in rule based systems

We propose that in large knowledge bases which are collections of atomic facts and general rules (Horn clauses), the rules should be allowed to occur in the answer for a query. We introduce a new concept of the answer for a query which includes both atomic facts and general rules. We provide a method of transforming rules by relational algebra expressions built from projection, join, and selection and demonstrate how the answers consisting of both facts and general rules can be generated.

Answering queries on embedded-complete database schemes

It has been observed that, for some database schemes, users may have difficulties retrieving correct information, even for simple queries. The problem occurs when some implicit “piece” of information, defined on some subset of a relation scheme, is not explicitly represented in the database state. In this situation, users may be required to know how the state and the constraints interact before they can retrieve the information correctly.In this paper, the formal notion ofembedded-completenessis proposed, and it is shown that schemes with this property avoid the problem described above. A polynomial-time algorithm is given to test whether a database scheme is independent and embedded-complete. Under the assumption of independence, it is shown that embedded-complete schemes allow efficient computation of optimal relational algebra expressions equivalent to theX-total projection, for any set of attributesX.

Embedding deductive capabilities in relational database systems

The basic idea of our work is to include deductive capabilities in a relational database system by fully using the operands of relational algebra and without using any theorem prover. We present a deductive database system where the user can formulate deduction laws as logical formulas, but which works on an underlying relational database system by fully using the data manipulation routines of such a system. Generally, it is not possible to express the deduction laws by means of expressions of relational algebra (see Ref. 1). In our system, a relational algebra expression is associated with every database deduction law. The result of its evaluation (with respect to an underlying database) is the set of all database entries that can be deduced from the database by applying the respective law once (Theorem 1). By successively applying all these relational expressions for all the database deduction laws it is possible to generate the whole extensional database (i.e. all tuples deducible from the database).

Membership problems for data dependencies in relational expressions

In relational databases, a query can be formulated in terms of a relational algebra expression using projection, selection, restriction, cross product and union. In this paper, we consider a problem, called the membership problem, of determining whether a given dependency d is valid in a given relational expression E over a given database scheme R that is, whether every instance of the view scheme defined by E satisfies d (assuming that the underlying constraints in R are always satisfied). Consider the case where each relation scheme in R is associated with functional dependencies (FDs) as constraints, and d is an FD. Then the complement of the membership problem is NP-complete. However, if E contains no union, then the membership problem can be solved in polynomial time. Furthermore, if E contains neither a union nor a projection, then we can construct in polynomial time a cover for valid FDs in E, that is, a set of FDs which implies every valid FD in E. Consider the case where each relation scheme in R is associated with multivalued dependencies (MVDs) as well as FDs, and d is an FD or an MVD. Even if E consists of selections and cross products only, the membership problem is NP-hard. However, if E contains no union, and each relation scheme name in R occurs in E at most once, then the membership problem can be solved in polynomial time. As a corollary of this result, it can be determined in polynomial time whether a given FD or MVD is valid in R 1⋈⋯⋈R s , where R 1,…, R s are relation schemes with FDs and MVDs, and R i⋈R j is the natural join of R i and R j .

Inferring a Tree from Lowest Common Ancestors with an Application to the Optimization of Relational Expressions

We present an algorithm for constructing a tree to satisfy a set of lineage constraints on common ancestors. We then apply this algorithm to synthesize a relational algebra expression from a simple tableau, a problem arising in the theory of relational databases.

Calculating constraints on relational expression

This paper deals with the problem of determining which of a certain class of constraints hold on a given relational algebra expression where the base relations come from a given schema. The class of constraints includes functional dependencies, equality of domains, and constancy of domains. The relational algebra consists of projection, selection, restriction, cross product, union, and difference. The problem as given is undecidable, but if set difference is removed from the algebra, there is a solution. Operators specifying a closure function (similar to functional dependency closure on one relation) are defined; these will generate exactly the set of constraints valid on the given relational algebra expression. We prove that the operators are sound and complete.