relation algebra

Although the relational model for databases provides a great range of advantages over other data models, it lacks a comprehensive way to handle incomplete and uncertain data. Uncertainty in data values, however, is pervasive in all real-world environments and has received much attention in the literature. Several methods have been proposed for incorporating uncertain data into relational databases. However, the current approaches have many shortcomings and have not established an acceptable extension of the relational model. In this paper, we propose a consistent extension of the relational model. We present a revised relational structure and extend the relational algebra. The extended algebra is shown to be closed, a consistent extension of the conventional relational algebra, and reducible to the latter.

ABSTRACT. For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T, V)-algebras is a quasitopos. This ...

Motivated by research on how topology may be a helpful foundation for building information modeling (BIM), a relational database version of the notions of chain complex and chain complex morphism is defined and used for storing cw-complexes and their morphisms, hence instances of building projects and different views upon them, into relational databases. In many cases, this can be done without loss of topological information. The equivalence of categories between sets with binary relations and Alexandrov spaces is proven and used to incorporate the relational complexes into the more general setting of topological databases. For the latter, a topological version of a relational query language is defined by transferring the usual relational algebra operators into topological constructions. In the end, it is proven that such a topological version of relational algebra in general must be able to compute the transitive closure of a relation.

Reiter has proposed extended relational theory to formulate relational databases with null values and presented a query evaluation algorithm for such databases. However, due to indefinite information brought in by null values, Reiter's algorithm is sound but not complete. In this paper, we first propose an extended relation to represent indefinite information in relational databases. Then, we define an extended relational algebra for extended relations. Based on Reiter's extended relational theory, and our extended relations and the extended relational algebra, we present a sound and complete query evaluation algorithm for relational databases with null values

This paper introduces a reflective extension of the relational algebra. Reflection is achieved by storing and manipulating relational algebra programs as relations and by adding a LISP-like evaluation operation to the algebra. We first show that this extension, which we call the reflective algebra, can serve as a unifying formalization of various forms of procedural data management which have been considered in database systems research. We then study the expressive power and complexity of the reflective algebra. In particular, we establish a close correspondence between reflection and bounded looping, and between tailrecursive reflection and unbounded looping. These correspondences yield new logical characterizations of PTIME and PSPACE.

Adjoint negations, whose definition is based on the implications of an adjoint triple, arise as a generalization of residuated negations. Recently, interesting properties of these negation operators have been introduced . In this paper, a comparative survey with weak negations studied by Trillas, Esteva and Domingo [10, 13] is presented. Moreover, the relationship between weak and strong negations, introduced by these authors, is extended to adjoint negations. These technical developments lead us to increase the number of applications of adjoint negations.

Assuming data domains are partially ordered, we apply Paredaens' and Bancilhon's Theorem to examine the expressiveness of the extended relational algebra (the PORA), which allows the ordering predicate to be used in formulae of the selection operator (σ). The PORA expresses exactly the set of all possible relations which are invariant under order-preserving automorphism of databases. Our main result shows that there is a one-toone correspondence between three hierarchies of: (1) computable queries, (2) query languages and (3) partially ordered domains.

This paper represents the concept of keys and relational algebra in database management system. Basically key is an attribute that is able to identify the records in given relation uniquely. Relational algebra is procedural query language. It takes instance of relations as an input and gives the desired output in form of relation. For evaluating desired output there may be several queries in relational algebra but based on record comparisons and block retrievals efficiency of query is calculated and after identifying efficient query, that query is applied to find out the desired set of records in relational algebra.

The join operation is one of the fundamental relational database query operations. It facilitates the retrieval of information from two different relations based on a Cartesian product of the two relations. The join is one of the most diffidult operations to implement efficiently, as no predefined links between relations are required to exist (as they are with network and hierarchical systems). The join is the only relational algebra operation that allows the combining of related tuples from relations on different attribute schemes. Since it is executed frequently and is expensive, much research effort has been applied to the optimization of join processing. In this paper, the different kinds of joins and the various implementation techniques are surveyed. These different methods are classified based on how they partition tuples from different relations. Some require that all tuples from one be compared to all tuples from another; other algorithms only compare some tuples from each....

Functional completeness of a relational language is the ability to express linear recursive queries. We present such a language that also has the property of relational completeness (relational algebra is, in fact, embedded in it). The transitive closure of a binary relation is carried out by means of the projection function and its inverse.

Data mining evolved as a collection of applicative problems and efficient solution algorithms relative to rather peculiar problems, all focused on the discovery of relevant information hidden in databases of huge dimensions. In particular, one of the most investigated topics is the discovery of association rules.

We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic.

We show that the variety of n-dimensional weakly higher order cylindric algebras, introduced in Németi [9], [8], is finitely axiomatizable when n > 2. Our result implies that in certain non-well-founded set theories the finitization problem of algebraic logic admits a positive solution; and it shows that this variety is a good candidate for being the cylindric algebra theoretic counterpart of Tarski's quasi-projective relation algebras.

We investigate V-free algebras on n generators, F n = Fr(V, n), where V is a discriminator variety and, more specifically, where V is a variety of relation algebras or of cylindric algebras. Sample questions are: (a) Is F, § embeddable in F~? (b) Does Fn contain an n-element set that generates it non-freely? The answer to (a) is affirmative in some varieties of relation algebras, but it is negative in every congruence extensile variety in which some nontrivial finite member is an absolute retract. The answer to (b) is affirmative in every variety of relation algebras that contains the full algebra of relations on an infinite set.

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property. The proof relies on the special property of 2-variable logic that each model is 2-equivalent to some rather homogeneous model that we dub 2-transitive.

We exhibit two relation algebra atom structures such that they are elementarily equivalent but their term algebras are not. This answers Problem 14.19 in the book Hirsch, R. and Hodkinson, I., "Relation Algebras by Games", North-Holland, 2002.

We show that the variety of n-dimensional weakly higher order cylindric algebras, introduced in Németi [9], [8], is finitely axiomatizable when n > 2. Our result implies that in certain non-well-founded set theories the finitization problem of algebraic logic admits a positive solution; and it shows that this variety is a good candidate for being the cylindric algebra theoretic counterpart of Tarski's quasi-projective relation algebras.

We discuss some new properties of the natural Galois connection among set relation algebras, permutation groups, and first order logic. In particular, we exhibit infinitely many permutational relation algebras without a Galois closed representation, and we also show that every relation algebra on a set with at most six elements is Galois closed and essentially unique. Thus, we obtain the surprising result that on such sets, logic with three variables is as powerful in expression as full first order logic.

We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic.

In this paper we try to initiate a search for an explicite and direct definition of ultraproducts in categories which would share some of the attractive properties of products, coproducts, limits, and related category theoretic notions. Consider products as a motivating example. ...

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CA n of n-dimensional cylindric algebras and the class of representable algebras in CA n for finite n > 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Beth-definability property fails for the finite-variable fragments of first order logic as long as the number n of variables available is > 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CA n are best possible. We raise several open problems.

A finite axiom set for the identity-flee equations valid in relation algebras is given. This is a simplification of the one given by Jrnsson, and confirms a conjecture of Tarski. An axiom set for the identity-free equations valid in the representable relation algebras is given, too. We show that in the class of representable relation algebras, both the operation of taking converse and the identity constant are finitely axiomatizable (over the rest of the operations).

Lindström theorems characterize logics in terms of model-theoretic conditions such as Compactness and the Löwenheim-Skolem property. Most existing characterizations of this kind concern extensions of first-order logic. But on the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindström theorems for these languages can be on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we provide Lindström theorems for several fragments of first-order logic, including the k-variable fragments for k > 2, Tarski's relation algebra, graded modal logic, and the binary guarded fragment. We use two different proof techniques. One is a modification of the original Lindström proof. The other involves the modal concepts of bisimulation, tree unraveling, and finite depth. Our results also imply semantic preservation theorems.

This paper relates an experiment in writing an algebraic specification of a rather complex example, namely a subset of the UNIX' file system. The PLUSS specification language, which is used for this experiment, provides a set of linguistic features which allow the modularization of such specifications and the definition of a flexible and convenient syntax for expressions and axioms (such as mixtix operators, overloading, coercions). This experiment was a way for evaluating the adequacy of these features to several criteria: mainly legibility and understandability, but also reusability of specifications. The paper presents the specification and discusses it with respect to these important points. * This work is partially supported by ESPRIT project No. 432 METEOR and CNRS GRECO de Programmation. ' UNIX is a trademark of Bell Laboratories.

Datatype-generic programs are programs that are parameterised by a datatype. We review the allegorical foundations of a methodology of designing datatype-generic programs. The notion of F-reductivity, where F parametrises a datatype, is reviewed and a number of its properties are presented. The properties are used to give concise, effective proofs of termination of a number of datatype-generic programming schemas. The paper concludes with a concise proof of the well-foundedness of a datatype-generic occurs-in relation.

Downey, R., Every recursive boolean algebra is isomorphic to one with incomplete atoms, Annals of Pure and Applied Logic 60 (1993) 193-206. The theorem of the title is proven, solving an old question of Remmel. The method of proof uses an algebraic technique of Remmel-Vaught combined with a complex tree of strategies argument where the true path is needed to figure out the final isomorphism.

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CA n of n-dimensional cylindric algebras and the class of representable algebras in CA n for finite n > 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Beth-definability property fails for the finite-variable fragments of first order logic as long as the number n of variables available is > 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CA n are best possible. We raise several open problems.

In the relational model it has been shown that the flat relational algebra has the same expressive power as the nested relational algebra, as far as queries over flat relations and with flat results are concerned [11]. Hence, for each query that uses the nested relational model and that, with a flat table as input always has a flat table as output, there exists an equivalent flat query that only uses the flat relational model. In [12] a very direct proof is given of this fact using a simulation technique. In analogy, we study a related flat- ...

The idea of a database as a central facility of a programming environment is explored, taking the Ada program Library as a starting point. The database used is based on a node model similar to the one proposed in the CAIS standard (KIT/KITIA 1984). The paper suggests an expansion of the scope of the program library into the area of configuration and version management in an environment with several ungoing projects sharing software objects. The authors furthermore present a technique for the manipulation of the relationships in a relational algebra inspired way - providing some of the powerful capabilities of the relational approach within the context of node models.

This paper proposes an approach for representing and querying semistructured Web data, which is based on nested tables allowing internal nested structural variations. Our motivation is to reduce the complexity found in typical query languages for semistructured data and to provide users with an alternative for quickly querying data obtained from multiple-record Web pages. We show the feasibility of our proposal by developing a prototype for a graphical query interface called QSByE (Querying Semistructured data By Example), which implements a set of QBE-like operations that extends typical nested-relational-algebra operations to handle semistructured data.

In this paper, we introduce a fuzzy language to extract information from the web extending the web query language WebSQL [ 11. These extensions are based on two observations: the inadequacy of traditional Boolean query languages for web documents, and the need to move beyond the notion of query providing just a set of answers in order to provide a better data presentation through answers' restructuring. In order to address the first issue, we consider fuzzy sets to express imprecision in data, queries and answers. In our case, data imprecision comes from the data classification provided by several search engines. Query imprecision occurs in weighting values provided at query definition time. Answer imprecision allows to filter and rank the answers. To address the second point, we provide an answer restructuring language to model the restructuring phase that follows the query phase. The restructuring language allows creation/deletion of iinks and page creation. Thus several answer organizations are possible as a result to the same query. The resulting language extends in a uniform framework WebSQL. Then we provide a mapping for the language constructs into an extended relational algebra called S A M E W [2] expressing similarity-based queries over imprecisely classified data, queries involving navigation among web pages and answer restructurings. Finally, we study the optimization of similarity-based queries using equivalence and containment rules holding for SAMEW and presenting several algorithms for query evaluation.

Categories of lax (T, V)-algebras are shown to have pullbackstable coproducts if T preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax (T, V)-algebras, with T preserving inverse images. Moreover, we show that any such category of (T, V)-algebras has a concrete, coproduct preserving functor into the category of topological spaces.

Lindström theorems characterize logics in terms of model-theoretic conditions such as Compactness and the Löwenheim-Skolem property. Most existing characterizations of this kind concern extensions of first-order logic. But on the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindström theorems for these languages can be on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we provide Lindström theorems for several fragments of first-order logic, including the k-variable fragments for k > 2, Tarski's relation algebra, graded modal logic, and the binary guarded fragment. We use two different proof techniques. One is a modification of the original Lindström proof. The other involves the modal concepts of bisimulation, tree unraveling, and finite depth. Our results also imply semantic preservation theorems.

This paper presents a probabilistic relational modelling (implementation) of the major probabilistic retrieval models. Such a high-level implementation is useful since it supports the ranking of any object, it allows for the reasoning across structured and unstructured data, and it gives the software (knowledge) engineer control over ranking and thus supports customisation. The contributions of this paper include the specification of probabilistic SQL (PSQL) and probabilistic relational algebra (PRA), a new relational operator for probability estimation (the relational Bayes), the probabilistic relational modelling of retrieval models, a comparison of modelling retrieval with traditional SQL versus modelling retrieval with PSQL, and a comparison of the performance of probability estimation with traditional SQL versus PSQL. The main findings are that the PSQL/PRA paradigm allows for the description of advanced retrieval models, is suitable for solving large-scale retrieval tasks, and outperforms traditional SQL in terms of abstraction and performance regarding probability estimation.

Large infrastructures become technologically more and more complex and interdependencies between different infrastructures increase. As a consequence complex infrastructures become more vulnerable to emergencies. The personnel in charge must be supported in the challenging task of identifying and assessing an emergency situation and in choosing an appropriate reaction. This calls for a system which continuously analyses the available sensory data for emergency situations, making predictions for the future development of an emergency and recommending proper reactions to the emergency.

The article presents the models of production-logical inference based on algebraic structures. The conducted researches of abstract algebra and production inference models have established the principle of decision-making support in the development and operation of telecommunication systems based on the so-called FS-system. Mathematically, the FSsystem is a partial case of algebraic lattices. It is proven that the FS-system, as an abstract structure, with the product-logical properties of its signature, has the advantages of the efficiency of computational implementation over other models. It was established experimentally that such an approach ensures minimization of appeals to external sources and, accordingly, reduction in computing costs. In particular, it was determined that the practical relevance of the findings is that the scientifically substantiated recommendations for an advanced communication management system are designed to consider the possibility of leveraging them in other complex technical systems. Such an approach allows to significantly increase the efficiency of the communication and management process gained by the modernization of hardware and software of the computer system. Thus, there are grounds to state that it is expedient to utilize the proposed set of mathematical models of the formal description of the production-logical inference of knowledge to automating the decision-making support in telecommunication systems at the design and operation stages.

The article presents the models of production-logical inference based on algebraic structures. The conducted researches of abstract algebra and production inference models have established the principle of decision-making support in the development and operation of telecommunication systems based on the so-called FS-system. Mathematically, the FSsystem is a partial case of algebraic lattices. It is proven that the FS-system, as an abstract structure, with the product-logical properties of its signature, has the advantages of the efficiency of computational implementation over other models. It was established experimentally that such an approach ensures minimization of appeals to external sources and, accordingly, reduction in computing costs. In particular, it was determined that the practical relevance of the findings is that the scientifically substantiated recommendations for an advanced communication management system are designed to consider the possibility of leveraging them i...

In this paper, a construction of a congruence having a given filter is presented. Also as a generalization of an BE-algebra homomorphism, the notion of a relation on BE-algebra, called an BE-relation is introduced and some fundamental properties to BE-algebras are discussed.

In this paper, a construction of a congruence having a given filter is presented. Also as a generalization of an BE-algebra homomorphism, the notion of a relation on BE-algebra, called an BE-relation is introduced and some fundamental properties to BE-algebras are discussed.

It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski’s other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski’s axiom system

We present classes of algebras which may be viewed as weak relation algebras, where a Boolean part is replaced by a not necessarily distributive lattice. For each of the classes considered in the paper we prove a relational representation theorem.

In this paper we study the q-commutator of Wick products on the CCR (canonical commutation relation) algebra and on the CAR (canonical anticommutation relation) algebra. We obtain a formula on q-commutator of Wick products in terms of Wick products of monomials of lower order. Moreover, we give a relation of double q-commutator.

More than 70 years ago, Jaques Riguet suggested the existence of an "analogie frappante" (striking analogy) between so-called "relations de Ferrers" and a class of difunctional relations, members of which we call "diagonals". Inspired by his suggestion, we formulate an "analogie frappante" linking the notion of a block-ordered relation and the notion of the diagonal of a relation. We formulate several novel properties of the core/index of a diagonal, and use these properties to rephrase our "analogie frappante". Loosely speaking, we show that a block-ordered relation is a provisional ordering up to isomorphism and reduction to its core. (Our theorems make this informal statement precise.) Unlike Riguet (and others who follow his example), we avoid almost entirely the use of nested complements to express and reason about properties of these notions: we use factors (aka residuals) instead. The only (and inevitable) exception to this is to show that our definition of a "staircase" relation is equivalent to Riguet's definition of a "relation de Ferrers". Our "analogie frappante" also makes it obvious that a "staircase" relation is not necessarily block-ordered, in spite of the mental picture of such a relation presented by Riguet.