Elena V. Ravve

Dependency preserving refinements and the fundamental problem of database design

We introduce a new point of view into database schemes by applying systematically an old logical technique: translation schemes, and their induced formula and structure transformations. This allows us to re-examine the notion of dependency preserving decomposition and its generalization refinement. We demonstrate the usefulness of this approach by recasting the theory of vertical and horizontal decompositions in our terminology. The most important aspect of this approach, however, lies in laying the groundwork for a formulation of the Fundamental Problem of Database Design, namely to exhibit desirable differences between translation equivalent presentations of data and to examine refinements of data presentations in a systematic way. The emphasis in this paper is not on results. The main line of thought is an exploration of the use of an old logical tool in addressing the Fundamental Problem. Translation schemes allow us also to have a new look at normal forms of database schemes and to suggest a new line of search for other normal forms. Illustrating our techniques we show that every scheme specified by functional and inclusion dependencies has a dependency-preserving refinement into BCNF. Furthermore we give a characterization of the embedded implicational dependencies (EIDs) using FDs and inclusion dependencies (IDs) and a basic class of refinements consisting of projections and natural joins.

Translation Schemes and the Fundamental Problem of Database Design

We introduce a new point of view into database schemes by applying systematically an old logical technique: translation schemes, and their induced formula and structure transformations. This allows us to re-examine the notion of dependency preserving decomposition and its generalization refinement.

Incremental model checking for decomposable structures

Assume we are given a transition system which is composed from several well identified components. We propose a method which allows us to reduce the model checking of Monadic Second Order formulas in the complex system to model checking of derived formulas in Monadic Second Order Logic in the components.

On the location of roots of graph polynomials

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.

On the Location of Roots of Graph Polynomials

Abstract Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.

A Systematic Approach to Computations on Decomposable Graphs

In this paper, we try to build a bridge between pure theoretical approach to computations on decomposable graphs and heuristics, used in practice for treatment of particular cases of them. In theory, Feferman and Vaught in 1959 proposed a method to reduce solution of First Order definable problems on Disjoint Union of structures to solutions of derived problems on the components with some post-processing of the obtained results. In practice, the literature is very reach in examples of particular methods to deal with different variations of graphs, built from components. From the theoretical point of view we adapt and generalize the Feferman-Vaught method. We define a new kind of decomposable graphs: sum-like graphs and propose a new systematic approach, which allows us to reduce the solution of Monadic Second Order definable problems on such graphs to the solution of effectively derivable Monadic Second Order definable problems on the components. From the practical point of view, we consider in great details one application of our approach in the field of parallel computations on distributed data.

A Computational Framework for the Study of Partition Functions and Graph Polynomials

Partition functions and graph polynomials have found many applications in combinatorics, physics, biology and even the mathematics of finance. Studying their complexity poses some problems. To capture the complexity of their combinatorial nature, the Turing model of computation and Valiants notion of counting complexity classes seem most natural. To capture the algebraic and numeric nature of partition functions as real or complex valued functions, the Blum-Shub-Smale (BSS) model of computation seems more natural. As a result many papers use a naive hybrid approach in discussing their complexity or restrict their considerations to sub-fields of C which can be coded in a way to allow dealing with Turing computability. In this paper we propose a unified natural framework for the study of computability and complexity of partition functions and graph polynomials and show how classical results can be cast in this framework.

BCNF via attribute splitting

Boyce-Codd-Heath introduced criteria for good database design, which can be formulated in terms of FDs only. Classical design decomposes relations iteratively using projections. BCNF can not be always achieved using projections alone. 3NF was introduced as a compromise. In this paper we summarize all the known characterizations of BCNF and formulate a new one. In [MR96], attribute splitting was suggested as a heuristics to achieve BCNF in case projections do not do the job. Here we show how attribute splitting can be used to restructure a database scheme iteratively such that the result will be in BCNF, is information preserving and preserves the functional dependencies.

The Universal Edge Elimination Polynomial and the Dichromatic Polynomial

The dichromatic polynomial Z(G;q,v) can be characterized as the most general C-invariant, i.e., a graph polynomial satisfying a linear recurrence with respect to edge deletion and edge contraction. Similarly, the universal edge elimination polynomial ξ(G;x,y,z) introduced in [Ilya Averbouch, Benny Godlin, and Johann A. Makowsky. An extension of the bivariate chromatic polynomial. Eur. J. Comb, 31(1):1–17, 2010] can be characterized as the most general EE-invariant, i.e., a graph polynomial satisfying a linear recurrence with respect to edge deletion, edge contraction and edge extraction. In this paper we examine substitution instances of ξ(G;x,y,z) and show that among these the dichromatic polynomial Z(G;q,v) plays a distinctive role.

Views and Updates over Distributed Databases

This contribution deals with systematic exploitation of logical reduction techniques to databases. The particular applications are views and updates over distributed databases. Logical reduction techniques come in two favors. The first one: the syntactically defined translation schemes, which describe transformations of database schemes. They give rise to two induced maps, translations and transductions. Transductions describe the induced transformation of database instances and the translations describe the induced transformations of queries. The second one: Feferman-Vaught reductions, which are applied in situations, where a relational structure is pieced together from a set of sub-structures. The reduction describes how the queries over the structure can be computed from queries over the components and queries over the index set. Combination and development of these techniques allow us to generalize the propagation technique for relational algebra and the incremental re-computation technique for some kinds of Data log programs to cases of definable sets of tuples to be deleted or inserted.