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Vadim Tropashko

Publications and source records attributed to Vadim Tropashko.

7 recordsLinked to original sources
arXiv2017

Ideal Databases

From algebraic geometry perspective database relations are succinctly defined as Finite Varieties. After establishing basic framework, we give analytic proof of Heath theorem from Database Dependency theory. Next, we leverage Algebra/Geometry dictionary and focus on algebraic counterparts of finite varieties, polynomial ideals. It is well known that intersection and sum of ideals are lattice operations. We generalize this fact to ideals from different rings, therefore establishing that algebra of ideals is Relational Lattice. The final stop is casting the framework into Linear Algebra, and traversing to Quantum Theory.

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arXiv2009

Relational Lattice Foundation for Algebraic Logic

Relational Lattice is a succinct mathematical model for Relational Algebra. It reduces the set of six classic relational algebra operators to two: natural join and inner union. In this paper we push relational lattice theory in two directions. First, we uncover a pair of complementary lattice operators, and organize the model into a bilattice of four operations and four distinguished constants. We take a notice a peculiar way bilattice symmetry is broken. Then, we give axiomatic introduction of unary negation operation and prove several laws, including double negation and De Morgan. Next we reduce the model back to two basic binary operations and twelve axioms, and exhibit a convincing argument that the resulting system is complete in model-theoretic sense. The final parts of the paper casts relational lattice perspective onto database dependency theory and into cylindric algebras.

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arXiv2008

Relational Lattice Axioms

Relational lattice is a formal mathematical model for Relational algebra. It reduces the set of six classic relational algebra operators to two: natural join and inner union. We continue to investigate Relational lattice properties with emphasis onto axiomatic definition. New results include additional axioms, equational definition for set difference (more generally anti-join), and case study demonstrating application of the relational lattice theory for query transformations.

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arXiv2006

First Steps in Relational Lattice

Relational lattice reduces the set of six classic relational algebra operators to two binary lattice operations: natural join and inner union. We give an introduction to this theory with emphasis on formal algebraic laws. New results include Spight distributivity criteria and its applications to query transformations.

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arXiv2005

Relational Algebra as non-Distributive Lattice

We reduce the set of classic relational algebra operators to two binary operations: natural join and generalized union. We further demonstrate that this set of operators is relationally complete and honors lattice axioms.

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arXiv2004

Nested Intervals with Farey Fractions

Relational Databases are universally conceived as an advance over their predecessors Network and Hierarchical models. Superior in every querying respect, they turned out to be surprisingly incomplete when modeling transitive dependencies. Almost every couple of months a question how to model a tree in the database surfaces at comp.database.theory newsgroup. This article completes a series of articles exploring Nested Intervals Model. Previous articles introduced tree encoding with Binary Rational Numbers. However, binary encoding grows exponentially, both in breadth and in depth. In this article, we'll leverage Farey fractions in order to overcome this problem. We'll also demonstrate that our implementation scales to a tree with 1M nodes.

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