Relational Algebra Research Articles

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.

Asymptotically Better Query Optimization Using Indexed Algebra

Query optimization is essential for the efficient execution of queries. The necessary analysis, if we can and should apply optimizations and transform the query plan, is already challenging. Traditional techniques focus on the availability of columns at individual operators, which does not scale for analysis of data flow through the query. Tracking available columns per operator takes quadratic space, which can result in multi-second optimization time for deep algebra trees. Instead, we need to re-think the naïve algebra representation to efficiently support data flow analysis. In this paper, we introduce Indexed Algebra , a novel representation of relational algebra that makes common optimization tasks efficient. Indexed Algebra enables efficient reasoning with an auxiliary index structure based on link/cut trees that support dynamic updates and queries in O (log n ). This approach not only improves the asymptotic complexity, but also allows elegant and concise formulations for the data flow questions needed for query optimization. While large queries see theoretically unbounded improvements, Indexed Algebra also improves optimization time of the relatively harmless queries of TPC-H and TPC-DS by more than 1.8×.

Achieving sequenced SQL with log-segmented timestamps

In a period-timestamped, relational temporal database, each tuple is timestamped with a period. The timestamp records when the tuple is “alive” in some temporal dimension. Sequenced semantics is a special semantics for evaluating a query in a temporal database. The semantics stipulates that the query must, in effect, be evaluated simultaneously in each time instant using the tuples alive at that instant. Previous research has proposed changes to a database management system's query evaluation engine to support sequenced semantics. We show how to achieve sequenced semantics without modifying a query evaluation engine. Our technique has two pillars. First we use log-segmented timestamps to record a tuple's lifetime. A log-segmented timestamp divides the time-line into segments of known length. Any temporal period can be represented by a small number of such segments. Second, by taking advantage of the properties of log-segmented timestamps, we translate a sequenced query to a non-temporal query, using the operations already present in an unmodified, non-temporal query evaluation engine. The paper provides translations for both relational algebra and SQL. We experimentally evaluate the SQL translation. The primary contribution of this paper is how to implement sequenced semantics using log-segmented timestamped tuples in a generic SQL DBMS.

Higher Localization and Higher Branching Laws

Abstract For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_{X}$-modules. We extend this construction to an equivariant and derived setting using the formalism of h-complexes due to Beilinson–Ginzburg, and show that the localizations of Harish-Chandra $(\mathfrak{g}, K)$-modules onto $X = H \backslash G$ have regular holonomic cohomologies when $H, K \subset G$ are both spherical reductive subgroups. The relative Lie algebra homologies and $\operatorname{Ext}$-branching spaces for $(\mathfrak{g}, K)$-modules are interpreted geometrically in terms of equivariant derived localizations. As direct consequences, we show that they are finite-dimensional under the same assumptions, and relate Euler–Poincaré characteristics to local index theorem; this recovers parts of the recent results of M. Kitagawa. Examples and discussions on the relation to Schwartz homologies are also included.

Разработка и апробация метода и алгоритма формирования коалиций экспертов при исследовании процессов функционирования сетей связи

The issues related to the use of expert assessment methods in order to increase the optimality and objectivity of developed management decisions are considered. The expediency of involving experts is formulated when a collegial management decision is necessary. The basic stages of the research related to the application of the expert assessment method are given. The relevance of the chosen topic is justified by the need to develop effective methods and algorithms aimed at automating the processes of analysis and processing of data obtained during the expert survey, as well as summarizing its results. The aim of the study is to develop and test an algorithm for finding coordinated groups (coalitions) of experts, as well as a method that makes it necessary to work out the final decision on the results of the expert survey. The proposed algorithm for finding coalitions is based on the application of the mathematical apparatus of the theory of sets and algebra of relations; it uses a strict criterion in their formation and replenishment of coalitions, but does not exclude the belonging of experts to several coalitions, which provides a high informative value of the results of expert assessment. The method includes the calculation of weighting coefficients assigned to the formed coalitions taking into account the degrees of experts' membership in them. The resulting decision is formed on the basis of the data obtained by the coalition, which has the maximum dimensionality and the required degree of consistency. The developed method and algorithm were tested on the results of an expert survey for determining the degrees of significance of the processes implemented in communication networks. As a result, coalitions of experts who agreed with each other were formed, as well as the experts, whose opinions are more deviated from the rest, were identified. The coherence of the coalitions is evaluated using the concordance coefficient, and the final decision is formed on the basis of the method of average arithmetic ranks assigned to the studied processes of functioning of communication networks by the experts belonging to the largest coalition.

On difunctions

The notion of a difunction was introduced by Jacques Riguet in 1948. Since then it has played a prominent role in database theory, type theory, program specification and process theory. The theory of difunctions is, however, less known in computing than it perhaps should be. The main purpose of the current paper is to give an account of difunction theory in relation algebra, with the aim of making the topic more mainstream.As is common with many important concepts, there are several different but equivalent characterisations of difunctionality, each with its own strength and practical significance. This paper compares different proofs of the equivalence of the characterisations.A well-known property is that a difunction is a set of completely disjoint rectangles. This property suggests the introduction of the (general) notion of the “core” of a relation; we use this notion to give a novel and, we believe, illuminating characterisation of difunctionality as a bijection between the classes of certain partial equivalence relations.

Computing the Difference of Conjunctive Queries Efficiently

We investigate how to efficiently compute the difference result of two (or multiple) conjunctive queries, which is the last operator in relational algebra to be unraveled. The standard approach in practical database systems is to materialize the results for every input query as a separate set, and then compute the difference of two (or multiple) sets. This approach is bottlenecked by the complexity of evaluating every input query individually, which could be very expensive, particularly when there are only a few results in the difference. In this paper, we introduce a new approach by exploiting the structural property of input queries and rewriting the original query by pushing the difference operator down as much as possible. We show that for a large class of difference queries, this approach can lead to a linear-time algorithm, in terms of the input size and (final) output size, i.e., the number of query results that survive from the difference operator. We complete this result by showing the hardness of computing the remaining difference queries in linear time. Although a linear-time algorithm is hard to achieve in general, we also provide some heuristics that can provably improve the standard approach. At last, we compare our approach with standard SQL engines over graph and benchmark datasets. The experiment results demonstrate order-of-magnitude speedups achieved by our approach over the vanilla SQL engine.

Edge colourings and qualitative representations of chromatic algebras

Conventional Ramsey-theoretic investigations for edge-colourings of complete graphs are framed around avoidance of certain configurations. Motivated by considerations arising in the field of Qualitative Reasoning, we explore edge colourings that in addition to forbidding certain triangle configurations also require others to be present. These conditions have natural combinatorial interest in their own right, but also correspond to qualitative representability of certain nonassociative relation algebras, which we will call chromatic.

Logical reduction of relations: From relational databases to Peirce’s reduction thesis

Abstract We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them and introduce a new characteristic of relations, ternarity, that measures their ‘complexity of relating’ and allows to refine reduction results. In particular, we refine Peirce’s controversial reduction thesis, and show that reducibility behaviour is dramatically different on finite and infinite domains.

Indexed Streams: A Formal Intermediate Representation for Fused Contraction Programs

We introduce indexed streams, a formal operational model and intermediate representation that describes the fused execution of a contraction language that encompasses both sparse tensor algebra and relational algebra. We prove that the indexed stream model is correct with respect to a functional semantics. We also develop a compiler for contraction expressions that uses indexed streams as an intermediate representation. The compiler is only 540 lines of code, but we show that its performance can match both the TACO compiler for sparse tensor algebra and the SQLite and DuckDB query processing libraries for relational algebra.

Terwilliger algebras and some related algebras defined by finite connected simple graphs

For a finite connected simple graph, the Terwilliger algebra is a matrix algebra generated by the adjacency matrix and idempotents corresponding to the distance partition with respect to a fixed vertex. We will consider algebras defined by two other partitions and the centralizer algebra of the stabilizer of the fixed vertex in the automorphism group of the graph. We will give some methods to compute such algebras and examples for various graphs.

Relative cluster categories and Higgs categories

Cluster categories were introduced in 2006 by Buan-Marsh-Reineke-Reiten-Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot (2009) [3] and Plamondon (2011) [50] to arbitrary cluster algebras associated with quivers. A higher dimensional generalization is due to Guo (2011) [17]. Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, etc. The work of Geiss-Leclerc-Schröer often yields Frobenius exact categories which allow us to categorify such cluster algebras.In this paper, we generalize the construction of (higher) cluster categories by Claire Amiot and by Lingyan Guo to the relative context. We prove the existence of an n-cluster tilting object in a Frobenius extriangulated category, namely the Higgs category (generalizing the Frobenius categories of Geiss-Leclerc-Schröer), which is stably n-Calabi–Yau and Hom-finite, arising from a left (n+1)-Calabi–Yau morphism. Our results apply in particular to relative Ginzburg dg algebras coming from ice quivers with potential and higher Auslander algebras associated to n-representation-finite algebras.

The Poisson spectrum of the symmetric algebra of the Virasoro algebra

Let $W = \mathbb {C}[t,t^{-1}]\partial _t$ be the Witt algebra of algebraic vector fields on $\mathbb {C}^\times$ and let $V\!ir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In this paper, we study the Poisson ideal structure of the symmetric algebras of $V\!ir$ and $W$, as well as several related Lie algebras. We classify prime Poisson ideals and Poisson primitive ideals of $\operatorname {S}(V\!ir)$ and $\operatorname {S}(W)$. In particular, we show that the only functions in $W^*$ which vanish on a nontrivial Poisson ideal (that is, the only maximal ideals of $\operatorname {S}(W)$ with a nontrivial Poisson core) are given by linear combinations of derivatives at a finite set of points; we call such functions local. Given a local function $\chi \in W^*$, we construct the associated Poisson primitive ideal through computing the algebraic symplectic leaf of $\chi$, which gives a notion of coadjoint orbit in our setting. As an application, we prove a structure theorem for subalgebras of $V\!ir$ of finite codimension and show, in particular, that any such subalgebra of $V\!ir$ contains the central element $z$, substantially generalising a result of Ondrus and Wiesner on subalgebras of codimension one. As a consequence, we deduce that $\operatorname {S}(V\!ir)/(z-\zeta )$ is Poisson simple if and only if $\zeta \neq ~0$.

Особливості інтеграції геопросторових даних автомобільних доріг із застосуванням операції з’єднання (JOIN)

The purpose of the article is to research the integration of attributes of the set of geospatial data “Roads”, which was created on the basis of the topographic database of the Main State Topographic Map, and the geoinformation system of roads of the State Agency of Roads of Ukraine (hereinafter – Ukravtodor) for the creation and maintenance of core reference data of roads of the national spatial data infrastructure (hereinafter – NSDI) using an operation Theta-JOIN in object-relational database management systems (hereinafter referred to as DBMS). Method. The basis of the research is the analysis of the possibilities of applying the theory of spatial databases and knowledge bases, international and national harmonized standards in the field of Geographic Information/Geomatics to solve the issue of integration of spatial data using the Theta-JOIN operation in object-relational database management systems. The results. The article analyzes the geo-information model of the road, based on the data of which it is possible to make management decisions during planning, design, operation, management, organization of road safety, etc. The requirements for the presentation of this model are defined in the GDF 5.1 standard, which is used for modeling road infrastructure in a geographic information environment using the Unified Modeling Language (UML) and Model Management Architecture (MDA) technologies. This allows for data interoperability at different levels, for example, national and pan-European. The geoinformation model of a public road of state importance in the topographic database is a continuous feature that is an aggregation of the features of the road and transport network through which this model passes. Also, on the basis of regulatory and technical support, the geographic identifier of the highway was established – the road index, the structure and principles of which are defined in the draft of the national standard DSTU xxx:20xx “Geoinformation system of public highways. Part 1. Composition, content and requirements for the database”. Two sets of spatial data TDB of the Main state topographic map and GIS roads of Ukravtodor were combined using the theta-JOIN operation in the PostgreSQL/PostGIS. The scientific novelty and practical significance is in the fact that the integration of sets of core reference and thematic spatial data of roads was investigated using the theta-join operation of relational algebra, which is implemented in modern GIS and DBMS for the development of the national spatial data infrastructure. This will make it possible to expand the attributes of road features contained in TDB, due to the official data of Ukravtodor, joining these sets with a geographic identifier, road index.

On Monotonic Determinacy and Rewritability for Recursive Queries and Views

A query Q is monotonically determined over a set of views V if Q can be expressed as a monotonic function of the view image. In the case of relational algebra views and queries, monotonic determinacy coincides with rewritability as a union of conjunctive queries, and it is decidable in important special cases, such as for conjunctive query views and queries. We investigate the situation for views and queries in the recursive query language Datalog. We give both positive and negative results about the ability to decide monotonic determinacy, and also about the co-incidence of monotonic determinacy with Datalog rewritability.

DBSP: Automatic Incremental View Maintenance for Rich Query Languages

Incremental view maintenance (IVM) has long been a central problem in database theory. Many solutions have been proposed for restricted classes of database languages, such as the relational algebra, or Datalog. These techniques do not naturally generalize to richer languages. In this paper we give a general, heuristic-free solution to this problem in 3 steps: (1) we describe a simple but expressive language called DBSP for describing computations over data streams; (2) we give a new mathematical definition of IVM and a general algorithm for solving IVM for arbitrary DBSP programs, and (3) we show how to model many rich database query languages using DBSP (including the full relational algebra, queries over sets and multisets, arbitrarily nested relations, aggregation, flatmap (unnest), monotonic and non-monotonic recursion, streaming aggregation, and arbitrary compositions of all of these). SQL and Datalog can both be implemented in DBSP. As a consequence, we obtain efficient incremental view maintenance algorithms for queries written in all these languages.

Catoids and modal convolution algebras

We show how modal quantales arise as convolution algebras Q^X of functions from catoids X, multisemigroups equipped with source and target maps, into modal quantales value or weight quantales Q. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in X, Q and Q^X. The catoids introduced generalise Schweizer and Sklar’s function systems and single-set categories to structures isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our correspondence results support a generic construction of weighted modal quantales from catoids. This construction is illustrated by many examples. We also relate our results to reasoning with stochastic matrices or probabilistic predicate transformers.

Comparison Analysis of Native Database Design with Object Oriented Design

Database design requires a structured database design, because the database contains data or information. The design method of the database design determines the structure of the designed design. Database design have two methods, either native or object-oriented method. Native database design has two stages, it is Data Flow Dia-gram and Entity Relationship Diagram, where as if it is object-oriented design using use case diagrams. It is ac-companied by class diagrams. Native designs tend to be more unstructured than object-oriented design. Native design focuses more on entity flow while object-oriented design focuses on database design entities. Another ad-vantage of using object-oriented design is the ease of explaining the database design to the client because of the simple design so that it can be easily understood. The method used in this research is prototype and relational algebra. The prototyping method is a technique to collect certain information about the user's information needs appropriately. This research focuses on comparing the native and object-oriented design.

A Multi Cluster Secure System for Out Sourced Encrypted Data Bases

Recently, database users have begun to use cloud database services to outsource theirdatabases. The reason for this is the high computation speed and the huge storage capacity thatcloud owners provide at low prices. However, despite the attractiveness of the cloud computingenvironment to database users, privacy issues remain a cause for concern for database ownerssince data access is out of their control. Encryption is the only way of assuaging users' fearssurroundingdataprivacy,butexecutingStructuredQueryLanguage (SQL)queriesoverencrypteddatais achallengingtask,especiallyif the data a re-encryptedby arandomizedencryptionalgorithm. Theexistingsystem, whichisanindexingschemetoencodetheoriginaltable’stuplesinto bit vectors (BVs) prior to the encryption. The resulting index is then usedto narrow therange of retrieved encrypted records from the cloud to a small set of records that are candidatesfor the user's query. Based on the indexing scheme, the designed system is used to execute SQLqueries over the encrypted data. The data are encrypted by a single randomized encryptionalgorithm,namely theAdvanced Encryption Standard-Cipher-Block Chaining (AES-CBC). Inthe existing scheme, theindex values (BVs) are stored atuser's side, and which is supportedmostof relational algebra operators, such as select, join,etc. In the existing system,the clouddata server is able to extract the ratio of two underlying data records, and key management servercandecryptthe ciphertextsofowners’data records,whichbreakthe privacysecurity.

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of , this provides an explicit connection with the generic superintegrable model on the -dimensional sphere and the related Racah algebra R(n). In particular, we show explicitly how the models on the two-sphere and three-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of and , respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.