Theoretical Basics of NTA

2018 ◽  
pp. 97-124
Keyword(s):  
General Theory ◽  
Cartesian Product ◽  
Semantic Networks ◽  
Basic Structure ◽  
Relational Algebra ◽  
Binary Relations ◽  
Mathematical Structures ◽  
Theoretical Generalization

The chapter introduces theoretical features of n-tuple algebra developed by the authors as a theoretical generalization of structures and methods applied in intelligence systems. NTA supports formalization of a wide set of logical problems (abductive and modified conclusions, modeling of graphs, semantic networks, expert rules, etc.). This chapter contains main definitions and theorems of NTA. Unlike relational algebra and theory of binary relations, NTA uses Cartesian product of sets rather than sequences of elements (elementary n-tuples) as a basic structure and implements a general theory of n-ary relations. Novelty of our approach is that we developed some new mathematical structures allowing to implement many techniques of semantic and logical analyses; these methods have no analogies in conventional theories.

Logical Inference and Defeasible Reasoning in N-tuple Algebra

2013 ◽  
pp. 102-128 ◽  
Author(s):  
Boris Kulik ◽  
Alexander Fridman ◽  
Alexander Zuenko
Keyword(s):  
Semantic Networks ◽  
Axiom System ◽  
Logical Analysis ◽  
Defeasible Reasoning ◽  
Algebraic Methods ◽  
Relational Algebra ◽  
Logical Inference ◽  
Theoretical Generalization ◽  
Inference Methods ◽  
Structure Of Knowledge

This chapter examines the usage potential of n-tuple algebra (NTA) developed by the authors as a theoretical generalization of structures and methods applied in intelligence systems. NTA supports formalization of a wide set of logical problems (abductive and modified conclusions, modelling graphs, semantic networks, expert rules, etc.). This chapter mostly focuses on implementation of logical inference and defeasible reasoning by means of NTA. Logical inference procedures in NTA can include, besides the known logical calculus methods, new algebraic methods for checking correctness of a consequence or for finding corollaries to a given axiom system. Inference methods consider (above feasibility of certain substitutions) inner structure of knowledge to be processed, thus providing faster solving of standard logical analysis tasks. Matrix properties of NTA objects allow decreasing the complexity of intellectual procedures. As for making databases more intelligent, NTA can be considered as an extension of relational algebra to knowledge processing.

Tense? (Re)lax!

2020 ◽  
Vol 67 (1) ◽  
pp. 53-71
Author(s):  
Markus A. Pöchtrager
Keyword(s):  
General Theory ◽  
Basic Structure ◽  
Government Phonology ◽  
Vowel Height

AbstractThis article looks at what is referred to as the tense/lax contrast in English and proposes that members of the two sets of vowel have the same basic structure but differ in how part of that structure is made use of by its neighbours. The proposal forms part of a general theory of the representation of vowel height within the framework of Government Phonology 2.0.

Cartesian Product & Binary Relations

2017 ◽  
pp. 39-54
Author(s):  
Claudia Menini ◽  
Freddy Van Oystaeyen
Keyword(s):  
Cartesian Product ◽  
Binary Relations

N-Tuple Algebra as a Unifying System to Process Data and Knowledge

2019 ◽  
pp. 602-615
Author(s):  
Boris Alexandrovich Kulik ◽  
Alexander Yakovlevich Fridman
Keyword(s):  
Mathematical Model ◽  
General Theory ◽  
Information Technologies ◽  
Heterogeneous Data ◽  
Process Data ◽  
Modeling Uncertainties ◽  
Unify Representation ◽  
Theoretical Generalization ◽  
Universal Structure

Information technologies for analysis and processing heterogeneous data often face the necessity to unify representation of such data. To solve this problem, it seems reasonable to search for a universal structure that would allow for reducing different formats of data and knowledge to a single mathematical model with unitized manipulation methods. The concept of relation looks very prospective in this sense. So, with a view to developing a general theory of relations, the authors propose n-tuple algebra (NTA) developed as a theoretical generalization of structures and methods applicable in intelligence systems. NTA allows for formalizing a wide set of logical problems (deductive, abductive and modified reasoning, modeling uncertainties, and so on).

N-Tuple Algebra as a Generalized Theory of Relations

2021 ◽  
pp. 685-700
Author(s):  
Boris A. Kulik ◽  
Alexander Y. Fridman
Keyword(s):  
Artificial Intelligence ◽  
Mathematical Model ◽  
General Theory ◽  
Relational Algebra ◽  
Heterogeneous Information ◽  
Mathematical Objects ◽  
Knowledge Models ◽  
Analysis Methods ◽  
Modeling Uncertainties ◽  
Universal Structure

In ITs, analysis of heterogeneous information often necessitates unification of presentation forms and processing procedures for such data. To solve this problem, one needs a universal structure, which allows reducing various data and knowledge models to a single mathematical model with unified analysis methods. Such a universal structure is the relation, which is mainly associated with relational algebra. However, relations can model as different, at first glance, mathematical objects as graphs, networks, artificial intelligence structures, predicates, logical formulas, etc. Representation and analysis of such structures and models requires for more expressive means and methods than relational algebra provides. So, with a view to developing a general theory of relations, the authors propose n-tuple algebra (NTA) that allows for formalizing a wide set of logical problems (deductive, abductive, and modified reasoning; modeling uncertainties; and so on). This paper considers matters of metrization and clustering for NTA objects with ordered domains of attributes.

A general theory of separation, regularity, and connectedness properties

1976 ◽  
Vol 80 (1) ◽  
pp. 81-90 ◽  
Author(s):  
D. M. G. McSherry
Keyword(s):  
General Theory ◽  
Regularity Conditions ◽  
Binary Relations ◽  
Separation Axiom ◽  
Separation Axioms ◽  
The Family

AbstractThe notion of basic topological binary relations is introduced, and corresponding to each such relation we define -regularity, -connectedness, and a separation axiom . The family of separation axioms so obtained is shown to include all the standard axioms T0, T1, …, T5, while familiar properties such as hyperconnectedness and ultraconnectedness are among the class of connectedness conditions. The regularity conditions include R0, R1, and regularity itself.

scholarly journals ANALISA OPTIMALISASI BAHASA SQL BERDASARKAN RELATIONAL ALGEBRA PADA KASUS REKAPITULASI MAHASISWA LAYAK WISUDA

2015 ◽  
Vol 6 (2) ◽  
pp. 405
Author(s):  
Eko Darmanto
Keyword(s):  
Cartesian Product ◽  
Relational Algebra ◽  
Query Tree

ABSTRAK Aljabar relasional sangat membantu adanya kemungkinan penggunaan sintaks bahasa SQL yang berlainan dalam masalah yang sama. Masalah yang digunakan sebagai eksperimen adalah kasus rekapitulasi data mahasiswa layak wisuda pada suatu perguruan tinggi. Penggunaan sintaks bahasa SQL yang optimal dapat dianalisa menggunakan teknik query tree untuk mendapatkan hal utama dalam pemrosesan basis data yaitu kecepatan dan ketepatan komputasi yang berhubungan dengan pemuatan data ke dalam memori komputer. Hasil dari penelitian menunjukkan bahwa pemuatan data ke dalam memori komputer lebih optimal jika menggunakan join dibandingankan dengan Cartesian product. Dimungkinkan gabungan keduanya menjadi lebih cepat jika aturan penggunaan seleksi kemudian proyeksi didahulukan ketimbang penggabungan menggunakan Join atau Cartesian product terlebih dulu, hal ini bertujuan untuk mengurangi pemuatan baris dan kolom data yang berlebih. Ditinjau dari kecepatan komputasi untuk operasi himpunan yang melibatkan lebih dari satu tabel, Cartesian product dengan didahului proyeksi dan seleksi lebih cepat dibandingkan dengan Join untuk kasus yang sama. Kata kunci: optimalisasi sintaks SQL, aljabar relasional, query-tree.

scholarly journals On the Satisfiability Problem for SPARQL Patterns

2016 ◽  
Vol 56 ◽  
pp. 403-428 ◽  
Author(s):  
Xiaowang Zhang ◽  
Jan Van den Bussche ◽  
François Picalausa
Keyword(s):  
Relational Algebra ◽  
Satisfiability Problem ◽  
Binary Relations ◽  
Bound Constraints ◽  
Np Complete ◽  
Difference Operation ◽  
Undecidability Result

The satisfiability problem for SPARQL 1.0 patterns is undecidable in general, since the relational algebra can be emulated using such patterns. The goal of this paper is to delineate the boundary of decidability of satisfiability in terms of the constraints allowed in filter conditions. The classes of constraints considered are bound-constraints, negated bound- constraints, equalities, nonequalities, constant-equalities, and constant-nonequalities. The main result of the paper can be summarized by saying that, as soon as inconsistent filter conditions can be formed, satisfiability is undecidable. The key insight in each case is to find a way to emulate the set difference operation. Undecidability can then be obtained from a known undecidability result for the algebra of binary relations with union, composition, and set difference. When no inconsistent filter conditions can be formed, satisfiability is decidable by syntactic checks on bound variables and on the use of literals. Although the problem is shown to be NP-complete, it is experimentally shown that the checks can be implemented efficiently in practice. The paper also points out that satisfiability for the so-called ‘well-designed’ patterns can be decided by a check on bound variables and a check for inconsistent filter conditions.

Temporal Data Management and Processing with Column Oriented NoSQL Databases

2015 ◽  
Vol 26 (3) ◽  
pp. 41-70 ◽  
Author(s):  
Yong Hu ◽  
Stefan Dessloch
Keyword(s):  
Cartesian Product ◽  
Relational Algebra ◽  
Temporal Data ◽  
Nosql Databases ◽  
Operator Model ◽  
Temporal Interval ◽  
Multiple Data ◽  
Aggregation Functions ◽  
Interval Representation ◽  
Temporal Operator

This article introduces how temporal data can be maintained and processed by utilizing Column-oriented NoSQL databases (CoNoSQLDBs). Although each column in a CoNoSQLDB can store multiple data versions with their corresponded timestamps, its implicit temporal interval representation can cause wrong or misleading results during temporal query processing. In consequence, the original table representation supported by CoNoSQLDBs is not suitable for storing temporal data. To maintain the temporal data in the CoNoSQLDB tables, two alternative table representations can be adopted, namely, explicit history representation (EHR) and tuple time-stamping representation (TTR) in which each tuple (data version) has an explicit temporal interval. For processing TTR, the temporal relational algebra is extended to TTRO operator model with minor modifications. For processing EHR, a novel temporal operator model called CTO is proposed. Both TTRO and CTO contain eight temporal data operators, namely, Union, Difference, Intersection, Project, Filter, Cartesian product, Theta-Join and Group by with a set of aggregation functions, such as SUM, AVG, MAX and etc. Moreover, the authors implement each temporal operator by utilizing MapReduce framework to indicate which temporal operator model is more suitable for temporal data processing in the context of CoNoSQLDBs.

Sign in / Sign up

Export Citation Format

Share Document