A complete natural deduction system for the relational calculus

A relational calculus is a formal system in which relation is the fundamental concept. The simplest relational calcu1us, that of ordinary binary relations, was introduced by Tarski in [4]. Tarski's system is essentially an algebra in which the operations are the usual Boolean operations (on sets of ordered pairs) together with the two special operations converse (denoted by u) and composition (denoted by ";" or by juxtaposition)

On insertion-deletion systems over relational words

We introduce a new notion of a relational word as a finite totally ordered set of positions endowed with three binary relations that describe which positions are labeled by equal data, by unequal data and those having an undefined relation between their labels. We define the operations of insertion and deletion on relational words generalizing corresponding operations on strings. We prove that the transitive and reflexive closure of these operations has a decidable membership problem for the case of short insertion-deletion rules (of size two/three and three/two). At the same time, we show that in the general case such systems can produce a coding of any recursively enumerable language leading to undecidabilty of reachability questions.Comment: 24 pages, 8 figure

Certain binary relations and operations and their use in research of bicentric polygons

In the article we consider certain binary relations and operations and their use in research of bicentric n-gons where n ≥ 3 is an odd integer. The considered binary relations and operations are defined on the set whose elements are integers 1, 2, . . . , (n−1)/2 which are relatively prime to n. We have found that some properties concerning bicentric n-gons can be a source or generator for many useful ideas and procedures in number theory and theory of groups. So using partition and ordering concerning bicentric n-gons, where n is an odd integer we have found some interesting relations concerning number theory

Axioms for signatures with domain and demonic composition

Demonic composition ∗ is an associative operation on binary relations, and demonic refinement ⊑ is a partial order on binary relations. Other operations on binary relations considered here include the unary domain operation D and the left restrictive multiplication operation ∘ given by s∘t=D(s)∗t. We show that the class of relation algebras of signature {⊑,D,∗}, or equivalently {⊆,∘,∗}, has no finite axiomatisation. A large number of other non-finite axiomatisability consequences of this result are also given, along with some further negative results for related signatures. On the positive side, a finite set of axioms is obtained for relation algebras with signature {⊑,∘,∗}, hence also for {⊆,∘,∗}

Compressed Data Structures for Binary Relations in Practice

[Abstract] Binary relations are commonly used in Computer Science for modeling data. In addition to classical representations using matrices or lists, some compressed data structures have recently been proposed to represent binary relations in compact space, such as the k 2 -tree and the Binary Relation Wavelet Tree (BRWT). Knowing their storage needs, supported operations and time performance is key for enabling an appropriate choice of data representation given a domain or application, its data distribution and typical operations that are computed over the data. In this work, we present an empirical comparison among several compressed representations for binary relations. We analyze their space usage and the speed of their operations using different (synthetic and real) data distributions. We include both neighborhood and set operations, also proposing algorithms for set operations for the BRWT, which were not presented before in the literature. We conclude that there is not a clear choice that outperforms the rest, but we give some recommendations of usage of each compact representation depending on the data distribution and types of operations performed over the data. We also include a scalability study of the data representations.Ministerio de Ciencia, Innovación y Universidades; TIN2016-77158-C4-3-RMinisterio de Ciencia, Innovación y Universidades; TIN2016-78011-C4-1-RMinisterio de Ciencia, Innovación y Universidades; RTC-2017-5908-7Consellería de Economía e Industria; IN852A 2018/14Xunta de Galicia; ED431C 2017/58Xunta de Galicia co-funded with ERDF; ED431G/01University of Bío-Bío; 192119 2/RUniversity of Bío-Bío; 195119 GI/V

Compositions of ternary relations

summary:In this paper, we introduce six basic types of composition of ternary relations, four of which are associative. These compositions are based on two types of composition of a ternary relation with a binary relation recently introduced by Zedam et al. We study the properties of these compositions, in particular the link with the usual composition of binary relations through the use of the operations of projection and cylindrical extension