Relational Calculus Research Articles

The Discrete Borel–Pompeiu Formula on a Rectangular Lattice

Discrete function theory is a natural extension of the continuous theory to functions defined on lattices. The idea of the discrete function theory is to work directly with discretised domains (lattices) and to transfer all important properties from the continuous case to the discrete level. In the field of boundary value problems it is more beneficial to work with rectangular lattices, i.e. allowing two different stepsizes. Thus, the aim of this paper is to present the extension of the discrete function theory to rectangular lattices. Particularly, we present the discrete analogue of the Borel–Pompeiu formula. Analogously to the continuous theory it is the core of the developed discrete function theory and the related operator calculus.

Resolução de situações-problemas do campo conceitual multiplicativo: o cálculo relacional e o numérico

Apresentamos os resultados de um estudo com o objetivo de identificar as dificuldades dos alunos do 6º ano do Ensino Fundamental no cálculo relacional e numérico na resolução de situações-problema do campo conceitual multiplicativo. Este estudo foi baseado na Teoria dos Campos Conceituais de Vergnaud (1990). Dentre os procedimentos metodológicos, 150 alunos de uma escola da rede pública estadual de Recife-PE foram solicitados a responder cinco situações-problemas; sendo duas de relação quaternária (proporção simples) e três de relação terciária (comparação multiplicativa e produto de medidas). Os resultados indicam um maior número de acertos na situação-problema do eixo proporção simples (partição). Enquanto as situações-problemas do eixo comparação multiplicativa (procura do referente) e a do eixo produto de medidas (combinatória) foram aquelas que os alunos apresentaram maior dificuldade em respondê-las. De modo geral, em todas as situações-problemas, as dificuldades dos alunos foram maiores no cálculo relacional que no cálculo numérico.

Class dependency of fuzzy relational database using relational calculus and conditional probability

In this paper, we propose a design of fuzzy relational database to deal with a conditional probability relation using fuzzy relational calculus. In the previous, there are several researches about equivalence class in fuzzy database using similarity or approximate relation. It is an interesting topic to investigate the fuzzy dependency using equivalence classes. Our goal is to introduce a formulation of a fuzzy relational database model using the relational calculus on the category of fuzzy relations. We also introduce general formulas of the relational calculus for the notion of database operations such as ’projection’, ’selection’, ’injection’ and ’natural join’. Using the fuzzy relational calculus and conditional probabilities, we introduce notions of equivalence class, redundant, and dependency in the theory fuzzy relational database.

Relational calculi in a deductive system

This work describes the addition of relational calculus languages in the deductive database system DES. Based on first-order logic, such languages admit a clean logical reading of queries, providing truly declarativeness, in contrast to other languages based on logic such as Prolog (a classical language used to build expert systems). Interesting properties as termination (for finite relations) and recursion are ensured because the DES deductive engine is used for solving relational calculus queries. Recursion in particular opens a brand new ream of applications (social networks, data warehouses, ...) for relational calculus languages which were unmanageable up to now. Since the DES system was targeted at teaching, we have also make a special emphasis on providing a practical system for students by providing appropriate syntax error feedback in a system supporting different languages (including relational algebra, SQL and Datalog).

Granular Encoders and Decoders: A Study in Processing Information Granules

Information granules are generic building blocks supporting the processing realized in granular computing and facilitating communication with the environment. In this paper, we are concerned with a fundamental problem of encoding–decoding of information granules. The essence of the problem is outlined as follows: given a finite collection of granular data X 1, X 2,…, XN (sets, fuzzy sets, etc.), construct an optimal codebook composed of information granules A 1 , A2 , …, Ac , where typically c N, so that any Xk represented in terms of A i 's and then decoded (reconstructed) with the help of this codebook leads to the lowest decoding error. A fundamental result is established, which states that in the proposed encoders and decoders, when encoding–decoding error is present, the information granule coming as a result of decoding is of a higher type than the original information granules (say, if Xk is information granule of type-1, then its decoded version becomes information granule of type-2). It would be beneficial to note that as the encoding–decoding process is not lossless (in general, with an exception of a few special cases), the lossy nature of the method is emphasized by the emergence of information granules of higher type (in comparison with the original data being processed). For instance, when realizing encoding–decoding of numeric data (viz., information granules of type-0), the losses occur and they are quantified in terms of intervals, fuzzy sets, probabilities, rough sets, etc., where, in fact, the result becomes an information granule of type-1. In light of the nature of the constructed result when Xk is an interval or a fuzzy set, an optimized performance index engages a distance between the bounds of the interval-valued membership function. We develop decoding and encoding mechanisms by engaging the theory of possibility and fuzzy relational calculus and show that the decoded information granule is either a granular interval or interval-valued fuzzy set. The optimization mechanism is realized with the aid of the particle swarm optimization (PSO). A series of experiments are reported with intent to illustrate the details of the encoding–decoding mechanisms and show that the PSO algorithm can efficiently optimize the granular codebook.

Relational similarity-based model of data part 1: foundations and query systems

We present a general rank-aware model of data which supports handling of similarity in relational databases. The model is based on the assumption that in many cases it is desirable to replace equalities on values in data tables by similarity relations expressing degrees to which the values are similar. In this context, we study various phenomena which emerge in the model, including similarity-based queries and similarity-based data dependencies. Central notion in our model is that of a ranked data table over domains with similarities which is our counterpart to the notion of relation on relation scheme from the classical relational model. Compared to other approaches which cover related problems, we do not propose a similarity-based or ranking module on top of the classical relational model. Instead, we generalize the very core of the model by replacing the classical, two-valued logic upon which the classical model is built by a more general logic involving a scale of truth degrees that, in addition to the classical truth degrees 0 and 1, contains intermediate truth degrees. While the classical truth degrees 0 and 1 represent nonequality and equality of values, and subsequently mismatch and match of queries, the intermediate truth degrees in the new model represent similarity of values and partial match of queries. Moreover, the truth functions of many-valued logical connectives in the new model serve to aggregate degrees of similarity. The presented approach is conceptually clean, logically sound, and retains most properties of the classical model while enabling us to employ new types of queries and data dependencies. Most importantly, similarity is not handled in an ad hoc way or by putting a “similarity module” atop the classical model in our approach. Rather, it is consistently viewed as a notion that generalizes and replaces equality in the very core of the relational model. We present fundamentals of the formal model and two equivalent query systems which are analogues of the classical relational algebra and domain relational calculus with range declarations. In the sequel to this paper, we deal with similarity-based dependencies.

Efficacy of a Kalanchoe gastonis-bonnieri extract to control bacterial biofilms and dental calculus in dogs

ABSTRACT: An aqueous leaf extract of the medicinal species Kalanchoe gastonis-bonnieri (here denominated KGB) has been found to be effective as an antimicrobial agent against canine oral cavity bacteria in in vitro assays. In this study, we investigated the effect of topic oral administration of KGB on the development of dental biofilm in Beagle dogs. The experiments were performed with an experimental group (0.2% of KGB extract), a negative control group (0.9% of saline solution) and a positive control group (0.12% chlorhexidine). Each treatment was sprayed into the oral cavity daily for 28 days. Thirty Beagle dogs with similar characteristics and kept under the same management and diet were used. The measurement of dental plaque and calculus was performed using a computerized analytical method. The phenolic profile of KGB extract was analyzed by HPLC-DAD. KGB extract at 0.2% showed efficacy in controlling the formation of plaque compared to the negative control group, and dental calculus in relation to the negative and positive control groups. A significant difference was observed among these three groups. Peaks attributed to flavonoids and phenolic acids were identified in the HPLC-DAD chromatogram of the KGB extract. The presence of these substances could be related to the activity observed. Our findings demonstrate that treatment with KGB is effective in controlling periodontal disease in dogs, providing new insights into the medicinal properties of this plant. KGB extract has a potential use as a supplemental agent in pharmaceutical products for the prevention of periodontal disease.

‘Horrent with Mysterious Spiculæ’. Augustus De Morgan’s Logic Notation of 1850 as a ‘Calculus of Opposite Relations’

The present paper expounds the logic notation proposed by Augustus De Morgan in 1850 from within the original context of De Morgan’s account of syllogistic logic and his approach to quantification. The notational system of 1850 is shown to be a flexible tool to state inferences, to prove their validity and to derive formulæ of the respective system by ‘blind’ application of transformation rules. These pertain to the swapping of operator signs, which are of inverse ‘character’ in a two-fold sense: The signs both represent inverse operations and exemplify oppositeness due to their very shapes. Thus, while De Morgan’s initial choice of symbol shapes is due to certain diagrammatic features, his notational system of 1850 leads to an account of logical operations as effected by manipulation of symbol tokens. It is concluded that as an instantiation of what De Morgan terms a ‘calculus of opposite relations’, his notation of 1850 may also be conceived of as a ‘graphicism’ affording for the working of a ‘symbolic machine’ on (intermediate) ‘de-semantification’.

ON TARSKI’S AXIOMATIC FOUNDATIONS OF THE CALCULUS OF RELATIONS

AbstractIt is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski’s other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski’s axiom system.

Invariance to ordinal transformations in rank-aware databases

We study influence of ordinal transformations on results of queries in rank-aware databases which derive their operations with ranked relations from totally ordered structures of scores with infima acting as aggregation functions. We introduce notions of ordinal containment and equivalence of ranked relations and prove that infima-based algebraic operations with ranked relations are invariant to ordinal transformations: Queries applied to original and transformed data yield results which are equivalent in terms of the order given by scores, meaning that top-k results of queries remain the same. We show this important property is preserved in alternative query systems based of relational calculi developed in context of Gödel logic. We comment on relationship to monotone query evaluation and show that the results can be attained in alternative rank-aware approaches.

On the TL-transitivity of fuzzy similarity measures

In fuzzy neural networks, it is usual to perform rule matching by employing fuzzy similarity measures. A fuzzy similarity measure is reserved for a binary fuzzy relation. In fuzzy relational calculus, the notion of T-transitivity is indispensable because it is strongly related to the concept of a metric. In this study, we consider whether the two types of fuzzy similarity measures given by Li et al. (2014) [24] satisfy TL-transitivity. We also investigate the TL-transitivity of fuzzy equivalencies.

Relational division in rank-aware databases

We present a survey of existing approaches to relational division in rank-aware databases, discuss issues of the present approaches, and outline generalizations of several types of classic division-like operations. We work in a model which generalizes the Codd model of data by considering tuples in relations annotated by ranks, indicating degrees to which tuples in relations match queries. The approach utilizes complete residuated lattices as the basic structures of degrees. We argue that unlike the classic model, relational divisions are fundamental operations which cannot in general be expressed by means of other operations. In addition, we compare the existing and proposed operations and identify those which are faithful counterparts of universally quantified queries formulated in relational calculi. We introduce Pseudo Tuple Calculus in the ranked model which is further used to show mutual definability of the various forms of divisions presented in the paper.

Fuzzy Logic-Based Formalisms for Gynecology Disease Diagnosis

AbstractThe very basis of the present article is the fact that the medical knowledge consisting of clinical presentation, diagnosis, and treatment of a disease is with imprecision and uncertainty. The overall approach in gynecological disease diagnosis could be divided into three distinct stages, and this was confirmed by seven experienced gynecologists. Stage 1 refers to an initial screening process in order to arrive at a single disease diagnosis for the patients, which is based only on the subjective information provided by patients to the physician. In stage 2, the patient who has not received a single diagnostic label in stage 1 is further investigated for a single disease diagnosis using past history criteria. If stage 2 fails to arrive at a single disease diagnosis for a patient, then physical examination and various tests like imaging tests, blood tests, etc., are conducted, and the test results are processed in stage 3. In stage 1, we have revisited fuzzy relational calculus and mathematically evaluated the perceptions of the domain experts (gynecologists) with respect to 31 gynecological diseases. The paper also presents the research findings with a case study focused on stage 2 using a type 1 fuzzy inference system. Out of 226 patients, 50 are correctly diagnosed for a single disease and 147 for multiple diseases in stage 1. The paper concludes that fuzzy relational calculus is an effective method as an “initial screening” process to arrive at a single disease diagnosis. We have identified 29 out of 226 patients satisfying past history criteria to achieve a single disease diagnosis by stage 2. Investigations for stage 3 are in progress.

Can Fuzzy Relational Calculus Bring Complex Issues in Selection of Examiners into Focus?

AbstractThe examinee and the examiner play pivotal roles in the educational grading system. Students’ academic performance evaluation by multiple experts involves epistemic uncertainty, which can be modeled using a fuzzy set theory. How many evaluators/experts are almost similar in their perceptual subjective evaluation of the students answer paper? In other words, how many experts are reliable for a particular evaluation task with a defined possibility level? In this paper, the focus is onobject’s features(students’ marks) as a basis in the subjective evaluation process to identify thedegree of similarityamong the domain experts. The case study reveals that 11 out of 20 evaluators are similar in their decision making of students’ academic performance with possibility (α-level cut, 0.98). The inter-rater reliability (κ-coefficient) among the selected 11 teachers is 0.41, which signifies a fair/moderate agreement in the evaluation process. This paper proposes an approach that is useful for the selection of experts having similar perceptions in judgment. This paper demonstrates a case study showing how it is useful to educational policy makers in the selection of examiners.

SQL query expressed in relational calculus

In this paper, we present some results on the equivalence between SQL queries and relation calculus expressions. These results are used to translate SQL queries into equivalent expressions of relational algebra.

SQL’s Three-Valued Logic and Certain Answers

The goal of the article is to bridge the difference between theoretical and practical approaches to answering queries over databases with nulls. Theoretical research has long ago identified the notion of correctness of query answering over incomplete data: one needs to find certain answers, which are true regardless of how incomplete information is interpreted. This serves as the notion of correctness of query answering, but carries a huge complexity tag. In practice, on the other hand, query answering must be very efficient, and to achieve this, SQL uses three-valued logic for evaluating queries on databases with nulls. Due to the complexity mismatch, the two approaches cannot coincide, but perhaps they are related in some way. For instance, does SQL always produce answers we can be certain about? This is not so: SQL’s and certain answers semantics could be totally unrelated. We show, however, that a slight modification of the three-valued semantics for relational calculus queries can provide the required certainty guarantees. The key point of the new scheme is to fully utilize the three-valued semantics, and classify answers not into certain or noncertain, as was done before, but rather into certainly true, certainly false, or unknown. This yields relatively small changes to the evaluation procedure, which we consider at the level of both declarative (relational calculus) and procedural (relational algebra) queries. These new evaluation procedures give us certainty guarantees even for queries returning tuples with null values.

Formalizing first-order logic in the equational theory of 3-dimensional diagonal-free cylindric algebras

In this paper, we exhibit a relation algebra reduct of any diagonal-free cylindric algebra of dimension 3 having sufficiently strong projection and equality parameters. We also offer a complete (and corrected) proof that full first-order logic can be formalized in the calculus of binary relations (a result due to Maddux and Tarski). Finally, we use these constructions to recursively define a translation function from the sentences of first-order logic to the equational theory of Df3, which preserves validity.

Kernels, in a nutshell

A classical result in algebraic specification states that a total function defined on an initial algebra is a homomorphism if and only if the kernel of that function is a congruence. We expand on the discussion of that result from an earlier paper: extending it from total to partial functions, simplifying the proofs using relational calculus, and generalising the setting to regular categories.

Relations Between Spatial Calculi About Directions and Orientations

Qualitative spatial descriptions characterize essential properties of spatial objects or configurations by relying on relative comparisons rather than measuring. Typically, in qualitative approaches only relatively coarse distinctions between configurations are made. Qualitative spatial knowledge can be used to represent incomplete and underdetermined knowledge in a systematic way. This is especially useful if the task is to describe features of classes of configurations rather than individual configurations. Although reasoning with them is generally NP-hard, relative directions are important because they play a key role in human spatial descriptions and there are several approaches how to represent them using qualitative methods. In these approaches directions between spatial locations can be expressed as constraints over infinite domains, e.g. the Euclidean plane. The theory of relation algebras has been successfully applied to this field. Viewing relation algebras as universal algebras and applying and modifying standard tools from universal algebra in this work, we (re)define notions of qualitative constraint calculus, of homomorphism between calculi, and of quotient of calculi. Based on this method we derive important properties for spatial calculi from corresponding properties of related calculi. From a conceptual point of view these formal mappings between calculi are a means to translate between different granularities.

Explaining the Formation of Zongfan Relations: Structure of Power, Domestic Influences and Ancient Sino-Korea Relations

The ancient regional order in Asia is believed to have existed on the basis of a Sino-centric tributary system (朝贡体系). This long-lasting inter-state system was considered hierarchical and built on the Chinese cultural superiority and egocentricity. Other scholars believe the tributary system came to existence because of the cultural or trade relations between China and neighboring states. I challenge the above traditional wisdom about the Chinese tributary system. I believe the existing literature has failed to provide a persuasive explanation of how the tributary system emerged. I do not think cultural superiority and economic relations alone can adequately explain the formation of the Chinese tributary system in ancient international relations in Asia. In this study, I first distinguish the zongfan relations (宗藩关系) from the tributary system (朝贡体系). While the tributary system emphasizes on rituals and moral authority of the Chinese emperor over other small states, the zongfan relations emphasize on China’s suzerainty over other small states. I argue that culture and economic relations alone cannot explain why small neighboring voluntarily came to zongfan relations with China. There were strong cultural and civilizational ties between the Han Chinese and neighboring small states and ethnical minorities. But the Chinese cultural influence, moral authority and economic relations were not the only reason for small neighboring state to accept the Chinese suzerainty, and there were also other reasons such as their calculus of power relations and domestic influences when it came to considerations of their relations with China. In the Chinese history, the Chinese Han regime did not always rule the whole China and there were constant changes of political regimes in the country. How did other neighboring states respond to these regime changes? I use the case of China-Korean relations from the Tang to the Qing dynasties to illustrate the formation of zongfan relations between China and the Korean peninsula.This study uses the theory of neoclassical realism to explain the recurrent patterns of Korea’s zongfan relations with successive dynasties of China. I argue that system-level variables, i.e., the distribution of material capabilities and the balance of military forces of great powers, played an important role in shaping both Chinese and Korean foreign policies. At the unit level, Korea would not automatically become a vassal state of China when there were regime changes in China. It would have to make power calculus in its relations with the old power and rising power in the Chinese regime changes. In the power rivalry over who would rule China, the rising power or the new power that established its rule in China tended to indispensably adopt a coercive policy to destroy the existing cultural and political ties with Korea and force Korea into a new zongfan relationship with itself. The balance of power between the old and new powers would determine what policy the new power would adopt in dealing with Korea. If the new power is weaker than or equal to the old power, the new power would adopt a highly coercive policy to make Korea yield to it. When the new power reduced the old power and became the hegemon, it would then adjust its policy towards Korea. When the new power’s policy became coercive, Korea had to make survival its first concern and thus chose to kowtow to the emperor of the new power. However, Korea would not completely align itself with the new power but adopt a hedging strategy between two sides until it believed that the new power was powerful enough to reduce or even annihilate the old power. After the new hegemon emerged, Korea would try to improve its position within the system and maintain its domestic stability; the establishment of zongfan relations with the new power would be conducive to the realization of these goals. Thus Korea established the zongfan relations with the new power.