Relational Calculus Research Articles

Synthesizing nested relational queries from implicit specifications: via model theory and via proof theory

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving two distinguished sets of relational symbols. One set of relations is for the "source data" I, and the other is for the "interface data" O. Such a specification is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a transformation (or "query" below) that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be converted to explicit definitions in the nested relational calculus (NRC). We first provide a model-theoretic argument for this result, which makes some additional connections that may be of independent interest, between NRC queries, interpretations, a standard mechanism for defining structure-to-structure translation in logic, and between interpretations and implicit to definability "up to unique isomorphism". The latter connection uses a variation of a result of Gaifman concerning "relatively categorical" theories. We also provide a proof-theoretic result that provides an effective argument: from a proof witnessing implicit definability, we can efficiently produce an NRC definition. This will involve introducing the appropriate proof system for reasoning with nested sets, along with some auxiliary Beth-type results for this system. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

Optimizing Sensor-Controlled Systems with Minimal Intervention: A Fuzzy Relational Calculus Approach

This article describes an approach for optimizing sensor-controlled systems through minimal intervention, utilizing fuzzy linear systems of equations (FLSEs). Starting with a generalized model of the system behavior, we incorporate an array of control units, environmental sensors, and an expert knowledge base. The described problems of detecting the level of intervention needed to change the system state to another is handled with the help of developed methods for solving the inverse problem faced by FLSEs. By achieving minimal intervention, we ensure that the system adjustments are effective, economically optimal, and non-intrusive. A MATLAB-based implementation is presented.

Student Performance Prediction

Abstract: Academic integration is now defined by how students are able to access, and effectively interact with, and are supported by engineering faculty and staff; Academic integration, which is a very important construct that contributes to a student’s persistence and ultimate graduation. But, several factors, either numeric or non-numeric can have varying degrees of influence on academic performance of students. Rankings of aforementioned factors, hopefully, will assist the management in order to initiate corrective action with a view to students' academic performance improvement. In an academic environment, teachers are frequently in close contact with students and therefore their belief about the degree of influence of factors on students’ academic performance could be of vital significance. The case study presented in this paper uses fuzzy relational calculus and the Dempster Shafer theory of evidence to examine and suggest a feasible and realistic ranking approach of only chosen factors!

On The Reasonable Effectiveness of Relational Diagrams: Explaining Relational Query Patterns and the Pattern Expressiveness of Relational Languages

Comparing relational languages by their logical expressiveness is well understood. Less well understood is how to compare relational languages by their ability to represent relational query patterns. Indeed, what are query patterns other than "a certain way of writing a query"? And how can query patterns be defined across procedural and declarative languages, irrespective of their syntax? To the best of our knowledge, we provide the first semantic definition of relational query patterns by using a variant of structure-preserving mappings between the relational tables of queries. This formalism allows us to analyze the relative pattern expressiveness of relational language fragments and create a hierarchy of languages with equal logical expressiveness yet different pattern expressiveness. Notably, for the non-disjunctive language fragment, we show that relational calculus can express a larger class of patterns than the basic operators of relational algebra. Our language-independent definition of query patterns opens novel paths for assisting database users. For example, these patterns could be leveraged to create visual query representations that faithfully represent query patterns, speed up interpretation, and provide visual feedback during query editing. As a concrete example, we propose Relational Diagrams, a complete and sound diagrammatic representation of safe relational calculus that is provably (i) unambiguous, (ii) relationally complete, and (iii) able to represent all query patterns for unions of non-disjunctive queries. Among all diagrammatic representations for relational queries that we are aware of, ours is the only one with these three properties. Furthermore, our anonymously preregistered user study shows that Relational Diagrams allow users to recognize patterns meaningfully faster and more accurately than SQL.

Efficient Evaluation of Arbitrary Relational Calculus Queries

The relational calculus (RC) is a concise, declarative query language. However, existing RC query evaluation approaches are inefficient and often deviate from established algorithms based on finite tables used in database management systems. We devise a new translation of an arbitrary RC query into two safe-range queries, for which the finiteness of the query's evaluation result is guaranteed. Assuming an infinite domain, the two queries have the following meaning: The first is closed and characterizes the original query's relative safety, i.e., whether given a fixed database, the original query evaluates to a finite relation. The second safe-range query is equivalent to the original query, if the latter is relatively safe. We compose our translation with other, more standard ones to ultimately obtain two SQL queries. This allows us to use standard database management systems to evaluate arbitrary RC queries. We show that our translation improves the time complexity over existing approaches, which we also empirically confirm in both realistic and synthetic experiments.

BC-system, absolute cyclotomy and the quantized calculus

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the “zeta sector” of the latter space as the Scaling Site. The new result that we present concerns the description of the BC-system as the universal Witt ring (i.e., K -theory of endomorphisms) of the “algebraic closure” of the absolute base \mathbb{S} . In this way we attain a conceptual meaning of the BC dynamical system at the most basic algebraic level. Furthermore, we define an invariant of Schwartz kernels in one dimension and relate the Fourier transform (in one dimension) to its role over the algebraic closure of \mathbb{S} . We implement this invariant to prove that, when applied to the quantized differential of a function, it provides its Schwarzian derivative. Finally, we survey the roles of the quantized calculus in relation to Weil's positivity, and that of spectral triples in relation to the zeros of the Riemann zeta function.

A Tutorial on Visual Representations of Relational Queries

Query formulation is increasingly performed by systems that need to guess a user's intent (e.g. via spoken word interfaces). But how can a user know that the computational agent is returning answers to the "right" query? More generally, given that relational queries can become pretty complicated, how can we help users understand existing relational queries , whether human-generated or automatically generated? Now seems the right moment to revisit a topic that predates the birth of the relational model: developing visual metaphors that help users understand relational queries. This lecture-style tutorial surveys the key visual metaphors developed for visual representations of relational expressions. We will survey the history and state-of-the art of relationally-complete diagrammatic representations of relational queries, discuss the key visual metaphors developed in over a century of investigating diagrammatic languages, and organize the landscape by mapping their used visual alphabets to the syntax and semantics of Relational Algebra (RA) and Relational Calculus (RC).

On the Term Set's Semantics for Pairwise Comparisons in Fuzzy Linguistic Preference Models.

The main objective of this paper is the definition of a membership function assignment procedure based on inherent features of linguistic terms to determine their semantics when they are used for preference modelling. For this purpose, we consider what linguists say about concepts such as language complementarity, the influence of context, or the effects of the use of hedges (modifiers) on adverbs meaning. As a result, specificity, entropy and position in the universe of discourse of the functions assigned to each linguistic term are mainly determined by the intrinsic meaning of the hedges concerned. We uphold that the meaning of weakening hedges is linguistically non-inclusive because their semantics are subordinated to the proximity to the indifference meaning, whereas reinforcement hedges are linguistically inclusive. Consequently, the membership function assignment rules are different: fuzzy relational calculus and the horizon shifting model derived from the Alternative Set Theory are used to handle weakening and reinforcement hedges, respectively. The proposed elicitation method provides for the term set semantics, non-uniform distributions of non-symmetrical triangular fuzzy numbers, depending on the number of terms used and the character of the hedges involved. (This article belongs to the section "Information Theory, Probability and Statistics").

Deconstructing the Calculus of Relations with Tape Diagrams

Rig categories with finite biproducts are categories with two monoidal products, where one is a biproduct and the other distributes over it. In this work we present tape diagrams, a sound and complete diagrammatic language for these categories, that can be intuitively thought as string diagrams of string diagrams. We test the effectiveness of our approach against the positive fragment of Tarski's calculus of relations.

Natural Language Processing in Translation of Relational Languages

Methods of data manipulation in combination with sturcture of the data and integrity constraints define data model used in the relational databases. This article focuses on the methods and processes of operation sets which are used for selection of data from relational database and translation between various formats of this manipulation. The article presents the design, implementation and experimental evaluation of tool for translating between relational algebra, tuple relational calculus, Structured Query Language and unrestricted natural language in all directions. Presented software tool translates from natural language with the use of artificial intelligence methods for natural language processing, syntactic analysis and parsing. Generative pretrained transformer of third generation (GPT3) based Codex – Davinci translation model was used as an artificial intelligence method used in the workflow of translation from English language to relational languages.

A Logic for Aristotle's Modal Syllogistic

ABSTRACT We propose a new modal logic endowed with a simple deductive system to interpret Aristotle's theory of the modal syllogism. While being inspired by standard propositional modal logic, it is also a logic of terms that admits a (sound) extensional semantics involving possible states-of-affairs in a given world. Applied to the analysis of Aristotle's modal syllogistic as found in the Prior Analytics A8-22, it sheds light on various fine-grained distinctions which when made allow us to clarify some ambiguities and obtain a completely consistent system and prove all of the modal syllogisms considered valid by Aristotle. This logic allows us also to make a connection with the axioms of modern propositional modal logic and to perceive to what extent these are implicit in Aristotle's reasoning. Further work will involve addressing the question of the completeness of this logic (or variants thereof) together with an extension of the logic to include a calculus of relations (for instance the relational syllogistic treated in Galen's Introduction to Logic) which Slomkowsky has argued is already found in the Topics.

Strongly-Normalizing Higher-Order Relational Queries

Language-integrated query is a powerful programming construct allowing database queries and ordinary program code to interoperate seamlessly and safely. Language-integrated query techniques rely on classical results about the nested relational calculus, stating that its queries can be algorithmically translated to SQL, as long as their result type is a flat relation. Cooper and others advocated higher-order nested relational calculi as a basis for language-integrated queries in functional languages such as Links and F#. However, the translation of higher-order relational queries to SQL relies on a rewrite system for which no strong normalization proof has been published: a previous proof attempt does not deal correctly with rewrite rules that duplicate subterms. This paper fills the gap in the literature, explaining the difficulty with a previous proof attempt, and showing how to extend the $\top\top$-lifting approach of Lindley and Stark to accommodate duplicating rewrites. We also show how to extend the proof to a recently-introduced calculus for heterogeneous queries mixing set and multiset semantics.

A Formalization of SQL with Nulls

SQL is the world’s most popular declarative language, forming the basis of the multi-billion-dollar database industry. Although SQL has been standardized, the full standard is based on ambiguous natural language rather than formal specification. Commercial SQL implementations interpret the standard in different ways, so that, given the same input data, the same query can yield different results depending on the SQL system it is run on. Even for a particular system, mechanically checked formalization of all widely-used features of SQL remains an open problem. The lack of a well-understood formal semantics makes it very difficult to validate the soundness of database implementations. Although formal semantics for fragments of SQL were designed in the past, they usually did not support set and bag operations, lateral joins, nested subqueries, and, crucially, null values. Null values complicate SQL’s semantics in profound ways analogous to null pointers or side-effects in other programming languages. Since certain SQL queries are equivalent in the absence of null values, but produce different results when applied to tables containing incomplete data, semantics which ignore null values are able to prove query equivalences that are unsound in realistic databases. A formal semantics of SQL supporting all the aforementioned features was only proposed recently. In this paper, we report about our mechanization of SQL semantics covering set/bag operations, lateral joins, nested subqueries, and nulls, written in the Coq proof assistant, and describe the validation of key metatheoretic properties. Additionally, we are able to use the same framework to formalize the semantics of a flat relational calculus (with null values), and show a certified translation of its normal forms into SQL.

The exact completion for regular categories enriched in posets

We construct the exact completion for regular categories enriched in the cartesian closed category Pos of partially ordered sets and monotone functions by employing a suitable calculus of relations. We then characterize the embedding of any regular category into its completion and use this to obtain examples of concrete categories which arise as such completions. In particular, we prove that the exact completion in this enriched sense of both the categories of Stone and Priestley spaces is the category of compact ordered spaces of L. Nachbin. Finally, we consider the relationship between the enriched exact completion and categories of internal posets in ordinary categories.

Translating canonical SQL to imperative code in Coq

SQL is by far the most widely used and implemented query language. Yet, on some key features, such as correlated queries and NULL value semantics, many implementations diverge or contain bugs. We leverage recent advances in the formalization of SQL and query compilers to develop DBCert, the first mechanically verified compiler from SQL queries written in a canonical form to imperative code. Building DBCert required several new contributions which are described in this paper. First, we specify and mechanize a complete translation from SQL to the Nested Relational Algebra which can be used for query optimization. Second, we define Imp, a small imperative language sufficient to express SQL and which can target several execution languages including JavaScript. Finally, we develop a mechanized translation from the nested relational algebra to Imp, using the nested relational calculus as an intermediate step.

Beyond dirty teeth: Integrating dental calculus studies with osteoarchaeological parameters

The study of ancient human dental calculus (mineralized dental plaque, also known as tartar) is becoming increasingly important in osteoarchaeology, human palaeoecology and environmental archaeology. Microremains of different origin (e.g. starch granules, pollen, phytoliths, feather barbules) as well as biomolecules and chemical compounds retrieved from its mineral matrix may represent an important link between past humans and their physical, biological and social environment, but they are rarely fully linked to the evidence from skeletal remains. This paper critically reviews the lines of evidence retrieved from dental calculus in relation to osteoarchaeological parameters, employing macroscopic, microscopic and biomolecular approaches, assessing synergy potential and limitations. The scope of this paper is also to contribute to the building of a much needed theoretical framework in this emerging subfield.

Fuzzy relational representation, modeling and interpretation of temporal data

In virtue of omnipresence of temporal data (time series), a quest for efficient and interpretable models of such data becomes of paramount relevance. The accuracy-interpretable dilemma becomes more visible in light of the current trend of producing accurate, yet comprehensible models supporting users in making sound and highly actionable conclusions. This study is positioned in the realm of human-centric granular representation of temporal data (time series) where the logic settings of relational calculus is realized at the level of information granules. We establish modeling pursuits and resulting constructs at a certain level of abstraction delivered by information granules expressed as fuzzy sets. While have been some studies involving a framework of granular computing applied to granular models of time series, in this study we advocate a unified and original environment of logic blueprint of relational dependencies among information granules articulated by means of weighted logic expressions where both the components of the expressions as well as their associations are quantified (calibrated) in the presence of available experimental data. An overall processing scheme is structured as the following sequence of main functional modules: granulation (encoding) – relational computing – logic interpretation – numeric characterization of results (produced through decoding). Relational computing engages a suite of architectures that capture temporal dependencies describing predictions at the level of granular amplitudes and granular trends. The quality of the relational model is quantified at the level of information granules (which emphasizes the interpretability aspect) as well as. A series of experiments is reported to demonstrate the performance of the relational models and their performance impacted by the parameters of the relational architectures.

A relation calculus for reasoning about t-probing security

In the context of side-channel attacks against cryptographic circuits, t-probing security characterizes the amount of information derivable about sensitive values (e.g., keys) by observing t output/internal values. Non-interference is a useful mathematical tool used by researchers to assess the probing security of a circuit which employs Boolean masking to protect itself from attacks. However, reasoning about non-interference still requires either difficult ratiocination or complex automatic tools. In this work, we propose a novel point of view to reason about non-interference, by exploiting the Walsh transform of a Boolean function. To this end, we introduce a calculus for mechanically reasoning about the shares of a variable and show that this formalism provides a lean algebraic explanation of known compositional patterns allowing for the discovery of new ones. Eventually, we show how this formalism can be applied to study the probing security of known cryptographic gadgets.

Expressive power and succinctness of the positive calculus of binary relations

In this paper, we study the expressive power and succinctness of the positive calculus of (binary) relations. We show that, for binary relations, (1) the calculus has the same expressive power as that of three-variable existential positive (first-order) logic, and (2) the calculus is exponentially less succinct than three-variable existential positive logic, namely, there is no subexponential-size translation from three-variable existential positive logic to the calculus. Additionally, we give a more fine-grained expressive power equivalence between the (full) calculus of relations and three-variable first-order logic in terms of the quantifier alternation hierarchy. It remains open whether the calculus of relations is also exponentially less succinct than three-variable first-order logic.

A relational theory of effects and coeffects

Graded modal types systems and coeffects are becoming a standard formalism to deal with context-dependent, usage-sensitive computations, especially when combined with computational effects. From a semantic perspective, effectful and coeffectful languages have been studied mostly by means of denotational semantics and almost nothing has been done from the point of view of relational reasoning. This gap in the literature is quite surprising, since many cornerstone results — such asnon-interference,metric preservation, andproof irrelevance— on concrete coeffects are inherently relational. In this paper, we fill this gap by developing a general theory and calculus of program relations for higher-order languages with combined effects and coeffects. The relational calculus builds upon the novel notion of acorelator(orcomonadic lax extension) to handle coeffects relationally. Inside such a calculus, we define three notions of effectful and coeffectful program refinements:contextual approximation,logical preorder, andapplicative similarity. These are the first operationally-based notions of program refinement (and, consequently, equivalence) for languages with combined effects and coeffects appearing in the literature. We show that the axiomatics of a corelator (together with the one of a relator) is precisely what is needed to prove all the aforementioned program refinements to be precongruences, this way obtaining compositional relational techniques for reasoning about combined effects and coeffects.