Relational Graph Models at Work

We study the relational graph models that constitute a natural subclass of relational models of lambda-calculus. We prove that among the lambda-theories induced by such models there exists a minimal one, and that the corresponding relational graph model is very natural and easy to construct. We then study relational graph models that are fully abstract, in the sense that they capture some observational equivalence between lambda-terms. We focus on the two main observational equivalences in the lambda-calculus, the theory H+ generated by taking as observables the beta-normal forms, and H* generated by considering as observables the head normal forms. On the one hand we introduce a notion of lambda-K\"onig model and prove that a relational graph model is fully abstract for H+ if and only if it is extensional and lambda-K\"onig. On the other hand we show that the dual notion of hyperimmune model, together with extensionality, captures the full abstraction for H*

Nominal Coalgebraic Data Types with Applications to Lambda Calculus

We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus

Flexible Correct-by-Construction Programming

Correctness-by-Construction (CbC) is an incremental program construction process to construct functionally correct programs. The programs are constructed stepwise along with a specification that is inherently guaranteed to be satisfied. CbC is complex to use without specialized tool support, since it needs a set of predefined refinement rules of fixed granularity which are additional rules on top of the programming language. Each refinement rule introduces a specific programming statement and developers cannot depart from these rules to construct programs. CbC allows to develop software in a structured and incremental way to ensure correctness, but the limited flexibility is a disadvantage of CbC. In this work, we compare classic CbC with CbC-Block and TraitCbC. Both approaches CbC-Block and TraitCbC, are related to CbC, but they have new language constructs that enable a more flexible software construction approach. We provide for both approaches a programming guideline, which similar to CbC, leads to well-structured programs. CbC-Block extends CbC by adding a refinement rule to insert any block of statements. Therefore, we introduce CbC-Block as an extension of CbC. TraitCbC implements correctness-by-construction on the basis of traits with specified methods. We formally introduce TraitCbC and prove soundness of the construction strategy. All three development approaches are qualitatively compared regarding their programming constructs, tool support, and usability to assess which is best suited for certain tasks and developers.Comment: arXiv admin note: substantial text overlap with arXiv:2204.0564

Application of the Variational Database Management System to Schema Evolution and Software Product Lines

As a general solution to the problem of managing structural and content variability in relational databases, in previous work we have introduced the Variational Database Management System (VDBMS). VDBMS consists of a representation of a variational database (VDB) and a corresponding typed query language (v-query). However, since this is a novel database representation, there are no existing instances of VDBs or v-queries that can be used to evaluate the VDBMS. In this project, we present two case studies to demonstrate the use of VDBMS and support its evaluation. The case studies were constructed by systematically encoding variability scenarios from prior work and generating corresponding VDBs by adapting existing widely-used data sets. The first case study shows how to use the VDBMS to manage database variants under a schema evolution scenario. The second case study demonstrates how to integrate the VDBMS with a database-backed software product line. Each case study provides a VDB and a set of v-queries that will be used to evaluate the VDBMS. Additionally, we provide some insights into generating VDBs from relational databases that could assist future VDBMS users

Theory and Implementation of a Variational Database Management System

In this thesis, I present the variational database management system, a formal framework and its implementation for representing variation in relational databases and managing variational information needs. A variational database is intended to support any kind of variation in a database. Specific kinds of variation in databases have already been studied and are well-supported, for example, schema evolution systems address the variation of a database’s schema over time and data integration systems address variation caused by accessing data from multiple data sources simultaneously. However, many other kinds of variation in databases arise in practice, and different kinds of variation often interact, but these scenarios are not well-supported by the existing work. For example, neither the schema evolution systems nor the database integration systems can address variation that arises when data sources combined in one database evolve over time. This thesis collects a large amount of work: It defines the variational database framework and the syntax and denotational semantics of the variational relational algebra, a query language for variational databases. It presents two use cases of the variational database framework that are based on existing datasets and scenarios that are partially supported by existing techniques. It presents the variational database management system which is a prototype of variational databases and variational relational algebra as an abstract layer written in Haskell on top of a traditional RDBMS. It also presents several theoretical results related to the framework and the query language, such as syntax-based equivalence rules that preserve the semantics of a query, a type system for ensuring that a variational query is well-formed with respect to the underlying variational schema, and a confluence property of the variational relational algebra type system with respect to the relational algebra type system and its denotational semantics

Nonmodularity results for lambda calculus

The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. In this paper we prove that the lattice of lambda theories is not modular and that the variety generated by the term algebra of a semi-sensible lambda theory is not congruence modular. Another result of the paper is that the Mal'cev condition for congruence modularity is inconsistent with the lambda theory generated by equating all the unsolvable lambda-terms