Consistency of Heterogeneously Typed Behavioural Models: A Coalgebraic Approach

Under embargo until: 2023-07-03Systematic and formally underpinned consistency checking of heterogeneously typed interdependent behavioural models requires a common metamodel, into which the involved models can be translated. And, if additional system properties are imposed on the behavioural models by modal logic formulae, the question arises, whether these formulae are faithfully translated, as well. In this paper, we propose a formal methodology based on natural transformations between coalgebraic specifications, which enables state-space preserving translations into a category of homogeneously typed systems, and we determine mild assumptions for the transformations to guarantee preservation and reflection of truth of translated formulae.acceptedVersio

A General Methodology for Internalising Multi-level Model Typing

Multilevel Modelling approaches allow for an arbitrary number of abstraction levels in typing chains. In this paper, a transformation of a multi-level typing chain into a single all-covering representing model is proposed. This comprehensive model is of equal size as the most concrete model in the chain and encodes all typing information in its labels, such that the typing chain can completely be restored. This guideline for maintaining multi-level typing chains in respective implementations of multi-level typing environments is based on a categorical equivalence theorem, which we generalize to a more convenient graph-oriented version.acceptedVersio

A Diagrammatic Logic for Object-Oriented Visual Modeling

Formal generalized sketches is a graph-based specification format that borrows its main ideas from categorical and ordinary first-order logic, and adapts them to software engineering needs. In the engineering jargon, it is a modeling language design pattern that combines mathematical rigor and appealing graphical appearance. The paper presents a careful motivation and justification of the applicability of generalized sketches for formalizing practical modeling notations. We extend the sketch formalism by dependencies between predicate symbols and develop new semantic notions based on the Instances-as-typed-structures idea. We show that this new framework fits in the general patterns of the institution theory and is well amenable to algebraic manipulations. Keywords: Diagrammatic modeling; model management; generic logic; categorical logic; diagram predicate; categorical sketchpublishedVersio

Indexed and Fibred Structures for Hoare Logic

Indexed and fibred categorical concepts are widely used in computer science as models of logical systems and type theories. Here we focus on Hoare logic and show that a comprehensive categorical analysis of its axiomatic semantics needs the languages of indexed category and fibred category theory. The structural features of the language are presented in an indexed setting, while the logical features of deduction are modeled in the fibred one. Especially, Hoare triples arise naturally as special arrows in a fibred category over a syntactic category of programs, while deduction in the Hoare calculus can be characterized categorically by the heuristic deduction = generation of cartesian arrows + composition of arrows.publishedVersio

Structural Operational Semantics for Heterogeneously Typed Coalgebras

Concurrently interacting components of a modular software architecture are heterogeneously structured behavioural models. We consider them as coalgebras based on different endofunctors. We formalize the composition of these coalgebras as specially tailored segments of distributive laws of the bialgebraic approach of Turi and Plotkin. The resulting categorical rules for structural operational semantics involve many-sorted algebraic specifications, which leads to a description of the components together with the composed system as a single holistic behavioural system. We evaluate our approach by showing that observational equivalence is a congruence with respect to the algebraic composition operation.publishedVersio

An Approach to Flexible Multilevel Modelling

Multilevel modelling approaches tackle issues related to lack of flexibility and mixed levels of abstraction by providing features like deep modelling and linguistic extension. However, the lack of a clear consensus on fundamental concepts of the paradigm has in turn led to lack of common focus in current multilevel modelling tools and their adoption. In this paper, we propose a formal framework, together with its corresponding tools, to tackle these challenges. The approach facilitates definition of flexible multilevel modelling hierarchies by allowing addition and deletion of intermediate abstraction levels in the hierarchies. Moreover, it facilitates separation of concerns by allowing integration of different multilevel modelling hierarchies as different aspects of the system to be modelled. In addition, our approach facilitates reusability of concepts and their behaviour by allowing definition of flexible transformation rules which are applicable to different hierarchies with a variable number of levels. As a proof of concept, a prototype tool and a domain-specific language for the definition of these rules is provided.publishedVersio

A Diagrammatic Logic for Object-Oriented Visual Modeling

Formal generalized sketches is a graph-based specification format that borrows its main ideas from categorical and ordinary first-order logic, and adapts them to software engineering needs. In the engineering jargon, it is a modeling language design pattern that combines mathematical rigor and appealing graphical appearance. The paper presents a careful motivation and justification of the applicability of generalized sketches for formalizing practical modeling notations. We extend the sketch formalism by dependencies between predicate symbols and develop new semantic notions based on the Instances-as-typed-structures idea. We show that this new framework fits in the general patterns of the institution theory and is well amenable to algebraic manipulations. Keywords: Diagrammatic modeling; model management; generic logic; categorical logic; diagram predicate; categorical sketc

Logics of Statements in Context-Category Independent Basics

Based on a formalization of open formulas as statements in context, the paper presents a freshly new and abstract view of logics and specification formalisms. Generalizing concepts like sets of generators in Group Theory, underlying graph of a sketch in Category Theory, sets of individual names in Description Logic and underlying graph-based structure of a software model in Software Engineering, we coin an abstract concept of context. We show how to define, in a category independent way, arbitrary first-order statements in arbitrary contexts. Examples of those statements are defining relations in Group Theory, commutative, limit and colimit diagrams in Category Theory, assertional axioms in Description Logic and constraints in Software Engineering. To validate the appropriateness of the newly proposed abstract framework, we prove that our category independent definitions and constructions give us a very broad spectrum of Institutions of Statements at hand. For any Institution of Statements, a specification (presentation) is given by a context together with a set of first-order statements in that context. Since many of our motivating examples are variants of sketches, we will simply use the term sketch for those specifications. We investigate exhaustively different kinds of arrows between sketches and their interrelations. To pave the way for a future development of category independent deduction calculi for sketches, we define arbitrary first-order sketch conditions and corresponding sketch constraints as a generalization of graph conditions and graph constraints, respectively. Sketch constraints are the crucial conceptual tool to describe and reason about the structure of sketches. We close the paper with some vital observations, insights and ideas related to future deduction calculi for sketches. Moreover, we outline that our universal method to define sketch constraints enables us to establish and to work with conceptual hierarchies of sketches.publishedVersio

A General Methodology for Internalising Multi-level Model Typing

Multilevel Modelling approaches allow for an arbitrary number of abstraction levels in typing chains. In this paper, a transformation of a multi-level typing chain into a single all-covering representing model is proposed. This comprehensive model is of equal size as the most concrete model in the chain and encodes all typing information in its labels, such that the typing chain can completely be restored. This guideline for maintaining multi-level typing chains in respective implementations of multi-level typing environments is based on a categorical equivalence theorem, which we generalize to a more convenient graph-oriented version

Multilevel Typed Graph Transformations

Multilevel modeling extends traditional modeling techniques with a potentially unlimited number of abstraction levels. Multilevel models can be formally represented by multilevel typed graphs whose manipulation and transformation are carried out by multilevel typed graph transformation rules. These rules are cospans of three graphs and two inclusion graph homomorphisms where the three graphs are multilevel typed over a common typing chain. In this paper, we show that typed graph transformations can be appropriately generalized to multilevel typed graph transformations improving preciseness, flexibility and reusability of transformation rules. We identify type compatibility conditions, for rules and their matches, formulated as equations and inequations, respectively, between composed partial typing morphisms. These conditions are crucial presuppositions for the application of a rule for a match—based on a pushout and a final pullback complement construction for the underlying graphs in the category GRAPH—to always provide a well-defined canonical result in the multilevel typed setting. Moreover, to formalize and analyze multilevel typing as well as to prove the necessary results, in a systematic way, we introduce the category CHAIN of typing chains and typing chain morphisms