Multilevel Modelling (MLM) approaches make it possible for designers and modellers to work with an unlimited number of abstraction levels to specify their domain-specific modelling languages (DSMLs). To fully exploit MLM techniques, we need powerful model composition operators. Indeed, the composition of DSMLs is becoming increasingly relevant to the modelling community either because some DSMLs may share commonalities that we want to make reusable, or because we want to facilitate interoperability between DSMLs. In this paper, we propose a composition mechanism for structure and behaviour of multilevel modelling hierarchies. Our approach facilitates the inclusion of additional features while keeping a clear separation of concerns that enhances modularity. We provide a formal semantics of the constructions based on category theory and graph transformations, and show their use in practice on a case study.

Abstract Hoare Logic has a long tradition in formal verification and has been continuously developed and used to verify a broad class of programs, including sequential, object-oriented, and concurrent programs. Here we focus on partial and total correctness assertions within the framework of Hoare logic and show that a comprehensive categorical analysis of its axiomatic semantics needs the languages of indexed and fibered category theory. We consider Hoare formulas with local, finite contexts, of program and logical variables. The structural features of Hoare assertions are presented in an indexed setting, while the logical features of deduction are modeled in the fibered one.

Fibred semantics is the foundation of the model-instance pattern of software engineering. Software models can often be formalized as objects of presheaf topoi, e.g. the category of directed graphs. Multimodeling requires to construct colimits of diagrams of single models and their instances, while decomposition of instances of the multimodel is given by pullback. Compositionality requires an exact interplay of these operations, i.e., the diagrams must enjoy the Van Kampen property. However, checking the validity of the Van Kampen property algorithmically based on its definition is often impossible. In this paper we state a necessary and sufficient yet easily checkable condition for the Van Kampen property to hold for diagrams in presheaf topoi. It is based on a uniqueness property of path-like structures within the defining congruence classes that make up the colimiting cocone of the models. We thus add to the statement "Being Van Kampen is a Universal Property" by Heindel and Sobocinski presented at CALCO 2009 the fact that the Van Kampen property reveals a set-based structural uniqueness feature.

Journal Article Towards a uniform presentation of logical systems by indexed categories and adjoint situations Get access U. Wolter, U. Wolter Department of Informatics, University of Bergen, Norway.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar A. Martini, A. Martini Faculdade de Informática, PUCRS, Brasil.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar E. H. Häusler E. H. Häusler Departamento de Ciência da Computação, PUC-Rio, Brasil.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Journal of Logic and Computation, Volume 25, Issue 1, February 2015, Pages 57–93, https://doi.org/10.1093/logcom/exs038 Published: 03 September 2012 Article history Received: 26 January 2012 Published: 03 September 2012

Abstract Metamodelling is one of the pillars of model-driven engineering, used for language engineering and domain modelling. Even though metamodelling is traditionally based on a two-metalevel approach, several researchers have pointed out limitations of this solution and proposed an alternative deep (also called multi-level ) approach to obtain simpler system specifications. However, this approach currently lacks a formalisation that can be used to explain fundamental concepts such as deep characterisation, double linguistic/ontological typing and linguistic extension. This paper provides such a formalisation based on the Diagram Predicate Framework, and discusses its practical realisation in the metaDepth tool.

The raise of the abstraction level of programming languages has resulted in the usage of models and model transformations in software development processes. As a consequence of the usage of models as input to model transformation tools, there is a need for formal modeling languages and formal transformation definition techniques which can be employed to automatically translate between (and integrate) models. Therefore, a major focus of our research is on the formalization of modeling and model transformation in the generic formalism, Diagrammatic Predicate Logic (DPL). In this paper, we discuss a formalization approach to model transformation definitions based on DPL. Then, based on this formalization, some features of model transformations such as traceability, bidirectionality and compositionality are discussed.

Categorical constructions inherent to a theory of algebras with strict partial operations are presented and exploited to provide a categorical deduction calculus for conditional existence equations and an alternative definition of such algebras based on the notion of syntactic categories. A compact presentation of the structural theory of parameterized (partial) specifications is given using the categorical approach. This theory is shown to be suitable for providing initial semantics as well as the compositionality results necessary for the definition of specification languages likeACT ONEandACT TWO

Inspired by the work of S. Kaplan on positive/negative conditional rewriting, we investigate initial semantics for algebraic specifications with Gentzen formulas. Since the standard initial approach is limited to conditional equations (i.e. positive Horn formulas), the notion of semi-initial and quasi-initial algebras is introduced, and it is shown that all specifications with (positive) Gentzen formulas admit quasi-initial models. The whole approach is generalized to the parametric case where quasi-initiality generalizes to quasi-freeness. Since quasi-free objects need not be isomorphic, the persistency requirement is added to obtain auniquesemantics for many interesting practical examples. Unique persistent quasi-free semantics can be described as a free construction if the homomorphisms of the parameter category are suitably restricted. Furthermore, it turns out that unique persistent quasi-free semantics applies especially to specifications where the Gentzen formulas can be interpreted as hierarchical positive/negative conditional equations. The data type constructor of finite function spaces is used as an example that does not admit a correct initial semantics, but does admit a correct unique persistent quasi-initial semantics. The example demonstrates that the concepts introduced in this paper might be of some importance in practical applications.