An Institutional Framework for Heterogeneous Formal Development in UML

We present a framework for formal software development with UML. In contrast to previous approaches that equip UML with a formal semantics, we follow an institution based heterogeneous approach. This can express suitable formal semantics of the different UML diagram types directly, without the need to map everything to one specific formalism (let it be first-order logic or graph grammars). We show how different aspects of the formal development process can be coherently formalised, ranging from requirements over design and Hoare-style conditions on code to the implementation itself. The framework can be used to verify consistency of different UML diagrams both horizontally (e.g., consistency among various requirements) as well as vertically (e.g., correctness of design or implementation w.r.t. the requirements)

Heterogeneous Theories and the Heterogeneous Tool Set

Heterogeneous multi-logic theories arise in different contexts: they are needed for the specification of large software systems, as well as for mediating between different ontologies. This is because large theories typically involve different aspects that are best specified in different logics (like equational logics, description logics, first-order logics, higher-order logics, modal logics), but also because different formalisms are in practical use (like RDF, OWL, EML). Using heterogeneous theories, different formalims being developed at different sites can be related, i.e. there is a formal interoperability among languages and tools. In many cases, specialized languages and tools have their strengths in particular aspects. Using heterogeneous theories, these strengths can be combined with comparably small effort. By contrast, a true combination of all the involved logics into a single logic would be too complex (or even inconsistent) in many cases. We propose to use emph{institutions} as a formalization of the notion of logical system. Institutions can be related by so-called institution morphsims and comorphisms. Any graph of institutions and (co)morphisms can be flattened to a so-called emph{Grothendieck institution}, which is kind of disjoint union of all the logics, enriched with connections via the (co)morphisms. This semantic basis for heterogeneous theories is complemented by the heterogeneous tool set, which provides tool support. Based on an object-oriented interface for institutions (using type classes in Haskell), it implements the Grothendieck institution and provides a heterogeneous parser, static analysis and proof support for heterogeneous theories. This is based on parsers, static analysers and proof support for the individual institutions, and on a heterogeneous proof calculus for theories in the Grothendieck institution. See also the Hets web page: http://www.tzi.de/cofi/het

The Tannakian Formalism and the Langlands Conjectures

Let H be a connected reductive group over an algebraically closed field of characteristic zero, and let G be an abstract group. In this note we show that every homomorphism from the Grothendieck semiring of H to that of G which maps irreducible representations to irreducibles, comes from a group homomorphism from G to H. We also connect this result with the Langlands conjectures.Comment: 15 page

A brief review of abelian categorifications

This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various representations of the symmetric group and its Hecke algebra via highest weight categories of modules over the Lie algebra sl(n). The review is intended to give non-experts in representation theory who are familiar with the topological aspects of categorification (lifting quantum link invariants to homology theories) an idea for the sort of categories that appear when link homology is extended to tangles.Comment: latex, 35 pages, 4 eps figure

The FOLE Table

This paper continues the discussion of the representation of ontologies in the first-order logical environment FOLE. According to Gruber, an ontology defines the primitives with which to model the knowledge resources for a community of discourse. These primitives, consisting of classes, relationships and properties, are represented by the entity-relationship-attribute ERA data model of Chen. An ontology uses formal axioms to constrain the interpretation of these primitives. In short, an ontology specifies a logical theory. A series of three papers by the author provide a rigorous mathematical representation for the ERA data model in particular, and ontologies in general, within FOLE. The first two papers, which provide a foundation and superstructure for FOLE, represent the formalism and semantics of (many-sorted) first-order logic in a classification form corresponding to ideas discussed in the Information Flow Framework (IFF). The third paper will define an interpretation of FOLE in terms of the transformational passage, first described in (Kent, 2013), from the classification form of first-order logic to an equivalent interpretation form, thereby defining the formalism and semantics of first-order logical/relational database systems. Two papers will provide a precise mathematical basis for FOLE interpretation: the current paper develops the notion of a FOLE relational table following the relational model of Codd, and a follow-up paper will develop the notion of a FOLE relational database. Both of these papers expand on material found in the paper (Kent, 2011). Although the classification form follows the entity-relationship-attribute data model of Chen, the interpretation form follows the relational data model of Codd. In general, the FOLE representation uses a conceptual structures approach, that is completely compatible with formal concept analysis and information flow.Comment: 48 pages, 21 figures, 9 tables, submitted to T.A.C. for review in August 201