Till Mossakowski

The heterogeneous tool set, HETS

Heterogeneous specification becomes more and more important because complex systems are often specified using multiple viewpoints, involving multiple formalisms (see Fig. 1). Moreover, a formal software development process may lead to a change of formalism during the development.

High-level nets with nets and rules as tokens

High-Level net models following the paradigm “nets as tokens” have been studied already in the literature with several interesting applications. In this paper we propose the new paradigm “nets and rules as tokens”, where in addition to nets as tokens also rules as tokens are considered. The rules can be used to change the net structure. This leads to the new concept of high-level net and rule systems, which allows to integrate the token game with rule-based transformations of P/T-systems. The new concept is based on algebraic high-level nets and on the main ideas of graph transformation systems. We introduce the new concept with the case study “House of Philosophers”, a dynamic extension of the well-known dining philosophers. In the main part we present a basic theory for rule-based transformations of P/T-systems and for high-level nets with nets and rules as tokens leading to the concept of high-level net and rule systems.

Relating CASL with other specification languages: the institution level

In this work, we investigate various specification languages and their relation to CASL, the recently developed Common Algebraic Specification Language. In particular, we consider the languages Larch, OBJ3 and functional CafeOBJ, ACT ONE, ASF, and HEP-theories, as well as various sublanguages of CASL. All these languages are translated to an appropriate sublanguage of CASL.The translation mainly concerns the level of specification in-the-small: the logics underlying the languages are formalized as institutions, and representations among the institutions are developed. However, it is also considered how these translations interact with specification in-the-large.Thus, we obtain, on the one hand, translations of any of the above-mentioned specification languages to an appropriate sublanguage of CASL. This allows us to take libraries and case studies that have been developed for other languages and re-use them in CASL.On the other hand, we set up institution representations going from the CASL institution (and some of its subinstitutions) to simpler subinstitutions. Given a theorem proving tool for such a simpler subinstitution, with the help of a representation, it can also be used for a more complex institution. Thus, first-order theorem provers and conditional term rewriting tools become usable for CASL.

Development graphs—Proof management for structured specifications

Development graphs are a tool for dealing with structured specifications in a formal program development in order to ease the management of change and reusing proofs. In this work, we extend development graphs with hiding (e.g. hidden operations). Hiding is a particularly difficult to realize operation, since it does not admit such a good decomposition of the involved specifications as other structuring operations do. We develop both a semantics and proof rules for development graphs with hiding. The rules are proven to be sound, and also complete relative to an oracle for conservative extensions. We also show that an absolutely complete set of rules cannot exist. The whole framework is developed in a way independent of the underlying logical system (and thus also does not prescribe the nature of the parts of a specification that may be hidden). We also show how various other logic independent specification formalisms can be mapped into development graphs; thus, development graphs can serve as a kernel formalism for management of proofs and of change.

What is a Logic

This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum algebra construction, its generalization to a Lindenbaum category construction that includes proofs, and model cardinality spectra; these are used in some examples to show logics inequivalent. Generalizations of familiar results from first order to arbitrary logics are also discussed, including Craig interpolation and Beth definability.

Carnap, Goguen, and the Hyperontologies: Logical Pluralism and Heterogeneous Structuring in Ontology Design

This paper addresses questions of universality related to ontological engineering, namely aims at substantiating (negative) answers to the following three basic questions: (i) Is there a ‘universal ontology’?, (ii) Is there a ‘universal formal ontology language’?, and (iii) Is there a universally applicable ‘mode of reasoning’ for formal ontologies? To support our answers in a principled way, we present a general framework for the design of formal ontologies resting on two main principles: firstly, we endorse Rudolf Carnap’s principle of logical tolerance by giving central stage to the concept of logical heterogeneity, i.e. the use of a plurality of logical languages within one ontology design. Secondly, to structure and combine heterogeneous ontologies in a semantically well-founded way, we base our work on abstract model theory in the form of institutional semantics, as forcefully put forward by Joseph Goguen and Rod Burstall. In particular, we employ the structuring mechanisms of the heterogeneous algebraic specification language HetCasl for defining a general concept of heterogeneous, distributed, highly modular and structured ontologies, called hyperontologies. Moreover, we distinguish, on a structural and semantic level, several different kinds of combining and aligning heterogeneous ontologies, namely integration, connection, and refinement. We show how the notion of heterogeneous refinement can be used to provide both a general notion of sub-ontology as well as a notion of heterogeneous equivalence of ontologies, and finally sketch how different modes of reasoning over ontologies are related to these different structuring aspects.

Comorphism-Based Grothendieck Logics

In order to obtain a semantic foundation for heterogeneous specification, we extend Diaconescus morphism-based Grothendieck institutions to the case of comorphisms. This is not just a dualization, because we obtain more general results, especially concerning amalgamation properties. We also introduce a proof calculus for structured heterogeneous specifications and study its soundness and completeness (where amalgamation properties play a role for obtaining the latter).

The Development Graph Manager MAYA

The Maya-system is mostly implemented in Common Lisp while parts of the GUI, shared with the OMEGA-system [9], are written in Mozart. The Caslparser is provided by the CoFI-group in Bremen. The Maya-system is available from the Maya-web-page at www.dfki.de/~inka/maya.html.

Project abstract: logic atlas and integrator (LATIN)

LATIN aims at developing methods, techniques, and tools for interfacing logics and related formal systems. These systems are at the core of mathematics and computer science and are implemented in systems like (semi-)automated theorem provers, model checkers, computer algebra systems, constraint solvers, or concept classifiers. Unfortunately, these systems have differing domains of applications, foundational assumptions, and input languages, which makes them non-interoperable and difficult to compare and evaluate in practice.

What is a Logic Translation

We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof- theoretic and model-theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quantifiers, although not present in the definition of logic and logic translation, can be recovered by their abstract properties and are preserved and reflected by translations under suitable conditions.