Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.

Csp-Casl is a combination of the process algebra Csp [11,22] and the algebraic specification language Casl [7,1]. Its novel aspects include the combination of denotational semantics in the process part and, in particular, loose semantics for the data types covering both concepts partiality and sub-sorting. Technically, this integration involves the development of a new co-called data-logic formulated as an institution. This data-logic serves as a link between the institution underlying Casl and the alphabet of communications necessary for the Csp semantics. Besides being generic in the various denotational Csp semantics, this construction leads also to an appropriate notion of refinement with clear relations to both data refinement in Casl and process refinement in Csp. Key words: Algebraic specification; institution; process algebra; CASL, CSP. 1

Abstract. Heterogeneous specification becomes more and more important because complex systems are often specified using multiple viewpoints, involving multiple formalisms. Moreover, a formal software development process may lead to a change of formalism during the development. However, current research in integrated formal methods only deals with ad-hoc integrations of different formalisms. The heterogeneous tool set (Hets) is a parsing, static analysis and proof management tool combining various such tools for individual specification languages, thus providing a tool for heterogeneous multi-logic specification. Hets is based on a graph of logics and languages (formalized as so-called institutions), their tools, and their translations. This provides a clean semantics of heterogeneous specification, as well as a corresponding proof calculus. For proof management, the calculus of development graphs (known from other large-scale proof management systems) has been adapted to heterogeneous specification. Development graphs provide an overview of the (heterogeneous) specification module hierarchy and the current proof state, and thus may be used for monitoring the overall correctness of a heterogeneous development. 1

Abstract. This paper describes the formal specification of a future banking system by abstract data types and process algebra. In contrast to previous exercises (e.g., [1]), the system’s description is an actual industrial standard which is being used to develop the next generation of automatic banking machines. The specification language Csp-Casl is particularly well suited to this type of problem, since it combines both control and data aspects and allows loose specification of data types for later refinement. During the formalisation, several inconsistencies and ambiguities were exhibited. The obtained specification serves as a starting point for further validation. 1

Coalgebra has in recent years been recognized as the framework of choice for the treatment of reactive systems at an appropriate level of generality. Proofs about the reactive behavior of a coalgebraic system typically rely on the method of coinduction. In comparison to ‘traditional ’ coinduction, which has the disadvantage of requiring the invention of a bisimulation relation, the method of circular coinduction allows a higher degree of automation. As part of an effort to provide proof support for the algebraic-coalgebraic specification language CoCasl, we develop a new coinductive proof strategy which iteratively constructs a bisimulation relation, thus arriving at a new variant of circular coinduction. Based on this result, we design and implement tactics for the theorem prover Isabelle which allow for both automatic and semiautomatic coinductive proofs. The flexibility of this approach is demonstrated by means of examples of (semi-)automatic proofs of consequences of Co-Casl specifications, automatically translated into Isabelle theories by means of the Bremen heterogeneous Casl tool set Hets.

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The central idea of the Heterogeneous Tool Set (HETS) is to provide a general framework for formal methods integration and proof management. One can think of HETS acting like a motherboard where different expansion cards can be plugged in, the expansion cards here being individual logics (with their analysis

The central idea of the Heterogeneous Tool Set (HETS) is to provide a general framework for formal methods integration and proof management. One can think of HETS acting like a motherboard where different expansion cards can be plugged in, the expansion cards here being individual logics (with their analysis