This paper gives a treatment of substitution for "parametric" objects in final coalgebras, and also presents principles of definition by corecursion for such objects. The substitution results are coalgebraic versions of well-known consequences of initiality, and the work on corecursion is a general formulation which allows one to specify elements of final coalgebras using systems of equations. One source of our results is the theory of hypersets, and at the end of this paper we sketch a development of that theory which calls upon the general work of this paper to a very large extent and particular facts of elementary set theory to a much smaller extent. 1 Introduction This paper has two overall goals. The first is a general theory of substitution and corecursion having to do with final coalgebras. To give an example of the kind of phenomena we have in mind, consider any set S and form the functor F on sets defined by Fa = S \Theta a \Theta a. F is defined on functions in the u...

Corecursive programs produce values of greatest fixpoint types, in contrast to recursive programs, which consume values of least fixpoint types. There are a number of widely used methods for proving properties of corecursive programs, including fixpoint induction, the take lemma, and coinduction. However, these methods are all rather low-level, in the sense that they do not exploit the common structure that is often present in corecursive definitions. We argue for a more structured approach to proving properties of corecursive programs. In particular, we show that by writing corecursive programs using an operator called unfold that encapsulates a common pattern of corecursive de nition, we can then use high-level algebraic properties of this operator to conduct proofs in a purely calculational style that avoids the use of either induction or coinduction.

This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimulations) which underly it. We introduce the syntax and semantics of basic modal logic, discuss its expressivity at the level of models, examine its computational properties, and then consider what it can say at the level of frames. We then move beyond the basic modal language, examine the kinds of expressivity offered by a number of richer modal logics, and try to pin down what it is that makes them all ‘modal’. We conclude by discussing an example which brings many of the ideas we discuss into play: games.

Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable back-and-forth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focused on L!1! and its fragments (see e.g. Keisler [19]), since countable formulas seemed best behaved. The past decade has seen a renewed interest in L1! and its finite variable fragments L (k) 1! (for 2 k ! !) and the modal fragment L \Pi 1! (see e.g. Ebbinghaus and Flum [17] on the former and Barwise and Moss [9] on the latter), due to various connections with topics in computer science. These logics form a hierarchy of increasingly powerful logics L \Pi 1! ae L (2) 1! ae L (3) 1! ae : : : ae L (k) 1! ae : : : ae L1! ; with each of these inclusions being proper. Moreover, there is a useful and elegant algebraic characterization of equivalence in L in each of these logics L, from b...

This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributive laws. 1

Would ye both eat your cake and have your cake?

. The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-wellfounded sets, or hypersets (cf. [17], [5]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [7]). Instead of the Foundation Axiom these set theories adopt the so-called Anti-Foundation Axiom, AFA, which gives rise to a rich universe of sets. AFA provides an elegant tool for modeling all sorts of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and pr...

and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of

The origins of bisimulation and bisimilarity are examined, in the three fields where they have been

Requirements Evolution is one of the main issues that affect development activities as well as system features (e.g., system dependability). Although researchers and practitioners recognise the importance of requirements evolution, research results and experience are still patchy. This points out a lack of methodologies that address requirements evolution. This thesis investigates the current understanding of requirements evolution and explores new directions in requirements evolution research. The empirical analysis of industrial case studies highlights software requirements evolution as an important issue. Unfortunately, traditional requirements engineering methodologies provide limited support to capture requirements evolution. Heterogeneous engineering provides a comprehensive account of system requirements. Heterogeneous engineering stresses a holistic viewpoint that allows us to understand the underlying mechanisms of evolution of socio-technical systems. Requirements, as mappings between socio-technical solutions and problems, represent an account of the history of socio-technical issues arising and being solved within industrial settings. The formal extension of a heterogeneous account of requirements provides a framework to model and capture requirements evolution. The application of the proposed framework provides further evidence that it is possible to capture and model evolutionary information about requirements. The discussion of scenarios of use stresses practical necessities for methodologies addressing requirements evolution. Finally, the identification of a broad spectrum of evolutions in socio-technical systems points out strong contingencies between system evolution and dependability. This thesis argues that the better our understanding of socio-techn...