We investigate the relation between the set-theoretical description of coinduction based on Tarski Fixpoint Theorem, and the categorical description of coinduction based on coalgebras. In particular, we examine set-theoretic generalizations of the coinduction proof principle, in the spirit of Milner's bisimulation "up-to", and we discuss categorical counterparts for these. Moreover, we investigate the connection between these and the equivalences induced by T -coiterative functions. These are morphisms into final coalgebras, satisfying the T -coiteration scheme, which is a generalization of both the coiteration and the corecursion scheme. We generalize Rutten's transformation from coalgebraic bisimulations to set-theoretic bisimulations, in order to cover also the case of bisimulations "up-to". A list of examples of set-theoretic coinductive specifications which appear not to be easily expressible in coalgebraic terms are discussed. Introduction Coinductive definitions and ...

We report on our experience in using the Isabelle/HOL theorem prover to mechanize proofs of observation equivalence for systems with infinitely many states, and for parameterized systems. We follow the direct approach: An infinite relation containing the pair of systems to be shown equivalent is defined, and then proved to be a weak bisimulation. The weak bisimilarity proof is split into many cases, corresponding to the derivatives of the pairs in the relation. Isabelle/HOL automatically proves simple cases, and guarantees that no case is forgotten. The strengths and weaknesses of the approach are discussed.

Bisimilarity is the most widespread notion of behavioral equivalence and hence algorithms for bisimulation checking are of fundamental importance for verifying that two systems are behaviorally equivalent (seen from the perspective of the environment). We investigate this problem in the context of behavioral equivalences of graphs and graph transformation systems, where the extension of the DPO approach to borrowed contexts provides us with a formal basis for reasoning about bisimilarity of graphs. In this paper we extend Hirschkoff’s on-the-fly algorithm for bisimulation checking, enabling it to verify whether two graphs are bisimilar with respect to a given set of productions. We then apply this framework to refactoring problems and verify instances of a model transformation which describes the minimization of deterministic finite automata.