Steps and traces

Abstract

In the theory of coalgebras, trace semantics can be defined in various distinct ways, including through algebraic logics, the Kleisli category of a monad or its Eilenberg–Moore category. This paper elaborates two new unifying ideas: (i) coalgebraic,draftrules trace semantics is naturally presented in terms of corecursive algebras, and (ii) all three approaches arise as instances of the same abstract setting. Our perspective puts the different approaches under a common roof and allows to derive conditions under which some of them coincide.

Preservation and Reflection of Bisimilarity via Invertible Steps

“…We briefly present the required theory of steps, first termed as such in [41]. This structure occurs already in work on coalgebraic modal logic [35,14,40,32,17,38] where a step gives the one-step semantics of a logic.…”

“…, where ε is the counit of the Eilenberg-Moore adjunction. In fact, this can be viewed as a step-induced lifting for BT which arises by composing δ and the counit, see [41].…”

“…Finally, the examples that we have studied in this paper do not yet exploit the full generality of invertible steps: our main motivating examples are based on an Eilenberg-Moore adjunction (or close, as in the example based on Stone spaces). In [41] it is shown that steps are relevant in a much wider setting, for instance when based on a Kleisli adjunction or on contravariant adjunctions and dualities. The latter type of steps are relevant for coalgebraic modal logics-we have studied a first instance in our example of deterministic and non-deterministic automata.…”

“…These basically replace the Eilenberg-Moore adjunction inherent in the weak liftings approach by a general adjunction. In this context, a step allows one to lift the left adjoint to coalgebras-this is a widely occurring phenomenon, for instance in the semantics of coalgebraic modal logic, testing semantics and trace semantics (see [41] for an overview). The key idea here is to assume a right inverse, allowing the lifting of the right adjoint, such that we generalise the two-stage construction above.…”