In this report we study the connection between two well known models for interactive systems. Reactive Systems à la Leifer and Milner allow to derive an interactive semantics from a reduction semantics guaranteeing, semantics (bisimilarity). Universal Coalgebra provides a categorical framework where bisimilarity can be characterized as final semantics, i.e., as the unique morphism to the final coalgebra. Moreover, if lifting a coalgebra to a structured setting is possible, then bisimilarity is compositional with respect to the lifted structure. Here we show that for every reactive system we can build a coalgebra. Furthermore, if bisimilarity is compositional in the reactive system, then we can lift this coalgebra to a structured coalgebra. 1

Abstract. The operational semantics of interactive systems is usually described by labeled transition systems. Abstract semantics (that is defined in terms of bisimilarity) is characterized by the final morphism in some category of coalgebras. Since the behaviour of interactive systems is for many reasons infinite, symbolic semantics were introduced as a mean to define smaller, possibly finite, transition systems, by employing symbolic actions and avoiding some sources of infiniteness. Unfortunately, symbolic bisimilarity has a different “shape ” with respect to ordinary bisimilarity, and thus the standard coalgebraic characterization does not work. In this paper, we introduce its coalgebraic models. 1

In this report we study the connection between two well known models for interactive systems. Reactive Systems à la Leifer and Milner allow to derive an interactive semantics from a reduction semantics guaranteeing, semantics (bisimilarity). Universal Coalgebra provides a categorical framework where bisimilarity can be characterized as final semantics, i.e., as the unique morphism to the final coalgebra. Moreover, if lifting a coalgebra to a structured setting is possible, then bisimilarity is compositional with respect to the lifted structure. Here we show that for every reactive system we can build a coalgebra. Furthermore, if bisimilarity is compositional in the reactive system, then we can lift this coalgebra to a structured coalgebra.

xtend Definition 13, in order to include also the new concept of algebraicity. In the first version it is was stated as follows. Definition 13 (tile functoriality). Let R = h\Sigma oe ; \Sigma ; N;Ri be an ars. A symmetric equivalence relation =f` A(\Sigma oe ) \Theta A(\Sigma oe ) is functorial for R if, whenever s =f t; s 0 =f t 0 for generic s; s 0 ; t; t 0 elements of A(\Sigma oe ), then s; s 0 =f t; t 0 (whenever defined) and s