Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambda-calculus terms. We show that lax logical relations coincide with the correspondences of Schoett, the algebraic relations of Mitchell and the pre-logical relations of Honsell and Sannella on Henkin models, but also generalise naturally to models in cartesian closed categories and to richer languages.

This paper addresses questions regarding the decidability of hybrid automata that may be constructed hierarchically and in a modular way, as is the case in many exemplar systems, be it natural or engineered. Since the basic fundamental step in such constructions is a product operation, which constructs a new product hybrid automaton by combining two simpler component hybrid automata, an essential property that would be desired is that the reachability property of the product hybrid automaton be decidable, provided that the component hybrid automata belong to a suitably restricted family of automata, for which the reachability property is provably decidable. Somewhat surprisingly, it does not appear that, under product operation, closure of decidability property for reachability condition could be guaranteed, in an arbitrarily general setting. Nonetheless, this paper establishes the decidability of the reachability condition over automata which are obtained by synchronizing two semialgebraic o-minimal systems. Such hybrid automata appear in systems biological modeling, and hence could be applied when one is interested in understanding a complex biological systems composed of smaller self-organizing systems.