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Abstract
We present an adequacy theorem for a concurrent extension of probabilistic GCL. The underlying denotational semantics is based on the so-called mixed powerdomains, which combine non-determinism with probabilistic behaviour. The theorem itself is formulated via M. Smyth's idea of treating observable properties as open sets of a topological space. The proof hinges on a 'topological generalisation' of Konig's lemma in the setting of probabilistic programming (a result that is proved in the paper as well). One application of the theorem is that it entails semi-decidability w.r.t. whether a concurrent program satisfies an observable property (written in a certain form). This is related to M. Escardo's conjecture about semi-decidability w.r.t. may and must probabilistic testing.

Abstract
The design and analysis of systems that combine computational behaviour with physical processes' continuous dynamics - such as movement, velocity, and voltage - is a famous, challenging task. Several theoretical results from programming theory emerged in the last decades to tackle the issue; some of which are the basis of a proof-of-concept tool, called Lince, that aids in the analysis of such systems, by presenting simulations of their respective behaviours. However being a proof-of-concept, the tool is quite limited with respect to usability, and when attempting to apply it to a set of common, concrete problems, involving autonomous driving and others, it either simply cannot simulate them or fails to provide a satisfactory user-experience. The current work complements the aforementioned theoretical approaches with a more practical perspective, by improving Lince along several dimensions: to name a few, richer syntactic constructs, more operations, more informative plotting systems and errors messages, and a better performance overall. We illustrate our improvements via a variety of examples that involve both autonomous driving and electrical systems.

Abstract
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others.Our main result is the introduction of a V-equational deductive system for linear lambda-calculus together with a proof that it is sound and complete. In fact we go further than this, by showing that linear lambda-theories based on this V-equational system form a category equivalent to a category of autonomous categories enriched over 'generalised metric spaces'. If we instantiate this result to inequations, we get an equivalence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an equivalence with autonomous categories enriched over (ultra)metric spaces. Additionally, we show that this syntax-semantics correspondence extends to the affine setting.We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing.