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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.

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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.

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