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We introduce a notion of real-valued reward testing for probabilistic processes by extending the traditional nonnegative-reward testing with negative rewards. In this richer testing framework, the may and must preorders turn out to be inverses. We show that for convergent processes with finitely many states and transitions, but not in the presence of divergence, the real-reward must-testing preorder coincides with the nonnegative-reward must-testing preorder. To prove this coincidence we characterise the usual resolution-based testing in terms of the weak transitions of processes, without having to involve policies, adversaries, schedulers, resolutions, or similar structures that are external to the process under investigation. This requires establishing the continuity of our function for calculating testing outcomes.
May and must testing were introduced by De Nicola and Hennessy to define semantic equivalences on processes. May-testing equivalence exactly captures safety properties, and must-testing equivalence liveness properties. This paper proposes reward test
In 1992 Wang & Larsen extended the may- and must preorders of De Nicola and Hennessy to processes featuring probabilistic as well as nondeterministic choice. They concluded with two problems that have remained open throughout the years, namely to fin
Two of the most studied extensions of trace and testing equivalences to nondeterministic and probabilistic processes induce distinctions that have been questioned and lack properties that are desirable. Probabilistic trace-distribution equivalence di
The combination of nondeterminism and probability in concurrent systems lead to the development of several interpretations of process behavior. If we restrict our attention to linear properties only, we can identify three main approaches to trace and
Probabilistic transition system specifications (PTSSs) in the $nt mu ftheta / ntmu xtheta$ format provide structural operational semantics for Segala-type systems that exhibit both probabilistic and nondeterministic behavior and guarantee that bisimi