Real Reward Testing for Probabilistic Processes

Abstract

We introduce a notion of real reward testing for probabilistic processes by extending the traditional nonnegative reward testing with negative rewards. In this testing framework, the may and must preorders turn out to be the inverse relations of each other. 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 involving 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.