Unifying Probability with Nondeterminism

摘要

Early support for reasoning about probabilistic system behaviour replaced nondeterminism with probabilism. Only relatively recently have formalisms been studied that combine the two, and hence facilitate reasoning about probabilistic systems at levels of abstraction more general than code. Such studies have revealed an unsuspected subtlety in the interaction between nondeterministic and probabilistic choices that can be summarised: the demon resolving the nondeterministic choice has memory of previous state changes, whilst the probabilistic choice is made spontaneously. As a result, assignments to distinct variables need no longer commute. This paper introduces a model with explicit control of the length of the demon's memory. It does so by expanding the standard (initial-final) state view of computation to incorporate a third state, the 'original' state which checkpoints the most recent nondeterministic choice. That enables a nondeterministic choice to be made on the basis of only certain past probabilistic choices and so facilitates independent nondeterministic combinations to be chosen against just those. Sound laws are presented and used to analyse first an example in which no new behaviour should result, and second one that lies beyond the scope of traditional models.