Proper Linear-time Specifications of Environment Behaviors in Nondeterministic Planning and Reactive Synthesis

To help it achieve its goal, an agent exploits assumptions it has about the behavior of its environment. The common view in planning and reactive synthesis is that such assumptions are sets of traces. This trace-centric view has the advantage of having well-understood specification formalisms, such as linear-time temporal logic. An alternative view, that we have promoted as being conceptually superior, is strategy-centric: assumptions are non-empty sets of environment strategies. In this work we relate these views and show that the strategy-centric view is a refinement of the trace-centric view. We thus address the following fundamental question: when should a set of traces be considered an assumption that the agent has about the environment's behavior? Our answer is in terms of coverability: every trace in the set should be consistent with some environment strategy that enforces it. We call such sets ``proper environment specifications''. Typical examples are given by (the traces consistent with a given) planning domain, and fairness constraints, but not arbitrary trace constraints. We provide an algorithm that, given a specification in linear-time temporal logic (LTL) decides whether or not it is a proper environment specification. Furthermore, we show that every set of traces has a ``proper environment core'', which excludes traces that the agent can ignore when devising its plan. We provide an algorithm for computing a representation of the core of an LTL formula, and prove that the core of an LTL-definable property is itself LTL-definable.