Small Undecidable Problems in Epistemic Planning

Abstract

Epistemic planning extends classical planning with knowledge and is based on dynamic epistemic logic (DEL). The epistemic planning problem is undecidable in general. We exhibit a small undecidable subclass of epistemic planning over 2-agent S5 models with a fixed repertoire of one action, 6 propositions and a fixed goal. We furthermore consider a variant of the epistemic planning problem where the initial knowledge state is an automatic structure, hence possibly infinite. In that case, we show the epistemic planning problem with 1 public action and 2 propositions to be undecidable, while it is known to be decidable with public actions over finite models. Our results are obtained by reducing the reachability problem over small universal cellular automata. While our reductions yield a goal formula that displays the common knowledge operator, we show, for each of our considered epistemic problems, a reduction into an epistemic planning problem for a common-knowledge-operator-free goal formula by using 2 additional actions.