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Consider an agent traversing a graph of "gadgets", where each gadget has local state that changes with each traversal by the agent according to specified rules. Prior work has studied the computational complexity of deciding whether the agent can reach a specified location, a problem we call reachability. This paper introduces new goals for the agent, aiming to characterize when the computational complexity of these problems is the same or differs from that of reachability. First we characterize the complexity of universal traversal—where the goal is to traverse every gadget at least once—for DAG gadgets (partially), one-state gadgets, and reversible deterministic gadgets. Then we study the complexity of reconfiguration—where the goal is to bring the system of gadgets to a specified state. We prove many cases PSPACE-complete, and show in some cases that reconfiguration is strictly harder than reachability, while in other cases, reachability is strictly harder than reconfiguration. [ABSTRACT FROM AUTHOR]

We show that several classes of polyhedra are joined by a sequence of O (1) refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into exactly the next polyhedron; in other words, a polyhedron is refoldable into another polyhedron if they share a common unfolding. Specifically, assuming equal surface area, we prove that (1) any two tetramonohedra are refoldable to each other, (2) any doubly covered triangle is refoldable to a tetramonohedron, (3) any (augmented) regular prismatoid and doubly covered regular polygon is refoldable to a tetramonohedron, (4) any tetrahedron has a 3-step refolding sequence to a tetramonohedron, and (5) the regular dodecahedron has a 4-step refolding sequence to a tetramonohedron. In particular, we obtain a ≤6-step refolding sequence between any pair of Platonic solids, applying (5) for the dodecahedron and (1) and/or (2) for all other Platonic solids. As far as the authors know, this is the first result about common unfolding involving the regular dodecahedron. [ABSTRACT FROM AUTHOR]