Abstract. In this article, we study Hartnell's Firefighter Problem through the group theoretic notions of growth and quasi-isometry. A graph has the ncontainment property if for every finite initial fire, there is a strategy to contain the fire by protecting n vertices at each turn. A graph has the constant containment property if there is an integer n such that it has the n-containment property. Our first result is that any locally finite connected graph with quadratic growth has the constant containment property; the converse does not hold. A second result is that in the class of graphs with bounded degree, having the constant containment property is closed under quasi-isometry. We prove analogous results for the {fn}-containment property, where fn is an integer sequence corresponding to the number of vertices protected at time n. In particular, we positively answer a conjecture by Develin and Hartke by proving that the d-dimensional square grid L d does not satisfy the cn d−3 -containment property for any constant c.
Previous work on story planning has lacked a knowledge representation for characters that attempt actions that fail because of the characters' misconceptions about the world state. This work describes HeadSpace, a state-space heuristic search planning system that generates stories that track and manipulate characters' beliefs about the story world. The planner produces story plans with actions that are attempted but fail. We show an example story plan that contains failed-action content that cannot be generated by typical planning-based approaches to story creation, and we provide an analytical evaluation that characterizes our planner's increased expressive range relative to other narrative planners addressing character belief and/or failed action execution.
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