The universal tangle for spatial reasoning

The topological $\mu$-calculus has gathered attention in recent years as a powerful framework for representation of spatial knowledge. In particular, spatial relations can be represented over finite structures in the guise of weakly transitive wK4 frames. In this paper we show that the topological $\mu$-calculus is equivalent to a simple fragment based on a variant of the `tangle' operator. Similar results were proven for transitive frames by Dawar and Otto, using modal characterisation theorems for the corresponding classes of frames. However, since these theorems are not available in our setting, which has the upshot of providing a more explicit translation and upper bounds on formula size.Comment: 20 page

Games for topological fixpoint logic

Topological fixpoint logics are a family of logics that admits topological models and where the fixpoint operators are defined with respect to the topological interpretations. Here we consider a topological fixpoint logic for relational structures based on Stone spaces, where the fixpoint operators are interpreted via clopen sets. We develop a game-theoretic semantics for this logic. First we introduce games characterising clopen fixpoints of monotone operators on Stone spaces. These fixpoint games allow us to characterise the semantics for our topological fixpoint logic using a two-player graph game. Adequacy of this game is the main result of our paper. Finally, we define bisimulations for the topological structures under consideration and use our game semantics to prove that the truth of a formula of our topological fixpoint logic is bisimulation-invariant

Modeling Narrative Discourse

This thesis describes new approaches to the formal modeling of narrative discourse. Although narratives of all kinds are ubiquitous in daily life, contemporary text processing techniques typically do not leverage the aspects that separate narrative from expository discourse. We describe two approaches to the problem. The first approach considers the conversational networks to be found in literary fiction as a key aspect of discourse coherence; by isolating and analyzing these networks, we are able to comment on longstanding literary theories. The second approach proposes a new set of discourse relations that are specific to narrative. By focusing on certain key aspects, such as agentive characters, goals, plans, beliefs, and time, these relations represent a theory-of-mind interpretation of a text. We show that these discourse relations are expressive, formal, robust, and through the use of a software system, amenable to corpus collection projects through the use of trained annotators. We have procured and released a collection of over 100 encodings, covering a set of fables as well as longer texts including literary fiction and epic poetry. We are able to inferentially find similarities and analogies between encoded stories based on the proposed relations, and an evaluation of this technique shows that human raters prefer such a measure of similarity to a more traditional one based on the semantic distances between story propositions