An Incremental Framework for Topological Dialogue Semantics: Efficient Reasoning in Discrete Spaces
This work addresses computational efficiency in formal semantics for dialogue systems, though it appears incremental as it builds on existing topological intuitions.
The paper tackles the problem of making topological dialogue semantics computationally tractable by introducing an incremental framework based on finite discrete semantic spaces, resulting in a provably correct algorithm for nerve updates with support for inconsistency tracking and entailment ranking.
We present a tractable, incremental framework for topological dialogue semantics based on finite, discrete semantic spaces. Building on the intuition that utterances correspond to open sets and their combinatorial relations form a simplicial complex (the dialogue nerve), we give a rigorous foundation, a provably correct incremental algorithm for nerve updates, and a reference implementation in the Wolfram Language. The framework supports negative nerve computation (inconsistency tracking), consequence extraction, and a transparent, set-theoretic ranking of entailments. We clarify which combinatorial properties hold in the discrete case, provide motivating examples, and outline limitations and prospects for richer logical and categorical extensions.