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.
View on arXiv@article{santacana2025_2506.00615,
title={ An Incremental Framework for Topological Dialogue Semantics: Efficient Reasoning in Discrete Spaces },
author={ Andreu Ballus Santacana },
journal={arXiv preprint arXiv:2506.00615},
year={ 2025 }
}