# An Incremental Framework for Topological Dialogue Semantics: Efficient Reasoning in Discrete Spaces

arXiv: https://arxiv.org/abs/2506.00615 · HTML: https://mlsanity.com/paper/2506.00615/

## Key facts

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- First posted on arXiv 31 May 2025.

## Abstract (arXiv)

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.