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arXiv · 2506.00615

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

Andreu Ballus Santacana
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Abstract

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.

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Andreu Ballus Santacana. 2025-05-31. An Incremental Framework for Topological Dialogue Semantics: Efficient Reasoning in Discrete Spaces. https://arxiv.org/abs/2506.00615

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