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

31 May 2025
Andreu Ballus Santacana
ArXiv (abs)PDFHTML
Main:12 Pages
Bibliography:3 Pages
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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@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 }
}
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