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

• URL: http://arxiv.org/abs/2506.00615v2
• Date: Sat, 14 Jun 2025 00:27:23 GMT
• Title: An Incremental Framework for Topological Dialogue Semantics: Efficient Reasoning in Discrete Spaces
• Authors: Andreu Ballus Santacana,
• Abstract summary: We present a tractable, incremental framework for dialogue semantics based on finite, discrete semantic spaces.<n>We give a rigorous foundation, a provably correct incremental algorithm for nerve updates, and a reference implementation in the Wolfram Language.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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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