An Incremental Framework for Topological Dialogue Semantics: Efficient Reasoning in Discrete Spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916793744883712 |
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| author | Santacana, Andreu Ballus |
| author_facet | Santacana, Andreu Ballus |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_00615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Incremental Framework for Topological Dialogue Semantics: Efficient Reasoning in Discrete Spaces Santacana, Andreu Ballus Logic in Computer Science Artificial Intelligence Algebraic Topology Logic 03B05, 55U10, 68T27 F.4.1; I.2.3; I.2.4 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. |
| title | An Incremental Framework for Topological Dialogue Semantics: Efficient Reasoning in Discrete Spaces |
| topic | Logic in Computer Science Artificial Intelligence Algebraic Topology Logic 03B05, 55U10, 68T27 F.4.1; I.2.3; I.2.4 |
| url | https://arxiv.org/abs/2506.00615 |