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

TL;DR

This paper introduces an efficient incremental framework for topological dialogue semantics using finite discrete spaces, enabling tractable reasoning about dialogue structures and entailments.

Contribution

It provides a rigorous, correct incremental algorithm for updating dialogue nerves and a reference implementation, advancing topological semantics in dialogue systems.

Findings

Supports negative nerve computation for inconsistency detection

Enables consequence extraction and ranking of entailments

Clarifies properties of discrete topological structures in dialogue semantics

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.