Imprecise concept boundaries with fuzzy sets

In my last blog post, I explained the underlying idea of my formalization of the conceptual spaces framework. At the end of the text, I made the remark:

There’s one little detail left, and I’ll cover it in my next blog post. It is the problem of imprecise concept boundaries.

This problem is probably best explained by an example: Suppose we would like to define the meaning of “tall person”. Let’s say we call everybody who has a height of at least 1.80 m “tall”, and everybody below this threshold “not tall”. Why is this problematic?

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Formalizing Conceptual Spaces With Simple Star-Shaped Sets

In the last two blog posts (see here and here), I have described the problem I address in my paper “A Thorough Formalization of Conceptual Spaces” [1]: Convex regions in a conceptual space are unable to encode correlations between domains. I have furthermore argued that star-shaped regions are a possible solution to this problem.

In this blog post, I would like to illustrate the basic idea of my formalization. I won’t address the underlying math too much (whoever is interested can check out the paper), because my main goal is to give you some intuitive understanding of what’s going on.

The question that was still left open last time is the following one: How exactly can we represent star-shaped regions in a conceptual space?

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Star-shaped regions can encode correlations

So in the last blog post, I argued why it is not a good idea to use convex regions for representing concepts in a conceptual space – we are not able to represent correlations between domains. This time, I will show you how star-shaped regions can save the day.

But first of all, what does “star-shaped” mean in this context?

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Convex regions in a conceptual space are problematic

Recently, another one of my papers [1] (look at the preprint here) has been accepted at the German Conference on Artificial Intelligence. It is a quite technical paper with a lot of formulas, but I’ll try to illustrate the overall high-level idea in this and one or two future blog posts.

Today I would like to talk about the starting point of the research presented in this paper: The observation that convex regions in a conceptual space are highly problematic if we want to represent correlations between domains.

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