## 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?

## 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?

## 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.

## Where do the dimensions of a conceptual space come from?

This week, the first paper of my PhD research [1] has been accepted for publication (you can take a look at the preprint here). I would like to seize the opportunity and explain here on a high level what this paper is about.

I’ve explained in a previous post what a conceptual space looks like. The aforementioned paper discusses the question posed in the title of this post: “Where do the dimensions of a conceptual space come from?”

## My PhD project in a nutshell

After having covered different topics in my little “What is …?” series, it is now time to put the parts together and explain what my PhD research is all about.

The preliminary title of my dissertation is “Concept Formation in Conceptual Spaces”. As you have probably already guessed, it involves both the area of concept formation and the framework of conceptual spaces.

The basic idea is the following one: