Regularity of Position Sequences

TL;DR

This paper introduces a mathematical model for predicting the continuation of position sequences on paper, capturing varying degrees of regularity and perception of arbitrariness through a valuation function.

Contribution

It proposes a novel valuation-based mathematical framework to analyze the regularity and perceived arbitrariness of position sequences.

Findings

Model reflects opinions and perceptions of sequence continuation

Uses invariant features to ensure model quality

Accommodates sequences perceived as arbitrary

Abstract

A person is given a numbered sequence of positions on a sheet of paper. The person is asked, "Which will be the next (or the next after that) position?" Everyone has an opinion as to how he or she would proceed. There are regular sequences for which there is general agreement on how to continue. However, there are less regular sequences for which this assessment is less certain. There are sequences for which every continuation is perceived to be arbitrary. I would like to present a mathematical model that reflects these opinions and perceptions with the aid of a valuation function. It is necessary to apply a rich set of invariant features of position sequences to ensure the quality of this model. All other properties of the model are arbitrary.