The language (and series) of Hammersley-type processes

TL;DR

This paper explores the formal languages and power series linked to variants of Hammersley's process, revealing their classification and providing algorithms that support conjectures about their scaling constants.

Contribution

It classifies languages associated with Hammersley's process variants and introduces algorithms for computing related power series, supporting the conjecture of the golden ratio as the scaling constant.

Findings

Hammersley process yields a regular language.

Hammersley tree process yields deterministic context-free languages.

Algorithms support the conjecture of the golden ratio as the scaling constant.

Abstract

We study languages and formal power series associated to (variants of) Hammersley's process. We show that the ordinary Hammersley process yields a regular language and the Hammersley tree process yields deterministic context-free (but non-regular) languages. For the extension to intervals of the Hammersley process we show that there are two relevant formal languages. One of them leads to the same class of languages as the ordinary Hammersley tree process. The other one yields non-context-free languages. The results are motivated by the problem of studying the analog of the famous Ulam-Hammersley problem for heapable sequences. Towards this goal we also give an algorithm for computing formal power series associated to the variants of Hammersley's process. We employ these algorithms to settle the nature of the scaling constant, conjectured in previous work to be the golden ratio. Our results provide experimental support to this conjecture.