String Compression in FA-Presentable Structures

TL;DR

This paper constructs FA-presentations of natural number structures with extremely fast growth in a numerical characteristic, introduces a general compressibility rate, and compares growth rates across different structures, including Turing machine configurations and groups.

Contribution

It introduces a new notion of compressibility rate for FA-presentations and demonstrates structures with super-exponential growth in this measure, advancing understanding of FA-presentability.

Findings

Existence of FA-presentations with growth faster than any fixed-height tower of exponents.

Bounded linear growth of compressibility rate in Presburger arithmetic.

FA-presentations of certain structures exhibit super-exponential growth in compressibility.

Abstract

We construct a FA-presentation $\psi: L \rightarrow \mathbb{N}$ of the structure $(\mathbb{N};\mathrm{S})$ for which a numerical characteristic $r(n)$ defined as the maximum number $\psi(w)$ for all strings $w \in L$ of length less than or equal to $n$ grows faster than any tower of exponents of a fixed height. This result leads us to a more general notion of a compressibility rate defined for FA-presentations of any FA-presentable structure. We show the existence of FA-presentations for the configuration space of a Turing machine and Cayley graphs of some groups for which it grows faster than any tower of exponents of a fixed height. For FA-presentations of the Presburger arithmetic $(\mathbb{N};+)$ we show that it is bounded from above by a linear function.