# On the Height of Towers of Subsequences and Prefixes

## TL;DR

This paper investigates the maximum height of finite towers of subsequences between two languages represented by automata, establishing bounds that depend on automata size and alphabet, and explores towers of prefixes.

## Contribution

It provides polynomial upper bounds and exponential lower bounds on tower heights, highlighting gaps and open problems in automata theory related to subsequence towers.

## Key findings

- Upper bound is polynomial in states, exponential in alphabet size
- Lower bound is exponential when alphabet size is proportional to automata states
- Gaps remain between bounds, open problem for optimal bounds

## Abstract

A tower is a sequence of words alternating between two languages in such a way that every word is a subsequence of the following word. The height of the tower is the number of words in the sequence. If there is no infinite tower (a tower of infinite height), then the height of all towers between the languages is bounded. We study upper and lower bounds on the height of maximal finite towers with respect to the size of the NFA (the DFA) representation of the languages. We show that the upper bound is polynomial in the number of states and exponential in the size of the alphabet, and that it is asymptotically tight if the size of the alphabet is fixed. If the alphabet may grow, then, using an alphabet of size approximately the number of states of the automata, the lower bound on the height of towers is exponential with respect to that number. In this case, there is a gap between the lower and upper bound, and the asymptotically optimal bound remains an open problem. Since, in many cases, the constructed towers are sequences of prefixes, we also study towers of prefixes.