# Re-pairing brackets

## TL;DR

This paper studies a bracket re-pairing game, establishing bounds on the complexity of re-pairing strategies and applying these results to automata theory, notably deriving quasi-polynomial lower bounds for automata translation.

## Contribution

It provides new lower bounds on re-pairing strategy width and applies these bounds to prove automata translation complexity results, answering an open question.

## Key findings

- Upper and lower bounds on re-pairing strategy width.
- Sub-exponential width strategies for complete binary tree sequences.
- Quasi-polynomial lower bounds on automata translation complexity.

## Abstract

Consider the following one-player game. Take a well-formed sequence of opening and closing brackets. As a move, the player can pair any opening bracket with any closing bracket to its right, erasing them. The goal is to re-pair (erase) the entire sequence, and the complexity of a strategy is measured by its width: the maximum number of nonempty segments of symbols (separated by blank space) seen during the play. For various initial sequences, we prove upper and lower bounds on the minimum width sufficient for re-pairing. (In particular, the sequence associated with the complete binary tree of height $n$ admits a strategy of width sub-exponential in $\log n$.) Our two key contributions are (1) lower bounds on the width and (2) their application in automata theory: quasi-polynomial lower bounds on the translation from one-counter automata to Parikh-equivalent nondeterministic finite automata. The latter result answers a question by Atig et al. (2016).