On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs

TL;DR

This paper investigates the randomized generation of slowly synchronizing automata, showing simple methods typically produce automata with small reset thresholds, and introduces advanced algorithms to generate automata with large reset thresholds, including new families with quadratic order.

Contribution

It demonstrates limitations of simple random procedures for generating automata with large reset thresholds and proposes a more sophisticated algorithm to produce such automata, including new quadratic threshold families.

Findings

Simple random procedures yield automata with O(n log n) reset threshold.

Uniformly sampled NDFAs have short directing words of length O(n log n).

New families of automata with reset threshold of order Ω(n^2/4) are constructed.

Abstract

We tackle the problem of the randomized generation of slowly synchronizing deterministic automata (DFAs) by generating random primitive sets of matrices. We show that when the randomized procedure is too simple the exponent of the generated sets is O(n log n) with high probability, thus the procedure fails to return DFAs with large reset threshold. We extend this result to random nondeterministic automata (NDFAs) by showing, in particular, that a uniformly sampled NDFA has both a 2-directing word and a 3-directing word of length O(n log n) with high probability. We then present a more involved randomized algorithm that manages to generate DFAs with large reset threshold and we finally leverage this finding for exhibiting new families of DFAs with reset threshold of order $ \Omega(n^2/4) $.