[Paper Review] Slowly Synchronizing Automata with Idempotent Letters of Low Rank

Mikhail V. Volkov|arXiv (Cornell University)|Jul 18, 2018

semigroups and automata theory23 references3 citations

TL;DR

This paper presents a semigroup-theoretic construction that transforms any synchronizing automaton into a new automaton with twice the number of states and an additional idempotent input letter of rank n/2. The method preserves synchronization while doubling the reset threshold, demonstrating that automata with idempotent letters of low rank can achieve asymptotically optimal reset thresholds of n²/2. This construction provides a systematic way to generate slowly synchronizing automata with specific structural properties.

ABSTRACT

We use a semigroup-theoretic construction by Peter Higgins in order to produce, for each even $n$, an $n$-state and 3-letter synchronizing automaton with the following two features: 1) all its input letters act as idempotent selfmaps of rank $\dfrac{n}2$; 2) its reset threshold is asymptotically equal to $\dfrac{n^2}2$. In the revised version a few inaccuracies (spotted by the anonymous referees of the previous version) have been removed and several relevant references have been added.

Motivation & Objective

• To address the scarcity of known examples of slowly synchronizing automata with non-binary alphabets and idempotent input letters.
• To construct infinite families of synchronizing n-automata with reset thresholds approaching the theoretical lower bound of (n−1)².
• To demonstrate that automata with idempotent letters of rank n/2 can achieve asymptotically optimal synchronization thresholds.
• To provide a systematic method for generating proper, non-binary synchronizing automata with high reset thresholds.

Proposed method

• Utilizes a semigroup-theoretic construction by Higgins to transform a given DFA into a new automaton with 2n states and k+1 idempotent input letters.
• The transformation preserves the synchronizing property and doubles the reset threshold of the original automaton.
• Each input letter in the new automaton acts as an idempotent selfmap of rank n/2 on the state set.
• The construction ensures that the resulting automaton is proper, meaning all letters in the alphabet are essential for synchronization.
• The method relies on defining a new transition function that extends the original automaton by adding a new state and modifying transitions to preserve idempotency and synchronization.
• The proof of correctness involves showing that the new automaton remains synchronizing and that its reset threshold is exactly twice that of the original.

Experimental results

Research questions

• RQ1 Can a systematic construction generate non-binary synchronizing automata with idempotent letters of low rank and high reset thresholds?
• RQ2 What is the maximum possible reset threshold achievable by n-state automata with idempotent input letters of rank n/2?
• RQ3 Does the transformation preserve key synchronization properties such as properness and strong connectivity?
• RQ4 Can the construction be used to generate infinite families of slowly synchronizing automata with asymptotically optimal thresholds?

Key findings

• The proposed construction generates, for each even n, an n-state, 3-letter synchronizing automaton where all input letters act as idempotent selfmaps of rank n/2.
• The reset threshold of the constructed automaton is asymptotically equal to n²/2, matching the theoretical lower bound for slowly synchronizing automata.
• The transformation H(A) doubles the reset threshold of any input automaton A while preserving its synchronizing property and adding one idempotent letter of rank n/2.
• The method produces proper automata, meaning every letter in the alphabet is essential for synchronization, which is rare in non-binary settings.
• The construction demonstrates that automata with idempotent letters of low rank can achieve asymptotically optimal synchronization performance, challenging assumptions about the difficulty of achieving high reset thresholds under such constraints.