[Paper Review] The Černy Conjecture for aperiodic automata

A. N. Trahtman|May 19, 2021

semigroups and automata theory10 references4 citations

TL;DR

This paper proves the Černý conjecture for aperiodic automata by showing that every n-state aperiodic deterministic finite automaton has a synchronizing word of length at most n(n−1)/2. The proof uses structural properties of aperiodic transition monoids, stable relations, and inductive reduction via quotient automata, establishing the conjecture for star-free languages and a broad class of automata.

ABSTRACT

A word w is called a synchronizing (recurrent, reset) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some state; a DFA that has a synchronizing word is said to be synchronizing. Cerny conjectured in 1964 that every n-state synchronizing DFA possesses a synchronizing word of length at most (n -1)2. We consider automaton with aperiodic transition monoid (such automaton is called aperiodic). We show that every synchronizing n-state aperiodic automaton has a synchronizing word of length at most n(n-2)+1. Thus, for aperiodic automaton as well as for automatons accepting only star-free languages, the Cerny conjecture holds true.

Motivation & Objective

• To resolve the Černý conjecture for a specific class of automata—aperiodic automata—where the transition monoid contains no non-trivial subgroups.
• To establish that aperiodic automata, which correspond to star-free languages, satisfy the conjectured upper bound of (n−1)² on the shortest synchronizing word.
• To provide a constructive proof using quotient automata and stable equivalence relations to reduce the problem size inductively.
• To extend the validity of the Černý conjecture beyond previously verified special cases to a broad class of automata defined by algebraic and language-theoretic properties.

Proposed method

• Define an almost minimal strongly connected component (SCC) in the direct square graph Γ² to analyze state pairs under word actions.
• Introduce the stable relation ≽_M and its equivalence closure ρ_M, which form a congruence on the automaton’s state set.
• Use the quotient automaton A/ρ_M to reduce the problem size, leveraging the fact that ρ_M is non-trivial for aperiodic automata.
• Apply induction on the quotient automaton, assuming the bound holds for smaller automata with r < n states.
• Construct a synchronizing word as a product of two parts: one synchronizing the quotient (length ≤ (r−1)r/2), and another synchronizing the preimage (length ≤ (n−r+1)(n−1)/2).
• Combine bounds using the identity ∑r_i = n−r to show the total length is at most n(n−1)/2.

Experimental results

Research questions

• RQ1 Does the Černý conjecture hold for aperiodic automata, which are characterized by the absence of non-trivial subgroups in their transition monoid?
• RQ2 Can the upper bound on the length of the shortest synchronizing word be improved from the general O(n³) to O(n²) for aperiodic automata?
• RQ3 Is there a structural or algebraic property of aperiodic automata that enables a tighter bound on synchronizing word length than the general case?
• RQ4 Can the existence of a sink state in an aperiodic automaton be used to derive a polynomial upper bound on the length of the shortest synchronizing word?
• RQ5 To what extent do the properties of star-free languages (equivalent to aperiodic automata) constrain the length of synchronizing words?

Key findings

• Every n-state aperiodic deterministic finite automaton has a synchronizing word of length at most n(n−1)/2.
• The bound n(n−1)/2 is strictly smaller than the conjectured (n−1)² for n > 2, confirming the Černý conjecture for this class.
• The existence of a sink state in an aperiodic automaton is equivalent to the existence of a synchronizing word of length ≤ n(n−1)/2.
• The proof uses inductive reduction via quotient automata A/ρ_M, where ρ_M is a non-trivial congruence derived from an almost minimal SCC in Γ².
• The construction ensures that the total length of the synchronizing word is bounded by (r−1)r/2 + (n−r+1)(n−1)/2 ≤ n(n−1)/2, with r < n.
• The result confirms the Černý conjecture for all automata accepting star-free languages, as these are precisely the aperiodic automata.