[Paper Review] The Cerny conjecture and 1-contracting automata

Henk Don|arXiv (Cornell University)|Jul 22, 2015

semigroups and automata theory6 references3 citations

TL;DR

This paper introduces aperiodically 1-contracting automata—DFAs where every (n−1)-subset of states is reachable via 1-deficient words, and the induced state map forms a cyclic permutation. It proves such automata are synchronizing and satisfy the Černý conjecture, with all k-subsets reachable by words of length at most n(n−k), confirming the conjecture for this class, including certain circular and generalized circular automata.

ABSTRACT

A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. Černý conjectured in 1964 that a synchronizing automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$. We introduce the notion of aperiodically $1-$contracting automata and prove that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Furthermore, we give a sufficient condition under which the Černý conjecture holds for aperiodically $1-$contracting automata. As a special case, we prove some results for circular automata.

Motivation & Objective

• To establish a new class of automata—aperiodically 1-contracting automata—that guarantees synchronization.
• To prove that for such automata, all k-subsets of states are reachable by words of length at most n(n−k), supporting the Černý conjecture.
• To extend the validity of the Černý conjecture to a broader class of automata, including circular and generalized circular types.
• To provide a constructive framework for identifying synchronizing words in this class via 1-deficient word collections.
• To explore whether the conditions in the framework can be weakened while preserving the core results on reachability and synchronization.

Proposed method

• Define 1-contracting automata as those where every (n−1)-subset of states is reachable via a 1-deficient word, i.e., a word mapping the full state set to a subset of size n−1.
• Introduce the state map σW induced by a collection W of 1-deficient words, mapping each excluded state to its unique contracting state.
• Define aperiodically 1-contracting automata as those where σW is a cyclic permutation on the state set.
• Prove that aperiodically 1-contracting automata are synchronizing and that all nonempty subsets of states are reachable.
• Establish that if the 1-deficient words in an efficient collection have length at most n, then the Černý conjecture holds, with k-subsets reachable in at most n(n−k) steps.
• Use the power automaton framework to analyze reachability and synchronize via concatenation of 1-deficient words.

Experimental results

Research questions

• RQ1 Does the existence of a cyclic state map induced by 1-deficient words guarantee that an automaton is synchronizing?
• RQ2 Can the Černý conjecture be proven for automata where all (n−1)-subsets are reachable via 1-deficient words with a cyclic state map?
• RQ3 To what extent can the conditions for 1-contracting automata be relaxed while preserving reachability and synchronization?
• RQ4 Can the reachability bound of n(n−k) for k-subsets be universally applied to all synchronizing automata?
• RQ5 Is there a constructive algorithm to generate a synchronizing word in an aperiodically 1-contracting automaton using only 1-deficient words?

Key findings

• Aperiodically 1-contracting automata are synchronizing, as the cyclic state map ensures that repeated application of 1-deficient words leads to a singleton state.
• All k-subsets of the state set in such automata are reachable by a word of length at most n(n−k), which implies the Černý conjecture holds for this class.
• For automata with an efficient 1-contracting collection where the 1-deficient words have length at most n, the shortest synchronizing word has length at most (n−1)², confirming the Černý conjecture.
• The class includes circular automata and generalized circular automata with multiple labels on a cycle, extending known results to broader families.
• An example is provided of a non-circular automaton (without a full-state cycle) that still satisfies the Černý conjecture due to its aperiodically 1-contracting structure.
• The paper conjectures that all k-subsets in any synchronizing automaton are reachable by a word of length at most n(n−k), which would imply the Černý conjecture in general.