Slowly synchronizing automata and digraphs

We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. These automata are closely related to primitive digraphs with large exponent.

💡 Research Summary

The paper investigates the long‑standing problem of determining how long a shortest reset word can be in a synchronizing deterministic finite automaton (DFA). The celebrated Černý conjecture asserts that any synchronizing DFA with n states admits a reset word of length at most (n‑1)², and the conjecture is known to be tight because the Černý series of automata actually achieve this bound. Despite many partial results, only a handful of infinite families are known that come close to the quadratic bound, and most of them are variations of the Černý construction itself.

The authors take a different route by linking synchronizing automata to primitive directed graphs (digraphs). A digraph is primitive if some power of its adjacency matrix is positive, i.e., for a sufficiently large integer k every vertex can reach every other vertex by a directed walk of length exactly k. The smallest such k is called the exponent of the digraph. Wielandt’s theorem gives a universal upper bound on the exponent of an n‑vertex primitive digraph: it is at most (n‑1)²+1, and this bound is known to be tight. The key observation of the paper is that the exponent of the transition digraph of a DFA provides a lower bound on the length of any reset word: if the digraph has exponent e, then any word that synchronizes the automaton must have length at least e‑1, because before reaching the exponent the digraph does not yet guarantee that all states can be merged.

Using this observation, the authors construct several infinite series of synchronizing automata whose underlying transition digraphs are primitive with exponents that are asymptotically quadratic in the number of states. The constructions are explicit and rely on a small alphabet (typically two or three letters). The main families are:

1. Cyclic‑plus‑shortcut series – The automaton has a cyclic permutation a on n states and a second letter b that maps a carefully chosen subset of states directly to a distinguished “sink” state. The transition digraph of the pair (a, b) is primitive with exponent (n‑1)²‑2, and the authors prove that the shortest reset word has length exactly (n‑1)²‑2.

2. Double‑cycle series – Two cyclic permutations a and c are interleaved, and a third letter b acts as a “bridge” that gradually collapses the state set. The exponent of the resulting digraph is n²‑3n+3, and a constructive argument shows that no shorter reset word exists.

3. De Bruijn‑type series – States are identified with binary strings of length k (so n=2ᵏ). The alphabet consists of a shift‑left operation and a bit‑flip operation that together generate a primitive digraph whose exponent is very close to (n‑1)². The reset length of the automaton is shown to be n²‑O(n), again matching the quadratic order.

For each family the paper supplies a rigorous proof that (a) the transition digraph is primitive, (b) the exponent attains the claimed value (often by adapting Wielandt’s extremal constructions), and (c) any reset word must be at least as long as the exponent minus one. Moreover, explicit reset words of that length are exhibited, establishing optimality within the family.

The authors compare their constructions with the classical Černý series. While the Černý automata achieve the exact bound (n‑1)², they use a very specific structure (a single cyclic permutation and a reset letter that maps one state to another). The new families demonstrate that the quadratic bound can be approached even when the alphabet is limited to two letters and the underlying digraph has a completely different topology. This widens the landscape of “slowly synchronizing” automata and suggests that the exponent of the transition digraph is a natural invariant governing synchronization speed.

In the discussion section the paper points out several open directions. First, the relationship between digraph exponent and reset length is proved only for the specific families; a general theorem characterising all synchronizing automata in terms of their digraph exponent remains elusive. Second, computing the exact exponent of a given primitive digraph is known to be computationally hard, and the authors propose investigating approximation algorithms that could give useful lower bounds on reset lengths. Third, extending the constructions to larger alphabets or to automata with additional structural constraints (e.g., aperiodic or monotonic automata) could lead to new extremal examples.

Overall, the paper makes a substantial contribution by bridging automata theory and algebraic graph theory. It provides concrete infinite families of synchronizing automata whose shortest reset words are asymptotically quadratic, thereby enriching the catalogue of known “hard” instances for the Černý conjecture. The use of primitive digraph exponents as a tool for analyzing synchronization speed opens a promising line of research that may eventually lead to a deeper understanding of the conjecture’s true bound.