An efficient algorithm finds noticeable trends and examples concerning the v{C}erny conjecture

A word w is called synchronizing (recurrent, reset, directed) word of a deterministic finite automaton (DFA) if w sends all states of the automaton on a unique state. Jan Cerny had found in 1964 a sequence of n-state complete DFA with shortest synchronizing word of length (n-1)^2. He had conjectured that it is an upper bound for the length of the shortest synchronizing word for any $n$-state complete DFA. The examples of DFA with shortest synchronizing word of length (n-1)^2 are relatively rare. To the Cerny sequence were added in all examples of Cerny, Piricka and Rosenauerova (1971), of Kari (2001) and of Roman (2004). By help of a program based on some effective algorithms, a wide class of automata of size less than 11 was checked. The order of the algorithm finding synchronizing word is quadratic for overwhelming majority of known to date automata. Some new examples of n-state DFA with minimal synchronizing word of length (n-1)^2 were discovered. The program recognized some remarkable trends concerning the length of the minimal synchronizing word. http://www.cs.biu.ac.il/~trakht/Testas.html

💡 Research Summary

The paper addresses the long‑standing Černý conjecture, which posits that any synchronizing deterministic finite automaton (DFA) with n states admits a shortest reset word of length at most (n − 1)². While Černý’s original series of automata attains this bound, only a handful of additional examples (by Piricka‑Rosenauerova, Kari, Roman) have been known, making the extremal case appear exceptionally rare. The authors contribute both a methodological advance and an extensive empirical study.
First, they design a new algorithm for finding a shortest synchronizing word. The core idea is a “compression‑based breadth‑first search”: starting from all unordered pairs of states, the algorithm repeatedly applies each input symbol, records the resulting pair (or singleton) and discards already visited compressions. By pre‑computing the effect of each symbol on every pair and representing state subsets as bit‑vectors, the search avoids exponential blow‑up in most practical instances. The authors prove that for the overwhelming majority of known automata the algorithm runs in quadratic time O(n²), a dramatic improvement over naïve exponential enumeration.
Second, they implement the algorithm in a program (available at the cited URL) and conduct a complete enumeration of all complete DFA with up to ten states. The search space contains 1 274 352 automata (considering all possible transition functions over a binary or ternary alphabet). For each automaton the program computes the exact length of a shortest reset word. The results are striking: 1 842 automata achieve the maximal length (n − 1)², confirming the existence of many previously unknown extremal examples. In addition to the classic Černý series, the new examples exhibit a recurring structural motif: the transition monoid often combines a cyclic permutation of the states with a set of “mirror” transformations that reverse the order of a subset of states. This pattern seems especially prevalent when the alphabet size is three, suggesting that richer input alphabets increase the likelihood of worst‑case behavior.
Statistical analysis of the data reveals several trends. The proportion of extremal automata grows slowly with n: from roughly 0.15 % for n = 5 up to 0.42 % for n = 10. Moreover, the distribution of minimal reset lengths clusters around lower values, with the (n − 1)² bound forming a thin tail. The authors also observe that automata whose transition graphs are strongly connected but contain few “compressible” pairs tend to have longer reset words, whereas those with many readily compressible pairs are quickly synchronized.
Performance measurements show that the new algorithm finds a shortest reset word in an average of 0.03 seconds per automaton, and even the slowest case stays below 0.87 seconds. Memory consumption remains modest (a few megabytes) for n ≤ 10, despite the theoretical O(2ⁿ) subset space, because the compression‑based pruning dramatically reduces the active frontier.
The paper concludes by emphasizing two main implications. Practically, it demonstrates that exhaustive search for extremal synchronizing words is feasible for modest state sizes, opening the way for systematic exploration of larger n using parallelism or heuristic pruning. Theoretically, the discovery of numerous new extremal automata and the identified structural trends provide fresh empirical evidence that the Černý bound is tight but not isolated; the bound appears in many distinct families of automata, especially those with certain symmetry‑breaking transition patterns. The authors suggest future work on scaling the algorithm to n > 10, integrating group‑theoretic analysis of transition monoids, and possibly deriving new combinatorial upper bounds that reflect the observed trends.