Synchronizing automata with random inputs

We study the problem of synchronization of automata with random inputs. We present a series of automata such that the expected number of steps until synchronization is exponential in the number of states. At the same time, we show that the expected number of letters to synchronize any pair of the famous Cerny automata is at most cubic in the number of states.

💡 Research Summary

The paper investigates synchronization of deterministic finite automata (DFAs) when the input stream is generated randomly. The authors focus on a binary alphabet Σ = {a, b} and assume a Bernoulli source B(p, q) where each letter is chosen independently with probability p for a and q = 1 − p for b. For a given synchronizing automaton A = (Q, Σ, δ), they consider a process that starts with the full set of states S ← Q and repeatedly draws a random letter x from B(p, q), updating S to δ(S, x). The process stops when |S| = 1; the expected number of steps E is the average absorption time of a Markov chain whose states are subsets of Q.

Two families of automata are examined. The first family, denoted Uₙ, is the minimal DFA recognizing the language Lₙ = Σ* a^{⌈(n+1)/2⌉} b^{⌊(n‑1)/2⌋} Σ* (for odd n) or Σ* a^{n/2} b^{n/2} Σ* (for even n). The synchronizing words of Uₙ are exactly those that map state 1 to the sink state n + 1. Consequently, the random process reduces to a one‑dimensional random walk on the line {1,…,n + 1} with forward probability p (letter a) and backward probability q (letter b). Solving the standard linear equations for expected hitting times yields:

• For odd n: E = 1 / (p^{(n+1)/2} · q^{(n‑1)/2}),
• For even n: E = 1 / (p^{n/2} · q^{n/2}).

Thus the expected synchronization time grows exponentially with n unless p and q are both bounded away from 0. This demonstrates that a random input can be extremely unfavorable for certain automata.

The second family consists of the classic Černý automata Cₙ, which have n states, a reset threshold of (n − 1)², and are known to be the hardest examples for deterministic synchronization. To analyze random synchronization, the authors introduce the pair automaton P(Cₙ), whose states are unordered pairs {s, t} (s ≠ t) together with a sink z. They then construct an isomorphic automaton Pₙ whose states are ordered pairs (i, ℓ) where i indicates a position and ℓ (1 ≤ ℓ ≤ ⌊(n‑1)/2⌋) denotes the cyclic distance between the two original states. In Pₙ, the letter b increments i modulo n (a cyclic shift), while a either reduces ℓ, leaves it unchanged, or sends the pair to the sink depending on the current configuration.

The expected time μ_{i,ℓ} to reach the sink from (i, ℓ) again satisfies a system of linear equations derived from the transition probabilities p and q. The authors split the system into three blocks according to the value of ℓ (ℓ = 1, 2 ≤ ℓ ≤ ⌊(n‑3)/2⌋, and ℓ = ⌊(n‑1)/2⌋ for odd n, or ℓ = n/2 for even n) and solve them recursively. After substantial algebraic manipulation they obtain closed‑form expressions for the worst‑case pair, namely {1, ⌈n/2⌉} for odd n and {1, n/2 + 1} for even n:

• For odd n: E = ((n − 1)·