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What is the minimum length of a string that is accepted by a DFA, shows that the language accepted by that DFA is infinite?

I checked this post How to determine if an automata (DFA) accepts an infinite or finite language? and in the answer provided by Rick Decker, he states, "The language accepted by a DFA M with n states is infinite if and only if M accepts a string of length k, where n≤k<2n." But, if I have a string of minimum length 'n-1' that gets accepted by a DFA having 'n' states, it proves that the language accepted by the DFA is infinite. So, is the minimum lower bound 'n' or 'n-1'?

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Possibly the off-by-one is due to the fact that sometimes we (slightly) abuse the definition of DFAs by using a transition function that is not total. Of course, such a DFA is equivalent to a DFA with a total transition function and one more state, so the distinction usually does not matter. However, since you are after an exact bound on the length of a string as a function of the numbers states, we have to be careful.

To avoid confusion let's consider total transition functions only.

Claim: Let $D$ be a DFA with $n$ states. $L(D)$ is infinite if and only if $D$ accepts some string $x$ with $|x| \ge n-1$.

Proof: Let $\Sigma$ and $\delta$ be the alphabet and the transition function of $D$, respectively.

One direction is trivial: if $L(D)$ is infinite, then $L' = L(D) \setminus \left( \cup_{i=0}^{n-2} \Sigma^i \right)$ is also infinite (since $\cup_{i=0}^{n-2} \Sigma^i $ is finite) and we can pick any $x \in L'$.

Suppose then that $D$ accepts some $x$ with $|x| \ge n-1$. There must be some (not necessarily simple) path $\pi_x$ from the initial state $q_0$ of to an accepting state $q_A$ of $D$. Our goal is to show that there are infinitely many such paths. In particular, it suffices to show that there is a path $\pi^*$ from $q_0$ to $q_A$ that is not simple (i.e., it has some repeated vertex), since then $\pi^*$ must contain a cycle.

If there is some $s \in \Sigma$ such that $\delta(q_A,s)$ can reach $q_A$ (possibly $\delta(q_A,s)=q_A)$ via some path $\pi'$, we can choose $\pi^* = \pi_x \circ \pi'$, where $\circ$ denotes concatenation. Otherwise, there is at least one state $q$ that cannot reach $q_A$, which means that $\pi_x$ does not traverse $q$. Since the length of $\pi_x$ is at least $n-1$, and at most $n-1$ vertices can be traversed, the pigeonhole principle ensures that $\pi_x$ cannot be simple (i.e., we can choose $\pi^* = \pi_x$). $\square$

The above bound is tight in the sense that, for every $n \ge 1$, you can build a DFA $D$ with $n$ states such that all strings of length at most $n-2$ are accepted but $L(D)$ is finite (the graph of the DFA is a directed path with vertices $q_0, \dots, q_{n-1}$ where $q_0, \dots, q_{n-2}$ are accepting, and $q_{n-1}$ is a catch-all rejecting trap).

Moreover, the following holds:

Claim: if $L(D)$ is infinite then there must be some string $x$ with $n-1 \le |x| \le 2n-2$ that is accepted by $D$.

Proof: Let $\pi$ be the shortest accepting (non-simple) path of length at least $2n-1$. Since $\pi$ traverses $2n > n$ states (counting repetitions), $\pi$ must include a cycle. Deleting the cycle from $C$ yields an accepting path of length at most $2n-2$ and at least $2n-1-n=n-1$. $\square$

This is tight in the sense that there are DFAs $D$ such that $L(D)$ is infinite but all strings $x$ with $n-1 \le |x| \le 2n-3$ are rejected (the graph of the DFA is a directed cycle with states $q_0, \dots, q_{n-1}$, only state $q_{n-2}$ is accepting).


To summarize

If $D$ be a DFA with $n$ states then $L(D)$ is infinite if and only if $D$ accepts some string $x$ with $n-1 \le |x| \le 2n-2$.

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    $\begingroup$ hey, thanks Steven for clarifying, and explaining it rigorously! $\endgroup$ Oct 23, 2023 at 14:49

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