Certificates for short extending words in a finite automaton

Abstract: Let $\mathcal A$ be a complete deterministic finite automaton on a state set $Q$ of size $n$ with $k$ letters, and for a proper nonempty subset $S$ of $Q$ let $\mathrm{minext}(S)$ be the length of a shortest word $u$ with $|Su<sup>{-1}|&gt;|S|$, where $Su^{-1}={q: q\cdot u\in S}$. To each state $q$ attach the integer $$β<sup>{\ast}<em>q=\sum</em>{t=1}<sup>{n-1}k<sup>{\,n-1-t}(\mathrm{indeg}_t(q)-k<sup>{t})$$, where $\mathrm{indeg}<em>t(q)$ counts the pairs $(p,u)$ with $|u|=t$ and $p\cdot u=q$, and let $B(S)=\sum</em>{q\in S}β<sup>{\ast}_q$. On every synchronizing automaton, $B(S)\ge0$ implies $\mathrm{minext}(S)\le n-1$, so, as $B(Q)=0$, one of $S$ and $Q\setminus S$ extends within $n-1$; when $B(S)&gt;0$ no hypothesis is needed. Kari's Eulerian extension lemma is the case $β<sup>{\ast}=0$, and $β<sup>{\ast}$, like every member of the family $\sum_{t=1}<sup>{n-1}c_tσ_t$, $c_t&gt;0$, vanishes identically if and only if the automaton is Eulerian, where $$σ<em>t(S)=\sum</em>{q\in S}(\mathrm{indeg}<em>t(q)-k<sup>{t})$$. On strongly connected automata $σ_t(S)/k<sup>{t}$ has Cesàro limit $n\,e(S)/e(Q)-|S|$ for Friedman's weight $e$; that limit certifies singletons but no larger subset in general. The hypothesis $B(S)\ge0$ cannot be relaxed by one integer unit, nor can the constant $n-1$ be improved. A second-moment test on the sizes $|Su<sup>{-1}|$ certifies 60 to 95 percent of the subsets with $B(S)&lt;0$ at $n\le7$. Along non-Eulerian automata whose words of length $n-1$ merge a fraction of the state pairs bounded below, with $\max_q\mathrm{indeg}</em>{n-1}(q)=o(nk<sup>{n-1})$, it certifies all but a vanishing share of them. The functional $B$ certifies half of the subsets outside ${B=0}$. At each subset size coprime to $n$ ($n\ge4$) some synchronizing Eulerian binary automaton attains the constant $n-1$; whether only there is open. No reset bound follows: Černý's automata have subsets not extending within $n-1$.