The number of languages with maximum state complexity

Bjørn Kjos-Hanssen, Lei Liu

Champarnaud and Pin (1989) found that the minimal deterministic automaton of a language $L\subset\Sigma^n$, where $\Sigma=\{0,1\}$, has at most \[ \sum_{i=0}^n \min(2^i, 2^{2^{n-i}}-1) \] states, and for each $n$ there exists $L$ attaining this bound. C\^{a}mpeanu and Ho (2004) have shown more generally that the tight upper bound for $\Sigma$ of cardinality $k$ and for complete automata is \[ \frac{k^r-1}{k-1} + \sum_{j=0}^{n-r}(2^{k^j}-1) + 1 \] where $r=\min\{m:k^m\ge 2^{k^{n-m}}-1\}$. (In these results, requiring totality of the transition function adds 1 to the state count.) C\^{a}mpeanu and Ho's result can be viewed as concerning functions $f:[k]^n\to [2]$ where $[k]=\{0,\dots,k-1\}$ is a set of cardinality $k$. We generalize their result to arbitrary function $f:[k]^n\to [c]$ where $c$ is a positive integer. Let $O_i$ be the number of functions from $[b^{i}]$ to $[c^{b^{n-i}}]$ that are onto $[c^{b^{n-i}}-1]$. C\^ampeanu and Ho stated that it is very difficult to determine the number of maximum-complexity languages. Here we show that it is equal to $O_i$, for the least $i$ such that $O_i>0$. For monotone languages a tightness result seems harder to obtain. However, we show that the following upper bound is attained for all $n\le 10$. \[ \sum_{i=0}^n \min(2^i, M(n-i)-1), \] where $M(k)$ is the $k$th Dedekind number.

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