The Number of Languages with Maximum State Complexity

Abstract

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. Câmpeanu 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.) Câmpeanu and Ho’s result can be viewed as concerning functions $f:[k]^n\rightarrow [2]$ where $[k]=\{0,\dots ,k-1\}$ is a set of cardinality k. We generalize their result to arbitrary function $f:[k]^n\rightarrow [c]$ 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 $[c^{b^{n-i}}-1]$. Câmpeanu 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 kth Dedekind number.