# The Number of Languages with Maximum State Complexity

• Bjørn Kjos-Hanssen
• Lei Liu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)

## 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.

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