The Number of Languages with Maximum State Complexity

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


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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Authors and Affiliations

  1. 1.University of Hawaii at ManoaHonoluluUSA

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