On the State Complexity of the Shuffle of Regular Languages
On the State Complexity of the Shuffle of Regular Languages
复制标题
论正则语言洗牌的状态复杂性
DOI：
10.1007/978-3-319-41114-9_6
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Marek Szykuła
中科院分区：
文献类型：
--
作者：
J. Brzozowski;Galina Jirásková;B. Liu;A. Rajasekaran;Marek Szykuła
我们研究了由完全确定有限自动机表示的正则语言上的混洗操作。本文证明了f（m，n）=2^{mn-1} + 2^{（m-1）（n-1）}（2^{m-1}-1）（2^{n-1}-1）$$是两个分别具有状态复形mdn的正则语言的shuffle的状态复杂度的上界.我们还陈述了关于这个界的紧性的部分结果。证明了存在满足有界ifand的证明语言，也存在满足有界ifand的证明语言。此外，我们证明，在接受洗牌的NFA的子集自动机，allstates可以区分，和一个字母表的大小为3就足够了。因此，如果所有f（m，n）个状态都是可达的，则可以满足该界。我们知道至少需要一个字母表的大小。可达性的问题，因此也是一般的boundf（m，n）的紧性的问题，仍然是开放的。
We investigate the shuffle operation on regular languages represented by complete deterministic finite automata. We prove that $$f(m,n)=2^{mn-1} + 2^{(m-1)(n-1)}(2^{m-1}-1)(2^{n-1}-1)$$ is an upper bound on the state complexity of the shuffle of two regular languages having state complexitiesmandn, respectively. We also state partial results about the tightness of this bound. We show that there exist witness languages meeting the bound ifand, and also if. Moreover, we prove that in the subset automaton of the NFA accepting the shuffle, allstates can be distinguishable, and an alphabet of size three suffices for that. It follows that the bound can be met if allf(m,n) states are reachable. We know that an alphabet of size at leastmnis required provided that. The question of reachability, and hence also of the tightness of the boundf(m,n) in general, remains open.