若干涉及洗牌问题的复杂性与可判定性
On the Complexity and Decidability of Some Problems Involving Shuffle
摘要：
研究了包含混洗操作的各种决策问题的复杂性和可判定性。下面三个问题都是$NP$ -完全的：给定一个非确定有限自动机（NFA）$M$，以及两个词$u$和$v$，$L(M)$不是$u$与$v$混洗的子集，$u$与$v$混洗的子集不是$L(M)$的子集，以及$L(M)$不等于$u$与$v$混洗的子集？它还表明，有一个多项式时间的算法来确定，$NFA$ s $M_1, M_2$和一个确定性下推自动机$M_3$，是否$L(M_1)$洗牌与$L(M_2)$是一个子集的$L(M_3)$。当$M_1, M_2,M_3$是单向非确定性$l$ -反向有界$k$ -计数器机器，而$M_3$是确定性的时，情况也是如此。其他的可判定性和复杂性的结果，测试是否给定的语言$L_1, L_2$和$R$从各种语言家族满足$L_1$洗牌与$L_2$是一个子集的$R$，和$R$是一个子集的$L_1$洗牌与$L_2$。还示出了关于shuffle的几个闭合结果。
英文摘要：
The complexity and decidability of various decision problems involving the shuffle operation are studied. The following three problems are all shown to be $NP$-complete: given a nondeterministic finite automaton (NFA) $M$, and two words $u$ and $v$, is $L(M)$ not a subset of $u$ shuffled with $v$, is $u$ shuffled with $v$ not a subset of $L(M)$, and is $L(M)$ not equal to $u$ shuffled with $v$? It is also shown that there is a polynomial-time algorithm to determine, for $NFA$s $M_1, M_2$ and a deterministic pushdown automaton $M_3$, whether $L(M_1)$ shuffled with $L(M_2)$ is a subset of $L(M_3)$. The same is true when $M_1, M_2,M_3$ are one-way nondeterministic $l$-reversal-bounded $k$-counter machines, with $M_3$ being deterministic. Other decidability and complexity results are presented for testing whether given languages $L_1, L_2$ and $R$ from various languages families satisfy $L_1$ shuffled with $L_2$ is a subset of $R$, and $R$ is a subset of $L_1$ shuffled with $L_2$. Several closure results on shuffle are also shown.
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