有限词集Kleene星的Frobenius和因子普适性问题
The Frobenius and factor universality problems of the Kleene star of a finite set of words
摘要：
我们解决了公开的问题，关于Kleene星$L^*$的有限集$L$的话在一个字母表$Σ$。\emph{Frobenius monoid}问题是对于给定的有限词集$L$，语言$L^*$是否是上有限的问题。我们证明它是PSPACE完备的。我们还展示了一个无限系列的集$L$，使最长的话，而不是在$L^*$（当$L^*$是上限时）的长度是指数的最长的话，在$L$和次指数的总和中的字的长度在$L$。\emph{factor universality}问题是对于给定的有限词集$L$，是否$Σ$上的每个词都是来自$L^*$的某个词的因子（子串）的问题。我们证明了它也是PSPACE完备的。除此之外，我们展示了一个集合$L$的无限族，使得不作为$L^*$中任何单词的因子的最短单词的长度在$L$中的最长单词的长度中是指数的，并且在$L$中的单词的长度的总和中是次指数的。这基本上否定了长期存在的Restivo猜想（1981）及其弱变式。我们所有的解决方案都基于一个共享的结构，作为一个辅助的通用工具，我们引入了\emph{set rewriting systems}的概念。最后，我们补充的结果与上界。
英文摘要：
We solve open problems concerning the Kleene star $L^*$ of a finite set $L$ of words over an alphabet $Σ$. The \emph{Frobenius monoid} problem is the question for a given finite set of words $L$, whether the language $L^*$ is cofinite. We show that it is PSPACE-complete. We also exhibit an infinite family of sets $L$ such that the length of the longest words not in $L^*$ (when $L^*$ is cofinite) is exponential in the length of the longest words in $L$ and subexponential in the sum of the lengths of words in $L$. The \emph{factor universality} problem is the question for a given finite set of words $L$, whether every word over $Σ$ is a factor (substring) of some word from $L^*$. We show that it is also PSPACE-complete. Besides that, we exhibit an infinite family of sets $L$ such that the length of the shortest words not being a factor of any word in $L^*$ is exponential in the length of the longest words in $L$ and subexponential in the sum of the lengths of words in $L$. This essentially settles in the negative the longstanding Restivo's conjecture (1981) and its weak variations. All our solutions base on one shared construction, and as an auxiliary general tool, we introduce the concept of \emph{set rewriting systems}. Finally, we complement the results with upper bounds.
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