We solve open problems concerning the Kleene star of a finite set of words over an alphabet . The Frobenius monoid problem is the question for a given finite set of words , whether the language is cofinite. We show that it is PSPACE-complete. We also exhibit an infinite family of sets such that the length of the longest words not in (when is cofinite) is exponential in the length of the longest words in and subexponential in the sum of the lengths of words in . The factor universality problem is the question for a given finite set of words , whether every word over is a factor (substring) of some word from . We show that it is also PSPACE-complete. Besides that, we exhibit an infinite family of sets such that the length of the shortest words not being a factor of any word in is exponential in the length of the longest words in and subexponential in the sum of the lengths of words in . This essentially settles in the negative the longstanding Restivo’s conjecture (1981) and its weak variations. All our solutions are based on one shared construction, and as an auxiliary general tool, we introduce the concept of set rewriting systems . Finally, we complement the results with upper bounds.

中文翻译：

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有限词集 Kleene 星的 Frobenius 和因子普遍性问题

我们解决了有关 Kleene 星的未解决问题 有限集的 字母表上的单词 . 这Frobenius 单体问题是给定有限词集的问题 , 是否语言 是有限的。我们证明它是 PSPACE 完备的。我们还展示了一个无限的集合家族 这样最长单词的长度不在 （什么时候 是余有限的）在最长词的长度上是指数的 和中单词长度总和的次指数 . 这因素普遍性问题是给定有限词集的问题 , 是否每一个字都结束 是某个单词的一个因子（子字符串） . 我们证明它也是 PSPACE 完备的。除此之外，我们展示了一个无限的集合族 使得最短单词的长度不是任何单词的因素 是最长单词长度的指数 和中单词长度总和的次指数 . 这基本上否定了长期存在的 Restivo 猜想（1981 年）及其微弱的变体。我们所有的解决方案都基于一个共享结构，并且作为辅助通用工具，我们引入了设置重写系统. 最后，我们用上限补充结果。