We study FO+, a fragment of first-order logic on finite words, where monadic predicates can only appear positively. We show that there is a FO-definable language that is monotone in monadic predicates but not definable in FO+. This provides a simple proof that Lyndon's preservation theorem fails on finite structures. We additionally show that given a regular language, it is undecidable whether it is definable in FO+.
翻译:我们研究FO+,这是关于限值单词的第一阶逻辑的碎片, 其中monadic 的前提只能呈阳性。 我们显示有一种可定义的FO语言, 它在monadic 的前提中是单调的,但在FO+ 中是无法定义的。 这简单证明Lyndon的保存定理在有限结构上失败了。 我们还表明,根据一种常规语言,它能否在FO+ 中被定义是不可量化的。