Linear Temporal Logic (LTL) interpreted on finite traces is a robust specification framework popular in formal verification. However, despite the high interest in the logic in recent years, the topic of their quantitative extensions is not yet fully explored. The main goal of this work is to study the effect of adding weak forms of percentage constraints (e.g. that most of the positions in the past satisfy a given condition, or that sigma is the most-frequent letter occurring in the past) to fragments of LTL. Such extensions could potentially be used for the verification of influence networks or statistical reasoning. Unfortunately, as we prove in the paper, it turns out that percentage extensions of even tiny fragments of LTL have undecidable satisfiability and model-checking problems. Our undecidability proofs not only sharpen most of the undecidability results on logics with arithmetics interpreted on words known from the literature, but also are fairly simple. We also show that the undecidability can be avoided by restricting the allowed usage of the negation, and briefly discuss how the undecidability results transfer to first-order logic on words.

中文翻译：

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“大多数”导致不可判定性：向 LTL 添加频率失败

在有限轨迹上解释的线性时间逻辑 (LTL) 是一种在形式验证中流行的强大规范框架。然而，尽管近年来对逻辑的兴趣很高，但它们的定量扩展的主题尚未得到充分探索。这项工作的主要目标是研究将弱形式的百分比约束（例如，过去的大多数位置满足给定条件，或者 sigma 是过去出现的最频繁的字母）添加到零担。这种扩展可能用于验证影响网络或统计推理。不幸的是，正如我们在论文中证明的那样，事实证明，即使是 LTL 的微小片段的百分比扩展也具有不可判定的可满足性和模型检查问题。我们的不可判定性证明不仅可以通过对文献中已知单词的算术解释来增强逻辑的大部分不可判定性结果，而且还相当简单。我们还表明可以通过限制否定的允许使用来避免不可判定性，并简要讨论不可判定性结果如何转移到单词的一阶逻辑。