"Most of" leads to undecidability: Failure of adding frequencies to LTL

TL;DR

This paper demonstrates that adding percentage-based frequency constraints to fragments of Linear Temporal Logic leads to undecidability in satisfiability and model-checking, highlighting fundamental limits in quantitative extensions of LTL.

Contribution

It proves undecidability results for frequency-extended LTL fragments and shows how restrictions on negation can avoid these issues.

Findings

Frequency constraints cause undecidability in LTL fragments.

Undecidability proofs are simple and sharpen existing results.

Restrictions on negation can prevent undecidability.

Abstract

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.