Zeno, Hercules and the Hydra: Downward rational termination is Ackermannian

Ranko Lazic, Joël Ouaknine, and James Worrell

Metric temporal logic (MTL) is one of the most prominent specification formalisms for real-time systems. Over infinite timed words, full MTL is undecidable, but satisfiability for its safety fragment was proved decidable several years ago [18]. The problem is also known to be equivalent to a fair termination problem for a class of channel machines with insertion errors. However, the complexity has remained elusive, except for a non-elementary lower bound. Via another equivalent problem, namely termination for a class of rational relations, we show that satisfiability for safety MTL is not primitive recursive, yet is Ackermannian, i.e., among the simplest non-primitive recursive problems. This is surprising since decidability was originally established using Higman's Lemma, suggesting a much higher non-multiply recursive complexity.

Proceedings of MFCS 13, LNCS 8087, 2013. 12 pages.

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