Zeno, Hercules, and the Hydra

Abstract

A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. 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 a syntactially defined safety fragment, called safety MTL, was proved decidable several years ago. Satisfiability for safety MTL is also known to be equivalent to a fair termination problem for a class of channel machines with insertion errors. However, hitherto its precise computational complexity has remained elusive, with only 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 Ackermann-complete, i.e., among the easiest non-primitive recursive problems. This is surprising since decidability was originally established using Higman's Lemma, suggesting a much higher non-multiply recursive complexity.

The Complexity of Coverability in ν-Petri Nets

“…The basic building blocks of this development are Ackermannian counters reminiscent of the construction of Schnoebelen (2010) for reset Petri nets. The catch is that we need to be able to mimick this construction for non-fixed dimensions and to combine it with an iteration operator-of the kind employed recently by Lazić et al (2016) in the context of channel systems with insertion errors to show Ackermann-hardness-, which led us to develop delicate indexing mechanisms by data values; see Section 5.…”

What Makes Petri Nets Harder to Verify: Stack or Data?

“…The difficulty is that the bootstrapping implementation in UDPNs of the hierarchy of counters requires an iteration operator on 'counter libraries'-of the kind employed recently in the context of channel systems with insertion errors to obtain F ω -hardness [16] and in the context of ν-Petri nets to obtain F ω·2 -hardness [17]-but UDPNs have fundamentally unordered configurations as well as no basic mechanism for creating fresh data values. To overcome that obstacle-and this is the key technical idea in the paper-we enrich the interfaces of the counter implementations by a local freshness test: verifying that a given data value is distinct from all data values that the implementation (and, recursively, the implementations of all lesser counters in the hierarchy) currently employs internally; see Sections 5 and 6.…”