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Zeno, Hercules, and the Hydra: Safety Metric Temporal Logic is A ckermann -Complete

Zeno, Hercules, and the Hydra: Safety Metric Temporal Logic is A ckermann -Complete Zeno, Hercules, and the Hydra: Safety Metric Temporal Logic is ACKERMANN-Complete ´ RANKO LAZIC, University of Warwick ¨ JOEL OUAKNINE and JAMES WORRELL, University of Oxford 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 nonelementary 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 nonprimitive recursive problems). This is surprising since decidability was originally established using Higman's Lemma, suggesting a much higher nonmultiply recursive complexity. Categories and Subject Descriptors: D.2.4 [Software Engineering]: Software/Program Verification; F.1.1 [Computation by Abstract Devices]: Models of Computation; F.1.3 [Computation by Abstract Devices]: Complexity Measures and Classes General Terms: Algorithms, Theory, Verification Additional Key Words and Phrases: Metric temporal logic, channel machines, complexity hierarchies ACM Reference Format: Ranko http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computational Logic (TOCL) Association for Computing Machinery

Zeno, Hercules, and the Hydra: Safety Metric Temporal Logic is A ckermann -Complete

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References (34)

Publisher
Association for Computing Machinery
Copyright
Copyright © 2016 by ACM Inc.
ISSN
1529-3785
DOI
10.1145/2874774
Publisher site
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Abstract

Zeno, Hercules, and the Hydra: Safety Metric Temporal Logic is ACKERMANN-Complete ´ RANKO LAZIC, University of Warwick ¨ JOEL OUAKNINE and JAMES WORRELL, University of Oxford 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 nonelementary 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 nonprimitive recursive problems). This is surprising since decidability was originally established using Higman's Lemma, suggesting a much higher nonmultiply recursive complexity. Categories and Subject Descriptors: D.2.4 [Software Engineering]: Software/Program Verification; F.1.1 [Computation by Abstract Devices]: Models of Computation; F.1.3 [Computation by Abstract Devices]: Complexity Measures and Classes General Terms: Algorithms, Theory, Verification Additional Key Words and Phrases: Metric temporal logic, channel machines, complexity hierarchies ACM Reference Format: Ranko

Journal

ACM Transactions on Computational Logic (TOCL) – Association for Computing Machinery

Published: Feb 17, 2016

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