MTL and TPTL for One-Counter Machines: Expressiveness, Model Checking, and Satisfiability
Article No.: 12, Pages 1 - 34
Abstract
Metric Temporal Logic (MTL) and Timed Propositional Temporal Logic (TPTL) are quantitative extensions of Linear Temporal Logic (LTL) that are prominent and widely used in the verification of real-timed systems. We study MTL and TPTL as specification languages for one-counter machines. It is known that model checking one-counter machines against formulas of Freeze LTL (FLTL), a strict fragment of TPTL, is undecidable. We prove that in our setting, MTL is strictly less expressive than TPTL, and incomparable in expressiveness to FLTL, so undecidability for MTL is not implied by the result for FLTL. We show, however, that the model-checking problem for MTL is undecidable. We further prove that the satisfiability problem for the unary fragments of TPTL and MTL are undecidable; for TPTL, this even holds for the fragment in which only one register and the finally modality is used. This is opposed to a known decidability result for the satisfiability problem for the same fragment of FLTL.
References
[1]
Luca Aceto and Anna Ingólfsdóttir (Eds.). 2006. Proceedings of Foundations of Software Science and Computation Structures, 9th International Conference, FOSSACS 2006, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2006, Vienna, Austria, March 25-31, 2006. Lecture Notes in Computer Science, Vol. 3921. Springer.
[2]
Rajeev Alur and David L. Dill. 1994. A theory of timed automata. Theoretical Computer Science 126, 2 (1994), 183--235.
[3]
Rajeev Alur, Tomás Feder, and Thomas A. Henzinger. 1996. The benefits of relaxing punctuality. Journal of the ACM 43, 1 (1996), 116--146.
[4]
Rajeev Alur and Thomas A. Henzinger. 1993. Real-time logics: Complexity and expressiveness. Information and Computation 104, 1 (1993), 35--77.
[5]
Rajeev Alur and Thomas A. Henzinger. 1994. A really temporal logic. Journal of the ACM 41, 1 (1994), 181--204.
[6]
Mikołaj Bojańczyk, Anca Muscholl, Thomas Schwentick, Luc Segoufin, and Claire David. 2006. Two-variable logic on words with data. In Proceedings of the 21st IEEE Symposium on Logic in Computer Science (LICS’06). Los Alamitos, CA, 7--16.
[7]
Benedikt Bollig, Karin Quaas, and Arnaud Sangnier. 2017. The complexity of flat Freeze LTL. In Proceedings of the 28th International Conference on Concurrency Theory (CONCUR’17). Article 33, 16 pages.
[8]
Ahmed Bouajjani, Javier Esparza, and Oded Maler. 1997. Reachability analysis of pushdown automata: Application to model-checking. In Proceedings of the Concurrency Theory (CONCUR’97). Lecture Notes in Computer Science, Vol. 1243. Springer, 135--150.
[9]
Ahmed Bouajjani, Peter Habermehl, Yan Jurski, and Mihaela Sighireanu. 2007. Rewriting systems with data. In Proceedings of the Fundamentals of Computation Theory (FCT'07). Lecture Notes in Computer Science, Vol. 4639. Springer, 1--22.
[10]
Patricia Bouyer, Fabrice Chevalier, and Nicolas Markey. 2010. On the expressiveness of TPTL and MTL. Information and Computation 208, 2 (2010), 97--116.
[11]
Patricia Bouyer, Kim Guldstrand Larsen, and Nicolas Markey. 2008. Model checking one-clock priced timed automata. Logical Methods in Computer Science 4, 2 (2008), 108--122.
[12]
Patricia Bouyer, Nicolas Markey, Joël Ouaknine, and James Worrell. 2007. The cost of punctuality. In Proceedings of the 22nd IEEE Symposium on Logic in Computer Science (LICS’07). Los Alamitos, CA, 109--120.
[13]
Patricia Bouyer, Antoine Petit, and Denis Thérien. 2003. An algebraic approach to data languages and timed languages. Information and Computation 182, 2 (2003), 137--162.
[14]
Cristiana Chitic and Daniela Rosu. 2004. On validation of XML streams using finite state machines. In Proceedings of the 7thInternational Workshop on the Web and Databases (WebDB’04), Colocated with ACM SIGMOD/PODS 2004. 85--90. http://webdb2004.cs.columbia.edu/papers/6-2.pdf.
[15]
Stéphane Demri and Ranko Lazic. 2009. LTL with the freeze quantifier and register automata. ACM Transactions on Computational Logic 10, 3 (2009), Article 16.
[16]
Stéphane Demri, Ranko Lazic, and David Nowak. 2007. On the freeze quantifier in Constraint LTL: Decidability and complexity. Information and Computation 205, 1 (2007), 2--24.
[17]
Stéphane Demri, Ranko Lazic, and Arnaud Sangnier. 2008. Model checking Freeze LTL over one-counter automata. In Proceedings of the Foundations on Software Science and Computational Structures (FoSSACS'08). Lecture Notes in Computer Science, Vol. 4962. Springer, 490--504.
[18]
Stéphane Demri, Ranko Lazic, and Arnaud Sangnier. 2010. Model checking memoryful linear-time logics over one-counter automata. Theoretical Computer Science 411, 22--24 (2010), 2298--2316.
[19]
Stéphane Demri and Arnaud Sangnier. 2010. When model-checking Freeze LTL over counter machines becomes decidable. In Proceedings of the Foundations of Software Science and Computational Structures (FoSSaCS'10). Lecture Notes in Computer Science, Vol. 6014. Springer, 176--190.
[20]
Javier Esparza. 1996. Decidability and complexity of Petri net problems—An introduction. In Proceedings of the Lectures on Petri Nets I: Basic Models (ACPN'96). Lecture Notes in Computer Science, Vol. 1491. Springer, 374--428.
[21]
Kousha Etessami and Thomas Wilke. 2000. An until hierarchy and other applications of an Ehrenfeucht-Fraïssé game for temporal logic. Information and Computation 160, 1--2 (2000), 88--108.
[22]
Shiguang Feng, Markus Lohrey, and Karin Quaas. 2017. Path checking for MTL and TPTL over data words. Logical Methods in Computer Science 13, 3 (2017), 1--34.
[23]
Diego Figueira, Santiago Figueira, Sylvain Schmitz, and Philippe Schnoebelen. 2011. Ackermannian and primitive-recursive bounds with Dickson’s lemma. In Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS’11). IEEE, Los Alamitos, CA, 269--278.
[24]
Diego Figueira and Luc Segoufin. 2009. Future-looking logics on data words and trees. In Proceedings of the Mathematical Foundations of Computer Science (MFCS'09). Lecture Notes in Computer Science, Vol. 5734. Springer, 331--343.
[25]
Stefan Göller, Christoph Haase, Joël Ouaknine, and James Worrell. 2010. Model checking succinct and parametric one-counter automata. In Proceedings of the Automata, Languages and Programming (ICALP'10). Lectures Notes in Computer Science, Vol. 6199. Springer, 575--586.
[26]
Stefan Göller, Christoph Haase, Joël Ouaknine, and James Worrell. 2012. Branching-time model checking of parametric one-counter automata. In Proceedings of the Foundations of Software Science and Computational Structures (FoSSaCS'12). Lecture Notes in Computer Science, Vol. 7213. Springer, 406--420.
[27]
Stefan Göller and Markus Lohrey. 2013. Branching-time model checking of one-counter processes and timed automata. SIAM Journal on Computing 42, 3 (2013), 884--923.
[28]
Christoph Haase, Joël Ouaknine, and James Worrell. 2016. Relating reachability problems in timed and counter automata. Fundamenta Informaticae 143, 3--4 (2016), 317--338.
[29]
Piotr Hofman, Slawomir Lasota, Richard Mayr, and Patrick Totzke. 2016. Simulation problems over one-counter nets. Logical Methods in Computer Science 12, 1 (2016), 1--46.
[30]
Michael Kaminski and Nissim Francez. 1994. Finite-memory automata. Theoretical Computer Science 134, 2 (1994), 329--363.
[31]
Ron Koymans. 1990. Specifying real-time properties with metric temporal logic. Real-Time Systems 2, 4 (1990), 255--299.
[32]
Pascal Lafourcade, Denis Lugiez, and Ralf Treinen. 2005. Intruder deduction for AC-like equational theories with homomorphisms. In Proceedings of the Term Rewriting Techniques and Applications (RTA'05). Lecture Notes in Computer Science, Vol. 3467. Springer, 308--322.
[33]
François Laroussinie, Nicolas Markey, and Ph. Schnoebelen. 2002. On model checking durational Kripke structures. In Proceedings of the Foundations of Software Science and Computation Structures (FoSSaCS'02). Lecture Notes in Computer Science, Vol. 2303. Springer, 264--279.
[34]
Ranko Lazic, Joël Ouaknine, and James Worrell. 2016. Zeno, Hercules, and the Hydra: Safety metric temporal logic is Ackermann-complete. ACM Transactions on Computational Logic 17, 3 (2016), Article 16, 27 pages.
[35]
Antonia Lechner, Richard Mayr, Joël Ouaknine, Amaury Pouly, and James Worrell. 2016. Model checking flat Freeze LTL on one-counter automata. In Proceedings of the 27th International Conference on Concurrency Theory (CONCUR’16). Article 29, 14 pages.
[36]
Alexei Lisitsa and Igor Potapov. 2005. Temporal logic with predicate lambda-abstraction. In Proceedings of the 12th International Symposium on Temporal Representation and Reasoning (TIME’05). IEEE, Los Alamitos, CA, 147--155.
[37]
Marvin L. Minsky. 1961. Recursive unsolvability of Post’s problem of “Tag” and other topics in theory of Turing machines. Annals of Mathematics 74, 3 (Nov. 1961), 437--455.
[38]
Frank Neven, Thomas Schwentick, and Victor Vianu. 2004. Finite state machines for strings over infinite alphabets. ACM Transactions on Computational Logic 5, 3 (2004), 403--435.
[39]
Joël Ouaknine and James Worrell. 2006a. On metric temporal logic and faulty Turing machines. In Proceedings of the 9th European Joint Conference on Foundations of Software Science and Computation Structures (FOSSACS’06). 217--230.
[40]
Joël Ouaknine and James Worrell. 2006b. Safety metric temporal logic is fully decidable. In Proceedings of the Tools and Algorithms for the Construction and Analysis of Systems (TACAS'06). Lecture Notes in Computer Science, Vol. 3920. Springer, 411--425.
[41]
Joël Ouaknine and James Worrell. 2007. On the decidability and complexity of Metric Temporal Logic over finite words. Logical Methods in Computer Science 3, 1 (2007), 1--27.
[42]
Paritosh K. Pandya and Simoni S. Shah. 2011. On expressive powers of timed logics: Comparing boundedness, non-punctuality, and deterministic freezing. In Proceedings of the Concurrency Theory (CONCUR'11). Lecture Notes in Computer Science, Vol. 6901. Springer, 60--75.
[43]
Hiroshi Sakamoto and Daisuke Ikeda. 2000. Intractability of decision problems for finite-memory automata. Theoretical Computer Science 231, 2 (2000), 297--308.
[44]
Sylvain Schmitz and Philippe Schnoebelen. 2011. Multiply-recursive upper bounds with Higman’s lemma. In Proceedings of the Automata, Languages and Programming (ICALP'11). Lecture Notes in Computer Science, Vol. 6756. Springer, 441--452.
[45]
Olivier Serre. 2006. Parity games played on transition graphs of one-counter processes. In Proceedings of the Foundations of Software Science and Computation (FoSSaCS'06). Lecture Notes in Computer Science, Vol. 3921. Springer, 337--351.
Index Terms
- MTL and TPTL for One-Counter Machines: Expressiveness, Model Checking, and Satisfiability
Recommendations
From Non-punctuality to Non-adjacency: A Quest for Decidability of Timed Temporal Logics with Quantifiers
Metric Temporal Logic (MTL) and Timed Propositional Temporal Logic (TPTL) are prominent real-time extensions of Linear Temporal Logic (LTL). In general, the satisfiability checking problem for these extensions is undecidable when both the future (Until, U)...
On parametric timed automata and one-counter machines
Two decades ago, Alur, Henzinger, and Vardi introduced the reachability problem for parametric timed automata. Their main results are that reachability is decidable for timed automata with a single parametric clock, and undecidable for timed automata ...
DP Lower bounds for equivalence-checking and model-checking of one-counter automata
We present a general method for proving DP-hardness of problems related to formal verification of one-counter automata. For this we show a reduction of the SAT-UNSAT problem to the truth problem for a fragment of (Presburger) arithmetic. The fragment ...
Comments
Information & Contributors
Information
Published In
April 2020
316 pages
Copyright © 2019 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 20 December 2019
Accepted: 01 September 2019
Revised: 01 March 2018
Received: 01 July 2016
Published in TOCL Volume 21, Issue 2
Check for updates
Author Tags
Qualifiers
- Research-article
- Research
- Refereed
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 156Total Downloads
- Downloads (Last 12 months)10
- Downloads (Last 6 weeks)1
Reflects downloads up to 15 Oct 2024
Other Metrics
Citations
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign inFull Access
View options
View or Download as a PDF file.
PDFeReader
View online with eReader.
eReaderHTML Format
View this article in HTML Format.
HTML Format