Abstract
This paper investigates the complexity of verifying livelock freedom, self-stabilization, and weak stabilization in parameterized unidirectional ring and bidirectional chain topologies. Specifically, we illustrate that verifying livelock freedom of parameterized rings consisting of self-disabling and deterministic processes is undecidable (specifically, \(\Pi^0_1\)-complete). This result implies that verifying self-stabilization and weak stabilization for parameterized rings of self-disabling processes is also undecidable. The results of this paper strengthen previous work on the undecidability of verifying temporal logic properties in symmetric ring protocols. The proof of undecidability is based on a reduction from the periodic domino problem.
Keywords
- Parameterized Ring
- Linear Temporal Logic
- Periodic Propagation
- Weak Stabilization
- Legitimate State
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This work was sponsored by the NSF grant CCF-1116546.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abdulla, P.A., Jonsson, B., Nilsson, M., Saksena, M.: A survey of regular model checking. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 35–48. Springer, Heidelberg (2004)
Abello, J., Dolev, S.: On the computational power of self-stabilizing systems. Theoretical Computer Science 182(1-2), 159–170 (1997)
Apt, K.R., Kozen, D.C.: Limits for automatic verification of finite-state concurrent systems. Information Processing Letters 22(6), 307–309 (1986)
Berger, R.: The Undecidability of the Domino Problem. Memoirs; No 1/66. American Mathematical Society (1966)
Bouajjani, A., Jonsson, B., Nilsson, M., Touili, T.: Regular model checking. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, Springer, Heidelberg (2000)
Cachera, D., Morin-Allory, K.: Verification of safety properties for parameterized regular systems. ACM Transactions on Embedded Computing Systems 4(2), 228–266 (2005)
Dijkstra, E.W.: Self-stabilizing systems in spite of distributed control. Communications of the ACM 17(11), 643–644 (1974)
Emerson, E.A.: Temporal and modal logic. In: Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics (B), pp. 995–1072. Elsevier (1990)
Emerson, E.A., Kahlon, V.: Reducing model checking of the many to the few. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 236–254. Springer, Heidelberg (2000)
Emerson, E.A., Namjoshi, K.S.: Reasoning about rings. In: Cytron, R.K., Lee, P. (eds.) POPL, pp. 85–94. ACM Press (1995)
Emerson, E.A., Namjoshi, K.S.: On reasoning about rings. International Journal of Foundations of Computer Science 14(4), 527–550 (2003)
Farahat, A.: Automated Design of Self-Stabilization. PhD thesis, Michigan Technological University (2012)
Farahat, A., Ebnenasir, A.: A lightweight method for automated design of convergence in network protocols. TAAS 7(4), 38 (2012)
Farahat, A., Ebnenasir, A.: Local reasoning for global convergence of parameterized rings. In: ICDCS, pp. 496–505. IEEE (2012)
Fribourg, L., Olsén, H.: Reachability sets of parameterized rings as regular languages. Electronic Notes in Theoretical Computer Science 9, 40 (1997)
Funk, P., Zinnikus, I.: Self-stabilization as multiagent systems property. In: AAMAS, pp. 1413–1414. ACM (2002)
Gouda, M.G.: The theory of weak stabilization. In: Datta, A.K., Herman, T. (eds.) WSS 2001. LNCS, vol. 2194, pp. 114–123. Springer, Heidelberg (2001)
Gouda, M.G., Acharya, H.B.: Nash equilibria in stabilizing systems. In: Guerraoui, R., Petit, F. (eds.) SSS 2009. LNCS, vol. 5873, pp. 311–324. Springer, Heidelberg (2009)
Gouda, M.G., Multari, N.J.: Stabilizing communication protocols. IEEE Trans. Computers 40(4), 448–458 (1991)
Gurevich, Y., Koriakov, I.O.: A remark on Berger’s paper on the domino problem. Siberian Mathematical Journal 13(2), 319–321 (1972)
Kari, J.: The nilpotency problem of one-dimensional cellular automata. SIAM Journal on Computing 21(3), 571–586 (1992)
Klinkhamer, A.P., Ebnenasir, A.: Verifying livelock freedom on parameterized rings. Technical Report CS-TR-13-01, Michigan Technological University (July 2013), http://www.cs.mtu.edu/html/tr/13/13-01.pdf
La, H.J., Kim, S.D.: A self-stabilizing process for mobile cloud computing. In: SOSE, pp. 454–462. Computer Society (2013)
Rogers, H.: Theory of recursive functions and effective computability. MIT Press (1987) (reprint from 1967)
Suzuki, I.: Proving properties of a ring of finite-state machines. Information Processing Letters 28(4), 213–214 (1988)
Touili, T.: Regular model checking using widening techniques. Electronic Notes in Theoretical Computer Science 50(4), 342–356 (2001)
Wang, H., Telephone, A., Company, T.: Proving Theorems by Pattern Recognition -II. American Telephone and Telegraph Company (1961)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this paper
Cite this paper
Klinkhamer, A.P., Ebnenasir, A. (2013). Verifying Livelock Freedom on Parameterized Rings and Chains. In: Higashino, T., Katayama, Y., Masuzawa, T., Potop-Butucaru, M., Yamashita, M. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2013. Lecture Notes in Computer Science, vol 8255. Springer, Cham. https://doi.org/10.1007/978-3-319-03089-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-03089-0_12
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03088-3
Online ISBN: 978-3-319-03089-0
eBook Packages: Computer ScienceComputer Science (R0)