Abstract
We describe pKA, a probabilistic Kleene-style algebra, based on a well known model of probabilistic/demonic computation [3,16,10]. Our technical aim is to express probabilistic versions of Cohen’s separation theorems.
Separation theorems simplify reasoning about distributed systems, where with purely algebraic reasoning they can reduce complicated interleaving behaviour to “separated” behaviours each of which can be analysed on its own. Until now that has not been possible for probabilistic distributed systems.
Algebraic reasoning in general is very robust, and easy to check: thus an algebraic approach to probabilistic distributed systems is attractive because in that “doubly hostile” environment (probability and interleaving) the opportunities for subtle error abound. Especially tricky is the interaction of probability and the demonic or “adversarial” scheduling implied by concurrency.
Our case study — based on Rabin’s Mutual exclusion with bounded waiting — is one where just such problems have already occurred: the original presentation was later shown to have subtle flaws [15]. It motivates our interest in algebras, where assumptions relating probability and secrecy are clearly exposed and, in some cases, can be given simple characterisations in spite of their intricacy.
This is a preview of subscription content, log in via an 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
Cohen, E.: Separation and reduction. In Mathematics of Program Construction. In: Backhouse, R., Oliveira, J.N. (eds.) MPC 2000. LNCS, vol. 1837, pp. 45–59. Springer, Heidelberg (2000)
Halpern, J., Tuttle, M.: Knowledge, probabilities and adversaries. J. ACM 40(4), 917–962 (1993)
He, J., Seidel, K., McIver, A.K.: Probabilistic models for the guarded command language. Science of Computer Programming 28, 171–192 (1997); Earlier appeared in Proc. FMTA 1995, Warsaw (May 1995). Available at [9, key HSM 1995]
Hurd, J.: A formal approach to probabilistic termination. In: Carreño, V.A., Muñoz, C.A., Tahar, S. (eds.) TPHOLs 2002. LNCS, vol. 2410, Springer, Heidelberg (2002), www.cl.cam.ac.uk/~jeh1004/research/papers
Kozen, D.: Kleene algebra with tests. ACM Transactions on Programming Languages and Systems (TOPLAS) 19(3), 427–443 (1997)
Kushilevitz, E., Rabin, M.O.: Randomized mutual exclusion algorithms revisited. In: Proc. 11th Annual ACM Symp. on Principles of Distributed Computing (1992)
Lehmann, D., Shelah, S.: Reasoning with time and chance. Information and Control 53(3), 165–198 (1982)
McIver, A., Weber, T.: Towards automated proof support for probabilistic distributed systems. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 534–548. Springer, Heidelberg (2005)
McIver, A.K., Morgan, C.C., Sanders, J.W., Seidel, K.: Probabilistic Systems Group: Collected reports, http://web.comlab.ox.ac.uk/oucl/research/areas/probs
McIver, A., Morgan, C.: Abstraction, Refinement and Proof for Probabilistic Systems. Technical Monographs in Computer Science. Springer, Heidelberg (2004)
Moeller, B.: Lazy Kleene Algebra. In: Kozen, D., Shankland, C. (eds.) MPC 2004. LNCS, vol. 3125, pp. 252–273. Springer, Heidelberg (2004)
Morgan, C.C., McIver, A.K., Seidel, K.: Probabilistic predicate transformers. ACM Transactions on Programming Languages and Systems 18(3), 325–353 (1996), doi.acm.org/10.1145/229542.229547
PRISM. Probabilistic symbolic model checker www.cs.bham.ac.uk/~dxp/prism
Rabin, M.O.: N-process mutual exclusion with bounded waiting by 4 log 2n-valued shared variable. Journal of Computer and System Sciences 25(1), 66–75 (1982)
Saias, I.: Proving probabilistic correctness statements: the case of Rabin’s algorithm for mutual exclusion. In: Proc. 11th Annual ACM Symp. on Principles of Distributed Computing (1992)
Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, MIT (1995)
Takai, T., Furusawa, H.: Monadic tree Kleene Algebra. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, Springer, Heidelberg (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
McIver, A.K., Cohen, E., Morgan, C.C. (2006). Using Probabilistic Kleene Algebra for Protocol Verification. In: Schmidt, R.A. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2006. Lecture Notes in Computer Science, vol 4136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11828563_20
Download citation
DOI: https://doi.org/10.1007/11828563_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37873-0
Online ISBN: 978-3-540-37874-7
eBook Packages: Computer ScienceComputer Science (R0)