Abstract
Early support for reasoning about probabilistic system behaviour replaced nondeterminism with probabilism. Only relatively recently have formalisms been studied that combine the two, and hence facilitate reasoning about probabilistic systems at levels of abstraction more general than code. Such studies have revealed an unsuspected subtlety in the interaction between nondeterministic and probabilistic choices that can be summarised: the demon resolving the nondeterministic choice has memory of previous state changes, whilst the probabilistic choice is made spontaneously. As a result, assignments to distinct variables need no longer commute. This paper introduces a model with explicit control of the length of the demon’s memory. It does so by expanding the standard (initial-final) state view of computation to incorporate a third state, the ‘original’ state which checkpoints the most recent nondeterministic choice. That enables a nondeterministic choice to be made on the basis of only certain past probabilistic choices and so facilitates independent nondeterministic combinations to be chosen against just those. Sound laws are presented and used to analyse first an example in which no new behaviour should result, and second one that lies beyond the scope of traditional models.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Chatzikokolakis, K., Palamidessi, C., Panangaden, P.: On the Bayes risk in information-hiding protocols. In: Proceedings of the 20th IEEE Computer Security Foundations, pp. 341–354. IEEE Computer Society, Los Alamitos (2007)
Fernandez, L., Piron, R.: Should She Switch? A Game Theoretic Analysis of the Monty Hall Problem. Mathematics Magizine 72(3), 214–217 (1999)
Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)
He, J., Seidel, K., McIver, A.K.: Probabilistic models for the guarded command language. Science of Computer Programming 28(2), 171–192 (1997)
Hoare, C.A.R., He, J.: Unifying Theories of Programming. Prentice Hall, Englewood Cliffs (1998)
McIver, A.K., Morgan, C.C.: Abstraction, Refinement and Proof for Probabilistic Systems. Monographs in Computer Science. Springer, Heidelberg (2005)
Mislove, M., Ouaknine, J., Worrell, J.: Axioms for probability and nondeterminism. ENTCS 96, 7–28 (2004)
Morgan, C.C., McIver, A.K., Seidel, K.: Probabilistic Predicate Transformers. ACM TOPLAS 18(3), 325–353 (1996)
Morgan, C.C.: The shadow knows: Refinement of ignorance in sequential programs. In: Uustalu, T. (ed.) MPC 2006. LNCS, vol. 4014, pp. 359–378. Springer, Heidelberg (2006)
Pananganden, P.: Probabilistic relations. In: Baier, C., Huth, M., Kwiatkowska, M.Z., Ryan, M. (eds.) PROBMIV 1998, pp. 59–74 (1998)
Rabin, M.O.: Probabilistic algorithms. In: Traub, J.F. (ed.) Algorithms and Complexity: New Directions and Recent Results, pp. 21–39. Academic Press, London (1976)
Robinson, D., Morgan, C.C.: Restricted Demonic Choice for Modular Probabilistic Programs. In: ESSLI PLRC Workshop 1998 (1998)
Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR 1994. LNCS, vol. 836, pp. 1–43. Springer, Heidelberg (1994)
Sokolova, A., de Vink, E.: Probabilistic automata: system types, parallel composition and comparison. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 1–43. Springer, Heidelberg (2004)
Tix, R., Keimel, K., Plotkin, G.: Semantic domains for combining probability and non-determinism. ENTCS 222, 3–99 (2009)
Varacca, D., Winskel, G.: Distributing probability over non-determinism. Mathematical Structures in Computer Science 16(1), 87–113 (2006)
Ying, M.: Reasoning about probabilistic sequential programs in a probabilistic logic. Acta Informatica 39, 315–389 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chen, Y., Sanders, J.W. (2009). Unifying Probability with Nondeterminism. In: Cavalcanti, A., Dams, D.R. (eds) FM 2009: Formal Methods. FM 2009. Lecture Notes in Computer Science, vol 5850. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05089-3_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-05089-3_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05088-6
Online ISBN: 978-3-642-05089-3
eBook Packages: Computer ScienceComputer Science (R0)