Abstract
Computing invariants is the key issue in the analysis of infinite-state systems whether analysis means testing, verification or parameter synthesis. In particular, methods that allow to treat combinations of loops are of interest. We present a set of algorithms and methods that can be applied to characterize over-approximations of the set of reachable states of combinations of self-loops. We present two families of complementary techniques. The first one identifies a number of basic cases of pair of self-loops for which we provide an exact characterization of the reachable states. The second family of techniques is a set of rules based on static analysis that allow to reduce n self-loops (n≥ 2) to n-1 independent pairs of self-loops. The results of the analysis of the pairs of self-loops can then be combined to provide an over-approximation of the reachable states of the n self-loops. We illustrate our methods by synthesizing conditions under which the Biphase Mark protocol works properly.
Work partially supported by Région Rhône-Alpes, France
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References
Abdulla, P., Bouajjani, A., Jonsson, B.: On-the-fly analysis of systems with unbounded, lossy fifb channels. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 305–318. Springer, Heidelberg (1998)
Abdulla, P.A., Bouajjani, A., Jonsson, B., Nilsson, M.: Handling Global Conditions in Parameterized System Verification. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 134–145. Springer, Heidelberg (1999)
Alur, R., Dill, D.: A Theory of Timed Automata. Theoretical Computer Science 126 (1994)
Apt, K.R., Meertens, L.G.L.T.: Completeness with finite systems of intermediate assertions for recursive program schemes. SIAM J. Comp. 9, 665–671 (1980)
Baukus, K., Bensalem, S., Lakhnech, Y., Stahl, K.: Abstracting wsis systems to verify parameterized networks. In: Schwartzbach, M.I., Graf, S. (eds.) TACAS 2000. LNCS, vol. 1785, p. 188. Springer, Heidelberg (2000)
Boigelot, B., Godefroid, P.: Symbolic verification of communication protocols with infinite state spaces using QDDs. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 1–12. Springer, Heidelberg (1996)
Bultan, Gerber, Pugh.: Symbolic model checking of infinite state systems using presburger arithmetic. In: CAV International Conference on Computer Aided Verification(1997)
Cousot, P., Cousot, R.: Static determination of dynamic properties of programs. In: Proc. 2nd Int. Symp. on Programming, pp. 106–130 (1976)
Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fix-points. In: 4th ACM symp. of Prog. Lang., pp. 238–252. ACM Press, New York (1977)
Cousot, P., Cousot, R.: Abstract interpretation frameworks. Logic and Comp. 2(4), 511–547 (1992)
Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among the variables of a program. In: 5th ACM symp. of Prog. Lang., pp. 84–97. ACM Press, New York (1978)
Caspi, P., Halbwachs, N., Pilaud, D., Plaice, J.: LUSTRE, adclarative language for programming synchronous systems. In: 14th Symposium on Principles of Programming Langiages (1987)
Fribourg, L., Olsen, H.: A decompositional approach for computing least fized-points of datalog programs with z-counters. Constraints 2(3/4), 305–335 (1997)
Gorelick, G.A.: A complete axiomatic system for proving assertions about recursive and non-recursive programs. Technical report, Toronto (1975)
Henzinger, T.A., Nicollin, X., Sifakis, J., Yovine, S.: Symbolic modelchecking for real-time systems. In: Seventh Annual IEEE Symposium on Logic in Computer Science, pp. 394–406. IEEE Computer Society Press, Los Alamitos (1992)
Halbwachs, N., Proy, Y.-E., Raymond, P.: Verification of linear hybrid systems by means of convex approximations. In: Proceedings of the International Symposium on Static Analysis. LNCS, vol. 818, pp. 223–237. Springer, Heidelberg (1994)
Ivanov, S., Griffioen, W.0.D.: Verification of a biphase mark protocol. Report CSI-R9915, Computing Science Institute, University of Nijmegen (August 1999)
Karr, M.: Affine relationships among variables of a program. Acta Informatica 6, 133–151 (1976)
Chandy, K.M., Misra, J.: Parallel Program Design, May 1989. Addison-Wesley, Austin (1989)
Kesten, Y., Maler, O., Marcus, M., Pnueli, A., Shahar, E.: Symbolic Model Checking with Rich Assertional Languages. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 424–435. Springer, Heidelberg (1997)
Moore, J.S.: A formal model of asynchronous communication and its use in mechanically verifying a biphase mark protocol. Formal Aspects of Computing 3(1) (1993)
Vaandrager, F.: Analysis of a biphase mark protocol with uppaal. Presentation at the meeting of the VHS-ESPRIT Project
Wolper, P., Boigelot, B.: Verifying systems with infinite but regular state spaces. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 88–97. Springer, Heidelberg (1998)
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Bensalem, S., Bozga, M., Fernandez, J.C., Ghirvu, L., Lakhnech, Y. (2000). A Transformational Approach for Generating Non-linear Invariants. In: Palsberg, J. (eds) Static Analysis. SAS 2000. Lecture Notes in Computer Science, vol 1824. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45099-3_4
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