Abstract
Boolean notions of correctness are formalized by preorders on systems. Quantitative measures of correctness can be formalized by real-valued distance functions between systems, where the distance between implementation and specification provides a measure of “fit” or “desirability.” We extend the simulation preorder to the quantitative setting, by making each player of a simulation game pay a certain price for her choices. We use the resulting games with quantitative objectives to define three different simulation distances. The correctness distance measures how much the specification must be changed in order to be satisfied by the implementation. The coverage distance measures how much the implementation restricts the degrees of freedom offered by the specification. The robustness distance measures how much a system can deviate from the implementation description without violating the specification. We consider these distances for safety as well as liveness specifications. The distances can be computed in polynomial time for safety specifications, and for liveness specifications given by weak fairness constraints. We show that the distance functions satisfy the triangle inequality, that the distance between two systems does not increase under parallel composition with a third system, and that the distance between two systems can be bounded from above and below by distances between abstractions of the two systems. These properties suggest that our simulation distances provide an appropriate basis for a quantitative theory of discrete systems. We also demonstrate how the robustness distance can be used to measure how many transmission errors are tolerated by error correcting codes.
This work was partially supported by the European Union project COMBEST and the European Network of Excellence ArtistDesign.
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References
Alur, R., Henzinger, T., Kupferman, O., Vardi, M.: Alternating refinement relations. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 163–178. Springer, Heidelberg (1998)
Caspi, P., Benveniste, A.: Toward an approximation theory for computerised control. In: Sangiovanni-Vincentelli, A.L., Sifakis, J. (eds.) EMSOFT 2002. LNCS, vol. 2491, pp. 294–304. Springer, Heidelberg (2002)
Černý, P., Henzinger, T.A., Radhakrishna, A.: Simulation distances. Technical Report IST-2010-0003, IST Austria (June 2010)
Chatterjee, K., Doyen, L., Henzinger, T.: Quantitative languages. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 385–400. Springer, Heidelberg (2008)
Chatterjee, K., Henzinger, T.A., Jurdzinski, M.: Mean-payoff parity games. In: LICS, pp. 178–187 (2005)
de Alfaro, L., Faella, M., Stoelinga, M.: Linear and branching system metrics. IEEE Trans. Software Eng. 35(2), 258–273 (2009)
de Alfaro, L., Henzinger, T., Majumdar, R.: Discounting the future in systems theory. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 1022–1037. Springer, Heidelberg (2003)
de Alfaro, L., Majumdar, R., Raman, V., Stoelinga, M.: Game refinement relations and metrics. Logical Methods in Computer Science 4(3) (2008)
Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labelled Markov processes. Theor. Comput. Sci. 318(3), 323–354 (2004)
Droste, M., Gastin, P.: Weighted automata and weighted logics. Theor. Comput. Sci. 380(1-2), 69–86 (2007)
Ehrenfeucht, A., Mycielski, J.: Positional strategies for mean payoff games. International Journal of Game Theory, 163–178 (1979)
Fenton, N.: Software Metrics: A Rigorous and Practical Approach, Revised (Paperback). Course Technology (1998)
Gurevich, Y., Harrington, L.: Trees, automata, and games. In: STOC, pp. 60–65 (1982)
Hamming, R.W.: Error detecting and error correcting codes. Bell System Tech. J. 29, 147–160 (1950)
Henzinger, T.A., Kupferman, O., Rajamani, S.K.: Fair simulation. Information and Computation, 273–287 (1997)
Lincke, R., Lundberg, J., Löwe, W.: Comparing software metrics tools. In: ISSTA, pp. 131–142 (2008)
Milner, R.: An algebraic definition of simulation between programs. In: IJCAI, pp. 481–489 (1971)
van Breugel, F.: An introduction to metric semantics: operational and denotational models for programming and specification languages. Theor. Comput. Sci. 258(1-2), 1–98 (2001)
van Breugel, F., Worrell, J.: Approximating and computing behavioural distances in probabilistic transition systems. Theo. Comp. Sci. 360(1-3), 373–385 (2006)
Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158(1&2), 343–359 (1996)
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Černý, P., Henzinger, T.A., Radhakrishna, A. (2010). Simulation Distances. In: Gastin, P., Laroussinie, F. (eds) CONCUR 2010 - Concurrency Theory. CONCUR 2010. Lecture Notes in Computer Science, vol 6269. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15375-4_18
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