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.
Theoretical Computer Science
This work was partially supported by the ERC Advanced Grant QUAREM, the FWF NFN Grant S11402-N23 (RiSE), the European Union project COMBEST and the European Network of Excellence Artist Design.
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Cerny P, Henzinger TA, Radhakrishna A. Simulation distances. Theoretical Computer Science. 2012;413(1):21-35. doi:10.1016/j.tcs.2011.08.002
Cerny, P., Henzinger, T. A., & Radhakrishna, A. (2012). Simulation distances. Theoretical Computer Science. Elsevier. https://doi.org/10.1016/j.tcs.2011.08.002
Cerny, Pavol, Thomas A Henzinger, and Arjun Radhakrishna. “Simulation Distances.” Theoretical Computer Science. Elsevier, 2012. https://doi.org/10.1016/j.tcs.2011.08.002.
P. Cerny, T. A. Henzinger, and A. Radhakrishna, “Simulation distances,” Theoretical Computer Science, vol. 413, no. 1. Elsevier, pp. 21–35, 2012.
Cerny P, Henzinger TA, Radhakrishna A. 2012. Simulation distances. Theoretical Computer Science. 413(1), 21–35.
Cerny, Pavol, et al. “Simulation Distances.” Theoretical Computer Science, vol. 413, no. 1, Elsevier, 2012, pp. 21–35, doi:10.1016/j.tcs.2011.08.002.