Stochastic Formal Methods for Rare Failure Events due to the Accumulation of Errors
Résumé
This paper provides an accurate bound on the number of numeric operations (fixed or floating point) that can safely be performed before accuracy is lost based on the assumption that accumulated errors are uniformly distributed in $\pm \frac12$ unit in the last place. This work has important implications for control systems with safety-critical software, as these systems are now running fast enough and long enough for their errors to impact on their functionality. Furthermore, worst-case analysis would blindly advise the replacement of existing systems that have been successfully running for years and that will continue running before software development practices evolve. We present here new theorems that we are currently validating with the PVS proof assistant. This theory will allow code analyzing tools to produce formal certificates of accurate behavior. FAA regulations for aircraft require that the probability of an error be below $10^{-9}$ for a 10 hour flight. Such a low failure rate is stretching the limits of generic calculations solely based on the standard deviation of random variables for the intermediate sums. We need many individual errors for the Central Limit Theorem approximation to be sufficiently accurate (distance well below $10^{-9}$). The precise bound presented here enhances the number of bits of the result that can safely be regarded as correct.