Sound compilation of reals

Abstract

Writing accurate numerical software is hard because of many sources of unavoidable uncertainties, including finite numerical precision of implementations. We present a programming model where the user writes a program in a real-valued implementation and specification language that explicitly includes different types of uncertainties. We then present a compilation algorithm that generates a finite-precision implementation that is guaranteed to meet the desired precision with respect to real numbers. Our compilation performs a number of verification steps for different candidate precisions. It generates verification conditions that treat all sources of uncertainties in a unified way and encode reasoning about finiteprecision roundoff errors into reasoning about real numbers. Such verification conditions can be used as a standardized format for verifying the precision and the correctness of numerical programs. Due to their non-linear nature, precise reasoning about these verification conditions remains difficult and cannot be handled using state-of-the art SMT solvers alone. We therefore propose a new procedure that combines exact SMT solving over reals with approximate and sound affine and interval arithmetic. We show that this approach overcomes scalability limitations of SMT solvers while providing improved precision over affine and interval arithmetic. Our implementation gives promising results on several numerical models, including dynamical systems, transcendental functions, and controller implementations.

Control Strategies for Self-Adaptive Software Systems

“…For example, modern SMT tools can be used at compile time to verify the occurrence of numerical problems and automatically provide fixes guaranteeing the final results of the procedures to achieve a target accuracy [Darulova and Kuncak 2014].…”

“…Although control theory developed techniques to validate system models experimentally and in turn the theoretical effectiveness of their controllers, implementation may differ from its design. Software engineering is playing a role in both producing correct-by-construction implementations from the mathematical description of controllers [Kowshik et al 2002;Darulova and Kuncak 2014;Cai 2002] and in the definition of specialized testing and verification procedures [Darulova and Kuncak 2013;Ismail et al 2015;Matinnejad et al 2016;Zuliani et al 2010;Villegas et al 2011;Camara et al 2013]. Implementation of distributed control.…”

Automatic Verification of Finite Precision Implementations of Linear Controllers

“…There has been static analysis techniques (e.g., [5,13,16]) developed for the analysis of finite precision numerical programs, but they focus on verifying properties such as numerical stability, the absence of buffer overflow and the absence of arithmetic exception rather than verifying the equivalence between code and a dynamical system model as the specification of the controller. Finally, there has been software verification work using the model extraction technique [8,19,20,34,29], and the floating-point roundoff error estimation has been studied in [11,9,33].…”

Developing Verified Software Using Leon

“…They manipulate unbounded integer and bitvector numerical quantities, algebraic data types expressed as case classes, lists, functional arrays, and maps. An ambitious research direction introduces a Real data type that compiles into a desired finite-precision data type that meets given precision guarantees [2,3]. The main challenge in this work is automatically computing the accumulation of worst-case error bounds though non-linear computations, which requires also precisely computing ranges of variables in programs using constraint solving.…”