Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program

Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring specific accuracy issues due to their massive use of floating-point computations. Yet, it is uncommon to guarantee their correctness. Indeed, we had to extend existing methods and tools for proving the correct behavior of programs to verify an existing numerical analysis program. This C program implements the second-order centered finite difference explicit scheme for solving the 1D wave equation. In fact, we have gone much further as we have mechanically verified the convergence of the numerical scheme in order to get a complete formal proof covering all aspects from partial differential equations to actual numerical results. To the best of our knowledge, this is the first time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with arXiv:1112.179

Advanced Probabilistic Couplings for Differential Privacy

Differential privacy is a promising formal approach to data privacy, which provides a quantitative bound on the privacy cost of an algorithm that operates on sensitive information. Several tools have been developed for the formal verification of differentially private algorithms, including program logics and type systems. However, these tools do not capture fundamental techniques that have emerged in recent years, and cannot be used for reasoning about cutting-edge differentially private algorithms. Existing techniques fail to handle three broad classes of algorithms: 1) algorithms where privacy depends accuracy guarantees, 2) algorithms that are analyzed with the advanced composition theorem, which shows slower growth in the privacy cost, 3) algorithms that interactively accept adaptive inputs. We address these limitations with a new formalism extending apRHL, a relational program logic that has been used for proving differential privacy of non-interactive algorithms, and incorporating aHL, a (non-relational) program logic for accuracy properties. We illustrate our approach through a single running example, which exemplifies the three classes of algorithms and explores new variants of the Sparse Vector technique, a well-studied algorithm from the privacy literature. We implement our logic in EasyCrypt, and formally verify privacy. We also introduce a novel coupling technique called \emph{optimal subset coupling} that may be of independent interest

Forward Invariant Cuts to Simplify Proofs of Safety

The use of deductive techniques, such as theorem provers, has several advantages in safety verification of hybrid sys- tems; however, state-of-the-art theorem provers require ex- tensive manual intervention. Furthermore, there is often a gap between the type of assistance that a theorem prover requires to make progress on a proof task and the assis- tance that a system designer is able to provide. This paper presents an extension to KeYmaera, a deductive verification tool for differential dynamic logic; the new technique allows local reasoning using system designer intuition about per- formance within particular modes as part of a proof task. Our approach allows the theorem prover to leverage for- ward invariants, discovered using numerical techniques, as part of a proof of safety. We introduce a new inference rule into the proof calculus of KeYmaera, the forward invariant cut rule, and we present a methodology to discover useful forward invariants, which are then used with the new cut rule to complete verification tasks. We demonstrate how our new approach can be used to complete verification tasks that lie out of the reach of existing deductive approaches us- ing several examples, including one involving an automotive powertrain control system.Comment: Extended version of EMSOFT pape

Collaborative Verification-Driven Engineering of Hybrid Systems

Hybrid systems with both discrete and continuous dynamics are an important model for real-world cyber-physical systems. The key challenge is to ensure their correct functioning w.r.t. safety requirements. Promising techniques to ensure safety seem to be model-driven engineering to develop hybrid systems in a well-defined and traceable manner, and formal verification to prove their correctness. Their combination forms the vision of verification-driven engineering. Often, hybrid systems are rather complex in that they require expertise from many domains (e.g., robotics, control systems, computer science, software engineering, and mechanical engineering). Moreover, despite the remarkable progress in automating formal verification of hybrid systems, the construction of proofs of complex systems often requires nontrivial human guidance, since hybrid systems verification tools solve undecidable problems. It is, thus, not uncommon for development and verification teams to consist of many players with diverse expertise. This paper introduces a verification-driven engineering toolset that extends our previous work on hybrid and arithmetic verification with tools for (i) graphical (UML) and textual modeling of hybrid systems, (ii) exchanging and comparing models and proofs, and (iii) managing verification tasks. This toolset makes it easier to tackle large-scale verification tasks

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi