A library of Taylor models for PVS automatic proof checker

We present in this paper a library to compute with Taylor models, a technique extending interval arithmetic to reduce decorrelation and to solve differential equations. Numerical software usually produces only numerical results. Our library can be used to produce both results and proofs. As seen during the development of Fermat's last theorem reported by Aczel 1996, providing a proof is not sufficient. Our library provides a proof that has been thoroughly scrutinized by a trustworthy and tireless assistant. PVS is an automatic proof assistant that has been fairly developed and used and that has no internal connection with interval arithmetic or Taylor models. We built our library so that PVS validates each result as it is produced. As producing and validating a proof, is and will certainly remain a bigger task than just producing a numerical result our library will never be a replacement to imperative implementations of Taylor models such as Cosy Infinity. Our library should mainly be used to validate small to medium size results that are involved in safety or life critical applications

Formal proof for delayed finite field arithmetic using floating point operators

Formal proof checkers such as Coq are capable of validating proofs of correction of algorithms for finite field arithmetics but they require extensive training from potential users. The delayed solution of a triangular system over a finite field mixes operations on integers and operations on floating point numbers. We focus in this report on verifying proof obligations that state that no round off error occurred on any of the floating point operations. We use a tool named Gappa that can be learned in a matter of minutes to generate proofs related to floating point arithmetic and hide technicalities of formal proof checkers. We found that three facilities are missing from existing tools. The first one is the ability to use in Gappa new lemmas that cannot be easily expressed as rewriting rules. We coined the second one ``variable interchange'' as it would be required to validate loop interchanges. The third facility handles massive loop unrolling and argument instantiation by generating traces of execution for a large number of cases. We hope that these facilities may sometime in the future be integrated into mainstream code validation.Comment: 8th Conference on Real Numbers and Computers, Saint Jacques de Compostelle : Espagne (2008

Quantification and Correction of Systematic Errors Due to Detector Time-Averaging in Single-Molecule Tracking Experiments

Single-molecule tracking is a powerful way to look at the dynamic organization of plasma membranes. However, there are some limitations to its use. For example, it was recently observed, using numerical simulation, that time-averaging effects inherent to the exposure time of detectors are likely to bias the apparent motion of molecules confined in microdomains. Here, we solve this apparently limiting issue analytically. We explore this phenomenon by calculating its effects on the observed diffusion coefficients and domain sizes. We demonstrate that the real parameters can be easily recovered from the measured apparent ones. Interestingly, we find that single-molecule tracking can be used to explore events occurring at a timescale smaller than the exposure time.Comment: 3 pages (Letter); 1 figur