Inner approximated reachability analysis

International audienceComputing a tight inner approximation of the range of a function over some set is notoriously di cult, way beyond obtaining outer approximations. We propose here a new method to compute a tight inner approximation of the set of reachable states of non-linear dynamical systems on a bounded time interval. This approach involves a ne forms and Kaucher arithmetic, plus a number of extra ingredients from set-based methods. An implementation of the method is discussed, and illustrated on representative numerical schemes, discrete-time and continuous-time dynamical systems

Latest Developments on the IEEE 1788 Effort for the Standardization of Interval Arithmetic

(Standardization effort supported by the INRIA D2T.)International audienceInterval arithmetic undergoes a standardization effort started in 2008 by the IEEE P1788 working group. The structure of the proposed standard is presented: the mathematical level is distinguished from both the implementation and representation levels. The main definitions are introduced: interval, mathematical functions, either arithmetic operations or trigonometric functions, comparison relations, set operations. While developing this standard, some topics led to hot debate. Such a hot topic is the handling of exceptions. Eventually, the system of decorations has been adopted. A decoration is a piece of information that is attached to each interval. Rules for the propagation of decorations have also been defined. Another hot topic is the mathematical model used for interval arithmetic. Historically, the model introduced by R. Moore in the 60s covered only non-empty and bounded intervals. The set-based model includes the empty set and unbounded intervals as well. Tenants of Kaucher arithmetic also insisted on offering "reverse" intervals. It has eventually been decided that an implementation must provide at least one of these flavors of interval arithmetic. The standard provides hooks for these different flavors. As the preparation of the draft should end in December 2013, no chapter is missing. However, a reference implementation would be welcome

Polynomial function intervals for floating-point software verification

The focus of our work is the verification of tight functional properties of numerical programs, such as showing that a floating-point implementation of Riemann integration computes a close approximation of the exact integral. Programmers and engineers writing such programs will benefit from verification tools that support an expressive specification language and that are highly automated. Our work provides a new method for verification of numerical software, supporting a substantially more expressive language for specifications than other publicly available automated tools. The additional expressivity in the specification language is provided by two constructs. First, the specification can feature inclusions between interval arithmetic expressions. Second, the integral operator from classical analysis can be used in the specifications, where the integration bounds can be arbitrary expressions over real variables. To support our claim of expressivity, we outline the verification of four example programs, including the integration example mentioned earlier. A key component of our method is an algorithm for proving numerical theorems. This algorithm is based on automatic polynomial approximation of non-linear real and real-interval functions defined by expressions. The PolyPaver tool is our implementation of the algorithm and its source code is publicly available. In this paper we report on experiments using PolyPaver that indicate that the additional expressivity does not come at a performance cost when comparing with other publicly available state-of-the-art provers. We also include a scalability study that explores the limits of PolyPaver in proving tight functional specifications of progressively larger randomly generated programs

Walking a Supramolecular Tightrope: A Self-Assembled Dodecamer from an 8-Aryl-2′-deoxyguanosine Derivative

Guanosine quadruplexes (GQs) have emerged in recent years as key players in the development of promising functional nanostruc-tures.1 GQs are formed by the self-assembly of guanosine subunits into planar tetramers (G-tetrads) that stack on each other, assisted by the complexation of a metal cation such as K+ or Na+. Alternatively, GQs can also form via the folding of G-rich oligonucleotides (e.g., DNA, RNA) leading to monomeric, dimeric, and tetrameric structures via the association of one, two, or four oligonucleotides, respectively.1d,2 In the latter, the number of G-tetrads is primarily controlled by the sequence (intrinsic param-eter) of the oligonucleotide, whereas, in the former, such control can be primarily achieved by adjusting extrinsic parameters (e.g., concentration, temperature, solvent,3 the cation template,4 and/or its counteranion5). Controlling the molecularity via intrinsic parameters (i.e., structural information in the supramolecula

The MPFI Library: Towards IEEE 1788-2015 Compliance

International audienceThe IEEE 1788-2015 has standardized interval arithmetic. However, few libraries for interval arithmetic are compliant with this standard. The main features of the IEEE 1788-2015 standard are detailed, namely the structure into 4 levels, the possibility to accomodate a new mathematical theory of interval arithmetic through the notion of flavor, and the mechanism of decoration for handling exceptions. These features were not present in the libraries developed prior to the elaboration of the standard. MPFI is such a library: it is a C library, based on MPFR, for arbitrary precision interval arithmetic. MPFI is not (yet) compliant with the IEEE 1788-2015 standard for interval arithmetic: the planned modifications are presented. Some considerations about performance and HPC on interval computations based on this standard, or on MPFI, conclude the paper

Post-assembly modification of kinetically metastable Fe(II)2L3 triple helicates.

We report the covalent post-assembly modification of kinetically metastable amine-bearing Fe(II)2L3 triple helicates via acylation and azidation. Covalent modification of the metastable helicates prevented their reorganization to the thermodynamically favored Fe(II)4L4 tetrahedral cages, thus trapping the system at the non-equilibrium helicate structure. This functionalization strategy also conveniently provides access to a higher-order tris(porphyrinatoruthenium)-helicate complex that would be difficult to prepare by de novo ligand synthesis.This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC). D.A.R. acknowledges the Gates Cambridge Trust for Ph.D. (Gates Cambridge Scholarship) and conference funding.This is the final published version. It first appeared at http://pubs.acs.org/doi/abs/10.1021/ja5042397