Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

The hard-core model on Z3\mathbb{Z}^3 and Kepler's conjecture

We study the hard-core model of statistical mechanics on a unit cubic lattice $\mathbb{Z}^3$, which is intrinsically related to the sphere-packing problem for spheres with centers in $\mathbb{Z}^3$. The model is defined by the sphere diameter $D>0$ which is interpreted as a Euclidean exclusion distance between point particles located at spheres centers. The second parameter of the underlying model is the particle fugacity $u$. For $u>1$ the ground states of the model are given by the dense-packings of the spheres. The identification of such dense-packings is a considerable challenge, and we solve it for $D^2=2, 3, 4, 5, 6, 8, 9, 10, 11, 12$ as well as for $D^2=2\ell^2$, where $\ell\in\mathbb{N}$. For the former family of values of $D^2$ our proofs are self-contained. For $D^2=2\ell^2$ our results are based on the proof of Kepler's conjecture. Depending on the value of $D^2$, we encounter three physically distinct situations: (i) finitely many periodic ground states, (ii) countably many layered periodic ground states and (iii) countably many not necessarily layered periodic ground states. For the first two cases we use the Pirogov-Sinai theory and identify the corresponding periodic Gibbs distributions for $D^2=2,3,5,8,9,10,12$ and $D^2=2\ell^2$, $\ell\in\mathbb{N}$, in a high-density regime $u>u_*(D^2)$, where the system is ordered and tends to fluctuate around some ground states. In particular, for $D^2=5$ only a finite number out of countably many layered periodic ground states generate pure phases

Formal Proofs for Nonlinear Optimization

We present a formally verified global optimization framework. Given a semialgebraic or transcendental function $f$ and a compact semialgebraic domain $K$, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of $f$ over $K$. This method allows to bound in a modular way some of the constituents of $f$ by suprema of quadratic forms with a well chosen curvature. Thus, we reduce the initial goal to a hierarchy of semialgebraic optimization problems, solved by sums of squares relaxations. Our implementation tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of multivariate transcendental inequalities. We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table

Does Time Smoothen Space? Implications for Space-Time Representation

The continuous nature of space and time is a fundamental tenet of many scientific endeavors. That digital representation imposes granularity is well recognized, but whether it is possible to address space completely remains unanswered. This paper argues Hales' proof of Kepler's conjecture on the packing of hard spheres suggests the answer to be "no", providing examples of why this matters in GIS generally and considering implications for spatio-temporal GIS in particular. It seeks to resolve the dichotomy between continuous and granular space by showing how a continuous space may be emergent over a random graph. However, the projection of this latent space into 3D/4D imposes granularity. Perhaps surprisingly, representing space and time as locally conjugate may be key to addressing a "smooth" spatial continuum. This insight leads to the suggestion of Face Centered Cubic Packing as a space-time topology but also raises further questions for spatio-temporal representation

Coverage and Connectivity in Three-Dimensional Networks

Most wireless terrestrial networks are designed based on the assumption that the nodes are deployed on a two-dimensional (2D) plane. However, this 2D assumption is not valid in underwater, atmospheric, or space communications. In fact, recent interest in underwater acoustic ad hoc and sensor networks hints at the need to understand how to design networks in 3D. Unfortunately, the design of 3D networks is surprisingly more difficult than the design of 2D networks. For example, proofs of Kelvin's conjecture and Kepler's conjecture required centuries of research to achieve breakthroughs, whereas their 2D counterparts are trivial to solve. In this paper, we consider the coverage and connectivity issues of 3D networks, where the goal is to find a node placement strategy with 100% sensing coverage of a 3D space, while minimizing the number of nodes required for surveillance. Our results indicate that the use of the Voronoi tessellation of 3D space to create truncated octahedral cells results in the best strategy. In this truncated octahedron placement strategy, the transmission range must be at least 1.7889 times the sensing range in order to maintain connectivity among nodes. If the transmission range is between 1.4142 and 1.7889 times the sensing range, then a hexagonal prism placement strategy or a rhombic dodecahedron placement strategy should be used. Although the required number of nodes in the hexagonal prism and the rhombic dodecahedron placement strategies is the same, this number is 43.25% higher than the number of nodes required by the truncated octahedron placement strategy. We verify by simulation that our placement strategies indeed guarantee ubiquitous coverage. We believe that our approach and our results presented in this paper could be used for extending the processes of 2D network design to 3D networks.Comment: To appear in ACM Mobicom 200