Possibility Theory, Probability Theory and Multiple-Valued Logics: A Clarification

There has been a long-lasting misunderstanding in the literature of artificial intelligence and uncertainty modeling, regarding the role of fuzzy set theory and many-valued logics. The recurring question is that of the mathematical and pragmatic meaningfulness of a compositional calculus and the validity of the excluded middle law. This confusion pervades the early developments of probabilistic logic, despite early warnings of some philosophers of probability. This paper tries to clarify this situation. It emphasizes three main points. First, it suggests that the root of the controversies lies in the unfortunate confusion between degrees of belief and what logicians call degrees of truth. The latter are usually compositional, while the former cannot be so. This claim is first illustrated by laying bare the non-compositional belief representation embedded in the standard propositional calculus. It turns out to be an all-or-nothing version of possibility theory. This framework is then extended to discuss the case of fuzzy logic versus graded possibility theory. Next, it is demonstrated that any belief representation where compositionality is taken for granted is bound to at worst collapse to a Boolean truth assignment and at best to a poorly expressive tool. Lastly, some claims pertaining to an alleged compositionality of possibility theory are refuted, thus clarifying a pervasive confusion between possibility theory axioms and fuzzy set basic connectives.     

Globally Optimal Estimates for Geometric Reconstruction Problems

We introduce a framework for computing statistically optimal estimates of geometric reconstruction problems. While traditional algorithms often suffer from either local minima or non-optimality—or a combination of both—we pursue the goal of achieving global solutions of the statistically optimal cost-function. Our approach is based on a hierarchy of convex relaxations to solve non-convex optimization problems with polynomials. These convex relaxations generate a monotone sequence of lower bounds and we show how one can detect whether the global optimum is attained at a given relaxation. The technique is applied to a number of classical vision problems: triangulation, camera pose, homography estimation and last, but not least, epipolar geometry estimation. Experimental validation on both synthetic and real data is provided. In practice, only a few relaxations are needed for attaining the global optimum.     

Credible autocoding of convex optimization algorithms

The efficiency of modern optimization methods, coupled with increasing computational resources, has led to the possibility of real-time optimization algorithms acting in safety-critical roles. There is a considerable body of mathematical proofs on on-line optimization algorithms which can be leveraged to assist in the development and verification of their implementation. In this paper, we demonstrate how theoretical proofs of real-time optimization algorithms can be used to describe functional properties at the level of the code, thereby making it accessible for the formal methods community. The running example used in this paper is a generic semi-definite programming solver. Semi-definite programs can encode a wide variety of optimization problems and can be solved in polynomial time at a given accuracy. We describe a top-down approach that transforms a high-level analysis of the algorithm into useful code annotations. We formulate some general remarks on how such a task can be incorporated into a convex programming autocoder. We then take a first step towards the automatic verification of the optimization program by identifying key issues to be addressed in future work.     

Trajectory planning/re-planning for satellite systems in rendezvous mission in the presence of actuator faults based on attainable efforts analysis

The objective of fault-tolerant control (FTC) is to minimise the effect of faults on system performance (stability, trajectory tracking, etc.). However, the majority of the existing FTC methods continue to force the system to follow the pre-fault trajectories without considering the reduction in available control resources caused by actuator faults. Forcing the system to follow the same trajectories as before fault occurrence may result in actuator saturation and system's instability. Thus, pre-fault objectives should be redefined in function of the remaining resources to avoid potential saturation. The main contribution of this paper is a flatness-based trajectory planning/re-planning method that can be combined with any active FTC approach. The work considers the case of over-actuated systems where a new idea is proposed to evaluate the severity of faults occurred. In addition, the trajectory planning/re-planning approach is formulated as an optimisation problem based on the analysis of attainable efforts domain in fault-free and fault cases. The proposed approach is applied to two satellite systems in rendezvous mission.     

Advanced LMI based analysis and design for Acrobot walking

This article aims to further improve previously developed design for Acrobot walking based on partial exact feedback linearisation of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4-dimensional linear time-varying system having three time-varying entries only, the remaining entries being either zero or one. In such a way, exponentially stable tracking can be obtained by quadratically stabilising a linear system with polytopic uncertainty. The current improvement is based on applying linear matrix inequalities (LMI) methods to solve this problem numerically. This careful analysis significantly improves previously known approaches. Numerical simulations of Acrobot walking based on the above-mentioned LMI design are demonstrated as well.     

Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximisation for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimisation problems with the help of convex semidefinite programming (optimisation over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach.     

Simple Views on Metal/Oxide Interfaces: Contributions of the Long-Range Interactions to the Adhesion Energy

The image and van der Waals contributions to the metal/oxide work of adhesion are compared through the extent to which they follow the known prevalent trends, i.e. the increase in work of adhesion (a) with narrowing oxide band gap and (b) with increasing conduction electron density of the metal. The van der Waals interaction is shown to follow both trends, while the image term is suggested to be significant only for dense metals in contact with very ionic oxides. The relative contribution of these long-range interactions to the overall metal/oxide work of adhesion is found to be maximized for systems involving metals with low electronic densities and oxides with wide band gaps. At variance, high metallic electronic densities and narrow oxide gaps likely favour short-range interactions arising from charge transfer.     

Fullerene-functionalized carbon nanotubes as improved optical limiting devices

We report the synthesis, characterization and optical limiting behavior of a nanohybrid built by grafting C₆₀-fullerenes on carbon nanotubes (CNTs). The nonlinear optical limiting properties of the CNT-C₆₀ complex were investigated at wavelengths where C₆₀ does not absorb. We found that the nanohybrid had superior performances to those of CNTs and fullerenes, either taken individually or as a mixture. This enhanced optical limitation of the nanohybrid suggests not only cooperative but also synergistic effects between the two carbon forms. A mechanism involving higher excitonic states of the CNTs formed by Auger recombination of low energy excitons is proposed.     

Al0.3Ga0.7As-GaAs microwave resonant tunneling oscillator

We describe the design of a microwave oscillator using resonant tunneling diodes. The devices are fabricated from Al_0.3Ga_0.7As-GaAs double barrier hetero-structures grown by molecular beam epitaxy. Design criteria improving current drivability are established from a theoretical study of tunneling transmission probabilities. Very high peak current densities up to 3.10⁴ A/cm², favorable for high frequency operation as an oscillator, have been achieved experimentally. The devices exhibit stable oscillations at liquid nitrogen temperature and at room temperature when the tunnel diode oscillator is constructed with a stabilizing network.