A Coding Theoretic Study on MLL proof nets

Coding theory is very useful for real world applications. A notable example is digital television. Basically, coding theory is to study a way of detecting and/or correcting data that may be true or false. Moreover coding theory is an area of mathematics, in which there is an interplay between many branches of mathematics, e.g., abstract algebra, combinatorics, discrete geometry, information theory, etc. In this paper we propose a novel approach for analyzing proof nets of Multiplicative Linear Logic (MLL) by coding theory. We define families of proof structures and introduce a metric space for each family. In each family, 1. an MLL proof net is a true code element; 2. a proof structure that is not an MLL proof net is a false (or corrupted) code element. The definition of our metrics reflects the duality of the multiplicative connectives elegantly. In this paper we show that in the framework one error-detecting is possible but one error-correcting not. Our proof of the impossibility of one error-correcting is interesting in the sense that a proof theoretical property is proved using a graph theoretical argument. In addition, we show that affine logic and MLL + MIX are not appropriate for this framework. That explains why MLL is better than such similar logics.Comment: minor modification

Inverse zero-sum problems and algebraic invariants

In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups of rank two. Also, the results obtained here enable us to improve, via the resolution of a linear integer program, a result of W. Gao and A. Geroldinger concerning the minimal number of elements with maximal order in a long zero-sumfree sequence of a finite Abelian group of rank two.Comment: 17 pages, to appear in Acta Arithmetic

Fast rate of convergence in high dimensional linear discriminant analysis

This paper gives a theoretical analysis of high dimensional linear discrimination of Gaussian data. We study the excess risk of linear discriminant rules. We emphasis on the poor performances of standard procedures in the case when dimension p is larger than sample size n. The corresponding theoretical results are non asymptotic lower bounds. On the other hand, we propose two discrimination procedures based on dimensionality reduction and provide associated rates of convergence which can be O(log(p)/n) under sparsity assumptions. Finally all our results rely on a theorem that provides simple sharp relations between the excess risk and an estimation error associated to the geometric parameters defining the used discrimination rule

Low-Complexity Quantized Switching Controllers using Approximate Bisimulation

In this paper, we consider the problem of synthesizing low-complexity controllers for incrementally stable switched systems. For that purpose, we establish a new approximation result for the computation of symbolic models that are approximately bisimilar to a given switched system. The main advantage over existing results is that it allows us to design naturally quantized switching controllers for safety or reachability specifications; these can be pre-computed offline and therefore the online execution time is reduced. Then, we present a technique to reduce the memory needed to store the control law by borrowing ideas from algebraic decision diagrams for compact function representation and by exploiting the non-determinism of the synthesized controllers. We show the merits of our approach by applying it to a simple model of temperature regulation in a building

On a combinatorial problem of Erdos, Kleitman and Lemke

In this paper, we study a combinatorial problem originating in the following conjecture of Erdos and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n. This conjecture was proved by Kleitman and Lemke, who then extended the original question to a problem on a zero-sum invariant in the framework of finite Abelian groups. Building among others on earlier works by Alon and Dubiner and by the author, our main theorem gives a new upper bound for this invariant in the general case, and provides its right order of magnitude.Comment: 15 page

On the existence of zero-sum subsequences of distinct lengths

In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of Gao for a variety of such groups. Our main result is deduced from a theorem of Alon, Friedland and Kalai, originally proved so as to study the existence of regular subgraphs in almost regular graphs. In the special case of elementary p-groups, Gao's conjecture is solved using Alon's Combinatorial Nullstellensatz. To conclude, we show that, assuming every integer satisfies Property B, this conjecture holds in the case of finite Abelian groups of rank two.Comment: 10 pages, to appear in Rocky Mountain Journal of Mathematic

Plugin procedure in segmentation and application to hyperspectral image segmentation

In this article we give our contribution to the problem of segmentation with plug-in procedures. We give general sufficient conditions under which plug in procedure are efficient. We also give an algorithm that satisfy these conditions. We give an application of the used algorithm to hyperspectral images segmentation. Hyperspectral images are images that have both spatial and spectral coherence with thousands of spectral bands on each pixel. In the proposed procedure we combine a reduction dimension technique and a spatial regularisation technique. This regularisation is based on the mixlet modelisation of Kolaczyck and Al

Unification and Logarithmic Space

We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof theory and more specifically linear logic and Geometry of Interaction. We show how unification can be used to build a model of computation by means of specific subalgebras associated to finite permutations groups. We then prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. We also show that the construction can naturally represent pointer machines, an intuitive way of understanding logarithmic space computing

Punishing Pharmaceutical Companies for Unlawful Promotion of Approved Drugs: Why the False Claims Act is the Wrong Rx

This article criticizes the shift in focus from correction and compliance to punishment of pharmaceutical companies allegedly violating the Food, Drug, & Cosmetic Act (FD&C Act) prohibitions on unlawful drug promotion. Traditionally, the Food and Drug Administration (FDA) has addressed unlawful promotional activities under the misbranding and new drug provisions of the FD&C Act. Recently though, the Justice Department (DOJ) has expanded the purview of the False Claims Act to include the same allegedly unlawful behavior on the theory that unlawful promotion “induces” physicians to prescribe drugs that result in the filing of false claims for reimbursement. Unchecked and unchallenged, the DOJ has negotiated criminal and civil settlements with individual pharmaceutical companies ranging from just under ten to hundreds of millions of dollars. In part, companies settle these cases to avoid the potential loss of revenue associated with the exclusion regime administered by the U.S. Department of Health and Human Services, under which companies risk losing the right to participate in federal health care programs. Even more disturbing, these settlements allow DOJ to circumvent judicial review of its enforcement approach, preventing any type of accountability for its legal theories or procedures. This article discusses the traditional enforcement methods employed by the FDA as well as the more recent DOJ prosecutions under the False Claims Act. Although it concludes that the FD&C Act should provide the sole means for prosecuting unlawful drug promotion, it also suggests that when prosecuting pharmaceutical companies under either Act, the government must avoid the temptation to mine companies for large settlements in lieu of developing a more coherent and responsible enforcement strategy