28 research outputs found
Clustering, Hamming Embedding, Generalized LSH and the Max Norm
We study the convex relaxation of clustering and hamming embedding, focusing
on the asymmetric case (co-clustering and asymmetric hamming embedding),
understanding their relationship to LSH as studied by (Charikar 2002) and to
the max-norm ball, and the differences between their symmetric and asymmetric
versions.Comment: 17 page
The Moment Problem for Continuous Positive Semidefinite Linear functionals
Let be a locally convex topology on the countable dimensional
polynomial -algebra \rx:=\reals[X_1,...,X_n]. Let be a closed
subset of , and let be a finitely generated
quadratic module in \rx. We investigate the following question: When is the
cone \Pos(K) (of polynomials nonnegative on ) included in the closure of
? We give an interpretation of this inclusion with respect to representing
continuous linear functionals by measures. We discuss several examples; we
compute the closure of M=\sos with respect to weighted norm- topologies.
We show that this closure coincides with the cone \Pos(K) where is a
certain convex compact polyhedron.Comment: 14 page
Polarizing Double Negation Translations
Double-negation translations are used to encode and decode classical proofs
in intuitionistic logic. We show that, in the cut-free fragment, we can
simplify the translations and introduce fewer negations. To achieve this, we
consider the polarization of the formul{\ae}{} and adapt those translation to
the different connectives and quantifiers. We show that the embedding results
still hold, using a customized version of the focused classical sequent
calculus. We also prove the latter equivalent to more usual versions of the
sequent calculus. This polarization process allows lighter embeddings, and
sheds some light on the relationship between intuitionistic and classical
connectives
ASMs and Operational Algorithmic Completeness of Lambda Calculus
We show that lambda calculus is a computation model which can step by step
simulate any sequential deterministic algorithm for any computable function
over integers or words or any datatype. More formally, given an algorithm above
a family of computable functions (taken as primitive tools, i.e., kind of
oracle functions for the algorithm), for every constant K big enough, each
computation step of the algorithm can be simulated by exactly K successive
reductions in a natural extension of lambda calculus with constants for
functions in the above considered family. The proof is based on a fixed point
technique in lambda calculus and on Gurevich sequential Thesis which allows to
identify sequential deterministic algorithms with Abstract State Machines. This
extends to algorithms for partial computable functions in such a way that
finite computations ending with exceptions are associated to finite reductions
leading to terms with a particular very simple feature.Comment: 37 page
Unified classical logic completeness: a coinductive pearl
Codatatypes are absent from many programming languages and proof assistants. We make a case for their importance by revisiting a classic result: the completeness theorem for first-order logic established through a Gentzen system. The core of the proof establishes an abstract property of possibly infinite derivation trees, independently of the concrete syntax or inference rules. This separation of concerns simplifies the presentation. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems as well as various flavors of first order logic. The corresponding Isabelle/HOL formalization demonstrates the recently introduced support for codatatypes and the Haskell code generator
On the Generation of Positivstellensatz Witnesses in Degenerate Cases
One can reduce the problem of proving that a polynomial is nonnegative, or
more generally of proving that a system of polynomial inequalities has no
solutions, to finding polynomials that are sums of squares of polynomials and
satisfy some linear equality (Positivstellensatz). This produces a witness for
the desired property, from which it is reasonably easy to obtain a formal proof
of the property suitable for a proof assistant such as Coq. The problem of
finding a witness reduces to a feasibility problem in semidefinite programming,
for which there exist numerical solvers. Unfortunately, this problem is in
general not strictly feasible, meaning the solution can be a convex set with
empty interior, in which case the numerical optimization method fails.
Previously published methods thus assumed strict feasibility; we propose a
workaround for this difficulty. We implemented our method and illustrate its
use with examples, including extractions of proofs to Coq.Comment: To appear in ITP 201
Antiretroviral-naive and -treated HIV-1 patients can harbour more resistant viruses in CSF than in plasma
Objectives The neurological disorders in HIV-1-infected patients remain prevalent. The HIV-1 resistance in plasma and CSF was compared in patients with neurological disorders in a multicentre study. Methods Blood and CSF samples were collected at time of neurological disorders for 244 patients. The viral loads were >50 copies/mL in both compartments and bulk genotypic tests were realized. Results On 244 patients, 89 and 155 were antiretroviral (ARV) naive and ARV treated, respectively. In ARV-naive patients, detection of mutations in CSF and not in plasma were reported for the reverse transcriptase (RT) gene in 2/89 patients (2.2%) and for the protease gene in 1/89 patients (1.1%). In ARV-treated patients, 19/152 (12.5%) patients had HIV-1 mutations only in the CSF for the RT gene and 30/151 (19.8%) for the protease gene. Two mutations appeared statistically more prevalent in the CSF than in plasma: M41L (P = 0.0455) and T215Y (P = 0.0455). Conclusions In most cases, resistance mutations were present and similar in both studied compartments. However, in 3.4% of ARV-naive and 8.8% of ARV-treated patients, the virus was more resistant in CSF than in plasma. These results support the need for genotypic resistance testing when lumbar puncture is performe