6,203 research outputs found
On the strength of proof-irrelevant type theories
We present a type theory with some proof-irrelevance built into the
conversion rule. We argue that this feature is useful when type theory is used
as the logical formalism underlying a theorem prover. We also show a close
relation with the subset types of the theory of PVS. We show that in these
theories, because of the additional extentionality, the axiom of choice implies
the decidability of equality, that is, almost classical logic. Finally we
describe a simple set-theoretic semantics.Comment: 20 pages, Logical Methods in Computer Science, Long version of IJCAR
2006 pape
Hard probes and the event generator EPOS
After a short presentation of the event generator EPOS, we discuss the
production of heavy quarks and prompt photons which has been recently
implemented. Whereas we have satisfying results for the charm, work on photons
is still in progress
Self tolerance in a minimal model of the idiotypic network
We consider the problem of self tolerance in the frame of a minimalistic
model of the idiotypic network. A node of this network represents a population
of B lymphocytes of the same idiotype which is encoded by a bit string. The
links of the network connect nodes with (nearly) complementary strings. The
population of a node survives if the number of occupied neighbours is not too
small and not too large. There is an influx of lymphocytes with random idiotype
from the bone marrow. Previous investigations have shown that this system
evolves toward highly organized architectures, where the nodes can be
classified into groups according to their statistical properties. The building
principles of these architectures can be analytically described and the
statistical results of simulations agree very well with results of a modular
mean field theory. In this paper we present simulation results for the case
that one or several nodes, playing the role of self, are permanently occupied.
We observe that the group structure of the architecture is very similar to the
case without self antigen, but organized such that the neighbours of the self
are only weakly occupied, thus providing self tolerance. We also treat this
situation in mean field theory which give results in good agreement with data
from simulation.Comment: 7 pages, 6 figures, 1 tabl
Proof-irrelevant model of CC with predicative induction and judgmental equality
We present a set-theoretic, proof-irrelevant model for Calculus of
Constructions (CC) with predicative induction and judgmental equality in
Zermelo-Fraenkel set theory with an axiom for countably many inaccessible
cardinals. We use Aczel's trace encoding which is universally defined for any
function type, regardless of being impredicative. Direct and concrete
interpretations of simultaneous induction and mutually recursive functions are
also provided by extending Dybjer's interpretations on the basis of Aczel's
rule sets. Our model can be regarded as a higher-order generalization of the
truth-table methods. We provide a relatively simple consistency proof of type
theory, which can be used as the basis for a theorem prover
La Vérité et la Machine
www.math.cnrs.frEn quelques pages nous présentons des avancées récentes où l'ordinateur permet d'établir des vérités mathématiques
Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation
We consider the problem of certifying an inequality of the form ,
, where is a multivariate transcendental function, and
is a compact semialgebraic set. We introduce a certification method, combining
semialgebraic optimization and max-plus approximation. We assume that is
given by a syntaxic tree, the constituents of which involve semialgebraic
operations as well as some transcendental functions like , ,
, etc. We bound some of these constituents by suprema or infima of
quadratic forms (max-plus approximation method, initially introduced in optimal
control), leading to semialgebraic optimization problems which we solve by
semidefinite relaxations. The max-plus approximation is iteratively refined and
combined with branch and bound techniques to reduce the relaxation gap.
Illustrative examples of application of this algorithm are provided, explaining
how we solved tight inequalities issued from the Flyspeck project (one of the
main purposes of which is to certify numerical inequalities used in the proof
of the Kepler conjecture by Thomas Hales).Comment: 7 pages, 3 figures, 3 tables, Appears in the Proceedings of the
European Control Conference ECC'13, July 17-19, 2013, Zurich, pp. 2244--2250,
copyright EUCA 201
Why is the Court of Justice of the European Union accepted? Three mechanisms of opposition abatement
The Court of Justice of the European Union (CJEU) played a very important role in the process of European integration. Its jurisprudence has again and again strengthened the competencies of the supranational level to the disadvantage of the member states. The CJEU has always been criticized for this pro-integrationist activism but that never had a serious impact on the court’s behavior. In recent years, however, the environment for legal integration has changed: The CJEU is increasingly treading on political sensitive issues; and that in a period when the integration project as such is becoming more and more contested. Scholars of legal integration have expected that this would lead to more criticism of, resistance to or even attacks on the court’s power and thus to a changing or less important role of the CJEU in the integration process. Yet, this expectation has not been fulfilled. Although there have been much controversy on some recent CJEU decisions, this criticism has never exceeded the local stage and led to attempts to recast the Court’s role. The present article approaches this puzzle by investigating the CJEU’s jurisprudence on Golden Shares, one of the most controversial lines of case law in recent years. By doing so, three mechanisms of opposition abatement are identified: Uncertainty about the future development of the case law, case-specific concessions made by the CJEU, and limited damage in the concrete cases at hand
Formal Proofs for Nonlinear Optimization
We present a formally verified global optimization framework. Given a
semialgebraic or transcendental function and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of 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
Certification of Real Inequalities -- Templates and Sums of Squares
We consider the problem of certifying lower bounds for real-valued
multivariate transcendental functions. The functions we are dealing with are
nonlinear and involve semialgebraic operations as well as some transcendental
functions like , , , etc. Our general framework is to use
different approximation methods to relax the original problem into polynomial
optimization problems, which we solve by sparse sums of squares relaxations. In
particular, we combine the ideas of the maxplus estimators (originally
introduced in optimal control) and of the linear templates (originally
introduced in static analysis by abstract interpretation). The nonlinear
templates control the complexity of the semialgebraic relaxations at the price
of coarsening the maxplus approximations. In that way, we arrive at a new -
template based - certified global optimization method, which exploits both the
precision of sums of squares relaxations and the scalability of abstraction
methods. We analyze the performance of the method on problems from the global
optimization literature, as well as medium-size inequalities issued from the
Flyspeck project.Comment: 27 pages, 3 figures, 4 table
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