145,649 research outputs found
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
Characterizing and Reasoning about Probabilistic and Non-Probabilistic Expectation
Expectation is a central notion in probability theory. The notion of
expectation also makes sense for other notions of uncertainty. We introduce a
propositional logic for reasoning about expectation, where the semantics
depends on the underlying representation of uncertainty. We give sound and
complete axiomatizations for the logic in the case that the underlying
representation is (a) probability, (b) sets of probability measures, (c) belief
functions, and (d) possibility measures. We show that this logic is more
expressive than the corresponding logic for reasoning about likelihood in the
case of sets of probability measures, but equi-expressive in the case of
probability, belief, and possibility. Finally, we show that satisfiability for
these logics is NP-complete, no harder than satisfiability for propositional
logic.Comment: To appear in Journal of the AC
A logic for reasoning about upper probabilities
We present a propositional logic %which can be used to reason about the
uncertainty of events, where the uncertainty is modeled by a set of probability
measures assigning an interval of probability to each event. We give a sound
and complete axiomatization for the logic, and show that the satisfiability
problem is NP-complete, no harder than satisfiability for propositional logic.Comment: A preliminary version of this paper appeared in Proc. of the 17th
Conference on Uncertainty in AI, 200
On Aerts' overlooked solution to the EPR paradox
The Einstein-Podolsky-Rosen (EPR) paradox was enunciated in 1935 and since
then it has made a lot of ink flow. Being a subtle result, it has also been
largely misunderstood. Indeed, if questioned about its solution, many
physicists will still affirm today that the paradox has been solved by the
Bell-test experimental results, which have shown that entangled states are
real. However, this remains a wrong view, as the validity of the EPR
ex-absurdum reasoning is independent from the Bell-test experiments, and the
possible structural shortcomings it evidenced cannot be eliminated. These were
correctly identified by the Belgian physicist Diederik Aerts, in the eighties
of last century, and are about the inability of the quantum formalism to
describe separate physical systems. The purpose of the present article is to
bring Aerts' overlooked result to the attention again of the physics'
community, explaining its content and implications
Quantum Team Logic and Bell's Inequalities
A logical approach to Bell's Inequalities of quantum mechanics has been
introduced by Abramsky and Hardy [2]. We point out that the logical Bell's
Inequalities of [2] are provable in the probability logic of Fagin, Halpern and
Megiddo [4]. Since it is now considered empirically established that quantum
mechanics violates Bell's Inequalities, we introduce a modified probability
logic, that we call quantum team logic, in which Bell's Inequalities are not
provable, and prove a Completeness Theorem for this logic. For this end we
generalise the team semantics of dependence logic [7] first to probabilistic
team semantics, and then to what we call quantum team semantics
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