704 research outputs found
On Quasi-Interpretations, Blind Abstractions and Implicit Complexity
Quasi-interpretations are a technique to guarantee complexity bounds on
first-order functional programs: with termination orderings they give in
particular a sufficient condition for a program to be executable in polynomial
time, called here the P-criterion. We study properties of the programs
satisfying the P-criterion, in order to better understand its intensional
expressive power. Given a program on binary lists, its blind abstraction is the
nondeterministic program obtained by replacing lists by their lengths (natural
numbers). A program is blindly polynomial if its blind abstraction terminates
in polynomial time. We show that all programs satisfying a variant of the
P-criterion are in fact blindly polynomial. Then we give two extensions of the
P-criterion: one by relaxing the termination ordering condition, and the other
one (the bounded value property) giving a necessary and sufficient condition
for a program to be polynomial time executable, with memoisation.Comment: 18 page
Spiking Neural P Systems with Structural Plasticity: Attacking the Subset Sum Problem
Spiking neural P systems with structural plasticity (in short,
SNPSP systems) are models of computations inspired by the function and
structure of biological neurons. In SNPSP systems, neurons can create
or delete synapses using plasticity rules. We report two families of solutions:
a non-uniform and a uniform one, to the NP-complete problem
Subset Sum using SNPSP systems. Instead of the usual rule-level nondeterminism
(choosing which rule to apply) we use synapse-level nondeterminism
(choosing which synapses to create or delete). The nondeterminism
due to plasticity rules have the following improvements from a
previous solution: in our non-uniform solution, plasticity rules allowed
for a normal form to be used (i.e. without forgetting rules or rules with
delays, system is simple, only synapse-level nondeterminism); in our uniform
solution the number of neurons and the computation steps are
reduced.Ministerio de Economía y Competitividad TIN2012-3743
Generic properties of subgroups of free groups and finite presentations
Asymptotic properties of finitely generated subgroups of free groups, and of
finite group presentations, can be considered in several fashions, depending on
the way these objects are represented and on the distribution assumed on these
representations: here we assume that they are represented by tuples of reduced
words (generators of a subgroup) or of cyclically reduced words (relators).
Classical models consider fixed size tuples of words (e.g. the few-generator
model) or exponential size tuples (e.g. Gromov's density model), and they
usually consider that equal length words are equally likely. We generalize both
the few-generator and the density models with probabilistic schemes that also
allow variability in the size of tuples and non-uniform distributions on words
of a given length.Our first results rely on a relatively mild prefix-heaviness
hypothesis on the distributions, which states essentially that the probability
of a word decreases exponentially fast as its length grows. Under this
hypothesis, we generalize several classical results: exponentially generically
a randomly chosen tuple is a basis of the subgroup it generates, this subgroup
is malnormal and the tuple satisfies a small cancellation property, even for
exponential size tuples. In the special case of the uniform distribution on
words of a given length, we give a phase transition theorem for the central
tree property, a combinatorial property closely linked to the fact that a tuple
freely generates a subgroup. We then further refine our results when the
distribution is specified by a Markovian scheme, and in particular we give a
phase transition theorem which generalizes the classical results on the
densities up to which a tuple of cyclically reduced words chosen uniformly at
random exponentially generically satisfies a small cancellation property, and
beyond which it presents a trivial group
Uniform solutions to SAT and Subset Sum by spiking neural P systems
We continue the investigations concerning the possibility of using spiking neural
P systems as a framework for solving computationally hard problems, addressing two
problems which were already recently considered in this respect: Subset Sum and SAT: For
both of them we provide uniform constructions of standard spiking neural P systems (i.e.,
not using extended rules or parallel use of rules) which solve these problems in a constant
number of steps, working in a non-deterministic way. This improves known results of this
type where the construction was non-uniform, and/or was using various ingredients added
to the initial definition of spiking neural P systems (the SN P systems as defined initially are
called here ‘‘standard’’). However, in the Subset Sum case, a price to pay for this
improvement is that the solution is obtained either in a time which depends on the value of
the numbers involved in the problem, or by using a system whose size depends on the same
values, or again by using complicated regular expressions. A uniform solution to 3-SAT is
also provided, that works in constant time.Ministerio de Educación y Ciencia TIN2006-13425Junta de Andalucía TIC-581Ministerio de Educación y Ciencia HI 2005-019
The complexity of quantified constraints using the algebraic formulation
Peer reviewedFinal Published versio
The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach
We consider the high-frequency Helmholtz equation with a given source term,
and a small absorption parameter \a>0. The high-frequency (or:
semi-classical) parameter is \eps>0. We let \eps and \a go to zero
simultaneously. We assume that the zero energy is non-trapping for the
underlying classical flow. We also assume that the classical trajectories
starting from the origin satisfy a transversality condition, a generic
assumption. Under these assumptions, we prove that the solution u^\eps
radiates in the outgoing direction, {\bf uniformly} in \eps. In particular,
the function u^\eps, when conveniently rescaled at the scale \eps close to
the origin, is shown to converge towards the {\bf outgoing} solution of the
Helmholtz equation, with coefficients frozen at the origin. This provides a
uniform version (in \eps) of the limiting absorption principle. Writing the
resolvent of the Helmholtz equation as the integral in time of the associated
semi-classical Schr\"odinger propagator, our analysis relies on the following
tools: (i) For very large times, we prove and use a uniform version of the
Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove
a uniform dispersive estimate that relies on a wave-packet approach, together
with the above mentioned transversality condition; (iii) for small times, we
prove that the semi-classical Schr\"odinger operator with variable coefficients
has the same dispersive properties as in the constant coefficients case,
uniformly in \eps
- …