Separation for dot-depth two

The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages of finite words. By a theorem of McNaughton and Papert, these are also the first-order definable languages. The dot-depth rose to prominence following the work of Thomas, who proved an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level in the dot-depth hierarchy consists of all languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to settle whether the membership problem is decidable for each level: is it possible to decide whether an input regular language belongs to this level? Despite a significant research effort, membership by itself has only been solved for low levels. A recent breakthrough was achieved by replacing membership with a more general problem: separation. Given two input languages, one has to decide whether there exists a third language in the investigated level containing the first language and disjoint from the second. The motivation is that: (1) while more difficult, separation is more rewarding (2) it provides a more convenient framework (3) all recent membership algorithms are reductions to separation for lower levels. We present a separation algorithm for dot-depth two. While this is our most prominent application, our result is more general. We consider a family of hierarchies that includes the dot-depth: concatenation hierarchies. They are built via a generic construction process. One first chooses an initial class, the basis, which is the lowest level in the hierarchy. Further levels are built by applying generic operations. Our main theorem states that for any concatenation hierarchy whose basis is finite, separation is decidable for level one. In the special case of the dot-depth, this can be lifted to level two using previously known results

Separating regular languages with two quantifier alternations

We investigate a famous decision problem in automata theory: separation. Given a class of language C, the separation problem for C takes as input two regular languages and asks whether there exists a third one which belongs to C, includes the first one and is disjoint from the second. Typically, obtaining an algorithm for separation yields a deep understanding of the investigated class C. This explains why a lot of effort has been devoted to finding algorithms for the most prominent classes. Here, we are interested in classes within concatenation hierarchies. Such hierarchies are built using a generic construction process: one starts from an initial class called the basis and builds new levels by applying generic operations. The most famous one, the dot-depth hierarchy of Brzozowski and Cohen, classifies the languages definable in first-order logic. Moreover, it was shown by Thomas that it corresponds to the quantifier alternation hierarchy of first-order logic: each level in the dot-depth corresponds to the languages that can be defined with a prescribed number of quantifier blocks. Finding separation algorithms for all levels in this hierarchy is among the most famous open problems in automata theory. Our main theorem is generic: we show that separation is decidable for the level 3/2 of any concatenation hierarchy whose basis is finite. Furthermore, in the special case of the dot-depth, we push this result to the level 5/2. In logical terms, this solves separation for $\Sigma_3$: first-order sentences having at most three quantifier blocks starting with an existential one

A New Algorithm for Supernova Neutrino Transport and Some Applications

We have developed an implicit, multi-group, time-dependent, spherical neutrino transport code based on the Feautrier variables, the tangent-ray method, and accelerated ${\bf \Lambda}$ iteration. The code achieves high angular resolution, is good to O($v/c$), is equivalent to a Boltzmann solver (without gravitational redshifts), and solves the transport equation at all optical depths with precision. In this paper, we present our formulation of the relevant numerics and microphysics and explore protoneutron star atmospheres for snapshot post-bounce models. Our major focus is on spectra, neutrino-matter heating rates, Eddington factors, angular distributions, and phase-space occupancies. In addition, we investigate the influence on neutrino spectra and heating of final-state electron blocking, stimulated absorption, velocity terms in the transport equation, neutrino-nucleon scattering asymmetry, and weak magnetism and recoil effects. Furthermore, we compare the emergent spectra and heating rates obtained using full transport with those obtained using representative flux-limited transport formulations to gauge their accuracy and viability. Finally, we derive useful formulae for the neutrino source strength due to nucleon-nucleon bremsstrahlung and determine bremsstrahlung's influence on the emergent $\nu_{\mu}$ and $\nu_{\tau}$ neutrino spectra.Comment: 58 pages, single-spaced LaTeX, 23 figures, revised title, also available at http://jupiter.as.arizona.edu/~burrows/papers, accepted for publication in the Ap.

Logic Meets Algebra: the Case of Regular Languages

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.Comment: 37 page

Population III Star Formation in a Lambda WDM Universe

In this paper we examine aspects of primordial star formation in a gravitino warm dark matter universe with a cosmological constant. We compare a set of simulations using a single cosmological realization but with a wide range of warm dark matter particle masses which have not yet been conclusively ruled out by observations. The addition of a warm dark matter component to the initial power spectrum results in a delay in the collapse of high density gas at the center of the most massive halo in the simulation and, as a result, an increase in the virial mass of this halo at the onset of baryon collapse. Both of these effects become more pronounced as the warm dark matter particle mass becomes smaller. A cosmology using a gravitino warm dark matter power spectrum assuming a particle mass of m_{WDM} ~ 40keV is effectively indistinguishable from the cold dark matter case, whereas the m_{WDM} ~ 15 keV case delays star formation by approx. 10^8 years. There is remarkably little scatter between simulations in the final properties of the primordial protostar which forms at the center of the halo, possibly due to the overall low rate of halo mergers which is a result of the WDM power spectrum. The detailed evolution of the collapsing halo core in two representative WDM cosmologies is described. At low densities (n_{b} <= 10^5 cm^{-3}), the evolution of the two calculations is qualitatively similar, but occurs on significantly different timescales, with the halo in the lower particle mass calculation taking much longer to evolve over the same density range and reach runaway collapse. Once the gas in the center of the halo reaches relatively high densities (n_{b} >= 10^5 cm^{-3}) the overall evolution is essentially identical in the two calculations.Comment: 36 pages, 12 figures (3 color). Astrophysical Journal, accepte

Optimized Two-Baseline Beta-Beam Experiment

We propose a realistic Beta-Beam experiment with four source ions and two baselines for the best possible sensitivity to theta_{13}, CP violation and mass hierarchy. Neutrinos from 18Ne and 6He with Lorentz boost gamma=350 are detected in a 500 kton water Cerenkov detector at a distance L=650 km (first oscillation peak) from the source. Neutrinos from 8B and 8Li are detected in a 50 kton magnetized iron detector at a distance L=7000 km (magic baseline) from the source. Since the decay ring requires a tilt angle of 34.5 degrees to send the beam to the magic baseline, the far end of the ring has a maximum depth of d=2132 m for magnetic field strength of 8.3 T, if one demands that the fraction of ions that decay along the straight sections of the racetrack geometry decay ring (called livetime) is 0.3. We alleviate this problem by proposing to trade reduction of the livetime of the decay ring with the increase in the boost factor of the ions, such that the number of events at the detector remains almost the same. This allows to substantially reduce the maximum depth of the decay ring at the far end, without significantly compromising the sensitivity of the experiment to the oscillation parameters. We take 8B and 8Li with gamma=390 and 656 respectively, as these are the largest possible boost factors possible with the envisaged upgrades of the SPS at CERN. This allows us to reduce d of the decay ring by a factor of 1.7 for 8.3 T magnetic field. Increase of magnetic field to 15 T would further reduce d to 738 m only. We study the sensitivity reach of this two baseline two storage ring Beta-Beam experiment, and compare it with the corresponding reach of the other proposed facilities.Comment: 17 pages, 3 eps figures. Minor changes, matches version accepted in JHE