Skolem Functions for Factored Formulas

Given a propositional formula F(x,y), a Skolem function for x is a function \Psi(y), such that substituting \Psi(y) for x in F gives a formula semantically equivalent to \exists F. Automatically generating Skolem functions is of significant interest in several applications including certified QBF solving, finding strategies of players in games, synthesising circuits and bit-vector programs from specifications, disjunctive decomposition of sequential circuits etc. In many such applications, F is given as a conjunction of factors, each of which depends on a small subset of variables. Existing algorithms for Skolem function generation ignore any such factored form and treat F as a monolithic function. This presents scalability hurdles in medium to large problem instances. In this paper, we argue that exploiting the factored form of F can give significant performance improvements in practice when computing Skolem functions. We present a new CEGAR style algorithm for generating Skolem functions from factored propositional formulas. In contrast to earlier work, our algorithm neither requires a proof of QBF satisfiability nor uses composition of monolithic conjunctions of factors. We show experimentally that our algorithm generates smaller Skolem functions and outperforms state-of-the-art approaches on several large benchmarks.Comment: Full version of FMCAD 2015 conference publicatio

The role of finite-size effects on the spectrum of equivalent photons in proton-proton collisions at the LHC

Photon-photon interactions represent an important class of physics processes at the LHC, where quasi-real photons are emitted by both colliding protons. These reactions can result in the exclusive production of a final state $X$, $p+p \rightarrow p+p+X$. When computing such cross sections, it has already been shown that finite size effects of colliding protons are important to consider for a realistic estimate of the cross sections. These first results have been essential in understanding the physics case of heavy-ion collisions in the low invariant mass range, where heavy ions collide to form an exclusive final state like a $J/\Psi$ vector meson. In this paper, our purpose is to present some calculations that are valid also for the exclusive production of high masses final states in proton-proton collisions, like the production of a pair of $W$ bosons or the Higgs boson. Therefore, we propose a complete treatment of the finite size effects of incident protons irrespective of the mass range explored in the collision. Our expectations are shown to be in very good agreement with existing experimental data obtained at the LHC.Comment: 14 pages, 7 figures, 1 table, submitted to Phys. Lett.

Numerical verification of Littlewood's bounds for ∣L(1,χ)∣\vert L(1,\chi)\vert

Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $\chi$ defined $\bmod\ q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $\vert L(1,\chi) \vert$ using the values of Euler's $\Gamma$ function. We also introduce an alternative way of computing $\log \Gamma(x)$ and $\psi(x)= \Gamma^\prime/\Gamma(x)$,$x\in(0,1)$. Using such algorithms we numerically verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan estimates on $\vert L(1,\chi) \vert$, where $\chi$ runs over the non trivial primitive Dirichlet characters $\bmod\ q$, for every odd prime $q$ up to $10^7$. The programs used and the results here described are collected at the following address \url{http://www.math.unipd.it/~languasc/Littlewood_ineq.html}.Comment: 21 pages, 2 tables, 12 figures. One reference update

Decay of the pseudoscalar glueball into vector, axial-vector, scalar and pseudoscalar mesons

We resume the investigation of the ground-state pseudoscalar glueball, $J^{PC}=0^{-+}$, by computing its two- and three-body decays into vector and axial-vector quark-antiquark meson fields additional to scalar and pseudoscalar mesons through the construction of an interaction chiral Lagrangian that produces these decays. We evaluate the branching ratio, via a parameter-free calculation, by setting the mass of the pseudoscalar glueball to $2.6$ GeV as predicted by lattice QCD simulations. We duplicate the computation for the branching ratios for a pseudoscalar glueball mass $2.37$ GeV which matches to the resonance $X(2370)$ mass in the BESIII experiment suppress measuring. We observe that the $\rho\pi$ decay mode is the largest, followed by the $K^*(892)K$ one. Both of them are sizable and could explain the $\rho\pi$ puzzle of the charmonium state $\psi(2S)$. The present channels and states are potentially reached and are interesting for the running BESIII and Belle-II experiments and the planned PANDA experiment at FAIR/GSI which will be able to detect the pseudoscalar glueball within the accessible energy range.Comment: 11 pages, 2 tables, 1 figur

Object-oriented implementations of the MPDATA advection equation solver in C++, Python and Fortran

Three object-oriented implementations of a prototype solver of the advection equation are introduced. The presented programs are based on Blitz++ (C++), NumPy (Python), and Fortran's built-in array containers. The solvers include an implementation of the Multidimensional Positive-Definite Advective Transport Algorithm (MPDATA). The introduced codes exemplify how the application of object-oriented programming (OOP) techniques allows to reproduce the mathematical notation used in the literature within the program code. A discussion on the tradeoffs of the programming language choice is presented. The main angles of comparison are code brevity and syntax clarity (and hence maintainability and auditability) as well as performance. In the case of Python, a significant performance gain is observed when switching from the standard interpreter (CPython) to the PyPy implementation of Python. Entire source code of all three implementations is embedded in the text and is licensed under the terms of the GNU GPL license