Limit Your Consumption! Finding Bounds in Average-energy Games

Energy games are infinite two-player games played in weighted arenas with quantitative objectives that restrict the consumption of a resource modeled by the weights, e.g., a battery that is charged and drained. Typically, upper and/or lower bounds on the battery capacity are part of the problem description. Here, we consider the problem of determining upper bounds on the average accumulated energy or on the capacity while satisfying a given lower bound, i.e., we do not determine whether a given bound is sufficient to meet the specification, but if there exists a sufficient bound to meet it. In the classical setting with positive and negative weights, we show that the problem of determining the existence of a sufficient bound on the long-run average accumulated energy can be solved in doubly-exponential time. Then, we consider recharge games: here, all weights are negative, but there are recharge edges that recharge the energy to some fixed capacity. We show that bounding the long-run average energy in such games is complete for exponential time. Then, we consider the existential version of the problem, which turns out to be solvable in polynomial time: here, we ask whether there is a recharge capacity that allows the system player to win the game. We conclude by studying tradeoffs between the memory needed to implement strategies and the bounds they realize. We give an example showing that memory can be traded for bounds and vice versa. Also, we show that increasing the capacity allows to lower the average accumulated energy.Comment: In Proceedings QAPL'16, arXiv:1610.0769

Average-energy games

Two-player quantitative zero-sum games provide a natural framework to synthesize controllers with performance guarantees for reactive systems within an uncontrollable environment. Classical settings include mean-payoff games, where the objective is to optimize the long-run average gain per action, and energy games, where the system has to avoid running out of energy. We study average-energy games, where the goal is to optimize the long-run average of the accumulated energy. We show that this objective arises naturally in several applications, and that it yields interesting connections with previous concepts in the literature. We prove that deciding the winner in such games is in NP inter coNP and at least as hard as solving mean-payoff games, and we establish that memoryless strategies suffice to win. We also consider the case where the system has to minimize the average-energy while maintaining the accumulated energy within predefined bounds at all times: this corresponds to operating with a finite-capacity storage for energy. We give results for one-player and two-player games, and establish complexity bounds and memory requirements.Comment: In Proceedings GandALF 2015, arXiv:1509.0685

The hermitian Wilson-Dirac operator in smooth SU(2) instanton backgrounds

We study the spectral flow of the hermitian Wilson-Dirac operator \ham(m) as a function of $m$ in smooth SU(2) instanton backgrounds on the lattice. For a single instanton background with Dirichlet boundary conditions on \ham(m), we find a level crossing in the spectral flow of \ham(m), and we find the shape of the crossing mode at the crossing point to be in good agreement with the zero mode associated with the single instanton background. With anti-periodic boundary conditions on \ham(m), we find that the instanton background in the singular gauge has the correct spectral flow but the one in regular gauge does not. We also investigate the spectral flows of two instanton and instanton-anti-instanton backgrounds.Comment: 18 pages, Latex file, 12 postscript figure

A Study of Practical Implementations of the Overlap-Dirac Operator in Four Dimensions

We study three practical implementations of the Overlap-Dirac operator $D_o= (1/2) [1 + \gamma_5\epsilon(H_w)]$ in four dimensions. Two implementations are based on different representations of $\epsilon(H_w)$ as a sum over poles. One of them is a polar decomposition and the other is an optimal fit to a ratio of polynomials. The third one is obtained by representing $\epsilon(H_w)$ using Gegenbauer polynomials and is referred to as the fractional inverse method. After presenting some spectral properties of the Hermitian operator $H_o=\gamma_5 D_o$, we study its spectrum in a smooth SU(2) instanton background with the aim of comparing the three implementations of $D_o$. We also present some results in SU(2) gauge field backgrounds generated at $\beta=2.5$ on an $8^4$ lattice. Chiral properties have been numerically verified.Comment: 23 pages latex with 9 postscript figures included by epsf. Some change in referencing and one figure modifie

γZ\gamma Z Pair Production at the Photon Linear Collider

$\gamma\gamma\to\gamma Z$ scattering at the Photon Linear Collider is considered. Explicit formulas for helicity amplitudes due to $W$ boson loops are presented. It is shown that the $Z\gamma$ pair production will be easily observable at PLC and separation of the $W$ loop contribution will be possible at $e^+e^-$ c.m. energy of 300~GeV or higher.Comment: 9 pages of standard LaTeX + 3 PostScript figures (uuencoded and compressed

Theoretical Analysis of Acceptance Rates in Multigrid Monte Carlo

We analyze the kinematics of multigrid Monte Carlo algorithms by investigating acceptance rates for nonlocal Metropolis updates. With the help of a simple criterion we can decide whether or not a multigrid algorithm will have a chance to overcome critial slowing down for a given model. Our method is introduced in the context of spin models. A multigrid Monte Carlo procedure for nonabelian lattice gauge theory is described, and its kinematics is analyzed in detail.Comment: 7 pages, no figures, (talk at LATTICE 92 in Amsterdam

Multigrid Monte Carlo Algorithms for SU(2) Lattice Gauge Theory: Two versus Four Dimensions

We study a multigrid method for nonabelian lattice gauge theory, the time slice blocking, in two and four dimensions. For SU(2) gauge fields in two dimensions, critical slowing down is almost completely eliminated by this method. This result is in accordance with theoretical arguments based on the analysis of the scale dependence of acceptance rates for nonlocal Metropolis updates. The generalization of the time slice blocking to SU(2) in four dimensions is investigated analytically and by numerical simulations. Compared to two dimensions, the local disorder in the four dimensional gauge field leads to kinematical problems.Comment: 24 pages, PostScript file (compressed and uuencoded), preprint MS-TPI-94-