Algorithms for Game Metrics

Simulation and bisimulation metrics for stochastic systems provide a quantitative generalization of the classical simulation and bisimulation relations. These metrics capture the similarity of states with respect to quantitative specifications written in the quantitative {\mu}-calculus and related probabilistic logics. We first show that the metrics provide a bound for the difference in long-run average and discounted average behavior across states, indicating that the metrics can be used both in system verification, and in performance evaluation. For turn-based games and MDPs, we provide a polynomial-time algorithm for the computation of the one-step metric distance between states. The algorithm is based on linear programming; it improves on the previous known exponential-time algorithm based on a reduction to the theory of reals. We then present PSPACE algorithms for both the decision problem and the problem of approximating the metric distance between two states, matching the best known algorithms for Markov chains. For the bisimulation kernel of the metric our algorithm works in time O(n^4) for both turn-based games and MDPs; improving the previously best known O(n^9\cdot log(n)) time algorithm for MDPs. For a concurrent game G, we show that computing the exact distance between states is at least as hard as computing the value of concurrent reachability games and the square-root-sum problem in computational geometry. We show that checking whether the metric distance is bounded by a rational r, can be done via a reduction to the theory of real closed fields, involving a formula with three quantifier alternations, yielding O(|G|^O(|G|^5)) time complexity, improving the previously known reduction, which yielded O(|G|^O(|G|^7)) time complexity. These algorithms can be iterated to approximate the metrics using binary search.Comment: 27 pages. Full version of the paper accepted at FSTTCS 200

On the equivalence of game and denotational semantics for the probabilistic mu-calculus

The probabilistic (or quantitative) modal mu-calculus is a fixed-point logic de- signed for expressing properties of probabilistic labeled transition systems (PLTS). Two semantics have been studied for this logic, both assigning to every process state a value in the interval [0,1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic games. The two semantics have been proved to coincide on all finite PLTS's, but the equivalence of the two semantics on arbitrary models has been open in literature. In this paper we prove that the equivalence indeed holds for arbitrary infinite models, and thus our result strengthens the fruitful connection between denotational and game semantics. Our proof adapts the unraveling or unfolding method, a general proof technique for proving result of parity games by induction on their complexity

Higher harmonic anisotropic flow measurements of charged particles in Pb-Pb collisions at 2.76 TeV

We report on the first measurement of the triangular $v_3$, quadrangular $v_4$, and pentagonal $v_5$ charged particle flow in Pb-Pb collisions at 2.76 TeV measured with the ALICE detector at the CERN Large Hadron Collider. We show that the triangular flow can be described in terms of the initial spatial anisotropy and its fluctuations, which provides strong constraints on its origin. In the most central events, where the elliptic flow $v_2$ and $v_3$ have similar magnitude, a double peaked structure in the two-particle azimuthal correlations is observed, which is often interpreted as a Mach cone response to fast partons. We show that this structure can be naturally explained from the measured anisotropic flow Fourier coefficients.Comment: 10 pages, 4 figures, published version, figures at http://aliceinfo.cern.ch/ArtSubmission/node/387

Measurement of charm production at central rapidity in proton-proton collisions at root s=7 TeV

The p(t)-differential inclusive production cross sections of the prompt charmed mesons D-0, D+, and D*(+) in the rapidity range vertical bar y vertical bar < 0.5 were measured in proton-proton collisions at root s = 7 TeV at the LHC using the ALICE detector. Reconstructing the decays D-0 -> K-pi(+), D+ -> K-pi(+)pi(+), D*(+) -> D-0 pi(+), and their charge conjugates, about 8,400 D-0, 2,900 D+, and 2,600 D*(+) mesons with 1 < p(t) < 24 GeV/c were counted, after selection cuts, in a data sample of 3.14 x 10(8) events collected with a minimum-bias trigger (integrated luminosity L-int = 5 nb(-1)). The results are described within uncertainties by predictions based on perturbative QCD

Charged-Particle Multiplicity Density at Midrapidity in Central Pb-Pb Collisions at root s(NN)=2.76 TeV

The first measurement of the charged-particle multiplicity density at midrapidity in Pb-Pb collisions at a center-of-mass energy per nucleon pair root s(NN) = 2.76 TeV is presented. For an event sample corresponding to the most central 5% of the hadronic cross section, the pseudorapidity density of primary charged particles at midrapidity is 1584 +/- 4(stat) +/- 76(syst), which corresponds to 8.3 +/- 0.4(syst) per participating nucleon pair. This represents an increase of about a factor 1.9 relative to pp collisions at similar collision energies, and about a factor 2.2 to central Au-Au collisions at root s(NN) = 0.2 TeV. This measurement provides the first experimental constraint for models of nucleus-nucleus collisions at LHC energies

Higher Harmonic Anisotropic Flow Measurements of Charged Particles in Pb-Pb Collisions at root s(NN)=2.76 TeV

We report on the first measurement of the triangular nu(3), quadrangular nu(4), and pentagonal nu(5) charged particle flow in Pb-Pb collisions at root s(NN) = 2.76 TeV measured with the ALICE detector at the CERN Large Hadron Collider. We show that the triangular flow can be described in terms of the initial spatial anisotropy and its fluctuations, which provides strong constraints on its origin. In the most central events, where the elliptic flow nu(2) and nu(3) have similar magnitude, a double peaked structure in the two-particle azimuthal correlations is observed, which is often interpreted as a Mach cone response to fast partons. We show that this structure can be naturally explained from the measured anisotropic flow Fourier coefficients

Measurement of event background fluctuations for charged particle jet reconstruction in Pb-Pb collisions at root s(NN)=2.76 TeV

The effect of event background fluctuations on charged particle jet reconstruction in Pb-Pb collisions at root s(NN) = 2.76 TeV has been measured with the ALICE experiment. The main sources of non-statistical fluctuations are characterized based purely on experimental data with an unbiased method, as well as by using single high p(t) particles and simulated jets embedded into real Pb-Pb events and reconstructed with the anti-k(t) jet finder. The influence of a low transverse momentum cut-off on particles used in the jet reconstruction is quantified by varying the minimum track p(t) between 0.15 GeV/c and 2 GeV/c. For embedded jets reconstructed from charged particles with p(t) > 0.15 GeV/c, the uncertainty in the reconstructed jet transverse momentum due to the heavy-ion background is measured to be 11.3 GeV/c (standard deviation) for the 10% most central Pb-Pb collisions, slightly larger than the value of 11.0 GeV/c measured using the unbiased method. For a higher particle transverse momentum threshold of 2 GeV/c, which will generate a stronger bias towards hard fragmentation in the jet finding process, the standard deviation of the fluctuations in the reconstructed jet transverse momentum is reduced to 4.8-5.0 GeV/c for the 10% most central events. A non-Gaussian tail of the momentum uncertainty is observed and its impact on the reconstructed jet spectrum is evaluated for varying particle momentum thresholds, by folding the measured fluctuations with steeply falling spectra

Centrality Dependence of the Charged-Particle Multiplicity Density at Midrapidity in Pb-Pb Collisions at root s(NN)=2.76 TeV

The centrality dependence of the charged-particle multiplicity density at midrapidity in Pb-Pb collisions at root s(NN) = 2: 76 TeV is presented. The charged-particle density normalized per participating nucleon pair increases by about a factor of 2 from peripheral (70%-80%) to central (0%-5%) collisions. The centrality dependence is found to be similar to that observed at lower collision energies. The data are compared with models based on different mechanisms for particle production in nuclear collisions

Measurement of inelastic, single- and double-diffraction cross sections in proton-proton collisions at the LHC with ALICE

Measurements of cross sections of inelastic and diffractive processes in proton-proton collisions at LHC energies were carried out with the ALICE detector. The fractions of diffractive processes in inelastic collisions were determined from a study of gaps in charged particle pseudorapidity distributions: for single diffraction (diffractive mass M-X < 200 GeV/c(2)) sigma(SD)/sigma(INEL) = 0.21 +/- 0.03, 0.20(-0.08)(+0.07,) and 0.20(-0.07)(+0.04), respectively at centre-of-mass energies root s = 0.9, 2.76, and 7 TeV; for double diffraction (for a pseudorapidity gap Delta eta > 3) sigma(DD)/sigma(INEL) = 0.11 +/- 0.03, 0.12 +/- 0.05, and 0.12(-0.04)(+0.05), respectively at root s = 0.9, 2.76, and 7 TeV. To measure the inelastic cross section, beam properties were determined with van der Meer scans, and, using a simulation of diffraction adjusted to data, the following values were obtained: sigma(INEL) = 62.8(-4.0)(+2.4)(model) +/- 1.2(lumi) mb at root s = 2.76 TeV and 73.2(-4.6)(+2.0)(model) +/- 2.6(lumi) mb at root s = 7 TeV. The single- and double-diffractive cross sections were calculated combining relative rates of diffraction with inelastic cross sections. The results are compared to previous measurements at proton-antiproton and proton-proton colliders at lower energies, to measurements by other experiments at the LHC, and to theoretical models

Transverse sphericity of primary charged particles in minimum bias proton-proton collisions at root s=0.9, 2.76 and 7 TeV

Measurements of the sphericity of primary charged particles in minimum bias proton-proton collisions at root s = 0.9, 2.76 and 7 TeV with the ALICE detector at the LHC are presented. The observable is measured in the plane perpendicular to the beam direction using primary charged tracks with p(T) > 0.5 GeV/c in vertical bar eta vertical bar < 0.8. The mean sphericity as a function of the charged particle multiplicity at mid-rapidity (N-ch) is reported for events with different p(T) scales (”soft” and “hard”) defined by the transverse momentum of the leading particle. In addition, the mean charged particle transverse momentum versus multiplicity is presented for the different event classes, and the sphericity distributions in bins of multiplicity are presented. The data are compared with calculations of standard Monte Carlo event generators. The transverse sphericity is found to grow with multiplicity at all collision energies, with a steeper rise at low N-ch, whereas the event generators show an opposite tendency. The combined study of the sphericity and the mean p(T) with multiplicity indicates that most of the tested event generators produce events with higher multiplicity by generating more back-to-back jets resulting in decreased sphericity (and isotropy). The PYTHIA6 generator with tune PERUGIA-2011 exhibits a noticeable improvement in describing the data, compared to the other tested generators