Memory lower bounds for deterministic self-stabilization

In the context of self-stabilization, a \emph{silent} algorithm guarantees that the register of every node does not change once the algorithm has stabilized. At the end of the 90's, Dolev et al. [Acta Inf. '99] showed that, for finding the centers of a graph, for electing a leader, or for constructing a spanning tree, every silent algorithm must use a memory of $\Omega(\log n)$ bits per register in $n$-node networks. Similarly, Korman et al. [Dist. Comp. '07] proved, using the notion of proof-labeling-scheme, that, for constructing a minimum-weight spanning trees (MST), every silent algorithm must use a memory of $\Omega(\log^2n)$ bits per register. It follows that requiring the algorithm to be silent has a cost in terms of memory space, while, in the context of self-stabilization, where every node constantly checks the states of its neighbors, the silence property can be of limited practical interest. In fact, it is known that relaxing this requirement results in algorithms with smaller space-complexity. In this paper, we are aiming at measuring how much gain in terms of memory can be expected by using arbitrary self-stabilizing algorithms, not necessarily silent. To our knowledge, the only known lower bound on the memory requirement for general algorithms, also established at the end of the 90's, is due to Beauquier et al.~[PODC '99] who proved that registers of constant size are not sufficient for leader election algorithms. We improve this result by establishing a tight lower bound of $\Theta(\log \Delta+\log \log n)$ bits per register for self-stabilizing algorithms solving $(\Delta+1)$-coloring or constructing a spanning tree in networks of maximum degree~$\Delta$. The lower bound $\Omega(\log \log n)$ bits per register also holds for leader election

Continuity, Deconfinement, and (Super) Yang-Mills Theory

We study the phase diagram of SU(2) Yang-Mills theory with one adjoint Weyl fermion on R^3xS^1 as a function of the fermion mass m and the compactification scale L. This theory reduces to thermal pure gauge theory as m->infinity and to circle-compactified (non-thermal) supersymmetric gluodynamics in the limit m->0. In the m-L plane, there is a line of center symmetry changing phase transitions. In the limit m->infinity, this transition takes place at L_c=1/T_c, where T_c is the critical temperature of the deconfinement transition in pure Yang-Mills theory. We show that near m=0, the critical compactification scale L_c can be computed using semi-classical methods and that the transition is of second order. This suggests that the deconfining phase transition in pure Yang-Mills theory is continuously connected to a transition that can be studied at weak coupling. The center symmetry changing phase transition arises from the competition of perturbative contributions and monopole-instantons that destabilize the center, and topological molecules (neutral bions) that stabilize the center. The contribution of molecules can be computed using supersymmetry in the limit m=0, and via the Bogomolnyi--Zinn-Justin (BZJ) prescription in the non-supersymmetric gauge theory. Finally, we also give a detailed discussion of an issue that has not received proper attention in the context of N=1 theories---the non-cancellation of nonzero-mode determinants around supersymmetric BPS and KK monopole-instanton backgrounds on R^3xS^1. We explain why the non-cancellation is required for consistency with holomorphy and supersymmetry and perform an explicit calculation of the one-loop determinant ratio.Comment: A discussion of the non-cancellation of the nonzero mode determinants around supersymmetric monopole-instantons in N=1 SYM on R^3xS^1 is added, including an explicit calculation. The non-cancellation is, in fact, required by supersymmetry and holomorphy in order for the affine-Toda superpotential to be reproduced. References have also been adde

Leader Election in Anonymous Rings: Franklin Goes Probabilistic

We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin, augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques can be employed to verify its correctness, that is, eventually a unique leader is elected with probability one. We also sketch a formal correctness proof of the algorithm for rings with arbitrary size

An Inflaton Mass Problem in String Inflation from Threshold Corrections to Volume Stabilization

Inflationary models whose vacuum energy arises from a D-term are believed not to suffer from the supergravity eta problem of F-term inflation. That is, D-term models have the desirable property that the inflaton mass can naturally remain much smaller than the Hubble scale. We observe that this advantage is lost in models based on string compactifications whose volume is stabilized by a nonperturbative superpotential: the F-term energy associated with volume stabilization causes the eta problem to reappear. Moreover, any shift symmetries introduced to protect the inflaton mass will typically be lifted by threshold corrections to the volume-stabilizing superpotential. Using threshold corrections computed by Berg, Haack, and Kors, we illustrate this point in the example of the D3-D7 inflationary model, and conclude that inflation is possible, but only for fine-tuned values of the stabilized moduli. More generally, we conclude that inflationary models in stable string compactifications, even D-term models with shift symmetries, will require a certain amount of fine-tuning to avoid this new contribution to the eta problem.Comment: 25 page

Enhanced Symmetries and the Ground State of String Theory

The ground state of string theory may lie at a point of ``maximally enhanced symmetry", at which all of the moduli transform under continuous or discrete symmetries. This hypothesis, along with the hypotheses that the theory at high energies has N=1 supersymmetry and that the gauge couplings are weak and unified, has definite consequences for low energy physics. We describe these, and offer some suggestions as to how these assumptions might be compatible.Comment: harvmac, 18 page

Update of D3/D7-Brane Inflation on K3 x T^2/Z_2

We update the D3/D7-brane inflation model on K3 x T^2/Z_2 with branes and fluxes. For this purpose, we study the low energy theory including g_s corrections to the gaugino condensate superpotential that stabilizes the K3 volume modulus. The gauge kinetic function is verified to become holomorphic when the original N=2 supersymmetry is spontaneously broken to N=1 by bulk fluxes. From the underlying classical N=2 supergravity, the theory inherits a shift symmetry which provides the inflaton with a naturally flat potential. We analyze the fate of this shift symmetry after the inclusion of quantum corrections. The field range of the inflaton is found to depend significantly on the complex structure of the torus but is independent of its volume. This allows for a large kinematical field range for the inflaton. Furthermore, we show that the D3/D7 model may lead to a realization of the recent CMB fit by Hindmarsh et al. with an 11% contribution from cosmic strings and a spectral index close to n_s=1. On the other hand, by a slight change of the parameters of the model one can strongly suppress the cosmic string contribution and reduce the spectral index n_s to fit the WMAP5 data in the absence of cosmic strings. We also demonstrate that the inclusion of quantum corrections allows for a regime of eternal D3/D7 inflation.Comment: LaTeX2e, 55 pages + appendices, 8 figures; v3: added appendix F and a note at the end of the conclusions in order to clarify the relation of our results to the recent work by Burgess et al. (arXiv:0811.1503

Supersymmetry Breaking and Dilaton Stabilization in String Gas Cosmology

In this Note we study supersymmetry breaking via gaugino condensation in string gas cosmology. We show that the same gaugino condensate which is introduced to stabilize the dilaton breaks supersymmetry. We study the constraints on the scale of supersymmetry breaking which this mechanism leads to.Comment: 11 page

Exploration of Finite 2D Square Grid by a Metamorphic Robotic System

We consider exploration of finite 2D square grid by a metamorphic robotic system consisting of anonymous oblivious modules. The number of possible shapes of a metamorphic robotic system grows as the number of modules increases. The shape of the system serves as its memory and shows its functionality. We consider the effect of global compass on the minimum number of modules necessary to explore a finite 2D square grid. We show that if the modules agree on the directions (north, south, east, and west), three modules are necessary and sufficient for exploration from an arbitrary initial configuration, otherwise five modules are necessary and sufficient for restricted initial configurations

General considerations of the cosmological constant and the stabilization of moduli in the brane-world picture

We argue that the brane-world picture with matter-fields confined to 4-d domain walls and with gravitational interactions across the bulk disallows adding an arbitrary constant to the low-energy, 4-d effective theory -- which finesses the usual cosmological constant problem. The analysis also points to difficulties in stabilizing moduli fields; as an alternative, we suggest scenarios in which the moduli motion is heavily damped by various cosmological mechanisms and varying ultra-slowly with time.Comment: 5 pages, no figure