On the Relative Strength of Pebbling and Resolution

The last decade has seen a revival of interest in pebble games in the context of proof complexity. Pebbling has proven a useful tool for studying resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs. The typical approach has been to encode the pebble game played on a graph as a CNF formula and then argue that proofs of this formula must inherit (various aspects of) the pebbling properties of the underlying graph. Unfortunately, the reductions used here are not tight. To simulate resolution proofs by pebblings, the full strength of nondeterministic black-white pebbling is needed, whereas resolution is only known to be able to simulate deterministic black pebbling. To obtain strong results, one therefore needs to find specific graph families which either have essentially the same properties for black and black-white pebbling (not at all true in general) or which admit simulations of black-white pebblings in resolution. This paper contributes to both these approaches. First, we design a restricted form of black-white pebbling that can be simulated in resolution and show that there are graph families for which such restricted pebblings can be asymptotically better than black pebblings. This proves that, perhaps somewhat unexpectedly, resolution can strictly beat black-only pebbling, and in particular that the space lower bounds on pebbling formulas in [Ben-Sasson and Nordstrom 2008] are tight. Second, we present a versatile parametrized graph family with essentially the same properties for black and black-white pebbling, which gives sharp simultaneous trade-offs for black and black-white pebbling for various parameter settings. Both of our contributions have been instrumental in obtaining the time-space trade-off results for resolution-based proof systems in [Ben-Sasson and Nordstrom 2009].Comment: Full-length version of paper to appear in Proceedings of the 25th Annual IEEE Conference on Computational Complexity (CCC '10), June 201

Pebbling and Branching Programs Solving the Tree Evaluation Problem

We study restricted computation models related to the Tree Evaluation Problem}. The TEP was introduced in earlier work as a simple candidate for the (*very*) long term goal of separating L and LogDCFL. The input to the problem is a rooted, balanced binary tree of height h, whose internal nodes are labeled with binary functions on [k] = {1,...,k} (each given simply as a list of k^2 elements of [k]), and whose leaves are labeled with elements of [k]. Each node obtains a value in [k] equal to its binary function applied to the values of its children, and the output is the value of the root. The first restricted computation model, called Fractional Pebbling, is a generalization of the black/white pebbling game on graphs, and arises in a natural way from the search for good upper bounds on the size of nondeterministic branching programs (BPs) solving the TEP - for any fixed h, if the binary tree of height h has fractional pebbling cost at most p, then there are nondeterministic BPs of size O(k^p) solving the height h TEP. We prove a lower bound on the fractional pebbling cost of d-ary trees that is tight to within an additive constant for each fixed d. The second restricted computation model we study is a semantic restriction on (non)deterministic BPs solving the TEP - Thrifty BPs. Deterministic (resp. nondeterministic) thrifty BPs suffice to implement the best known algorithms for the TEP, based on black (resp. fractional) pebbling. In earlier work, for each fixed h a lower bound on the size of deterministic thrifty BPs was proved that is tight for sufficiently large k. We give an alternative proof that achieves the same bound for all k. We show the same bound still holds in a less-restricted model, and also that gradually weaker lower bounds can be obtained for gradually weaker restrictions on the model.Comment: Written as one of the requirements for my MSc. 29 pages, 6 figure

Dynamics of pebbles in the vicinity of a growing planetary embryo: hydro-dynamical simulations

Understanding the growth of the cores of giant planets is a difficult problem. Recently, Lambrechts and Johansen (2012; LJ12) proposed a new model in which the cores grow by the accretion of pebble-size objects, as the latter drift towards the star due to gas drag. Here, we investigate the dynamics of pebble-size objects in the vicinity of planetary embryos of 1 and 5 Earth masses and the resulting accretion rates. We use hydrodynamical simulations, in which the embryo influences the dynamics of the gas and the pebbles suffer gas drag according to the local gas density and velocities. The pebble dynamics in the vicinity of the planetary embryo is non-trivial, and it changes significantly with the pebble size. Nevertheless, the accretion rate of the embryo that we measure is within an order of magnitude of the rate estimated in LJ12 and tends to their value with increasing pebble-size. We conclude that the model by LJ12 has the potential to explain the rapid growth of giant planet cores. The actual accretion rates however, depend on the surface density of pebble size objects in the disk, which is unknown to date.Comment: In press in Astronomy and Astrophysic

Growing Pebbles and Conceptual Prisms - Understanding The Source of Student Misconceptions About Rock Formation

Provides pedagogical insight concerning learners' pre-conceptions and misconceptions about the rock cycle The resource being annotated is: http://www.dlese.org/dds/catalog_NASA-Edmall-535.htm

Does the use of nest materials in a ground-nesting bird result from a compromise between the risk of egg overheating and camouflage?

Many studies addressing the use of nest materials by animals have focused on only one factor to explain its function. However, the consideration of more than one factor could explain the apparently maladaptive choice of nest materials that make nests conspicuous to predators. We experimentally tested whether there is a trade-off in the use of nest materials between the risks of egg predation versus protection from overheating. We studied the ground-nesting Kentish plover, Charadrius alexandrinus, in southern Spain. We added materials differing in thermal properties and coloration to the nests, thus affecting rates of egg heating, nest temperature and camouflage. Before these manipulations, adults selected materials that were lighter than the microhabitat, probably to buffer the risk of egg overheating. However, the adults did not keep the lightest experimental materials, probably because they reduced camouflage, and this could make the nests even more easily detectable to predators. In all nests, adults removed most of the experimental materials independently of their properties, so that egg camouflage returned to the original situation within a week of the experimental treatments. Although the thermal environment may affect the choice of nest materials by plovers, ambient temperatureswere not so high at our study site as to determine the acceptance of the lightest experimental materials

Pebbling Arguments for Tree Evaluation

The Tree Evaluation Problem was introduced by Cook et al. in 2010 as a candidate for separating P from L and NL. The most general space lower bounds known for the Tree Evaluation Problem require a semantic restriction on the branching programs and use a connection to well-known pebble games to generate a bottleneck argument. These bounds are met by corresponding upper bounds generated by natural implementations of optimal pebbling algorithms. In this paper we extend these ideas to a variety of restricted families of both deterministic and non-deterministic branching programs, proving tight lower bounds under these restricted models. We also survey and unify known lower bounds in our "pebbling argument" framework

On Characterizing the Data Movement Complexity of Computational DAGs for Parallel Execution

Technology trends are making the cost of data movement increasingly dominant, both in terms of energy and time, over the cost of performing arithmetic operations in computer systems. The fundamental ratio of aggregate data movement bandwidth to the total computational power (also referred to the machine balance parameter) in parallel computer systems is decreasing. It is there- fore of considerable importance to characterize the inherent data movement requirements of parallel algorithms, so that the minimal architectural balance parameters required to support it on future systems can be well understood. In this paper, we develop an extension of the well-known red-blue pebble game to develop lower bounds on the data movement complexity for the parallel execution of computational directed acyclic graphs (CDAGs) on parallel systems. We model multi-node multi-core parallel systems, with the total physical memory distributed across the nodes (that are connected through some interconnection network) and in a multi-level shared cache hierarchy for processors within a node. We also develop new techniques for lower bound characterization of non-homogeneous CDAGs. We demonstrate the use of the methodology by analyzing the CDAGs of several numerical algorithms, to develop lower bounds on data movement for their parallel execution

Pebbles in palms: Counter‐practices against despair

© 2019 John Wiley & Sons, Ltd. This is the accepted manuscript version of an article which has been published in final form at https://doi.org/10.1002/ppi.1481With ongoing news of hardship and suffering in the United Kingdom and throughout the world, and in the context of austerity, shrinking public services and increasing social inequalities, it is sometimes difficult not to fall into despair, to feel hopeless or ineffectual. In this paper we consider counter‐practices to such despair and hopelessness that we hope will be helpful to all clinicians.Peer reviewe