Counting Partial Orders with a Fixed Number of Comparable Pairs

This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.In 1978, Dhar suggested a model of a lattice gas whose states are partial orders. In this context he raised the question of determining the number of partial orders with a fixed number of comparable pairs. Dhar conjectured that in order to find a good approximation to this number, it should suffice to enumerate families of layer posets. In this paper we prove this conjecture and thereby prepare the ground for a complete answer to the question.Peer Reviewe

Capturing Polynomial Time using Modular Decomposition

The question of whether there is a logic that captures polynomial time is one of the main open problems in descriptive complexity theory and database theory. In 2010 Grohe showed that fixed point logic with counting captures polynomial time on all classes of graphs with excluded minors. We now consider classes of graphs with excluded induced subgraphs. For such graph classes, an effective graph decomposition, called modular decomposition, was introduced by Gallai in 1976. The graphs that are non-decomposable with respect to modular decomposition are called prime. We present a tool, the Modular Decomposition Theorem, that reduces (definable) canonization of a graph class C to (definable) canonization of the class of prime graphs of C that are colored with binary relations on a linearly ordered set. By an application of the Modular Decomposition Theorem, we show that fixed point logic with counting captures polynomial time on the class of permutation graphs. Within the proof of the Modular Decomposition Theorem, we show that the modular decomposition of a graph is definable in symmetric transitive closure logic with counting. We obtain that the modular decomposition tree is computable in logarithmic space. It follows that cograph recognition and cograph canonization is computable in logarithmic space.Comment: 38 pages, 10 Figures. A preliminary version of this article appeared in the Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS '17

Power counting with one-pion exchange

Techniques developed for handing inverse-power-law potentials in atomic physics are applied to the tensor one-pion exchange potential to determine the regions in which it can be treated perturbatively. In S-, P- and D-waves the critical values of the relative momentum are less than or of the order of 400 MeV. The RG is then used to determine the power counting for short-range interaction in the presence of this potential. In the P-and D-waves, where there are no low-energy bound or virtual states, these interactions have half-integer RG eigenvalues and are substantially promoted relative to naive expectations. These results are independent of whether the tensor force is attractive or repulsive. In the 3S1 channel the leading term is relevant, but it is demoted by half an order compared to the counting for the effective-range expansion with only a short-range potential. The tensor force can be treated perturbatively in those F-waves and above that do not couple to P- or D-waves. The corresponding power counting is the usual one given by naive dimensional analysis.Comment: 18 pages, RevTeX (further details, explanation added

Random structures for partially ordered sets

This thesis is presented in two parts. In the first part, we study a family of models of random partial orders, called classical sequential growth models, introduced by Rideout and Sorkin as possible models of discrete space-time. We analyse a particular model, called a random binary growth model, and show that the random partial order produced by this model almost surely has infinite dimension. We also give estimates on the size of the largest vertex incomparable to a particular element of the partial order. We show that there is some positive probability that the random partial order does not contain a particular subposet. This contrasts with other existing models of partial orders. We also study "continuum limits" of sequences of classical sequential growth models. We prove results on the structure of these limits when they exist, highlighting a deficiency of these models as models of space-time. In the second part of the thesis, we prove some correlation inequalities for mappings of rooted trees into complete trees. For T a rooted tree we can define the proportion of the total number of embeddings of T into a complete binary tree that map the root of T to the root of the complete binary tree. A theorem of Kubicki, Lehel and Morayne states that, for two binary trees with one a subposet of the other, this proportion is larger for the larger tree. They conjecture that the same is true for two arbitrary trees with one a subposet of the other. We disprove this conjecture by analysing the asymptotics of this proportion for large complete binary trees. We show that the theorem of Kubicki, Lehel and Morayne can be thought of as a correlation inequality which enables us to generalise their result in other directions

POPE: Partial Order Preserving Encoding

Recently there has been much interest in performing search queries over encrypted data to enable functionality while protecting sensitive data. One particularly efficient mechanism for executing such queries is order-preserving encryption/encoding (OPE) which results in ciphertexts that preserve the relative order of the underlying plaintexts thus allowing range and comparison queries to be performed directly on ciphertexts. In this paper, we propose an alternative approach to range queries over encrypted data that is optimized to support insert-heavy workloads as are common in "big data" applications while still maintaining search functionality and achieving stronger security. Specifically, we propose a new primitive called partial order preserving encoding (POPE) that achieves ideal OPE security with frequency hiding and also leaves a sizable fraction of the data pairwise incomparable. Using only O(1) persistent and $O(n^\epsilon)$ non-persistent client storage for $0<\epsilon<1$, our POPE scheme provides extremely fast batch insertion consisting of a single round, and efficient search with O(1) amortized cost for up to $O(n^{1-\epsilon})$ search queries. This improved security and performance makes our scheme better suited for today's insert-heavy databases.Comment: Appears in ACM CCS 2016 Proceeding