The first order convergence law fails for random perfect graphs

We consider first order expressible properties of random perfect graphs. That is, we pick a graph $G_n$ uniformly at random from all (labelled) perfect graphs on $n$ vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that $G_n$ satisfies it does not converge as $n\to\infty$.Comment: 11 pages. Minor corrections since last versio

Pattern Reduction in Paper Cutting

A large part of the paper industry involves supplying customers with reels of specified width in specifed quantities. These 'customer reels' must be cut from a set of wider 'jumbo reels', in as economical a way as possible. The first priority is to minimize the waste, i.e. to satisfy the customer demands using as few jumbo reels as possible. This is an example of the one-dimensional cutting stock problem, which has an extensive literature. Greycon have developed cutting stock algorithms which they include in their software packages. Greycon's initial presentation to the Study Group posed several questions, which are listed below, along with (partial) answers arising from the work described in this report. (1) Given a minimum-waste solution, what is the minimum number of patterns required? It is shown in Section 2 that even when all the patterns appearing in minimum-waste solutions are known, determining the minimum number of patterns may be hard. It seems unlikely that one can guarantee to find the minimum number of patterns for large classes of realistic problems with only a few seconds on a PC available. (2) Given an n → n-1 algorithm, will it find an optimal solution to the minimum- pattern problem? There are problems for which n → n - 1 reductions are not possible although a more dramatic reduction is. (3) Is there an efficient n → n-1 algorithm? In light of Question 2, Question 3 should perhaps be rephrased as 'Is there an efficient algorithm to reduce n patterns?' However, if an algorithm guaranteed to find some reduction whenever one existed then it could be applied iteratively to minimize the number of patterns, and we have seen this cannot be done easily. (4) Are there efficient 5 → 4 and 4 → 3 algorithms? (5) Is it worthwhile seeking alternatives to greedy heuristics? In response to Questions 4 and 5, we point to the algorithm described in the report, or variants of it. Such approaches seem capable of catching many higher reductions. (6) Is there a way to find solutions with the smallest possible number of single patterns? The Study Group did not investigate methods tailored specifically to this task, but the algorithm proposed here seems to do reasonably well. It will not increase the number of singleton patterns under any circumstances, and when the number of singletons is high there will be many possible moves that tend to eliminate them. (7) Can a solution be found which reduces the number of knife changes? The algorithm will help to reduce the number of necessary knife changes because it works by bringing patterns closer together, even if this does not proceed fully to a pattern reduction. If two patterns are equal across some of the customer widths, the knives for these reels need not be changed when moving from one to the other

Algorithmic Analysis of Qualitative and Quantitative Termination Problems for Affine Probabilistic Programs

In this paper, we consider termination of probabilistic programs with real-valued variables. The questions concerned are: 1. qualitative ones that ask (i) whether the program terminates with probability 1 (almost-sure termination) and (ii) whether the expected termination time is finite (finite termination); 2. quantitative ones that ask (i) to approximate the expected termination time (expectation problem) and (ii) to compute a bound B such that the probability to terminate after B steps decreases exponentially (concentration problem). To solve these questions, we utilize the notion of ranking supermartingales which is a powerful approach for proving termination of probabilistic programs. In detail, we focus on algorithmic synthesis of linear ranking-supermartingales over affine probabilistic programs (APP's) with both angelic and demonic non-determinism. An important subclass of APP's is LRAPP which is defined as the class of all APP's over which a linear ranking-supermartingale exists. Our main contributions are as follows. Firstly, we show that the membership problem of LRAPP (i) can be decided in polynomial time for APP's with at most demonic non-determinism, and (ii) is NP-hard and in PSPACE for APP's with angelic non-determinism; moreover, the NP-hardness result holds already for APP's without probability and demonic non-determinism. Secondly, we show that the concentration problem over LRAPP can be solved in the same complexity as for the membership problem of LRAPP. Finally, we show that the expectation problem over LRAPP can be solved in 2EXPTIME and is PSPACE-hard even for APP's without probability and non-determinism (i.e., deterministic programs). Our experimental results demonstrate the effectiveness of our approach to answer the qualitative and quantitative questions over APP's with at most demonic non-determinism.Comment: 24 pages, full version to the conference paper on POPL 201

On the chromatic number of random geometric graphs

Given independent random points X_1,...,X_n\in\eR^d with common probability distribution $\nu$, and a positive distance $r=r(n)>0$, we construct a random geometric graph $G_n$ with vertex set $\{1,...,n\}$ where distinct $i$ and $j$ are adjacent when \norm{X_i-X_j}\leq r. Here \norm{.} may be any norm on \eR^d, and $\nu$ may be any probability distribution on \eR^d with a bounded density function. We consider the chromatic number $\chi(G_n)$ of $G_n$ and its relation to the clique number $\omega(G_n)$ as $n \to \infty$. Both McDiarmid and Penrose considered the range of $r$ when $r \ll (\frac{\ln n}{n})^{1/d}$ and the range when $r \gg (\frac{\ln n}{n})^{1/d}$, and their results showed a dramatic difference between these two cases. Here we sharpen and extend the earlier results, and in particular we consider the `phase change' range when $r \sim (\frac{t\ln n}{n})^{1/d}$ with $t>0$ a fixed constant. Both McDiarmid and Penrose asked for the behaviour of the chromatic number in this range. We determine constants $c(t)$ such that $\frac{\chi(G_n)}{nr^d}\to c(t)$ almost surely. Further, we find a "sharp threshold" (except for less interesting choices of the norm when the unit ball tiles $d$-space): there is a constant $t_0>0$ such that if $t \leq t_0$ then $\frac{\chi(G_n)}{\omega(G_n)}$ tends to 1 almost surely, but if $t > t_0$ then $\frac{\chi(G_n)}{\omega(G_n)}$ tends to a limit $>1$ almost surely.Comment: 56 pages, to appear in Combinatorica. Some typos correcte

A measure of majorisation emerging from single-shot statistical mechanics

The use of the von Neumann entropy in formulating the laws of thermodynamics has recently been challenged. It is associated with the average work whereas the work guaranteed to be extracted in any single run of an experiment is the more interesting quantity in general. We show that an expression that quantifies majorisation determines the optimal guaranteed work. We argue it should therefore be the central quantity of statistical mechanics, rather than the von Neumann entropy. In the limit of many identical and independent subsystems (asymptotic i.i.d) the von Neumann entropy expressions are recovered but in the non-equilbrium regime the optimal guaranteed work can be radically different to the optimal average. Moreover our measure of majorisation governs which evolutions can be realized via thermal interactions, whereas the nondecrease of the von Neumann entropy is not sufficiently restrictive. Our results are inspired by single-shot information theory.Comment: 54 pages (15+39), 9 figures. Changed title / changed presentation, same main results / added minor result on pure bipartite state entanglement (appendix G) / near to published versio

Online Multi-Coloring with Advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

Excited states of linear polyenes

We present density matrix renormalisation group calculations of the Pariser- Parr-Pople-Peierls model of linear polyenes within the adiabatic approximation. We calculate the vertical and relaxed transition energies, and relaxed geometries for various excitations on long chains. The triplet (3Bu+) and even- parity singlet (2Ag+) states have a 2-soliton and 4-soliton form, respectively, both with large relaxation energies. The dipole-allowed (1Bu-) state forms an exciton-polaron and has a very small relaxation energy. The relaxed energy of the 2Ag+ state lies below that of the 1Bu- state. We observe an attraction between the soliton-antisoliton pairs in the 2Ag+ state. The calculated excitation energies agree well with the observed values for polyene oligomers; the agreement with polyacetylene thin films is less good, and we comment on the possible sources of the discrepencies. The photoinduced absorption is interpreted. The spin-spin correlation function shows that the unpaired spins coincide with the geometrical soliton positions. We study the roles of electron-electron interactions and electron-lattice coupling in determining the excitation energies and soliton structures. The electronic interactions play the key role in determining the ground state dimerisation and the excited state transition energies.Comment: LaTeX, 15 pages, 9 figure

Robustness and Generalization

We derive generalization bounds for learning algorithms based on their robustness: the property that if a testing sample is "similar" to a training sample, then the testing error is close to the training error. This provides a novel approach, different from the complexity or stability arguments, to study generalization of learning algorithms. We further show that a weak notion of robustness is both sufficient and necessary for generalizability, which implies that robustness is a fundamental property for learning algorithms to work

'Education, education, education' : legal, moral and clinical

This article brings together Professor Donald Nicolson's intellectual interest in professional legal ethics and his long-standing involvement with law clinics both as an advisor at the University of Cape Town and Director of the University of Bristol Law Clinic and the University of Strathclyde Law Clinic. In this article he looks at how legal education may help start this process of character development, arguing that the best means is through student involvement in voluntary law clinics. And here he builds upon his recent article which argues for voluntary, community service oriented law clinics over those which emphasise the education of students

Phylogeny of snakes (Serpentes): combining morphological and molecular data in likelihood Bayesian and parsimony analyses

Copyright © 2007 The Natural history MuseumThe phylogeny of living and fossil snakes is assessed using likelihood and parsimony approaches and a dataset combining 263 morphological characters with mitochondrial (2693 bp) and nuclear (1092 bp) gene sequences. The ‘no common mechanism’ (NCMr) and ‘Markovian’ (Mkv) models were employed for the morphological partition in likelihood analyses; likelihood scores in the NCMr model were more closely correlated with parsimony tree lengths. Both models accorded relatively less weight to the molecular data than did parsimony, with the effect being milder in the NCMr model. Partitioned branch and likelihood support values indicate that the mtDNA and nuclear gene partitions agree more closely with each other than with morphology. Despite differences between data partitions in phylogenetic signal, analytic models, and relative weighting, the parsimony and likelihood analyses all retrieved the following widely accepted groups: scolecophidians, alethinophidians, cylindrophiines, macrostomatans (sensu lato) and caenophidians. Anilius alone emerged as the most basal alethinophidian; the combined analyses resulted in a novel and stable position of uropeltines and cylindrophiines as the second-most basal clade of alethinophidians. The limbed marine pachyophiids, along with Dinilysia and Wonambi, were always basal to all living snakes. Other results stable in all combined analyses include: Xenopeltis and Loxocemus were sister taxa (fide morphology) but clustered with pythonines (fide molecules), and Ungaliophis clustered with a boine-erycine clade (fide molecules). Tropidophis remains enigmatic; it emerges as a basal alethinophidian in the parsimony analyses (fide molecules) but a derived form in the likelihood analyses (fide morphology), largely due to the different relative weighting accorded to data partitions.Michael S. Y. Lee, Andrew F. Hugall, Robin Lawson & John D. Scanlo