Presburger arithmetic, rational generating functions, and quasi-polynomials

Presburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p=(p_1,...,p_n) are a subset of the free variables in a Presburger formula, we can define a counting function g(p) to be the number of solutions to the formula, for a given p. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.Comment: revised, including significant additions explaining computational complexity results. To appear in Journal of Symbolic Logic. Extended abstract in ICALP 2013. 17 page

Computing the period of an Ehrhart quasi-polynomial

If P is a rational polytope in R^d, then i_P(t):=#(tP\cap Z^d) is a quasi-polynomial in t, called the Ehrhart quasi-polynomial of P. A period of i_P(t) is D(P), the smallest positive integer D such that D*P has integral vertices. Often, D(P) is the minimum period of i_P(t), but, in several interesting examples, the minimum period is smaller. We prove that, for fixed d, there is a polynomial time algorithm which, given a rational polytope P in R^d and an integer n, decides whether n is a period of i_P(t). In particular, there is a polynomial time algorithm to decide whether i_P(t) is a polynomial. We conjecture that, for fixed d, there is a polynomial time algorithm to compute the minimum period of i_P(t). The tools we use are rational generating functions.Comment: 15 page

Bounds on the number of inference functions of a graphical model

Directed and undirected graphical models, also called Bayesian networks and Markov random fields, respectively, are important statistical tools in a wide variety of fields, ranging from computational biology to probabilistic artificial intelligence. We give an upper bound on the number of inference functions of any graphical model. This bound is polynomial on the size of the model, for a fixed number of parameters, thus improving the exponential upper bound given by Pachter and Sturmfels. We also show that our bound is tight up to a constant factor, by constructing a family of hidden Markov models whose number of inference functions agrees asymptotically with the upper bound. Finally, we apply this bound to a model for sequence alignment that is used in computational biology.Comment: 19 pages, 7 figure

Neighborhood complexes and generating functions for affine semigroups

Given a_1,a_2,...,a_n in Z^d, we examine the set, G, of all non-negative integer combinations of these a_i. In particular, we examine the generating function f(z)=\sum_{b\in G} z^b. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the Scarf complex) on a particular lattice in Z^n. In the generic case, this follows from algebraic results of D. Bayer and B. Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice

Nutritional labelling in restaurants : whose responsibility is it anyway?

To explore consumer attitudes towards the potential implementation of compulsory nutritional labelling on commercial restaurant menus in the UK. This research was approached from the perspective of the consumer with the intention of gaining an insight into personal attitudes towards nutritional labelling on commercial restaurant menus and three focus groups consisting of participants with distinctly differing approaches to eating outside the home were conducted. The research suggests that while some consumers might welcome the introduction of nutritional labelling it is context dependent and without an appropriate education the information provided may not be understood anyway. The issue of responsibility for public health is unresolved although some effort could be made to provide greater nutritional balance in menus. Following this research up with a quantitative investigation, the ideas presented could be verified with the opinions of a larger sample. For example, a study into the reactions to nutritionally labelled menus in various restaurant environments. Consumers would react differently to the information being presented in a fine-dining restaurant than they would in popular catering or fast food. The obstacles faced by restaurants to provide not only nutritional information, but attractive, nutritious food are significant. Prior to this research there were few, if any, studies into the effects of food labelling on consumer choice behaviour in the context of hospitality management

Neighborhood Complexes and Generating Functions for Affine Semigroups

Given a_{1}; a_{2},...a_{n} in Z^{d}, we examine the set, G, of all nonnegative integer combinations of these ai. In particular, we examine the generating function f(z) = Sum_{b in G}z^{b}. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the Scarf complex) on a particular lattice in Z^{n}. In the generic case, this follows from algebraic results of D. Bayer and B. Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice.Integer programming, Complex of maximal lattice free bodies, Generating functions

Lengths of Systoles on Tileable Hyperbolic Surfaces

The same triangle may tile geometrically distinct surfaces of the same genus, and these tilings may determine isomorphic tiling groups. We determine if there are geometric differences in the surfaces that can be found using group theoretic methods. Specifically, we determine if the systole, the shortest closed geodesic on a surface, can distinguish a certain families of tilings. For example, there are three tilings of surfaces of genus 14 by the hyperbolic triangle with angles π/2 , π/3 , and π/7 whose tiling groups are all PSL2(13). These tilings can be distinguished by the lengths of their systoles

The generalized Frobenius problem via restricted partition functions

Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been generalized: given a nonnegative integer $k$, what is the largest integer with at most $k$ such representations? Other classical values can be generalized too: for example, how many nonnegative integers are representable in at most $k$ ways? For sufficiently large $k$, we give a complete answer to these questions by understanding how the output of the restricted partition function (the function $f(t)$ giving the number of representations of $t$) "interlaces" with itself. Furthermore, we give the full asymptotics of all of these values, as well as reprove formulas for some special cases (such as the $n=2$ case and a certain extremal family from the literature). Finally, we obtain the first two leading terms of the restricted partition function as a so-called quasi-polynomial.Comment: 18 page

Parametric inference of recombination in HIV genomes

Recombination is an important event in the evolution of HIV. It affects the global spread of the pandemic as well as evolutionary escape from host immune response and from drug therapy within single patients. Comprehensive computational methods are needed for detecting recombinant sequences in large databases, and for inferring the parental sequences. We present a hidden Markov model to annotate a query sequence as a recombinant of a given set of aligned sequences. Parametric inference is used to determine all optimal annotations for all parameters of the model. We show that the inferred annotations recover most features of established hand-curated annotations. Thus, parametric analysis of the hidden Markov model is feasible for HIV full-length genomes, and it improves the detection and annotation of recombinant forms. All computational results, reference alignments, and C++ source code are available at http://bio.math.berkeley.edu/recombination/.Comment: 20 pages, 5 figure

Long term sustainable product development at the packaging sector

This paper outlines the importance of sustainable product developments and their role in securing a sustainable future through current practices and procedures. It discusses the difficulties faced within organisations through the complexities and swamping of regulations when considering sustainability and the problems in policing such a system to ensure compliance. Focus is centred on the design stage, where large numbers of standards and interests must be factored in to create specifications that are highly compliant. Where there is a limited understanding of the complexities that are presented at this stage, less optimum specifications will be dispatched. This presents the need to think strategically with new systems and approaches which adapt to company behaviour, where decisions that are made at a design stage have impacts up and down the supply chain, changes that are made must be in line with company strategic objectives and provide influential returns on investment