Automatic Average-case Analysis of Algorithms

International audienceMany probabilistic properties of elementary discrete combinatorial structures of interest for the average-case analysis of algorithms prove to be decidable. This paper presents a general framework in which such decision procedures can be developed. It is based on a combination of generating function techniques for counting, and complex analysis techniques for asymptotic estimations. We expose here the theory of exact analysis in terms of generating functions for four different domains: the iterative/recursive and unlabelled/labelled data type domains. We then present some major components of the associated asymptotic theory and exhibit a class of naturally arising functions that can be automatically analyzed. A fair fragment of this theory is also incorporated into a system called Lambda-Upsilon-Omega. In this way, using computer algebra, one can produce automatically non-trivial average-case analyses of algorithms operating over a variety of "decomposable" combinatorial structures. At a fundamental level, this paper is part of a global attempt at understanding why so many elementary combinatorial problems tend to have elementary asymptotic solutions. In several cases, it proves possible to relate entire classes of elementary combinatorial problems whose structure is well defined with classes of elementary "special" functions and classes of asymptotic forms relative to counting, probabilities, or average-case complexity

Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as in its successor, the DLMF, but their proofs were lost. We use generating functions and symbolic summation techniques to produce new proofs for them.Comment: Final version, some typos were corrected. 21 pages, uses svmult.cl

Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model

We first study the properties of the Fuchsian ordinary differential equations for the three and four-particle contributions $\chi^{(3)}$ and $\chi^{(4)}$ of the square lattice Ising model susceptibility. An analysis of some mathematical properties of these Fuchsian differential equations is sketched. For instance, we study the factorization properties of the corresponding linear differential operators, and consider the singularities of the three and four-particle contributions $\chi^{(3)}$ and $\chi^{(4)}$, versus the singularities of the associated Fuchsian ordinary differential equations, which actually exhibit new ``Landau-like'' singularities. We sketch the analysis of the corresponding differential Galois groups. In particular we provide a simple, but efficient, method to calculate the so-called ``connection matrices'' (between two neighboring singularities) and deduce the singular behaviors of $\chi^{(3)}$ and $\chi^{(4)}$. We provide a set of comments and speculations on the Fuchsian ordinary differential equations associated with the $n$-particle contributions $\chi^{(n)}$ and address the problem of the apparent discrepancy between such a holonomic approach and some scaling results deduced from a Painlev\'e oriented approach.Comment: 21 pages Proceedings of the Counting Complexity conferenc

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

Lattice statistical mechanics, often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in a general mathematical framework, be too complex, or could not be defined. Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau ODEs, associated with double hypergeometric series, we show that holonomic functions are actually a good framework for actually finding the singular manifolds. We, then, analyse the singular algebraic varieties of the n-fold integrals $\chi^{(n)}$, corresponding to the decomposition of the magnetic susceptibility of the anisotropic square Ising model. We revisit a set of Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find a first set of non-Nickelian singularities for $\chi^{(3)}$ and $\chi^{(4)}$, that also turns out to be rational or ellipic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model. We address, from a birational viewpoint, the emergence of families of elliptic curves, and of Calabi-Yau manifolds on such problems. We discuss the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility.Comment: 36 page

Rigorous Polynomial Approximation using Taylor Models in Coq

International audienceOne of the most common and practical ways of representing a real function on machines is by using a polynomial approximation. It is then important to properly handle the error introduced by such an approximation. The purpose of this work is to offer guaranteed error bounds for a specific kind of rigorous polynomial approximation called Taylor model. We carry out this work in the Coq proof assistant, with a special focus on genericity and efficiency for our implementation. We give an abstract interface for rigorous polynomial approximations, parameter- ized by the type of coefficients and the implementation of polynomials, and we instantiate this interface to the case of Taylor models with inter- val coefficients, while providing all the machinery for computing them. We compare the performances of our implementation in Coq with those of the Sollya tool, which contains an implementation of Taylor models written in C. This is a milestone in our long-term goal of providing fully formally proved and efficient Taylor models

The Effects of Playing with Thin Dolls on Body Image and Food Intake in Young Girls

This study experimentally tested the effects of playing with thin dolls on body image and food intake in 6- to 10-year-old Dutch girls (N = 117). Girls were randomly assigned to play with a thin doll, an average-sized doll, or Legos in a no doll control condition. After 10 min, they participated in a taste-test and completed questionnaires about body image. No differences were found between conditions for any of the body image variables. However, girls who played with the average-sized doll ate significantly more food than girls in other exposure conditions. Although no support was found for the assumption that playing with thin dolls influences body image, the dolls directly affected actual food intake in these young girls

Write, draw, show, and tell: a child-centred dual methodology to explore perceptions of out-of-school physical activity

Background Research to increase children’s physical activity and inform intervention design has, to date, largely underrepresented children’s voices. Further, research has been limited to singular qualitative methods that overlook children’s varied linguistic ability and interaction preference. The aim of this study was to use a novel combination of qualitative techniques to explore children’s current views, experiences and perceptions of out-of-school physical activity as well as offering formative opinion about future intervention design. Methods Write, draw, show and tell (WDST) groups were conducted with 35 children aged 10–11 years from 7 primary schools. Data were analysed through a deductive and inductive process, firstly using the Youth Physical Activity Promotion Model as a thematic framework, and then inductively to enable emergent themes to be further explored. Pen profiles were constructed representing key emergent themes. Results The WDST combination of qualitative techniques generated complimentary interconnected data which both confirmed and uncovered new insights into factors relevant to children’s out-of-school physical activity. Physical activity was most frequently associated with organised sports. Fun, enjoyment, competence, and physical activity provision were all important predictors of children’s out-of-school physical activity. Paradoxically, parents served as both significant enablers (i.e. encouragement) and barriers (i.e. restricting participation) to physical activity participation. Some of these key findings would have otherwise remained hidden when compared to more traditional singular methods based approaches. Conclusions Parents are in a unique position to promote health promoting behaviours serving as role models, physical activity gatekeepers and choice architects. Given the strong socialising effect parents have on children’s physical activity, family-based physical activity intervention may offer a promising alternative compared to traditional school-based approaches. Parents' qualitative input is important to supplement children’s voices and inform future family-based intervention design. The WDST method developed here is an inclusive, interactive and child-centred methodology which facilitates the exploration of a wide range of topics and enhances data credibility

Unpacking the behavioural components and delivery features of early childhood obesity prevention interventions in the TOPCHILD Collaboration: a systematic review and intervention coding protocol.

INTRODUCTION: Little is known about how early (eg, commencing antenatally or in the first 12 months after birth) obesity prevention interventions seek to change behaviour and which components are or are not effective. This study aims to (1) characterise early obesity prevention interventions in terms of target behaviours, delivery features and behaviour change techniques (BCTs), (2) explore similarities and differences in BCTs used to target behaviours and (3) explore effectiveness of intervention components in preventing childhood obesity. METHODS AND ANALYSIS: Annual comprehensive systematic searches will be performed in Epub Ahead of Print/MEDLINE, Embase, Cochrane (CENTRAL), CINAHL, PsycINFO, as well as clinical trial registries. Eligible randomised controlled trials of behavioural interventions to prevent childhood obesity commencing antenatally or in the first year after birth will be invited to join the Transforming Obesity in CHILDren Collaboration. Standard ontologies will be used to code target behaviours, delivery features and BCTs in both published and unpublished intervention materials provided by trialists. Narrative syntheses will be performed to summarise intervention components and compare applied BCTs by types of target behaviours. Exploratory analyses will be undertaken to assess effectiveness of intervention components. ETHICS AND DISSEMINATION: The study has been approved by The University of Sydney Human Research Ethics Committee (project no. 2020/273) and Flinders University Social and Behavioural Research Ethics Committee (project no. HREC CIA2133-1). The study's findings will be disseminated through peer-reviewed publications, conference presentations and targeted communication with key stakeholders. PROSPERO REGISTRATION NUMBER: CRD42020177408