Exhaustible sets in higher-type computation

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces. We also show that, in the non-empty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications

The Biequivalence of Locally Cartesian Closed Categories and Martin-L\"of Type Theories

Seely's paper "Locally cartesian closed categories and type theory" contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-L\"of type theories with Pi-types, Sigma-types and extensional identity types. However, Seely's proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seely's theorem: that the B\'enabou-Hofmann interpretation of Martin-L\"of type theory in locally cartesian closed categories yields a biequivalence of 2-categories. To facilitate the technical development we employ categories with families as a substitute for syntactic Martin-L\"of type theories. As a second result we prove that if we remove Pi-types the resulting categories with families are biequivalent to left exact categories.Comment: TLCA 2011 - 10th Typed Lambda Calculi and Applications, Novi Sad : Serbia (2011

Applying psychological theory to evidence-based clinical practice : identifying factors predictive of managing upper respiratory tract infections without antibiotics

Background: Psychological models can be used to understand and predict behaviour in a wide range of settings. However, they have not been consistently applied to health professional behaviours, and the contribution of differing theories is not clear. The aim of this study was to explore the usefulness of a range of psychological theories to predict health professional behaviour relating to management of upper respiratory tract infections (URTIs) without antibiotics. Methods: Psychological measures were collected by postal questionnaire survey from a random sample of general practitioners (GPs) in Scotland. The outcome measures were clinical behaviour (using antibiotic prescription rates as a proxy indicator), behavioural simulation (scenario-based decisions to managing URTI with or without antibiotics) and behavioural intention (general intention to managing URTI without antibiotics). Explanatory variables were the constructs within the following theories: Theory of Planned Behaviour (TPB), Social Cognitive Theory (SCT), Common Sense Self-Regulation Model (CS-SRM), Operant Learning Theory (OLT), Implementation Intention (II), Stage Model (SM), and knowledge (a non-theoretical construct). For each outcome measure, multiple regression analysis was used to examine the predictive value of each theoretical model individually. Following this 'theory level' analysis, a 'cross theory' analysis was conducted to investigate the combined predictive value of all significant individual constructs across theories. Results: All theories were tested, but only significant results are presented. When predicting behaviour, at the theory level, OLT explained 6% of the variance and, in a cross theory analysis, OLT 'evidence of habitual behaviour' also explained 6%. When predicting behavioural simulation, at the theory level, the proportion of variance explained was: TPB, 31%; SCT, 26%; II, 6%; OLT, 24%. GPs who reported having already decided to change their management to try to avoid the use of antibiotics made significantly fewer scenario-based decisions to prescribe. In the cross theory analysis, perceived behavioural control (TPB), evidence of habitual behaviour (OLT), CS-SRM cause (chance/bad luck), and intention entered the equation, together explaining 36% of the variance. When predicting intention, at the theory level, the proportion of variance explained was: TPB, 30%; SCT, 29%; CS-SRM 27%; OLT, 43%. GPs who reported that they had already decided to change their management to try to avoid the use of antibiotics had a significantly higher intention to manage URTIs without prescribing antibiotics. In the cross theory analysis, OLT evidence of habitual behaviour, TPB attitudes, risk perception, CS-SRM control by doctor, TPB perceived behavioural control and CS-SRM control by treatment entered the equation, together explaining 49% of the variance in intention. Cnclusion: The study provides evidence that psychological models can be useful in understanding and predicting clinical behaviour. Taking a theory-based approach enables the reation of a replicable methodology for identifying factors that predict clinical behaviour. Hwever, a number of conceptual and methodological challenges remain

Characterisation of chemical component migration in automotive paint by synchrotron infrared imaging

Synchrotron infrared chemical imaging was employed to examine and assess the extent of interlayer component migration within multilayer automotive paint samples, with a particular emphasis on the cross-linking additive melamine. Two dimensional infrared chemical images revealed that melamine consistently diffuses in select paint samples from the underlying basecoat into the outermost clear coat layer. Pigments from the basecoat were also found to migrate into the adjoining layers. This is significant as the relative abundance of both melamine and pigments will vary greatly depending upon the region of the layer analysed. This component migration will undoubtedly impact the information gleaned from a questioned sample via library searching software or multivariate statistics. As a result, appropriate analytical protocols will need to be utilised to mitigate the effects of interlayer pigment and melamine diffusion, so as to afford a true representation of the composition of the coating for forensic identification purposes

An Improved Implementation and Abstract Interface for Hybrid

Hybrid is a formal theory implemented in Isabelle/HOL that provides an interface for representing and reasoning about object languages using higher-order abstract syntax (HOAS). This interface is built around an HOAS variable-binding operator that is constructed definitionally from a de Bruijn index representation. In this paper we make a variety of improvements to Hybrid, culminating in an abstract interface that on one hand makes Hybrid a more mathematically satisfactory theory, and on the other hand has important practical benefits. We start with a modification of Hybrid's type of terms that better hides its implementation in terms of de Bruijn indices, by excluding at the type level terms with dangling indices. We present an improved set of definitions, and a series of new lemmas that provide a complete characterization of Hybrid's primitives in terms of properties stated at the HOAS level. Benefits of this new package include a new proof of adequacy and improvements to reasoning about object logics. Such proofs are carried out at the higher level with no involvement of the lower level de Bruijn syntax.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

Type Classes for Mathematics in Type Theory

The introduction of first-class type classes in the Coq system calls for re-examination of the basic interfaces used for mathematical formalization in type theory. We present a new set of type classes for mathematics and take full advantage of their unique features to make practical a particularly flexible approach formerly thought infeasible. Thus, we address both traditional proof engineering challenges as well as new ones resulting from our ambition to build upon this development a library of constructive analysis in which abstraction penalties inhibiting efficient computation are reduced to a minimum. The base of our development consists of type classes representing a standard algebraic hierarchy, as well as portions of category theory and universal algebra. On this foundation we build a set of mathematically sound abstract interfaces for different kinds of numbers, succinctly expressed using categorical language and universal algebra constructions. Strategic use of type classes lets us support these high-level theory-friendly definitions while still enabling efficient implementations unhindered by gratuitous indirection, conversion or projection. Algebra thrives on the interplay between syntax and semantics. The Prolog-like abilities of type class instance resolution allow us to conveniently define a quote function, thus facilitating the use of reflective techniques

A multi-wavelength classification method for polar stratospheric cloud types using infrared limb spectra

The Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) instrument on board the ESA Envisat satellite operated from July 2002 until April 2012. The infrared limb emission measurements represent a unique dataset of daytime and night-time observations of polar stratospheric clouds (PSCs) up to both poles. Cloud detection sensitivity is comparable to space-borne lidars, and it is possible to classify different cloud types from the spectral measurements in different atmospheric windows regions. Here we present a new infrared PSC classification scheme based on the combination of a well-established two-colour ratio method and multiple 2-D brightness temperature difference probability density functions. The method is a simple probabilistic classifier based on Bayes' theorem with a strong independence assumption. The method has been tested in conjunction with a database of radiative transfer model calculations of realistic PSC particle size distributions, geometries, and composition. The Bayesian classifier distinguishes between solid particles of ice and nitric acid trihydrate (NAT), as well as liquid droplets of super-cooled ternary solution (STS). The classification results are compared to coincident measurements from the space-borne lidar Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) instrument over the temporal overlap of both satellite missions (June 2006–March 2012). Both datasets show a good agreement for the specific PSC classes, although the viewing geometries and the vertical and horizontal resolution are quite different. Discrepancies are observed between the CALIOP and the MIPAS ice class. The Bayesian classifier for MIPAS identifies substantially more ice clouds in the Southern Hemisphere polar vortex than CALIOP. This disagreement is attributed in part to the difference in the sensitivity on mixed-type clouds. Ice seems to dominate the spectral behaviour in the limb infrared spectra and may cause an overestimation in ice occurrence compared to the real fraction of ice within the PSC area in the polar vortex. The entire MIPAS measurement period was processed with the new classification approach. Examples like the detection of the Antarctic NAT belt during early winter, and its possible link to mountain wave events over the Antarctic Peninsula, which are observed by the Atmospheric Infrared Sounder (AIRS) instrument, highlight the importance of a climatology of 9 Southern Hemisphere and 10 Northern Hemisphere winters in total. The new dataset is valuable both for detailed process studies, and for comparisons with and improvements of the PSC parameterizations used in chemistry transport and climate models