First-Order Logical Duality

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2. In the present work, we generalize the entire arrangement from propositional to first-order logic. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed first in the form of a contravariant adjunction, is established by homming into a common dualizing object, now \Sets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology. The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we construct the classifying topos of a decidable coherent theory out of its groupoid of models via a simplified covering theorem for coherent toposes.Comment: Final pre-print version. 62 page

On Equivalence and Cores for Incomplete Databases in Open and Closed Worlds

Data exchange heavily relies on the notion of incomplete database instances. Several semantics for such instances have been proposed and include open (OWA), closed (CWA), and open-closed (OCWA) world. For all these semantics important questions are: whether one incomplete instance semantically implies another; when two are semantically equivalent; and whether a smaller or smallest semantically equivalent instance exists. For OWA and CWA these questions are fully answered. For several variants of OCWA, however, they remain open. In this work we adress these questions for Closed Powerset semantics and the OCWA semantics of [Leonid Libkin and Cristina Sirangelo, 2011]. We define a new OCWA semantics, called OCWA*, in terms of homomorphic covers that subsumes both semantics, and characterize semantic implication and equivalence in terms of such covers. This characterization yields a guess-and-check algorithm to decide equivalence, and shows that the problem is NP-complete. For the minimization problem we show that for several common notions of minimality there is in general no unique minimal equivalent instance for Closed Powerset semantics, and consequently not for the more expressive OCWA* either. However, for Closed Powerset semantics we show that one can find, for any incomplete database, a unique finite set of its subinstances which are subinstances (up to renaming of nulls) of all instances semantically equivalent to the original incomplete one. We study properties of this set, and extend the analysis to OCWA*

Very low calorie diet without aspartame in obese subjects: improved metabolic control after 4 weeks treatment

Abstract Background: Very low calorie diet (VLCD) is routinely used in programs for treatment of obesity and before bariatric surgery in order to reduce risk of postoperative complications. Aspartame, an artificial sweetener, is commonly used in VLCD and is well approved as a food additive without any adverse effects. The development of a new fructose containing VLCD formula without aspartame raises questions as to effects on glucose and lipid control. Methods: As part of an ongoing study of a novel bariatric surgery procedure, twenty-five obese subjects with mean body mass index (BMI) 39.8 kg/m 2 and mean age of 48.8 years enrolled in a single center observational study. Seven subjects presented with type 2 diabetes mellitus. The subjects underwent four weeks dietary treatment with VLCD Slanka (Slanka®). Blood samples including fasting plasma glucose, HbA1c, cholesterol and triglycerides were performed at start and after four weeks of diet. Blood pressure and weight were noted. Results: All subjects completed the diet without any adverse events. Mean weight reduction was 8.2 kg with 95% confidence interval 7.1-9.2 kg (p = 0.001). Excess weight (i.e. proportion of weight exceeding BMI 25) loss decreased by median 19.5% (inter quartile range (IQR) 16,8-24,2). Median fasting plasma glucose was at inclusion 5,6 mmol/l (IQR 5,3-6,8) and after diet 4.8 mmol/l (IQR 4,6-5,2) (p = 0.001). Median HbA1c changed from 39 mmol/mol (IQR 37-44) to 37 mmol/mol (IQR 35-43) (p = 0.001). There was also significant reduction in cholesterol and triglyceride levels as well as in systolic blood pressure. Changes in other monitored blood chemistry values were without clinical importance. Conclusion: Four weeks treatment with fructose containing VLCD of obese subjects preparing for bariatric surgery gave a substantial weight reduction without any significant negative metabolic effects