Uncertain Reasoning in Justification Logic

This thesis studies the combination of two well known formal systems for knowledge representation: probabilistic logic and justification logic. Our aim is to design a formal framework that allows the analysis of epistemic situations with incomplete information. In order to achieve this we introduce two probabilistic justification logics, which are defined by adding probability operators to the minimal justification logic J. We prove soundness and completeness theorems for our logics and establish decidability procedures. Both our logics rely on an infinitary rule so that strong completeness can be achieved. One of the most interesting mathematical results for our logics is the fact that adding only one iteration of the probability operator to the justification logic J does not increase the computational complexity of the logic

Dynamic Complexity Meets Parameterised Algorithms

Dynamic Complexity studies the maintainability of queries with logical formulas in a setting where the underlying structure or database changes over time. Most often, these formulas are from first-order logic, giving rise to the dynamic complexity class DynFO. This paper investigates extensions of DynFO in the spirit of parameterised algorithms. In this setting structures come with a parameter k and the extensions allow additional "space" of size f(k) (in the form of an additional structure of this size) or additional time f(k) (in the form of iterations of formulas) or both. The resulting classes are compared with their non-dynamic counterparts and other classes. The main part of the paper explores the applicability of methods for parameterised algorithms to this setting through case studies for various well-known parameterised problems

Probabilistic justification logic

We present a probabilistic justification logic, PPJ⁠, as a framework for uncertain reasoning about rational belief, degrees of belief and justifications. We establish soundness and strong completeness for PPJ with respect to the class of so-called measurable Kripke-like models and show that the satisfiability problem is decidable. We discuss how PPJ provides insight into the well-known lottery paradox

Assessment of Bone Mineral Density in Male Patients with Chronic Obstructive Pulmonary Disease by DXA and Quantitative Computed Tomography

The purpose of this study is to identify the prevalence of osteoporosis in male patients with chronic obstructive pulmonary disease (COPD) by dual-energy X-ray absorptiometry (DXA) and quantitative computed tomography (QCT) and to compare the diagnostic abilities of the above methods. Thirty-seven male patients with established COPD were examined with DXA and standard QCT in lumbar spine, including L1, L2, and L3 vertebrae. T-scores and bone mineral density values were calculated by DXA and QCT method, respectively. Comparative assessment of the findings was performed and statistical analysis was applied. QCT measurements found more COPD patients with impaired bone mineral density compared to DXA, namely, 13 (35.1%) versus 12 (32.4%) patients with osteopenia and 16 (43.2%) versus 9 (16.2%) patients with osteoporosis (p=0.04). More vertebrae were found with osteoporosis by QCT compared to DXA (p=0.03). The prevalence of osteoporosis among male patients with COPD is increased and DXA may underestimate this risk. QCT measurements have an improved discriminating ability to identify low BMD compared to DXA measurements because QCT is able to overcome diagnostic pitfalls including aortic calcifications and degenerative spinal osteophytes

Uncertain Reasoning in Justification Logic

This thesis studies the combination of two well known formal systems for knowledge representation: probabilistic logic and justification logic. Our aim is to design a formal framework that allows the analysis of epistemic situations with incomplete information. In order to achieve this we introduce two probabilistic justification logics, which are defined by adding probability operators to the minimal justification logic J. We prove soundness and completeness theorems for our logics and establish decidability procedures. Both our logics rely on an infinitary rule so that strong completeness can be achieved. One of the most interesting mathematical results for our logics is the fact that adding only one iteration of the probability operator to the justification logic J does not increase the computational complexity of the logic