Lecture Hall Theorems, q-series and Truncated Objects

We show here that the refined theorems for both lecture hall partitions and anti-lecture hall compositions can be obtained as straightforward consequences of two q-Chu Vandermonde identities, once an appropriate recurrence is derived. We use this approach to get new lecture hall-type theorems for truncated objects. We compute their generating function and give two different multivariate refinements of these new results : the q-calculus approach gives (u,v,q)-refinements, while a completely different approach gives odd/even (x,y)-refinements. From this, we are able to give a combinatorial characterization of truncated lecture hall partitions and new finitizations of refinements of Euler's theorem

Reasoning by analogy in the generation of domain acceptable ontology refinements

Refinements generated for a knowledge base often involve the learning of new knowledge to be added to or replace existing parts of a knowledge base. However, the justifiability of the refinement in the context of the domain (domain acceptability) is often overlooked. The work reported in this paper describes an approach to the generation of domain acceptable refinements for incomplete and incorrect ontology individuals through reasoning by analogy using existing domain knowledge. To illustrate this approach, individuals for refinement are identified during the application of a knowledge-based system, EIRA; when EIRA fails in its task, areas of its domain ontology are identified as requiring refinement. Refinements are subsequently generated by identifying and reasoning with similar individuals from the domain ontology. To evaluate this approach EIRA has been applied to the Intensive Care Unit (ICU) domain. An evaluation (by a domain expert) of the refinements generated by EIRA has indicated that this approach successfully produces domain acceptable refinements

Testing refinements by refining tests

One of the potential benefits of formal methods is that they offer the possibility of reducing the costs of testing. A specification acts as both the benchmark against which any implementation is tested, and also as the means by which tests are generated. There has therefore been interest in developing test generation techniques from formal specifications, and a number of different methods have been derived for state based languages such as Z, B and VDM. However, in addition to deriving tests from a formal specification, we might wish to refine the specification further before its implementation. The purpose of this paper is to explore the relationship between testing and refinement. As our model for test generation we use a DNF partition analysis for operations written in Z, which produces a number of disjoint test cases for each operation. In this paper we discuss how the partition analysis of an operation alters upon refinement, and we develop techniques that allow us to refine abstract tests in order to generate test cases for a refinement. To do so we use (and extend existing) methods for calculating the weakest data refinement of a specification

A global approach to the refinement of manifold data

A refinement of manifold data is a computational process, which produces a denser set of discrete data from a given one. Such refinements are closely related to multiresolution representations of manifold data by pyramid transforms, and approximation of manifold-valued functions by repeated refinements schemes. Most refinement methods compute each refined element separately, independently of the computations of the other elements. Here we propose a global method which computes all the refined elements simultaneously, using geodesic averages. We analyse repeated refinements schemes based on this global approach, and derive conditions guaranteeing strong convergence.Comment: arXiv admin note: text overlap with arXiv:1407.836