Second Set of Spaces

This document describes the Gloss infrastructure supporting implementation of location-aware services. The document is in two parts. The first part describes software architecture for the smart space. As described in D8, a local architecture provides a framework for constructing Gloss applications, termed assemblies, that run on individual physical nodes, whereas a global architecture defines an overlay network for linking individual assemblies. The second part outlines the hardware installation for local sensing. This describes the first phase of the installation in Strathclyde University

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3