The main purpose of this note is to give a short proof of Thomas’s theorem that every finite graph has a linked tree-decomposition of width no greater than its tree-width [ 9 ]. This is a useful tool in the theory of tree-decompositions; for example, it is a key lemma in Robertson & Seymour’s proof of the fact that every set of graphs of bounded tree-width is well-quasi-ordered [ 7 ]. This latt...

We review results concerning the problem of ‘Stability of Matter’. Non-relativistic, relativistic quantum mechanics as well as matter interacting with classical fields is discussed and the strategies of the various proofs of these results is given in some detail. This is followed by a short discussion of the corresponding problem of matter interacting with the radiation field.

One of the main differences between pen-and-paper proofs and computer proofs is in the number of simple facts derived. In automated proof generation, the number of simple facts can be large. We are addressing this problem by preprocessing of the axiomatic system that should enable reduction in the number of simple and redundant facts to some extent. We implemented two types of preprocessing tec...

We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined.