We briefly investigate the history of computational group theory and automated deduction in the context of group theory, indicating problems and methods which are shared by the two communities. We compare heuristics for computational group theory and Knuth-Bendix completion, using Tietze transformations. We develop a complex induction proof of a theorem about a wreath product of two groups, and report the success of a theorem prover in automatically solving a challenge problem, in the process reducing a ring theoretic problem to a semigroup problem, and so discovering a more general result. We give a collection of examples which illustrate different phenomena which can occur, and indicate some of the natural questions which arise. In particular we investigate how these results extend to subgroups and quotients.
Any simplification ordering on a monadic term algebra is an extension of an ordering by weight.
We extend this result to simplification orderings over arbitrary terms, by considering a related monadic term algebra called the spine. We show that any simplification ordering on the spine lifts to an ordering on the full term algebra. Conversely a simplification ordering on the term algebra defines a weight function on the spine which in turn can be lifted to a real valued function on the original ground terms, with the property that the orginal ordering is an extension of the weight ordering thus defined on ground terms. Thus we may associate numeric weights to any simplification ordering on the term algebra, and hence determine a classifying spavce. We also investigate the Knuth Bendix and polynomial orderings in this light, provide a general framework for orderings terms by counting embedded patterns, and investigate linear orderings related to the recursive path ordering.