431 research outputs found
The Andrews-Curtis Conjecture, Term Rewriting and First-Order Proofs
The Andrews-Curtis conjecture (ACC) remains one of the outstanding open problems in combinatorial group theory. In short, it states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of simple transformations. It is generally believed that the conjecture may be false and there are several series of potential counterexamples for which required simplifications are not known. Finding simplifications poses a challenge for any computational approach - the search space is unbounded and the lower bound on the length of simplification sequences is known to be at least superexponential. Various specialised search algorithms have been used to eliminate some of the potential counterexamples. In this paper we present an alternative approach based on automated reasoning. We formulate a term rewriting system ACT for AC-transformations, and its translation(s) into the first-order logic. The problem of finding AC-simplifications is reduced to the problem of proving first-order formulae, which is then tackled by the available automated theorem provers. We report on the experiments demonstrating the efficiency of the proposed method by finding required simplifications for several new open cases
Measuring sets in infinite groups
We are now witnessing a rapid growth of a new part of group theory which has
become known as "statistical group theory". A typical result in this area would
say something like ``a random element (or a tuple of elements) of a group G has
a property P with probability p". The validity of a statement like that does,
of course, heavily depend on how one defines probability on groups, or,
equivalently, how one measures sets in a group (in particular, in a free
group). We hope that new approaches to defining probabilities on groups
outlined in this paper create, among other things, an appropriate framework for
the study of the "average case" complexity of algorithms on groups.Comment: 22 page
- …