474 research outputs found
Generic-case complexity, decision problems in group theory and random walks
We give a precise definition of ``generic-case complexity'' and show that for
a very large class of finitely generated groups the classical decision problems
of group theory - the word, conjugacy and membership problems - all have
linear-time generic-case complexity. We prove such theorems by using the theory
of random walks on regular graphs.Comment: Revised versio
Uniform decision problems in automatic semigroups
We consider various decision problems for automatic semigroups, which involve
the provision of an automatic structure as part of the problem instance. With
mild restrictions on the automatic structure, which seem to be necessary to
make the problem well-defined, the uniform word problem for semigroups
described by automatic structures is decidable. Under the same conditions, we
show that one can also decide whether the semigroup is completely simple or
completely zero-simple; in the case that it is, one can compute a Rees matrix
representation for the semigroup, in the form of a Rees matrix together with an
automatic structure for its maximal subgroup. On the other hand, we show that
it is undecidable in general whether a given element of a given automatic
monoid has a right inverse.Comment: 19 page
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