We present a deterministic polynomial-time algorithm for the A B C problem, which is the membership problem for 2-generated commutative linear semigroups over an algebraic number field. We also obtain a polynomial-time algorithm for the (easier) membership problem for 2-generated abelian linear groups. Furthermore, we provide a polynomial-sized encoding for the set of all solutions.

We present a deterministic polynomial-time algorithm for the ABC problem, which is the membership problem for 2-generated commutative linear semigroups over an algebraic number field. We also obtain a polynomial time algorithm, for the (easier) membership problem, for 2-generated abelian linear groups. Furthermore, we provide a polynomial-sized encoding for the set of all solutions.<>

A polynomial time isomorphism test for a class of groups, properly containing the class of abelian groups, given either by multiplication tables or by generators and relators, is described. It is also shown that graph isomorphism testing is uniformly reducible to a word problem of a finitely presented group.

The Turing complexity of the word problems of a class of groups introduced by Grigorchuk (1985) is examined. In particular, it is shown that such problems of permutation groups of the infinite complete binary tree yield natural complete sets that separate time and space complexity classes if they are distinct. A refinement of Savitch's translation theorem as well as a similar result restricted for time complexity follow. New families of nonfinitely presented groups are shown to have word problems uniformly solvable in simultaneous logspace and quadratic time. A new family of public-key cryptosystems based on these word problems is constructed.

: The uniform word problem for finite groups presented by their multiplication tables is considered. Upper bounds of 0(k-squared) for arbitrary group and 0(n log-squared n) for arbitrary semigroup and 0(n log n) for abelian groups are shown where n is the length of the presentation. (Author)

: While there are by now well-established criteria for evaluating serial algorithms, such as space and time measures, these criteria cannot be readily applied to asynchronous algorithms. A method is proposed for the evaluation of the performance of an asynchronous algorithm. This method is based on the study of delays that are often introduced when one solves a synchronization problem. This method is illustrated by proving results about the efficiency of various solutions to synchronization problems.

A new algebraic structure, the R monoid, is associated with a discrete-time, time-invariant linear dynamical system over acommutative ring R, and its properties are investigated.