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, is presented. Isomorphism testing of group presentations for (a subclass of) the same class of groups is shown to be (graph) isomorphism complete. These seem to be the first known isomorphism complete problems in group theory. Subexponential tests are presented as well for rings and algebras.

A basic question m the area of asynchronous computation is.Given a synchromzaUon problem, what synchromzation primitives are needed for an "efficient" solution9 This paper is directed toward answering this question by providing characterizations of those synchronization problems solvable by DIjkstra's PV system of primitives and its various generalizations including PVgeneral, PVmultlple, PVchunk, Vector Addmon, and Loopless Penn Net systems These characterizations form the foundations of a formal synthesis procedure for determining effioent solutions to synchronization problems

The foundations are laid for a theory of multiplicative complexity of algebras and it is shown how “multiplication problems” such as multiplication of matrices, polynomials, quaternions, etc., are instances of this theory. The usefulness of the theory is then demonstrated by utilizing algebraic ideas and results to derive complexity bounds. In particular linear upper and lower bounds for the complexity of certain types of algebras are established.

Extending a result of Rabin, It Is shown that the word problem for hnear groups (groups of matrices) over a field of characteristic 0 is solvable in (deterministic) logspace As an apphcatlon of this result, it follows that the word problem for free groups and hence the membership problem for the two-sided Dyck language are solvable in logspace

Many of the process synchronization problems studied in the literature are of the form of a conjunction of finitely many conditions of the type “process $p_i$ blocks process $p_j$”. Such problems may be expressed as directed graphs whose nodes represent the processes and where there is an edge from node i to node j if and only if process $p_i$ blocks process $p_j$. We characterize the class of graphs which correspond to the system of synchronizing primitives of Vantilborgh and van Lamsweerde in terms of a normal form representation and present an efficient algorithm for determining whether an arbitrary graph is in this class.

A basic question in the area of asynchronous computation is: Given a synchronization problem, what synchronization primitives are needed for a solution? This paper is directed toward answering this question by characterizing the "behavior" of synchronization systems incorporating PV, PV multiple, PV chunk and PV general synchronization primitives.

This note presents a new algorithm for computing the cyclic convolution of two vectors over a commutative ring. The algorithm requires n(n1+1)...(nk+1)/2kmultiplications for the convolution of two n-vectors, where n=n1...nkis a factorization of n into factors which are pairwise relatively prime.