Logspace computations in graph products

We consider three important and well-studied algorithmic problems in group theory: the word, geodesic, and conjugacy problem. We show transfer results from individual groups to graph products. We concentrate on logspace complexity because the challenge is actually in small complexity classes, only. The most difficult transfer result is for the conjugacy problem. We have a general result for graph products, but even in the special case of a graph group the result is new. Graph groups are closely linked to the theory of Mazurkiewicz traces which form an algebraic model for concurrent processes. Our proofs are combinatorial and based on well-known concepts in trace theory. We also use rewriting techniques over traces. For the group-theoretical part we apply Bass-Serre theory. But as we need explicit formulae and as we design concrete algorithms all our group-theoretical calculations are completely explicit and accessible to non-specialists

Asynchronous Games over Tree Architectures

We consider the task of controlling in a distributed way a Zielonka asynchronous automaton. Every process of a controller has access to its causal past to determine the next set of actions it proposes to play. An action can be played only if every process controlling this action proposes to play it. We consider reachability objectives: every process should reach its set of final states. We show that this control problem is decidable for tree architectures, where every process can communicate with its parent, its children, and with the environment. The complexity of our algorithm is l-fold exponential with l being the height of the tree representing the architecture. We show that this is unavoidable by showing that even for three processes the problem is EXPTIME-complete, and that it is non-elementary in general

Solving word equations modulo partial commutations

AbstractIt is shown that it is decidable whether an equation over a free partially commutative monoid has a solution. We give a proof of this result using normal forms. Our method is a direct reduction of a trace equation system to a word equation system with regular constraints. Hereby we use the extension of Makanin's theorem on the decidability of word equations to word equations with regular constraints, which is due to Schulz

Solution sets for equations over free groups are EDT0L languages

© World Scientific Publishing Company. We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jez, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to NSPACE(n log n) here

Parallel Program Schemata and Maximal Parallelism I. Fundamental Results

The phenomenon of maximal parallelism is investigated in the framework of a class of parallel program schemata. Part I presents the basic properties of this model. Two types of equivalence relation on computations are presented, to each of which there corresponds a concept of determinacy and equivalence for schemata. The correspondence between these relations is shown and related to other properties of schemata. Then the concept of maximal parallelism using one of the relations as a basis is investigated. A partial order on schemata is defined which relates their inherent parallelism. The results presented are especially concerned with schemata which are maximal with respect to this order, i.e. maximally parallel schemata. Several other properties are presented and shown to be equivalent to the property of maximal parallelism. It is then shown that for any schema of a certain class, there exists a unique equivalent schema which is maximally parallel. We call such a schema the “closure” of the original schema