Logspace computations in graph products. (July 2016)

• Record Type: Journal Article
• Authors: Diekert, Volker; Kausch, Jonathan
• Is Part Of: Journal of symbolic computation. Volume 75(2016)
• Page Start: 94
• Page End: 109
• Publication Date: 2016-07
• Subjects: Algorithmic group theory -- Graph products -- Logspace complexity -- Word problem -- Geodesics -- Conjugacy problem; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.11.009
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: 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.