Membership problems for regular and context free trace languages

Trace languages have been introduced in order to describe the behaviour of concurrent systems in the same way as usual formal languages do for sequential system. They can be defined as subsets of a free partially commutative monoid and a theory of trace languages can be developed, generalizing the usual formal languages theory. In this paper, the time complexity of membership problems for regular and context-free trace languages is investigated. It is proved that the membership problem for context free trace languages can be solved in time O(BM(nα)), where α is the dimension of the greatest clique of the concurrency relation C and BM(n) is the time required for multiplying two arbitrary n × n boolean matrices. For regular trace languages, our method gives an algorithm which requires O(nα) time. Finally, the uniform membership problem is shown to be NP-complete. © 1989.