On the prefixes of a random trace and the membership problem for context-free trace languages

Abstract

In this paper we study the complexity of algorithms for problems on free partially commutative monoids (f.p.c.m.). These monoids are widely studied in the literature and their properties are used in several research areas. The elements of a f.p.c.m. and its subsets are called traces and trace languages respectively.We first present some probabilistic estimations of the number of prefixes of a trace. The proof technique is based on the analysis of the generating functions of particular languages and is related to classical combinatorial methods. Then, we apply these probabilistic results to the analysis of algorithms for problems on trace languages. In particular, we describe an algorithm for the Membership Problem of context-free trace languages and determine its time complexity both in the worst and in the average case, assuming that all the input strings of given length have the same probability. Moreover, we show that, with probability tending to 1, the time complexity of our algorithm has the same order of growth of its mean value. At last we emphasize the difference between worst case and average case behaviour on some general examples.KeywordsTime ComplexityRandom GraphMaximum CliqueTransitive ClosureFinite AutomatonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.