SOME LIMIT DISTRIBUTIONS IN ANALYSIS OF ALGORITHMS FOR PROBLEMS ON TRACE LANGUAGES

Abstract

We study the limit distribution of the sequence of random variables [Formula: see text] where, for each n, [Formula: see text] is the number of prefixes of a trace of length n chosen in a given free partially commutative monoid under the assumption that all representative strings of length n are equiprobable. We determine such a limit distribution for every free partially commutative monoid defined on a concurrent alphabet <Σ,C> such that the complementary relation Cc is transitive. We prove that if the complementary alphabet <Σ,Cc> has k connected components of equal size, then [Formula: see text] (suitably normalized) converges in distribution to a chi-square with k-1 degrees of freedom. On the other hand, if at least two connected components of <Σ,Cc> have a different number of vertices, then [Formula: see text] converges in distribution to a normal random variable. These probabilistic results yield the limit distribution of the time complexity of several algorithms for problems on trace languages, including algorithms for the Membership Problem of regular and context-free trace languages.