Probabilistic estimation of the number of prefixes of a trace

Abstract

We present a generating function technique to evaluate the number of strings of a given length recognized by a particular kind of finite-state automaton. Using the method we determine some asymptotic estimations of the number of prefixes in free partially commutative monoids. More precisely, we prove that for every concurrent alphabet 〈 Σ, C〉, assuming equiprobable strings, all of length n, the average number of prefixes of a trace of length n is ηn k +O( n k−1 ) and its variance is O( n 2 k−1 ), where k is the number of components of the dependency graph 〈 Σ, C c〉 and η is a rational constant depending only on 〈 Σ, C〉. These combinatorial results allow to determine the probabilistic behaviour of some algorithms for problems on trace languages, including the Membership Problems for regular and context-free trace languages.