Rational transductions and complexity of counting problems

This work presents an algebraic method, based on rational transductions, to study the sequential and parallel complexity of counting problems for regular and context-free languages. This approach allows us to obtain old and new results on the complexity of ranking and unranking as well as on other problems concerning the number of prefixes, suffixes, subwords, and factors of a word which belongs to a fixed language. Other results concern a suboptimal compression of finitely ambiguous context-free languages, the complexity of the value problem for rational and algebraic formal series in noncommuting variables, and a characterization of regular and Z-algebraic languages by means of ranking functions.