Efficiently Computing the Density of Regular Languages [chapter]

Manuel Bodirsky, Tobias Gärtner, Timo von Oertzen, Jan Schwinghammer
2004 Lecture Notes in Computer Science  
A regular language L is called dense if the fraction fm of words of length m over some fixed signature that are contained in L tends to one if m tends to infinity. We present an algorithm that computes the number of accumulation points of (fm) in polynomial time, if the regular language L is given by a finite deterministic automaton, and can then also efficiently check whether L is dense. Deciding whether the least accumulation point of (fm) is greater than a given rational number, however, is
more » ... oNP-complete. If the regular language is given by a non-deterministic automaton, checking whether L is dense becomes PSPACE-hard. We will formulate these problems as convergence problems of partially observable Markov chains, and reduce them to combinatorial problems for periodic sequences of rational numbers.
doi:10.1007/978-3-540-24698-5_30 fatcat:bukd7lwxa5behbvzvaiyq5saqm