On Sparseness, Ambiguity and other Decision Problems for Acceptors and Transducers

We consider some decision problems on sparseness, degrees of ambiguity and multiple valuedness concerning finite-state and pushdown acceptors and transducers. A language L is sparse if there is a polynomial P such that the number of strings of length n in L is atmost P(n). A recognizer (transducer) is of polynomial ambiguity (valued) if there exists a polynomial P such that the number of derivations (outputs) for any input of length n is at most P(n). We relate these problems and show that they are decidable for finite-state devices. For cfl's, only the sparseness problem is decidable. We also study some properties of structure generating function defined as f L (z)=Σa n z n , where a n is the number of strings of length n in a language L. Our results are useful in proving the non-regularity/non-context-freeness of some languages.