Recognizing group languages with OBDDs
Résumé
Let X be a subset of the free monoid {0,1}^* which is the inverse image of the unit in a morphism which maps {0,1}^* into a finite group. For each integer n, let Xn consist of all the words in X of length n. Identifying Xn with a Boolean function on n variables in the natural way, allows one to use an ordered binary decision diagram (OBDD) to recognize it. Such a diagram can be viewed as a finite deterministic automaton where the letters, instead of being read from left to right, are being read in a predetermined order. For a given Boolean function, the resulting size of the OBDD depends on the choice of the order. The authors proved that for a wide variety of subsets X, under the uniform distribution hypothesis over all orderings of n elements, there exists a real r > 1 such that with probability 1 when n tends to infinity, the size of the reduced OBDD computing Xn grows at least as fast as r^n.