### Abstract:

How likely is that a randomly given (non-) deterministic finite automaton recognizes no
word? A quick reflection seems to indicate that not too many finite automata accept no word;
but, can this intuition be confirmed?
In this paper we offer a statistical approach which allows us to conclude that for automata,
with a large enough number of states, the probability that a given (non-) deterministic finite
automaton recognizes no word is close to zero. More precisely, we will show, with a high
degree of accuracy (i.e., with precision higher than 99% and level of confidence 0.9973), that for
both deterministic and non-deterministic finite automata: a) the probability that an automaton
recognizes no word tends to zero when the number of states and the number of letters in the
alphabet tend to infinity, b) if the number of states is fixed and rather small, then even if the
number of letters of the alphabet of the automaton tends to infinity, the probability is strictly
positive. The result a) is obtained via a statistical analysis; for b) we use a combinatorial and
statistical analysis.
The present analysis shows that for all practical purposes the fraction of automata recognizing
no words tends to zero when the number of states and the number of letters in the alphabet
grow indefinitely.
In the last section we critically discuss the method and result obtained in this paper. From
a theoretical point of view, the result can motivate the search for “certitude”, that is, a proof
of the fact established here in probabilistic terms. In fact, the method used is much more
important than the result itself. The method is “general” in the sense that it can be applied to
a variety of questions in automata theory, certainly some more difficult than the problem solved
in this note.