  Angluin's L$^*$ algorithm learns the minimal deterministic finite automaton
(DFA) of a regular language using membership and equivalence queries. Its
probabilistic approximatively correct (PAC) version substitutes an equivalence
query by numerous random membership queries to get a high level confidence to
the answer. Thus it can be applied to any kind of device and may be viewed as
an algorithm for synthesizing an automaton abstracting the behavior of the
device based on observations. Here we are interested on how Angluin's PAC
learning algorithm behaves for devices which are obtained from a DFA by
introducing some noise. More precisely we study whether Angluin's algorithm
reduces the noise and produces a DFA closer to the original one than the noisy
device. We propose several ways to introduce the noise: (1) the noisy device
inverts the classification of words w.r.t. the DFA with a small probability,
(2) the noisy device modifies with a small probability the letters of the word
before asking its classification w.r.t. the DFA, (3) the noisy device combines
the classification of a word w.r.t. the DFA and its classification w.r.t. a
counter automaton, and (4) the noisy DFA is obtained by a random process from
two DFA such that the language of the first one is included in the second one.
Then when a word is accepted (resp. rejected) by the first (resp. second) one,
it is also accepted (resp. rejected) and in the remaining cases, it is accepted
with probability 0.5. Our main experimental contributions consist in showing
that: (1) Angluin's algorithm behaves well whenever the noisy device is
produced by a random process, (2) but poorly with a structured noise, and, that
(3) is able to eliminate pathological behaviours specified in a regular way.
Theoretically, we show that randomness almost surely yields systems with
non-recursively enumerable languages.
