Lecture Notes in Computer Science 3472

19 Model Checking 579
Expanding a Pack Let us now discuss how to extend a pack, evolving to the
automaton to learn. We will see the importance of the fact that the function γ
can be partial. Let us start by stating that for a given pack we say that a word
z is escaping when γ(z) is undefined.
We can use the knowledge of some escaping string z to adjust the observation
pack and get closer to U. In this case we can collect the appropriate observations
and expand the pack by adding a new component to it.
The fact that z escapes implies that, for each access string s in the pack,
there is a word sw ∈ Cs providing the suffix w that distinguishes z from s, inthe
sense that sw ∈ U ⇐⇒ zw �∈ U. A set formed by all such words zw, additionally
including z itself, and each labeled by the corresponding +/− label, forms a
component that can be added to the pack preserving two mentioned necessary
properties OP1 and OP2. Each expansion by one component brings the pack
one state closer to the minimal automaton representing U.
Now we want to have a so called closed pack, which means that for an access
string s and letter a there is no such word sa that escapes. The transition on a
from the state represented by s would be undefined if sa escaped. If we discover
an escaping word we expand the pack and when there are no more escaping
words of this kind we say that the pack is closed.
Definition 19.14. ApackO agreeing with U is closed for U if:
• γ(ε) is defined;
•∀s ∈ S, ∀ a ∈ Σ,γ(sa) is defined.
Note that the definition depends on U since γ = γ O,U depends on U. Definition
19.14 actually gives rise to a deterministic finite automaton, whose states
are the access strings of the pack, the initial state is γ(ε) and the accepting are
states those access strings labeled +. We define the transition function to be
δ(s, a) =γ(sa) and we extend the transition function δ in a rather common way
by, δ(s,ε)=s, andδ(s, wa) =δ(δ(s, w), a), for s ∈ S, w ∈ Σ ∗ ,anda ∈ Σ.
Theorem 19.15. If O is a pack, U is regular and agrees with O, andO has as
many components as M (the minimal DFA that recognizes U) has states, then
O is closed for U so that an automaton can be obtained from O in the manner
above, and furthermore this automaton is isomorphic to M.
Proof. By Lemma 19.11, no pack agreeing with U canhavemorethan|M|
components. Therefore, it is not possible to expand O preserving the agreement
with U, and thus it must be closed. Besides, by the cardinality condition, the
function f defined and used in Lemma 19.11 becomes bijective. It is routine to
show that this function is an isomorphism, that is: 1) it maps γ(ε) to the initial
state of M; 2) it maps exactly those s ∈ S with s ∈ U to the final states; and
3) it commutes with M’s transition function. ⊓⊔
Once we are able to form an automaton from our pack, we can make an
equivalence query to the Oracle to find out if it is equivalent to M. If we receive
the answer ’yes’, we are finished and the automaton is the minimal automaton