Introduction to Computational Linguistics

13. Complexity and Minimal Automata 47
It is easy to check that [b] M = [a] M . We shall see that this difference means that
there is an automaton checking M can based on less states than any automaton
checking membership in L.
a
Given two index set I and J, put I → J if and only if J = a\I. This is
well-defined. For let I = [⃗x] L . Then suppose that ⃗xa is a prefix of an accepted
string. Then [⃗xa] L = {⃗y : ⃗xa⃗y ∈ L} = {⃗y : a⃗y ∈ I} = a\I. This defines a
deterministic automaton with initial element L. Accepting sets are those which
contain ε. We call this the index automaton and denote it by I(L). (Often it is
called the Myhill-Nerode automaton.)
Theorem 15 (Myhill-Nerode) L(I(L)) = L.
Proof. By induction on ⃗x we show that L ⃗x → I if and only if [⃗x] L = I. If ⃗x = ε
the claim reads L = L if and only if [ε] L = L. But [ε] L = L, so the claim holds.
Next, let ⃗x = ⃗ya. By induction hypothesis, L ⃗y → J if and only if [⃗y] L = I. Now,
J a → J/a = [⃗ya] L . So, L ⃗x → J/a = [⃗x] L , as promised.
Now, ⃗x is accepted by I(L) if and only if there is a computation from L to a set
[⃗y] L containing ε. By the above this is equivalent to ε ∈ [⃗x] L , which means ⃗x ∈ L.
□
Given an automaton A and a state q put
(129) [q] := {⃗x : there is q ′ ∈ F: q ⃗x → q ′ }
It is easy to see that for every q there is a string ⃗x such that [q] ⊆ [⃗x] L . Namely, let
⃗x
⃗x be such that i 0 → q. Then for all ⃗y ∈ [q], ⃗x⃗y ∈ L(A), by definition of [q]. Hence
[q] ⊆ [⃗x] L . Conversely, for every [⃗x] L there must be a state q such that [q] ⊆ [⃗x] L .
⃗x
Again, q is found as a state such that i 0 → q. Suppose now that A is deterministic
and total. Then for each string ⃗x there is exactly one state [q] such that [q] ⊆ [⃗x] L .
⃗x⃗y
Then obviously [q] = [⃗x] L . For if ⃗y ∈ [⃗x] L then ⃗x⃗y ∈ L, whence i 0 −→ q ′ ∈ F for
some q ′ ⃗x
. Since the automaton is deterministic, i 0 → q → ⃗y q ′ , whence ⃗y ∈ [q].
It follows now that the index automaton is the smallest deterministic and total
automaton that recognizes the language. The next question is: how do we make
that automaton? There are two procedures; one starts from a given automaton,