Introduction to Computational Linguistics

23. Some Metatheorems 93
This proof can be simplified using another result. Consider a map µ : A → B ∗ ,
which assigns to each letter a ∈ A a string (possibly empty) of letters from B. We
extend µ to strings as follows.
(231) µ(x 0 x 1 x 2 · · · x n−1 ) = µ(x 0 )µ(x 1 )µ(x 2 ) · · · µ(x n−1 )
For example, let B = {c, d}, and µ(c) := anti ⌢ □ and µ(d) := missile ⌢ □. Then
(232)
(233)
(234)
µ(d) = missile ⌢ □
µ(cdd) = anti missile missile ⌢ □
µ(ccddd) = anti anti missile missile missile ⌢ □
So if M = {c k d k+1 : k ∈ N} then the language above is the µ–image of M (modulo
the blank at the end).
Theorem 36 Let µ : A → B ∗ and L ⊆ A ∗ be a regular language. Then the set
{µ(⃗v) : ⃗v ∈ L} ⊆ B ∗ also is regular.
However, we can also do the following: let ν be the map
(235) a ↦→ c, n ↦→ ε, t ↦→ ε, i ↦→ ε, m ↦→ d, s ↦→ ε, l ↦→ ε, e ↦→ ε, □ ↦→ ε
Then
(236) ν(anti) = c, ν(missile) = d, ν(□) = ε
So, M is also the image of L under ν. Now, to disprove that L is regular it is enough
to show that M is not regular. The proof is similar. Choose a number k. We show
that the conditions are not met for this k. And since it is arbitrary, the condition
is not met for any number. We take the string c k d k+1 . We try to decompose it into
⃗u⃗v⃗w such that ⃗u⃗v j ⃗w ∈ M for any j. Three cases are to be considered. (Case a)
⃗v = c p for some p (which must be > 0):
(237) cc · · · c • c · · · c • c · · · cdd · · · d
Then ⃗u⃗w = c k−p d k+1 , which is not in M. (Case b) ⃗v = d p for some p > 0. Similarly.
(Case c) ⃗v = c p d q , with p, q > 0.
(238) cc · · · c • c · · · cd · · · d • dd · · · d