Introduction to Computational Linguistics

12. Finite State Automata 43
Proof. First, we show that if ε → ⃗x u then ⃗x ∈ L(u). This is done by induction on
the length of ⃗x. If ⃗x = ε the claim trivially follows. Now let ⃗x = ⃗ya for some
a. Assume that ε → ⃗x u. Then there is t such that ε → ⃗y t → u. By inductive
assumption, ⃗y ∈ L(t), and by definition of the transition relation, L(t)a ⊆ L(u).
Whence the claim follows. Now for the converse, the claim that if ⃗x ∈ L(u) then
ε ⃗x → u. Again we show this by induction on the length of ⃗x. If ⃗x = ε we are done.
Now let ⃗x = ⃗ya for some a ∈ A. We have to show that there is a t ∈ s † such that
ta ⊆ u. For then by inductive assumption, ε ⃗y → t, and so ε ⃗ya → u, by definition of
the transition relation.
Now for the remaining claim: if ⃗ya ∈ L(u) then there is a t ∈ s † such that
⃗y ∈ L(t) and L(ta) ⊆ L(u). Again induction this time on u. Notice right away that
u † ⊆ s † , a fact that will become useful. Case 1. u = b for some letter. Clearly,
ε ∈ s † , and putting t := ε will do. Case 2. u = u 1 ∪ u 2 . Then u 1 and u 2 are
both in s † . Now, suppose ⃗ya ∈ L(u 1 ). By inductive hypothesis, there is t such
that L(ta) ⊆ L(u 1 ) ⊆ L(u), so the claim follows. Similarly if ⃗ya ∈ u 2 . Case 3.
u = u 1 u 2 . Subcase 2a. ⃗y = ⃗y 1 ⃗y 2 , with ⃗y 1 ∈ L(u 1 ) and ⃗y 2 ∈ L(u 2 ). Now, by inductive
hypothesis, there is a t 2 such that L(t 2 a) ⊆ L(u 2 ). Then t := u 1 t 2 is the desired
term. Since it is in u † , it is also in s † . And L(ta) = L(u 1 t 2 a) ⊆ L(u 1 u 2 ) = L(u).
Case 4. u = u ∗ 1 . Suppose ⃗ya ∈ L(u). Then ⃗y has a decomposition ⃗z 0⃗z 1 · · ·⃗z n−1 ⃗v such
that ⃗z i ∈ L(u 1 ) for all i < n, and also ⃗va is in L(u 1 ). By inductive hypothesis, there
is a t 1 such that L(t 1 a) ⊆ L(u 1 ), and ⃗v ∈ L(t 1 ). And ⃗z 0 ⃗z 1 · · ·⃗z n−1 ∈ L(u). Now put
t := ut 1 . This has the desired properties.
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Now, as a consequence, if L is regular, so is L p . Namely, take the automaton
A(s), where s is a regular term of L. Now change this automaton to make every
state accepting. This defines a new automaton which accepts every string that falls
under some t ∈ s † , by the previous results. Hence, it accepts all prefixes of string
from L.
We discuss an application. Syllables of a given language are subject to certain
conditions. One of the most famous constraints (presumed to be universal) is the
sonoricity hierarchy. It states that the sonoricity of phonemes in a syllable must
rise until the nucleus, and then fall. The nucleus contains the sounds of highest
sonoricity (which do not have to be vowels). The rising part is called the onset and
the falling part the rhyme. The sonoricity hierarchy translates into a finite state
automaton as follows. It has states 〈o, i〉, and 〈r, i〉, where i is a number smaller
a