Introduction to Computational Linguistics

12. Finite State Automata 39
There is a lot of repetition involved in this definition, so it is necessary only to
look at part of this. First, notice that prior to this definition, the program contains
definitions of sets of states, which are simply sets of integers. The type is called
StateSet. Similarly, the type CharSet is defined. Also there is the type of
transition, which has three entries, first, symbol and second. They define the
state prior to scanning the symbol, the symbol, and the state after scanning the
symbol, respectively. Then a type of sets of transitions is defined. These are
now used in the definition of the object type automaton. These things need to
be formally introduced. The attribute mutable shows that their values can be
changed. After the equation sign is a value that is set by default. You do not have
to set the value, but to prevent error it is wise to do so. You need a method to even
get at the value of the objects involved, and a method to change or assign a value
to them. The idea behind an object type is that the objects can be changed, and
you can change them interactively. For example, after you have loaded the file
fsa.ml typing #use "fsa.ml" you may issue
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# let a = new automaton;;
Then you have created an object identified by the letter a, which is a finite state
automaton. The initial state is 0, the state set, the alphabet, the set of accepting
states is empty, and so is the set of transitions. You can change any of the values
using the appropriate method. For example, type
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# a#add_alph ’z’;;
and you have added the letter ‘z’ to the alphabet. Similarly with all the other
components. By way of illustration let me show you how to add a transition.
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# a#add_transitions {first = 0; second = 2; symbol = ’z’};;
So, effectively you can define and modify the object step by step.
If A is not partial it is called total. It is an easy matter to make an automaton
total. Just add a state q ♯ and add a transition 〈q, a, q ♯ 〉 when there was no transition
〈q, a, q ′ 〉 for any q ′ . Finally, add all transitions 〈q ♯ , a, q ♯ 〉, q ♯ is not accepting.
Theorem 5 Let A be a finite state automaton. Then L(A) is regular.
The idea is to convert the automaton into an equation. To this end, for every state
x
q let T q := {x : i 0 −→ q}. Then if 〈r, a, q〉, we have T q ⊇ a · T r . In fact, if i i 0 we