Introduction to Computational Linguistics

21. Pushdown Automata 85
pushdown. Once you hit on a b you start popping the as from the pushdown, one
for each b that you find. If the pushdown is emptied before the string is complete,
then you have more bs than a. If the pushdown is not emptied but the string is
complete, then you have more as than bs. So, you can tell whether you have a
string of the required form if you can tell whether you have an empty storage. We
assume that this is the case. In fact, typically what one does is to fill the storage
before we start with a special symbol #, the end-of-storage marker. The storage
is represented as a string over an alphabet D that contains #, the storage alphabet.
Then we are only in need of the following operations and predicates:
➊ For each d ∈ D we have an operation push d : ⃗x ↦→ ⃗x ⌢ d.
➋ For each d ∈ D we have a predicate top d which if true of ⃗x iff ⃗x = ⃗y ⌢ d.
➌ We have an operation pop : ⃗x ⌢ d ↦→ ⃗x.
(If we do not have an end of stack marker, we also need a predicate ‘empty’, which
is true of a stack ⃗x iff ⃗x = ε.
Now, notice that the control structure is a finite state automaton. It schedules
the actions using the stack as a storage. This is done as follows. We have two
alphabets, A, the alphabet of letters read from the tape, and I, the stack alphabet,
which contains a special symbol, #. Initially, the stack contains one symbol, #. A
transition instruction is a a quintuple 〈s, c, t, s ′ , p〉, where s and s ′ are states, c is a
character or empty (the character read from the string), and t is a character (read
from the top of the stack) and finally p an instruction to either pop from the stack
or push a character (different from #) onto it. A PDA contains a set of instructions.
Formally, it is defined to be a quintuple 〈A, I, Q, i 0 , F, σ〉, where A is the input
alphabet, I the stack alphabet, Q the set of states, i 0 ∈ Q the start state, F ⊆ Q the
set of accepting states, and σ a set of instructions. If A is reading a string, then it
does the following. It is initialized with the stack containing # and the initial state
i 0 . Each instruction is an option for the machine to proceed. However, it can use
that option only if it is in state s, if the topmost stack symbol is t and if c ε, the
next character must match c (and is then consumed). The next state is s ′ and the
stack is determined from p. If p = pop, then the topmost symbol is popped, if it
is push a , then a is pushed onto stack. PDAs can be nondeterministic. For a given
situation we may have several options. If given the current stack, the current state
and the next character there is at most one operation that can be chosen, we call