Introduction to Computational Linguistics

22. Shift–Reduce–Parsing 89
different automata that recognize the languages based on that grammar. The PDA
implements what is often called a parsing strategy. The parsing strategy makes
use of the rules of the grammar, but depending on the grammar in quite different
ways. One very popular method of parsing is the following. We scan the string,
putting the symbols one by one on the stack. Every time we hit a right hand side
of a rule we undo the rule. This is to do the following: suppose the top of the
stack contains the right hand side of a rule (in reverse order). Then we pop that
sequence and push the left–hand side of the rule on the stack. So, if the rule is
A → a and we get a, we first push it onto the stack. The top of the stack now
matches the right hand side, we pop it again, and then push A. Suppose the top of
the stack contains the sequence BA and there is a rule S → AB, then we pop twice,
removing that part and push S. To do this, we need to be able to remember the top
of the stack. If a rule has two symbols to its right, then the top two symbols need
to be remembered. We have seen earlier that this is possible, if the machine is
allowed to do some empty transitions in between. Again, notice that the strategy
is nondeterministic in general, because several options can be pursued at each
step. (a) Suppose the top part of the stack is BA, and we have a rule S → AB. Then
either we push the next symbol onto the stack, or we use the rule that we have. (b)
Suppose the top part of the stack is BA and we have the rules S → AB and C → A.
Then we may either use the first or the second rule.
The strongest variant is to always reduce when the right hand side of a rule is
on top of the stack. Despite not being always successful, this strategy is actually
useful in a number of cases. The condition under which it works can be spelled out
as follows. Say that a CFG is transparent if for every string ⃗α and nonterminal
X ⇒ ∗ ⃗α, if there is an occurrence of a substring ⃗γ, say ⃗α = ⃗κ 1 ⃗γ⃗κ 2 , and if there is
a rule Y → ⃗γ, then 〈⃗κ 1 ,⃗κ2〉 is a constituent occurrence of category Y in ⃗α. This
means that up to inessential reorderings there is just one parse for any given string
if there is any. An example of a transparent language is arithmetical terms where
no brackets are omitted. Polish Notation and Reverse Polish Notation also belong
to this category. A broader class of languages where this strategory is successful
is the class of NTS–languages. A grammar has the NTS–property if whenever
there is a string S ⇒ ∗ ⃗κ 1 ⃗γ⃗κ 2 and if Y → ⃗γ then S ⇒ ∗ ⃗κ 1 Y⃗κ 2 . (Notice that this does
not mean that the constituent is a constituent under the same parse; it says that one
can find a parse that ends in Y being expanded in this way, but it might just be a
different parse.) Here is how the concepts differ. There is a grammar for numbers
that runs as follows.
(219)
< digit > → 0 | 1 | · · · | 9