Now the npda performs nondeterministically one of the

Now the npda performs nondeterministically one of the following:
• Push the input symbol onto the pushdown.
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, Λ, S
Λ,-
S,
S,
a,a,
b,b,
c,c,
c
S, a
b b, S
ε,
ε,ε, ε,
ε,
ε,
ε,ε,
ε
$,
Figure 6.1: npda for CFG G (by top-down construction)
• If A → α is any production and the top of the pushdown is α
R
then pop off α
R
one
symbol at a time and then push the nonterminal A .
At the end, when $ is encountered, if the pushdown contains the start symbol S the input
is accepted.
The correctness of this construction relies on showing that this procedure backtracks a right
most derivation.
The pda for the above grammar is shown in Figure 6.2.
b, a,
b,*,
c
b
c,*,
a,*,
a
1
S ,
S
c,
S ,
1
S
S
1
$,S,-
ε,
ε, ε,
ε,*,
ε,*,
ε,
ε,
1
ε,*,
Figure 6.2: npda for CFG G (by bottom-up construction)
In summary, a top-down recognizer can apply a production A → α by replacing the
topmost symbol A by α
R
or can match the input symbol with the pushdown symbol and
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pop the pushdown. A bottom-up recognizer can apply the above production by replacing
α
R
(by popping off one symbol at a time) by A or can push the input symbol read on
the pushdown. Extensive literature exists on these two types of recognition. Under some
conditions, the resulting pushdowns become deterministic. The subclass of CFGs for which
the top-down procedure yields a deterministic pda are known as LL grammars (the first
L specifies that the input string is read left to right and the second L specifies that the
recognition is done by following the leftmost derivation), and the subclass of CFGs for which
the bottom-up procedure results in a deterministic pda are known as LR grammars (the
first symbol L specifies that the input is read left to right and the second symbol R specifies
that the recognition is done by following the rightmost derivation).
For simplicity, assume that there is no ǫ production. Let the input string be x =
a
1
a
2
··· a
n
$. Suppose the current input symbol read is a
i
and the topmost symbol of the
pushdown is A . There can be several productions each with A as the left hand symbol. If the
pda can deterministically choose the production to apply based on a
i
and A then the gram-
mar is an LL (1) grammar. (Our standard pda model needs a minor change. We assumed
that if a transition is labeled a
i
, A then the pda automatically advances to the next input
symbol a
i +1
. For this construction, the pda stays on a
i
and the only time it advances is when
it matches and pops a
i
from the pushdown.) If the grammar can deterministically choose
the production based on a
i
a
i +1
··· a
i + k − 1
and A , then the grammar is an LL ( k ) grammar.
(An LL ( k ) grammar permits length k look ahead on the input tape.) ZZZ: Explain LR ( k )
grammars.
Before establishing the next result that npda’s can be converted into CFGs, we simplify
npdas to a special class of one state npdas. Initially the pushdown contains a special symbol
U in it. In these automata, the push operation is based on only the input symbol; i.e. unlike
in usual npdas, the operation is independent of the top symbol of the pushdown. Instead of
the normal pop, based on the input symbol, the pda can replace up to 2 top symbols of the
pushdown by a single symbol. Acceptance is based on the pushdown ending up in a special
symbol S when $ is encountered on the input. The main advantages gained by introducing
these special pdas are two fold: there is only one state and the push operation is independent
of the top symbol of the pushdown.