It is the process of analyzing a continuous stream of input in order to determine its grammatical structure with respect to a given formal grammar.

**PARSING**

It is the process of
analyzing a continuous stream of input in order to determine its grammatical
structure with respect to a given formal grammar.

**Parse tree:**

Graphical
representation of a derivation or deduction is called a parse tree. Each
interior node of the parse tree is a non-terminal; the children of the node can
be terminals or non-terminals.

**Types of parsing:**

1.
Top down parsing

2.
Bottom up parsing

Ø Top-down
parsing : A parser can start with the start symbol and try to transform it to
the input string. Example : LL Parsers.

Ø
Bottom-up parsing : A parser can start
with input and attempt to rewrite it

into the start symbol. Example : LR Parsers.

**TOP-DOWN PARSING**

It can be viewed as an
attempt to find a left-most derivation for an input string or an attempt to
construct a parse tree for the input starting from the root to the leaves.

**Types -downoftopparsing :**

1.
Recursive descent parsing

2.
Predictive parsing

**RECURSIVE DESCENT PARSING**

Typically, top-down
parsers are implemented as a set of recursive functions that descent through a
parse tree for a string. This approach is known as recursive descent parsing,
also known as LL(k) parsing where the first L stands for left-to-right, the
second L stands for leftmost-derivation, and k indicates k-symbol lookahead.

Therefore,
a parser using the single-symbol look-ahead method and top-down parsing without
backtracking is called LL(1) parser. In the following sections, we will also
use an extended BNF notation in which some regulation expression operators are
to be incorporated.

This parsing method may involve backtracking.

__Example for__**
:backtracking**

Consider
the grammar G : S → cAd

A→ab|a

and
the input string w=cad.

The
parse tree can be constructed using the following top-down approach :

**Step1:**

Initially create a tree
with single node labeled S. An input pointer points to ‘c’, the first symbol of
w. Expand the tree with the production of S.

**Step2:**

The leftmost leaf ‘c’
matches the first symbol of w, so advance the input pointer to the second
symbol of w ‘a’ and consider the next leaf ‘A’. Expand A using the first
alternative.

**Step3:**

The second symbol ‘a’
of w also matches with second leaf of tree. So advance the input pointer to
third symbol of w ‘d’**.**But the third leaf of tree is b which does not
match with the input symbol **d.**Hence discard the chosen production and
reset the pointer to second **backtracking.**

**Step4:**

Now try the second alternative for A.

Now we can halt and announce the successful
completion of parsing.

**Predictive parsing**

It is possible to build
a nonrecursive predictive parser by maintaining a stack explicitly, rather than
implicitly via recursive calls. The key problem during predictive parsing is
that of determining the production to be applied for a nonterminal . The
nonrecursive parser in figure looks up the production to be applied in parsing
table. In what follows, we shall see how the table can be constructed directly
from certain grammars.

**Fig.
2.4 Model of a nonrecursive predictive parser**

A table-driven
predictive parser has an input buffer, a stack, a parsing table, and an output
stream. The input buffer contains the string to be parsed, followed by $, a
symbol used as a right endmarker to indicate the end of the input string. The
stack contains a sequence of grammar symbols with $ on the bottom, indicating
the bottom of the stack. Initially, the stack contains the start symbol of the
grammar on top of $. The parsing table is a two dimensional array M[A,a] where
A is a nonterminal, and a is a terminal or the symbol $. The parser is
controlled by a program that behaves as follows. The program considers X, the
symbol on the top of the stack, and a, the current input symbol. These two
symbols determine the action of the parser. There are three possibilities.

1
If X= a=$, the parser halts and announces successful completion of parsing.

2 If
X=a!=$, the parser pops X off the stack and advances the input pointer to the
next input symbol.

3 If
X is a nonterminal, the program consults entry M[X,a] of the parsing table M.
This entry will be either an X-production of the grammar or an error entry. If,
for example, M[X,a]={X- >UVW}, the parser replaces X on top of the stack by
WVU( with U on top). As output, we shall assume that the parser just prints the
production used; any other code could be executed here. If M[X,a]=error, the
parser calls an error recovery routine

**Algorithm for Nonrecursive predictive
arsing.**

Input. A string w and a parsing table M for grammar
G.

Output. If w is in L(G), a leftmost derivation of w;
otherwise, an error indication.

Method. Initially, the
parser is in a configuration in which it has $S on the stack with S, the start
symbol of G on top, and w$ in the input buffer. The program that utilizes the
predictive parsing table M to produce a parse for the input is shown in Fig.

set ip to point to the first symbol of w$. repeat

let X be the top stack symbol and a the symbol
pointed to by ip. if X is a terminal of $ then

if
X=a then

pop X from the stack and advance ip else
error()

else

if M[X,a]=X->Y1Y2...Yk then begin pop
X from the stack;

push Yk,Yk-1...Y1 onto the stack, with Y1 on top;
output the production X-> Y1Y2...Yk

end

else
error()

until
X=$ /* stack is empty */

**Predictive
parsing table construction:**

The
construction of a predictive parser is aided by two functions associated with a
grammar G :

3. FIRST

4. FOLLOW

**Rules
for first( ):**

1. If
X is terminal, then FIRST(X) is {X}.

2. If
X → ε is a production, then add ε to FIRST(X).

3. If
X is non-terminal and X → aα is a production then add a to FIRST(X).

4. If
X is non-terminal and X → Y1 Y2…Yk is a production, then place a in FIRST(X) if
for some i, a is in FIRST(Yi), and ε is in all of FIRST(Y1),…,FIRST(Yi-1);that
is, Y1,….Yi-1=> ε. If ε is in FIRST(Yj) for all j=1,2,..,k, then add ε to
FIRST(X).

**Rules
for follow( ):**

1. If
S is a start symbol, then FOLLOW(S) contains $.

2. If
there is a production A → αBβ, then everything in FIRST(β) except ε is placed
in

follow(B).

3.
If there is a production A → αB, or a production A → αBβ where FIRST(β)
contains

ε,
then everything in FOLLOW(A) is in FOLLOW(B).

**Algorithm for construction of predictive
parsing table:**

Input : Grammar G

Output : Parsing table M

Method :

1. For
each production A → α of the grammar, do steps 2 and 3.

2. For
each terminal a in FIRST(α), add A → α to M[A, a].

3. If
ε is in FIRST(α), add A → α to M[A, b] for each terminal b in FOLLOW(A). If ε
is in FIRST(α) and $ is in FOLLOW(A) , add A → α to M[A, $].

4. Make
each undefined entry of M be error.

Example:

Consider the following grammar :

E→E+T|T

T→T*F|F

F→(E)|id

After eliminating left-recursion the grammar is

E →TE’

E’ → +TE’ | ε

T →FT’

T’ → *FT’ | ε

F → (E)|id

First( ) :

FIRST(E) = { ( , id}

FIRST(E’) ={+ , ε }

FIRST(T) = { ( , id}

FIRST(T’) = {*, ε }

FIRST(F) = { ( , id }

Follow( ):

FOLLOW(E) = { $, ) }

FOLLOW(E’) = { $, ) }

FOLLOW(T) = { +, $, ) }

FOLLOW(T’) = { +, $, ) }

FOLLOW(F) = {+, * , $ , ) }

**Predictive parsing Table**

**Stack Implementation**

**LL(1) grammar:**

The parsing table
entries are single entries. So each location has not more than one entry. This
type of grammar is called LL(1) grammar.

Consider this following
grammar:

S→iEtS | iEtSeS| a

E→b

After eliminating left factoring, we have

S→iEtSS’|a

S’→ eS | ε

E→b

To construct a parsing table, we need FIRST() and
FOLLOW() for all the non-terminals.

FIRST(S) = { i, a }

FIRST(S’) = {e, ε }

FIRST(E) = { b}

FOLLOW(S) = { $ ,e }

FOLLOW(S’) = { $ ,e }

FOLLOW(E) = {t}

**Parsing table:**

Since there are more than one production, the
grammar is not LL(1) grammar.

Actions performed in predictive parsing:

1. Shift

2. Reduce

3.
Accept

4.
Error

**Implementation of predictive parser:**

1.
Elimination of left recursion, left
factoring and ambiguous grammar.

2. Construct
FIRST() and FOLLOW() for all non-terminals.

3. Construct
predictive parsing table.

4. Parse
the given input string using stack and parsing table

**BOTTOM-UP PARSING**

Constructing a parse
tree for an input string beginning at the leaves and going towards the root is
called bottom-up parsing. A general type of bottom-up parser is a shift-reduce
parser.

**SHIFT-REDUCE PARSING**

Shift-reduce parsing is
a type of bottom-up parsing that attempts to construct a parse tree for an
input string beginning at the leaves (the bottom) and working up towards the
root (the top).

Example:

Consider the grammar:

S → aABe

A → Abc | b

B → d

The sentence to be recognized is abbcde.

**REDUCTION
(LEFTMOST) RIGHTMOST DERIVATION**

abbcde (A → b) S
→ aABe

aAbcde(A → Abc) →
aAde

aAde (B → d) → aAbcde

aABe (S → aABe) → abbcde

S

The reductions trace out the right-most derivation
in reverse.

**Handles:**

A handle of a string is
a substring that matches the right side of a production, and whose reduction to
the non-terminal on the left side of the production represents one step along
the reverse of a rightmost derivation.

Example:

Consider the grammar:

E→E+E

E→E*E

E→(E)

E→id

And the input string id1+id2*id3

The rightmost derivation is :

E→E+E

→ E+E*E

→ E+E*id3

→ E+id2*id3

→ id1+id2*

In the above derivation the underlined substrings
are called handles.

**Handle pruning:**

A rightmost derivation
in reverse can be obtained by “handle pruning”. (i.e.) if w is a sentence or
string of the grammar at hand, then w = γn, where γn is the nth rightsentinel
form of

some rightmost derivation.

**Actions in shift-reduce parser:**

•
shift - The next input symbol is shifted
onto the top of the stack.

•
reduce - The parser replaces the handle
within a stack with a non-terminal.

•
accept - The parser announces successful
completion of parsing.

•
error - The parser discovers that a
syntax error has occurred and calls an error recovery routine.

**Conflicts in shift-reduce parsing:**

There are two conflicts that occur in shift-reduce
parsing:

1. Shift-reduce
conflict: The parser cannot decide whether to shift or to reduce.

2. Reduce-reduce
conflict: The parser cannot decide which of several reductions to make.

**Stack implementation of shift-reduce
parsing :**

1. Shift-reduce conflict:

Example:

Consider the grammar:

E→E+E | E*E | id and input id+id*id

2. Reduce-reduce conflict:

Consider the grammar:
M→R+R|R+c|R

R→c

and input c+c

**Viable prefixes:**

α is a viable prefix of the grammar if there is w
such that αw is a right

The set of prefixes of right sentinel forms that can
appear on the stack of a shift-reduce parser are called viable prefixes

The set of viable prefixes is a regular language.

**OPERATOR-PRECEDENCE PARSING**

An efficient way of
constructing shift-reduce parser is called operator-precedence parsing.
Operator precedence parser can be constructed from a grammar called
Operator-grammar. These grammars have the property that no production on right
side is ɛor has two adjacent non-terminals.

Example:

Consider the grammar:

E→EAE|(E)|-E|id

A→+|-|*|/|↑

Since the right side
EAE has three consecutive non-terminals, the grammar can be written as follows:
E→E+E|E-E|E*E|E/E|E↑E |-E|id

**Operator precedence relations:**

There are three disjoint precedence
relations namely

<. - less than

=
- equal to

.>
- greater than

The relations give the following meaning:

a<.b
- a yields precedence to b

a=b
- a has the same precedence as b

a.>b
- a takes precedence over b

Rules for binary operations:

1. If
operator θ1 has higher precedence than operator θ2, then make

θ1 . > θ2 and θ2 < . θ1

2. If
operators θ1 and θ2, are of equal precedence, then make

θ1 . > θ2 and θ2 . > θ1 if operators are left
associative

θ1 < . θ2 and θ2 < . θ1 if right associative

3. Make
the following for all operators θ:

θ <.
id ,id.>θ

θ <.(,
(<.θ

).>θ, θ.>)

θ.>$ , $<. θ

Also make

( = ) , ( <. ( , )
.> ) , (<. id, id .>) , $ <. id , id .> $ , $ Example:

Operator-precedence relations for the grammar

E→E+E|E-E|E*E|E/E|E↑E |(E)|-E|idis given in the
following table assuming

1.
↑ is of highest precedence and
right-associative

2.
* and / are of next higher precedence
and left-associative, and

3.
+ and - are of lowest precedence and
left- Note that the blanks in the table denote error entries.

**Table
: Operator-precedence relations**

**Operator precedence parsing algorithm:**

Input : An input string w and a table of
precedence relations.

Output : If w is well
formed, a skeletal parse tree ,with a placeholder non-terminal E labeling all
interior nodes; otherwise, an error indication.

Method : Initially the
stack contains $ and the input buffer the string w $. To parse, we execute the
following program :

(1) Set
ip to point to the first symbol of w$;

(2)
repeat forever

(3) if $ is on top of
the stack and ip points to $ then

(4)
return else begin

(5)
let a be the topmost terminal symbol on
the stack and let b be the symbol pointed to by ip;

(6)
if a <. b or a = b then begin

(7)
push b onto the stack;

(8)
advance ip to the next input symbol;
end;

(9) else if a . > b
then /*reduce*/

(10)
repeat

(11)
pop the stack

(12)
until the top stack terminal is related
by <.to the terminal most recently popped

(13)
else error( ) end

**Stack implementation of operator
precedence parsing:**

Operator precedence
parsing uses a stack and precedence relation table for its implementation of
above algorithm. It is a shift-reduce parsing containing all four actions
shift, reduce, accept and error.

The initial configuration of an operator precedence
parsing is

STACK
: $

INPUT
: w$

where w is the input string to be parsed.

Example:

Consider the grammar E →
E+E | E-E | E*E | E/E | E↑E | (E) | id. Input string is id+id*id .The
implementation is as follows:

**Advantages of operator precedence
parsing:**

1.
It is easy to implement.

2. Once
an operator precedence relation is made between all pairs of terminals of a
grammar, the grammar can be ignored. The grammar is not referred anymore during
implementation.

**Disadvantages of operator precedence
parsing:**

1.
It is hard to handle tokens like the
minus sign (-) which has two different precedence.

2.
Only a small class of grammar can be
parsed using operator-precedence parser.

**LR PARSERS**

An efficient bottom-up
syntax analysis technique that can be used CFG is called LR(k) parsing. The ‘L’
is for left-to-right scanning of the input, the ‘R’ for constructing a rightmost
derivation in reverse, and the ‘k’ for the number of input symbols. When ‘k’ is
omitted, it is

assumed to be 1.

**Advantages of LR parsing:**

1.
It recognizes virtually all programming language constructs for which CFG can
be written.

2.
It is an efficient non-backtracking
shift-reduce parsing method.

3.
A grammar that can be parsed using LR
method is a proper superset of a grammar that can be parsed with predictive
parser

4.
4. It detects a syntactic error as soon
as possible.

**Drawbacks of LR method:**

It is too much of work
to construct a LR parser by hand for a programming language grammar. A
specialized tool, called a LR parser generator, is needed. Example: YACC.

**Types of LR parsing method:**

1. SLR- Simple LR

Easiest to implement,
least powerful.

2. CLR- Canonical LR

Most powerful, most
expensive.

3. LALR- Look-Ahead LR

Intermediate
in size and cost between the other two methods.

**The LR parsing algorithm:**

The schematic form of an LR parser is as follows:

**Fig.
2.5 Model of an LR parser**

It consists of an input, an output, a stack, a
driver program, and a pa parts (action and goto).

Ø
The driver program is the same for all
LR parser.

Ø
The parsing program reads characters
from an input buffer one at a time.

Ø
The program uses a stack to store a
string of the form s0X1s1X2s2…Xmsm, where sm is on top. Each Xi is a grammar
symbol and each si is a state.

Ø
The parsing table consists of two parts
: action and goto functions.

**Action : **The
parsing program determines sm, the state currently on top of stack, and ai, the**
**current input symbol. It then consults action[sm,ai] in the action table
which can have one of four values:

1. shift
s, where s is a state,

2. reduce
by a grammar production A → β,

3. accept,

4. 4.
error.

**Goto : **The
function goto takes a state and grammar symbol as arguments and produces a
state.

**LR Parsing algorithm:**

Input: An input string
w and an LR parsing table with functions action and goto for grammar G. Output:
If w is in L(G), a bottom-up-parse for w; otherwise, an error indication.

Method: Initially, the
parser has s0 on its stack, where s0 is the initial state, and w$ in the input
buffer. The parser then executes the following program:

set ip to point to the first input symbol of w$;
repeat forever begin

let s be the state on top of the stack and

a the symbol pointed to by ip;

if action[s, a] = shift s’ then begin

push a then s’ on top of the stack; advance ip to
the next input symbol end

else if action[s, a] = reduce A→β then
begin pop 2* | β | symbols off the stack;

let s’ be the state now on top of the stack; push A
then goto[s’, A] on top of the stack; output the production A→ β

end

else if action[s, a] = accept then

return

else error( )

end

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