Backtracking In Top Down Parsing

Overview:

Step 1: Keep a pointer just before the first symbol in the input string to be parsed

Step 2: Expand the start symbol

Step 3: Compare the terminal after the pointer with the unmatched leaf(from left to right, initially it is the first leaf).

Here there may be three cases,

• The leaf is a non-terminal: Then expand it by its production
• A matching terminal: Advance the pointer by one position
• A non-matching terminal: Do not advance pointer, but

Undo the current expansion/s and

Move pointer backwards basing on the matched symbols in the undone expansions and               Expand the non-terminal by its other alternative if it exists

Step 4: Repeat step 2 until the entire parse tree is constructed or failure occurs

Example:

Consider the grammar S→ cAd  A → ab | a           and parsing the string w = cad

Initially we maintain an input pointer to c in w and expand S by cAd as mentioned below

Now, the left most leaf c is matched with first symbol in w. So, we advance the input pointer to a, the second symbol in w and consider the next leaf
As the next leaf is a non-terminal A, expand it as displayed below

Now, we have a match for the second symbol that is a in w. So, we advance the input pointer to d.

Example:

Now, there is a mismatch between leaf b and d. So, we go back to A to expand it by another alternative
In going back to A, we must reset the pointer in w to point again to a

Now, we have a match for the second symbol that is a in w and third symbol d in w
So, we halt and announce successful completion of parsing of string w

Limitations:

It can't tolerate left recursion, which leads the parser into an infinite loop

The major problem is backtracking, which involves undoing of semantic effects like removing entries from the symbol table

The order in which the alternatives are tried can effect the language accepted

For Example, if we try alternative a for A first for string cabd

As S has no other alternative we report a failure even though cabd is in the language

When a failure is reported we have a little idea about where it actually occurred

Left Recursion:

A grammar G is said to be left recursive if it has an non-terminal A such that there is a derivation of the form A=> Aa for some ∝

Examples for Left Recursive Grammars

S → Sa | b

S → Ab A → Ad | Sa | e

S → Ac  A→Ab | Bc | a   B → Bb | Aa | Sc | b

Types of Left Recursion:

 Left Recursion which is immediate

This is indicated by production of the form A→A∝

Example: S →Sa | b

Left Recursion Which is not immediate

Some productions cause left recursion after two or more steps in the derivation process

Example:

Grammar S →Ab             A → Ad | Sa | e

                S ⇒ A    b ⇒ Sab (S is left recursive)

Elimination of Left Recursion which is immediate:
The immediate left recursion among some A productions of the form,

A→A∝₁| A∝₂ | ……. | A∝_m | β₁ | β₂| ….| β_n

Where no βi begins with A can be eliminated by replacing A productions by the following two productions
Where no βi begins with A can be eliminated by replacing A productions by the following two productions,
A→β1A'|β2A'|......|βnA'
A'→∝1A'|∝2A'|....|∝mA'|∈

Where A' is anew non-terminal introduced

Example 1: S→ Sa |b

After eliminating left recursion S→bS' S'→ aS'|∈

Example 2: E→E+T | T T→T*F|F F→(E)|id

Here only E and T are left recursive, after eliminating left recursion

E→TE' E'→+TE'|∈

T→FT' T'→FT'|∈

F→(E)|id

Elimination of Left Recursion which is not Immediate:

The following algorithm is used to eliminate left recursion which is not guide

1) arrange the non-terminals in some order A₁,A₂,....A_n

2) for(each i from 1 to n) {

3) for(each j from 1 to i-1) [

4) replace each production of the form A_i →A_jy by the

Ai→δ_1y| δ_2y|……| δ_ky, where

Aj→δ₁| δ₂|…..| δ_k are all current A_j-productions

5) }

6) eliminate the immediate left recursion among the A_i-productions

7) }

How to eliminate Backtracking?
The solution is, one must know that the given current input symbol a and the non terminal A are to be expanded in which one of the alternates of the production
A→β₁|β₂|......|β_n
is the unique alternate that derives a string beginning with a
In other terms, the proper alternate is detectable by looking at the first symbol it derives
Example 1:
Consider the grammar S→aS | bS | c
If the current input symbol is a, then we can easily say that the production S→aS has to be used
Example 2:
Consider the grammar S→aS | aaS | aSa | b
If the current input symbol is a, then we can't decide uniquely which production is to be used

Left Factoring:
The left factored grammar for
A→∝β₁|∝β₂|......|∝β_n|γ
where γ represents all alternates that do not begin with ∝ is,
A→∝A'|γ
A'→β₁|β₂|......|β_n
where A' is the new non terminal

Example 1: Grouping of common prefixes of alternative

                                Consider the grammar, which is not left factored

                                                S → aS | aaS | aSa | b

                                Common prefix is a

                                After left factoring,

                                                S → aS' | b

                                                S'→ S | aS | Sa

Example 2: Consider the grammar (dangling else problem), which is not left factored

                                S → iEts | iEtSeS | a

                                E → b

                Common prefix is iEtS

                After left factoring          S → iEtSS' | a

                                                S' → eS | ε

                                                E → b

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About Data Sciences by Venu

Hi, My name is Venugopala Chary and I'm Currently working as Associate Professor in Reputed Engineerng College, Hyderabad. I have B.Tech and M.tech in regular from JNTU Hyderabad. I have 11 Years of Teaching Experience for both B.Tech and M.Tech Courses.