Construct a Predictive Parsing table for the following grammar & also check whether string
id + id * id is accepted or not.

Problem − Consider the following grammar −

E → TE′

E′ → +TE′|ε

T′ → FT′

T′ → FT′|ε

F → (E)|id

Solution −

Step1− Elimination of Left Recursion & perform Left Factoring

As there is no left recursion in Grammar so, we will proceed as it is. Also, there is no need for Left Factoring.

Step2− Computation of FIRST

FIRST(E) = FIRST(T) = FIRST(F) = {(, id}

FIRST (E′) = {+, ε}

FIRST (T′) = {*, ε}

Step3− Computation of FOLLOW

FOLLOW (E) = FOLLOW(E′) = {), $}

FOLLOW (T) = FOLLOW(T′) = {+, ), $}

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

Step4− Construction of Predictive Parsing Table

Create the table, i.e., write all non-terminals row-wise & all terminal column-wise.

Now, fill the table by applying rules from the construction of the Predictive Parsing Table.

• E → TE′

Comparing E → TE′with A → α

∴ A = E, α = TE′

∴ FIRST(α) = FIRST(TE′) = FIRST(T) = {(, id}

Applying Rule (1) of Predictive Parsing Table

∴ ADD E → TE′ to M[E, ( ] and M[E, id]

∴ write E → TE′in front of Row (E)and Columns {(, id} (1)

• E′ → +TE′|𝛆

Comparing it with A → α

∴ A = E′ α = +TE′

∴ FIRST(α) = FIRST(+TE′) = {+}

∴ ADD E → +TE′ to M[E′, +]

∴ write production E′ → +TE′ in front of Row (E′)and Column (+) (2)

• E′ → ε

Comparing it with A → α

∴ α = ε

∴ FIRST(α) = {ε}

∴ Applying Rule (2)of the Predictive Parsing Table.

Find FOLLOW (E′) = { ), $}

∴ ADD Production E′ → ε to M[E′, )] and M[E′, $]

∴ write E′ → ε in front of Row (E′)and Column {$, )}                                                         (3)

• T → FT′

Comparing it with A → α

∴ A = T, α = FT′

∴ FIRST(α) = FIRST (FT′) = FIRST (F) = {(, id}

∴ ADD Production T → FT′ to M[T, (] and M[T, id]

∴ write T → FT′ in front of Row (T)and Column {(, id}                                                       (4)

• T′ →*FT′

Comparing it with A → α

∴ FIRST(α) = FIRST (* FT′) = {*}

∴ ADD Production T → +FT′ to M[T′,*]

∴ write T′ →∗ FT′ in front of Row (T′)and Column {*} (5)

• T′ → ε

Comparing it with A → α

∴ A = T′

α = ε

∴ FIRST(α) = FIRST {𝜀} = {ε}

∴ Applying Rule (2)of the Predictive Parsing Table.

Find FOLLOW (A) = FOLLOW (T′) = {+, ), $}

∴ ADD T′ → ε to M[T′, +], M[T′, )] and M[T′, $]

∴ write T′ → ε in front of Row (T′)and Column {+, ), $} (6)

• F →(E)

Comparing it with A → α

∴ A = F, α = E

∴ FIRST(α) = FIRST ((E)) = {(}

∴ ADD F → (E) to M[F, (]

∴ write F → (E) in front of Row (F)and Column (( ) (7)

• F → id

Comparing it with A → α

∴ FIRST(α) = FIRST (id) = {id}

∴ ADD F → id to M[F, id]

∴ write F → id in front of Row (F)and Column (id) (8)

Combining all statements (1) to (8) will generate the following Predictive or LL (1) Parsing Table −

Step5− Checking Acceptance of String id + id * id using Predictive Parsing Program

Initially, the stack will contain the starting symbol E and $ at the bottom of the stack. Input Buffer will contain a string attached with $ at the right end.

If the top of stack = Current Input Symbol, then symbol from the top of the stack will be popped, and also input pointer advances or reads next symbol.

The sequence of Moves by Predictive Parser