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What is Left Recursion and how it is eliminated?
A Grammar G (V, T, P, S) is left recursive if it has a production in the form.
A → A α |β.
The above Grammar is left recursive because the left of production is occurring at a first position on the right side of production. It can eliminate left recursion by replacing a pair of production with
A → βA′
A → αA′|ϵ
Elimination of Left Recursion
Left Recursion can be eliminated by introducing new non-terminal A such that.
This type of recursion is also called Immediate Left Recursion.
In Left Recursive Grammar, expansion of A will generate Aα, Aαα, Aααα at each step, causing it to enter into an infinite loop
The general form for left recursion is
A → Aα1|Aα2| … . |Aαm|β1|β2| … . . βn
can be replaced by
A → β1A′|β2A′| … . . | … . . |βnA′
A → α1A′|α2A′| … . . |αmA′|ε
Example1 − Consider the Left Recursion from the Grammar.
E → E + T|T
T → T * F|F
F → (E)|id
Eliminate immediate left recursion from the Grammar.
Solution
Comparing E → E + T|T with A → A α |β
| E | → | E | +T | | | T |
| A | → | A | α | | | Β |
∴ A = E, α = +T, β = T
∴ A → A α |β is changed to A → βA′and A′ → α A′|ε
∴ A → βA′ means E → TE′
A′ → α A′|ε means E′ → +TE′|ε
Comparing T → T ∗ F|F with A → Aα|β
| T | → | T | *F | | | F |
| A | → | A | α | | | β |
∴ A = T, α =∗ F, β = F
∴ A → β A′ means T → FT′
A → α A′|ε means T′ →* FT′|ε
Production F → (E)|id does not have any left recursion
∴ Combining productions 1, 2, 3, 4, 5, we get
E → TE′ E′ → +TE′| ε T → FT′ T →* FT′|ε F → (E)| id
Example2 − Eliminate the left recursion for the following Grammar.
S → a|^|(T)
T → T, S|S
Solution
We have immediate left recursion in T-productions.
Comparing T → T, S|S With A → A α | β where A = T, α =, S and β = S
∴ Complete Grammar will be
S→ a|^(T) T→ ST′ T′ →,ST′| ε
Example3 − Eliminate the left recursion from the grammar
E → E + T|T
T → T * F|F
F → (E)|id
Solution
The production after removing the left recursion will be
E → TE′ E′ → +TE′| ∈ T → FT′ T′ →∗ FT′| ∈ F → (E)|id
Example4 − Remove the left recursion from the grammar
E → E(T)|T
T → T(F)|F
F → id
Solution
Eliminating immediate left-recursion among all Aα productions, we obtain
E → TE′ E → (T)E′|ε T → FT′ T′ → (F)T′|ε F → id
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