Consider the ambiguous grammar.\\nE → E + E\\nE → E * E\\nE → (E)\\nE → id\\n(a) Construct LR (0) items for above grammar.\\n(b) Construct SLR parsing table for grammar.\\n(c) Parse the input string id + id * id.

Problem Statement

Consider the ambiguous grammar.
E → E + E
E → E * E
E → (E)
E → id
(a) Construct LR (0) items for above grammar.
(b) Construct SLR parsing table for grammar.
(c) Parse the input string id + id * id.

Solution

Step1− Construct Augmented Grammar

(0) E′ → S

(1) E → E + E

(2) E → E ∗ E

(3) E → (E)

(4) E → id

Step2− Find closure & goto functions to construct LR (0) items.

Closure (E′ → ? E) =

Applying goto on I₉

? goto cannot be applied on I₉, as the dot in E → (E). is on the last position.

Step3− Computation of FOLLOW

Applying Rule (1) FOLLOW

FOLLOW(B) = {$}                                                         (1)

• E → E + E

Comparing E → E + E with A → α B β

∴ α = ε, B = E, β = +E

? FIRST(β) = FIRST(+E) = {+}

∴ Rule (2a)of FOLLOW

FOLLOW(E) = {+}                                                         (2)

Applying Rule (3)

Comparing E → E + E with A → α B

∴ A = E, α = E+, B = E

FOLLOW(E) = {FOLLOW(E)}                                        (3)

•  E → E ∗ E

Applying Rule (2) and Rule (3) of FOLLOW, we get

FOLLOW(E) = {*}                                                          (4)

FOLLOW(E) = {FOLLOW(E)}                                        (5)

•  E → (E)

Applying Rule (2)

Comparing E → (E) with A → α B β

∴ FOLLOW (E) = {)}                                                       (6)

Rule (3) cannot be applied to this production

As E → (E) cannot be compared with A → α B

•  E → id

Rule (2) and (3) of FOLLOW cannot be applied on E → id. As E → id cannot be compared with A → α B β and A → α B.

Combining (1) to (6), we get

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

Step4 − Construction of SLR Parsing Table

In the parsing table, conflict occurs at Row state 7, 8, and column *, +.

In Action [7, +],                          Action [7, *]

Action [8, +],                             Action [8, *] there occurs a shift-reduce conflict.

The Association and precedence rules can remove this conflict.

Parsing the string id + id * id

Stack	Input	Action
0	id + id * id $	Shift
0 id 3	+ id * id $	Reduce by E → id
0 E 1	+id * id $	Shift
0 E 1 + 4	id * id $	Shift
0 E 1 + 4 id 3	* id $	Reduce by E → id
0 E 1 + 4 E 7	* id $	Conflict i. e. , s5 or r1 ∴ * has higher precedence then + ∴ Action [7,∗] = s5 So, shift-reduce, i.e., s5
0 E 1 + 4 E 7 * 5	$	Shift
0 E 1 + 4 E 7 * 5 id 3	$	Reduce by E → id
0 E 1 + 4 E 7 * 5 E 8	$	Reduce by E → E * E
0 E 1 + 4 E 7	$	Reduce by E → E + E
0 E 1	$	accept

The above parsing solves the conflicting problem in Action [7, *].

So, Action [7, *] = s5 instead of r1.

Similarly on Parsing the string id + id + id.

Stack	Input	Action
0	id + id + id $	Shift
0 id 3	+ id + id $	Reduce by E → id
0 E 1	+id + id $	Shift
0 E 1 + 4	id + id $	Shift
0 E 1 + 4 id 3	+id $	Reduce by E → id
0 E 1 + 4 E 7	+id $	Conflict i. e. , Action [7, +] = s4 or r1. ∴ + is Associative (left). 0E1 + 4E7 will be reduced before shifting + ∴ Action [7, +] = r1 ∴ Reduce by E → E + E
0 E 1	+id $	Shift
0 E + 4	id $	Shift
0 E + 4 id 3	$	Reduce by E → id
0 E + 4 E 7	$	Reduce by E → E + E
0 E 1	$	accept

So, the Above parsing shows how to resolve shift Reduce conflict at Action [7, +]

So, Action [7, +] = r1 instead of s4

Similarly, other entries such as Action [8, +] and Action [8, *] can be solved by taking strings.

id * id * id and id * id + id

Resolution is                 Action [8, +] = r2 and

                                     Action [8, *] = r2.