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- Regular Grammar
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- Pumping Lemma for Regular Grammar
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- Context-Free Grammars
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- CFG Simplification
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- Pushdown Automata
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- Decidability
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- Rice Theorem
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- Automata Theory Useful Resources
- Automata Theory - Quick Guide
- Automata Theory - Useful Resources
- Automata Theory - Discussion

In a CFG, it may happen that all the production rules and symbols are not needed for the derivation of strings. Besides, there may be some null productions and unit productions. Elimination of these productions and symbols is called **simplification of CFGs**. Simplification essentially comprises of the following steps −

- Reduction of CFG
- Removal of Unit Productions
- Removal of Null Productions

CFGs are reduced in two phases −

**Phase 1** − Derivation of an equivalent grammar, **G’**, from the CFG, **G**, such that each variable derives some terminal string.

**Derivation Procedure** −

Step 1 − Include all symbols, **W _{1}**, that derive some terminal and initialize

Step 2 − Include all symbols, **W _{i+1}**, that derive

Step 3 − Increment **i** and repeat Step 2, until **W _{i+1} = W_{i}**.

Step 4 − Include all production rules that have **W _{i}** in it.

**Phase 2** − Derivation of an equivalent grammar, **G”**, from the CFG, **G’**, such that each symbol appears in a sentential form.

**Derivation Procedure** −

Step 1 − Include the start symbol in **Y _{1}** and initialize

Step 2 − Include all symbols, **Y _{i+1}**, that can be derived from

Step 3 − Increment **i** and repeat Step 2, until **Y _{i+1} = Y_{i}**.

Find a reduced grammar equivalent to the grammar G, having production rules, P: S → AC | B, A → a, C → c | BC, E → aA | e

**Phase 1** −

T = { a, c, e }

W_{1} = { A, C, E } from rules A → a, C → c and E → aA

W_{2} = { A, C, E } U { S } from rule S → AC

W_{3} = { A, C, E, S } U ∅

Since W_{2} = W_{3}, we can derive G’ as −

G’ = { { A, C, E, S }, { a, c, e }, P, {S}}

where P: S → AC, A → a, C → c , E → aA | e

**Phase 2** −

Y_{1} = { S }

Y_{2} = { S, A, C } from rule S → AC

Y_{3} = { S, A, C, a, c } from rules A → a and C → c

Y_{4} = { S, A, C, a, c }

Since Y_{3} = Y_{4}, we can derive G” as −

G” = { { A, C, S }, { a, c }, P, {S}}

where P: S → AC, A → a, C → c

Any production rule in the form A → B where A, B ∈ Non-terminal is called **unit production.**.

**Step 1** − To remove **A → B**, add production **A → x** to the grammar rule whenever **B → x** occurs in the grammar. [x ∈ Terminal, x can be Null]

**Step 2** − Delete **A → B** from the grammar.

**Step 3** − Repeat from step 1 until all unit productions are removed.

**Problem**

Remove unit production from the following −

S → XY, X → a, Y → Z | b, Z → M, M → N, N → a

**Solution** −

There are 3 unit productions in the grammar −

Y → Z, Z → M, and M → N

**At first, we will remove M → N.**

As N → a, we add M → a, and M → N is removed.

The production set becomes

S → XY, X → a, Y → Z | b, Z → M, M → a, N → a

**Now we will remove Z → M.**

As M → a, we add Z→ a, and Z → M is removed.

The production set becomes

S → XY, X → a, Y → Z | b, Z → a, M → a, N → a

**Now we will remove Y → Z.**

As Z → a, we add Y→ a, and Y → Z is removed.

The production set becomes

S → XY, X → a, Y → a | b, Z → a, M → a, N → a

Now Z, M, and N are unreachable, hence we can remove those.

The final CFG is unit production free −

S → XY, X → a, Y → a | b

In a CFG, a non-terminal symbol **‘A’** is a nullable variable if there is a production **A → ε** or there is a derivation that starts at **A** and finally ends up with

ε: A → .......… → ε

**Step 1** − Find out nullable non-terminal variables which derive ε.

**Step 2** − For each production **A → a**, construct all productions **A → x** where **x** is obtained from **‘a’** by removing one or multiple non-terminals from Step 1.

**Step 3** − Combine the original productions with the result of step 2 and remove **ε - productions**.

**Problem**

Remove null production from the following −

S → ASA | aB | b, A → B, B → b | ∈

**Solution** −

There are two nullable variables − **A** and **B**

**At first, we will remove B → ε.**

After removing **B → ε**, the production set becomes −

S→ASA | aB | b | a, A ε B| b | &epsilon, B → b

**Now we will remove A → ε.**

After removing **A → ε**, the production set becomes −

S→ASA | aB | b | a | SA | AS | S, A → B| b, B → b

This is the final production set without null transition.

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