For the given Context free grammar reduce the following grammar SaBDh A Bbc | b DdD

1. Break Down the Problem

We need to reduce the given context-free grammar to eliminate unnecessary productions or symbols. The grammar is:

• S → aBDh
• A → Bbc | b
• D → dD | ε

2. Relevant Concepts

• Eliminating ambiguous productions.
• Removing useless symbols.
• Factoring and eliminating ε-productions if necessary.

3. Analysis and Detail

1. Identify the Non-Terminals:

   • The non-terminals in the grammar are S, A, B, and D.

2. Examine Each Production:

   • Start with D:

     • The production D → dD | ε allows us to generate strings consisting of zero or more occurrences of "d".
     • Since we have D → ε, we can rewrite D such that we can replace occurrences of D.

   • Next, look at A:

     • The production A → Bbc | b indicates that A can produce b or something that starts with B followed by bc.

3. Remove Unreachable Symbols:

   • Construct a derivation tree using S. We can see that all symbols (A, B, D) can be reached from S.

4. Épsilon Removal:

   • Let's eliminate ε from D:
     • Replace D in S with ε: S → aBh | aBdh.
     • For A: It will remain as it is since it does not depend on ε.

5. Simplifying Productions:

   • We cannot eliminate any non-terminals because B is still required for A.

4. Verify and Summarize

The reduced grammar would be:

• S → aBd h | aB d h
• A → Bbc | b
• D → dD | d

This grammar has removed unnecessary ε-production and retains necessary productions while keeping non-terminal dependencies clear.

Final Answer

The reduced context-free grammar is:

• S → aB d h
• A → Bbc | b
• D → dD | d

This grammar should be easier to work with while generating languages recognized by the original grammar.