Database System Concepts
Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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Question
**Prove whether the grammar below is LL(1) or not:** S → aSbS | bSaS | ε --- ### Analysis: To determine whether the given grammar is LL(1), we need to analyze the **FIRST** and **FOLLOW** sets for each non-terminal, specifically focusing on the non-terminal `S`. #### Steps to Determine if the Grammar is LL(1): 1. **Calculate the FIRST set** for each production: - FIRST(aSbS) = {a} - FIRST(bSaS) = {b} - FIRST(ε) = {ε} 2. **Calculate the FOLLOW set** for `S`: - Since `S` can derive ε, everything that follows `S` in any production must be in FOLLOW(S). Depending on the grammar in which `S` is used, this could be any symbol. 3. **Check for LL(1) conditions:** - For a grammar to be LL(1), the FIRST sets of different productions concerning a single non-terminal must be disjoint. - Check if there is any intersection between the different FIRST sets. - Also, check if ε is in FIRST(S), then FIRST(S) ∩ FOLLOW(S) must be empty. #### Conclusion: After determining the FIRST and FOLLOW sets, if they meet the LL(1) conditions (no intersection between FIRST sets of alternate productions for a single non-terminal and ε conditions as described), the grammar qualifies as LL(1). Otherwise, it does not.
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Transcribed Image Text:**Prove whether the grammar below is LL(1) or not:** S → aSbS | bSaS | ε --- ### Analysis: To determine whether the given grammar is LL(1), we need to analyze the **FIRST** and **FOLLOW** sets for each non-terminal, specifically focusing on the non-terminal `S`. #### Steps to Determine if the Grammar is LL(1): 1. **Calculate the FIRST set** for each production: - FIRST(aSbS) = {a} - FIRST(bSaS) = {b} - FIRST(ε) = {ε} 2. **Calculate the FOLLOW set** for `S`: - Since `S` can derive ε, everything that follows `S` in any production must be in FOLLOW(S). Depending on the grammar in which `S` is used, this could be any symbol. 3. **Check for LL(1) conditions:** - For a grammar to be LL(1), the FIRST sets of different productions concerning a single non-terminal must be disjoint. - Check if there is any intersection between the different FIRST sets. - Also, check if ε is in FIRST(S), then FIRST(S) ∩ FOLLOW(S) must be empty. #### Conclusion: After determining the FIRST and FOLLOW sets, if they meet the LL(1) conditions (no intersection between FIRST sets of alternate productions for a single non-terminal and ε conditions as described), the grammar qualifies as LL(1). Otherwise, it does not.
Expert Solution
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Step 1

To check the grammar whether it is LL(1) or not. you need to know first and follow of the grammar. There are some rules which are used to find first and follow.

Rules to find First of the grammar

  • For the production rule X->∈, First(X)={∈}
  • For the any terminal symbol a, the First(a)={a}
  • For the production rule X->Y1Y2Y3

Calculating the First(X)

  • If ∈∉First(Y1), then the First(X)=First(Y1)
  • If ∈ ∈ First(Y1), then the First(X)={First(Y1) – ∈}∪First(Y2Y3)

Calculating the first (Y2Y3)

  • If ∈ ∉ First(Y2), then the First(Y2Y3)=First(Y2)
  • If ∈ ∈ First(Y2), then the First(Y2Y3)={First(Y2) –∈}∪First(Y3)

Rules to find Follow of the grammar:-

  • For the start symbol S, place the $ in the Follow(S).
  • For the any production rule A->αB, then the Follow(B)=Follow(A)

For the any production rule A->αBβ, 

  1. If the ∈ ∉ First(β), then the Follow(B)=First(β)
  2. If ∈ ∈ First(β), then the Follow(B)={First(β)– ∈}∪Follow(A)

 

 

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