CFG Calculator: Compute FIRST and FOLLOW Sets for Context-Free Grammars

Understanding the Importance of FIRST and FOLLOW Sets in Compiler Design

Background Knowledge

In Compiler Design and Syntax Analysis, understanding how symbols are derived from a given Context-Free Grammar (CFG) is crucial. The FIRST and FOLLOW sets are essential tools used in constructing LL(1) parsers and ensuring that grammars are unambiguous.

• FIRST Set: Represents the set of terminal symbols that begin the strings derivable from a non-terminal.
• FOLLOW Set: Represents the set of terminal symbols that can appear immediately after a non-terminal in some sentential form.

Calculation Formula

To compute the FIRST and FOLLOW sets:

1. FIRST Set:

   • If a symbol derives ε (epsilon), include ε in its FIRST set.
   • For a sequence of symbols, compute the FIRST set by concatenating their individual FIRST sets until a non-nullable symbol is encountered.

2. FOLLOW Set:

   • Initialize the start symbol's FOLLOW set with $ (end of input).
   • Propagate FOLLOW information based on production rules where the non-terminal appears.

Example Calculation

Consider the following CFG:

S -> A B C
A -> a | ε
B -> b | ε
C -> c

Step 1: Compute FIRST Sets

• FIRST(A) = {a, ε}
• FIRST(B) = {b, ε}
• FIRST(C) = {c}

Step 2: Compute FOLLOW Sets

• FOLLOW(S) = {$}
• FOLLOW(A) = {b, c, $} (derived from S -> A B C)
• FOLLOW(B) = {c, $} (derived from S -> A B C)
• FOLLOW(C) = {$} (derived from S -> A B C)

FAQs

Q1: What happens if a grammar is not LL(1)? If a grammar is not LL(1), it means that there are conflicts in predicting which production rule to apply. This can be resolved by modifying the grammar or using more advanced parsing techniques like SLR or LALR.

Q2: Why are FIRST and FOLLOW sets important? FIRST and FOLLOW sets help determine the next possible token during parsing, ensuring that the parser can correctly predict and process input strings.

Vocabulary

• LL(1) Grammar: A context-free grammar that can be parsed by an LL(1) parser, meaning it looks ahead one token at a time.
• Nullable Non-Terminal: A non-terminal that can derive the empty string ε.
• Sentential Form: Any string derived from the start symbol by applying production rules.

Interesting Facts About CFGs

1. Ambiguity: Not all CFGs are unambiguous. Some require transformations to remove ambiguity for practical use in compilers.
2. Chomsky Normal Form: Every CFG can be converted into Chomsky Normal Form, simplifying parsing algorithms.
3. Applications: CFGs are widely used in programming languages, natural language processing, and even DNA sequencing analysis.