First Follow Sets Calculator

First and Follow sets are fundamental concepts in formal language theory, particularly in the construction of parsing tables for top-down parsers like LL(1) parsers. This calculator helps you compute these sets for any given context-free grammar.

What Are First and Follow Sets?

In formal language theory, First and Follow sets are used to determine the parsing table for top-down parsers. These sets help in predicting which production rule to apply during parsing.

First Set: For a given grammar symbol, the First set is the set of terminals that begin the strings derived from that symbol.

Follow Set: For a given non-terminal, the Follow set is the set of terminals that can appear immediately to the right of that non-terminal in any sentential form derived from the start symbol.

These sets are crucial for constructing LL(1) parsing tables, which are used in compilers and interpreters to analyze the syntax of programming languages.

How to Calculate First Sets

To calculate the First set for a grammar symbol, follow these steps:

If the symbol is a terminal, the First set is the symbol itself.
If the symbol is a non-terminal, the First set is the union of the First sets of all symbols that can appear at the beginning of any production for that non-terminal.
If the production starts with a non-terminal, include its First set, excluding ε (epsilon) if there are more symbols in the production.
If all symbols in a production can derive ε, include ε in the First set.

Note: The empty string ε is included in the First set only if the entire production can derive ε.

How to Calculate Follow Sets

To calculate the Follow set for a non-terminal, follow these steps:

Place $ (end of input marker) in the Follow set of the start symbol.
If there is a production A → αBβ, then everything in First(β) except ε is placed in Follow(B).
If there is a production A → αB, or a production A → αBβ where First(β) contains ε, then everything in Follow(A) is in Follow(B).

These rules are applied iteratively until no more symbols can be added to any Follow set.

Example Calculation

Consider the following grammar:

S → aA
A → bB | ε
B → c

Using the calculator, you can compute:

First(S) = {a}
First(A) = {b, ε}
First(B) = {c}
Follow(S) = {$}
Follow(A) = {$, c}
Follow(B) = {$, c}

This example demonstrates how First and Follow sets are calculated for a simple grammar.

FAQ

What is the difference between First and Follow sets?: The First set contains the terminals that can appear at the beginning of strings derived from a grammar symbol, while the Follow set contains the terminals that can appear immediately after a non-terminal in any sentential form.
When are First and Follow sets used?: First and Follow sets are primarily used in the construction of LL(1) parsing tables for top-down parsers in compilers and interpreters.
How do I handle ε (epsilon) in First sets?: ε is included in the First set of a non-terminal only if that non-terminal can derive the empty string. This is determined by checking if all productions for the non-terminal can derive ε.
What happens if a grammar is not LL(1)?: If a grammar is not LL(1), it means there are cases where the parser cannot determine which production to use based solely on the First and Follow sets. Such grammars may need to be rewritten or parsed using a different parsing technique.