Calculate Follow Sets

Follow sets are a fundamental concept in formal language theory, particularly in the construction of predictive parsers. They help determine which terminals can appear immediately after a given non-terminal in any valid string of the language. This guide explains how to calculate follow sets for context-free grammars.

What Are Follow Sets?

In formal language theory, follow sets are used to determine the set of terminals that can appear immediately to the right of a given non-terminal in any valid string of the language. They are essential for constructing predictive parsers and determining the lookahead sets needed for parsing decisions.

Follow sets are defined recursively and depend on the grammar's production rules. The follow set of a non-terminal A, denoted FOLLOW(A), is the set of terminals that can appear immediately to the right of A in any derivation of the grammar.

Follow sets are distinct from first sets, which determine the set of terminals that can appear immediately to the left of a given non-terminal in any valid string.

How to Calculate Follow Sets

Calculating follow sets involves several steps that are applied iteratively until no more terminals can be added to any follow set. Here's the general procedure:

Initialize: For each non-terminal A in the grammar, initialize FOLLOW(A) to the empty set.
Add end-of-input marker: For the start symbol S, add the end-of-input marker ($) to FOLLOW(S).
Process production rules: For each production rule of the form A → αBβ:
- Add FIRST(β) to FOLLOW(B), excluding ε (the empty string).
- If β can derive ε, add FOLLOW(A) to FOLLOW(B).
Repeat: Continue processing until no more terminals can be added to any follow set.

FOLLOW(A) = { a | S ⇒* αAaβ, a ∈ T ∪ {$} }

This process is typically implemented using an algorithm that iteratively applies these rules until a fixed point is reached.

Example Calculation

Consider the following simple grammar:

S → aA
S → bB
A → a
A → aS
B → b
B → bS

To calculate FOLLOW(S), we follow these steps:

Initialize FOLLOW(S) = { }
Add $ to FOLLOW(S) because S is the start symbol
From production S → aA, add FIRST(A) to FOLLOW(A)
From production A → aS, add FOLLOW(A) to FOLLOW(S)
Continue until no more changes occur

The final follow sets for this grammar are:

Non-terminal	Follow Set
S	{$, a, b}
A	{$, a, b}
B	{$, a, b}

Applications

Follow sets are primarily used in:

Predictive parsing: To determine which production rule to apply during parsing.
Compiler construction: In the design of parsers that need to make decisions based on lookahead symbols.
Language theory: To analyze the properties of formal grammars and languages.

Understanding follow sets is crucial for designing efficient parsers and compilers for programming languages.

FAQ

What is the difference between first and follow sets?: First sets determine the set of terminals that can appear immediately to the left of a given non-terminal, while follow sets determine the set of terminals that can appear immediately to the right.
When is the end-of-input marker ($) added to follow sets?: The end-of-input marker is added to the follow set of the start symbol of the grammar.
How are follow sets used in predictive parsing?: Follow sets are used to determine which production rule to apply during parsing by providing the lookahead symbols needed for decision-making.
Can follow sets be empty?: Yes, follow sets can be empty if there are no terminals that can appear to the right of the non-terminal in any valid string of the language.