Calculating First and Follow

In formal language theory, FIRST and FOLLOW sets are fundamental concepts used in parsing algorithms like LL(1) parsing. These sets help determine the validity of input strings and guide the parsing process. This guide explains how to calculate these sets and their importance in compiler design.

What are FIRST and FOLLOW sets?

FIRST and FOLLOW sets are used in the construction of parsing tables for top-down parsers. They help determine which production rule to apply at each step of the parsing process.

Key Concepts:

FIRST(X) - The set of terminals that begin the strings derived from symbol X.
FOLLOW(X) - The set of terminals that can appear immediately to the right of symbol X in any sentential form.

These sets are essential for:

Determining the validity of input strings
Guiding the parsing process in top-down parsers
Constructing LL(1) parsing tables
Identifying left recursion and other grammar issues

How to calculate FIRST sets

To calculate FIRST sets for a grammar, follow these steps:

For each terminal symbol, FIRST(X) = {X}
For each non-terminal symbol X:
- If X → ε (epsilon production), add ε to FIRST(X)
- For each production X → Y₁Y₂...Yₙ:
  - Add FIRST(Y₁) to FIRST(X)
  - If Y₁ can derive ε, add FIRST(Y₂) to FIRST(X)
  - Continue this process until you can't add any more terminals
Repeat the process until no more changes occur

Formal Definition:

For a grammar G, FIRST(X) is defined as:

FIRST(X) = {a | X ⇒* a...} ∪ {ε | X ⇒* ε}

FIRST sets help identify which terminals can start strings derived from each non-terminal symbol.

How to calculate FOLLOW sets

To calculate FOLLOW sets, follow these steps:

FOLLOW(S) = {$} where S is the start symbol and $ is the end marker
For each production A → αBβ:
- Add FIRST(β) to FOLLOW(B)
- If β can derive ε, add FOLLOW(A) to FOLLOW(B)
Repeat the process until no more changes occur

Formal Definition:

For a grammar G, FOLLOW(X) is defined as:

FOLLOW(X) = {a | S ⇒* αXaβ}

FOLLOW sets help identify which terminals can appear immediately after each non-terminal symbol in any valid string.

Example calculation

Consider the following grammar:

S → aB | bA

A → a | ε

B → b | ε

Let's calculate FIRST and FOLLOW sets for this grammar:

FIRST Sets

FIRST(S) = {a, b}
FIRST(A) = {a, ε}
FIRST(B) = {b, ε}

FOLLOW Sets

FOLLOW(S) = {$}
FOLLOW(A) = {$}
FOLLOW(B) = {$}

This example demonstrates how FIRST and FOLLOW sets can be calculated for a simple grammar.

Practical applications

FIRST and FOLLOW sets have several practical applications in compiler design:

LL(1) Parsing: These sets are used to construct parsing tables for LL(1) parsers.
Grammar Analysis: They help identify left recursion and other grammar issues.
Error Recovery: They can be used to implement more sophisticated error recovery strategies.
Optimization: They can help optimize the parsing process by reducing the number of choices at each step.

Understanding FIRST and FOLLOW sets is essential for anyone working with formal grammars and parsing algorithms.

FAQ

What is the difference between FIRST and FOLLOW sets?

FIRST sets identify the terminals that can begin strings derived from a symbol, while FOLLOW sets identify the terminals that can appear immediately after a symbol in any valid string.

When would I use FIRST and FOLLOW sets?

You would use FIRST and FOLLOW sets when designing or analyzing parsing algorithms, particularly top-down parsers like LL(1) parsers. They help construct parsing tables and guide the parsing process.

Can FIRST and FOLLOW sets be calculated for any grammar?

FIRST sets can be calculated for any grammar, but FOLLOW sets require that the grammar is unambiguous and does not contain left recursion. Some grammars may not have well-defined FOLLOW sets.

How do FIRST and FOLLOW sets relate to parsing tables?

FIRST and FOLLOW sets are used to populate parsing tables for top-down parsers. The FIRST set helps determine which production to use when encountering a non-terminal, while the FOLLOW set helps determine when to stop parsing a non-terminal.