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First and Follow Calculator

Reviewed by Calculator Editorial Team

FIRST and FOLLOW sets are fundamental concepts in formal language theory, particularly in the study of context-free grammars. These sets help in parsing algorithms and syntax analysis. This guide explains how to compute FIRST and FOLLOW sets for a given grammar.

What Are FIRST and FOLLOW Sets?

In formal language theory, FIRST and FOLLOW sets are used to determine the structure of a context-free grammar. These sets help in predicting which productions to use during parsing.

The FIRST Set

The FIRST set of a string of grammar symbols is the set of terminals that begin the strings derived from those symbols. For a non-terminal A, FIRST(A) is the set of terminals that appear as the first symbol of any string derived from A.

The FOLLOW Set

The FOLLOW set of a non-terminal A is the set of terminals that can appear immediately to the right of A in any sentential form derived from the start symbol of the grammar.

Key Points

FIRST and FOLLOW sets are essential for constructing parsing tables in top-down parsers like LL(1) parsers. They help in resolving parsing conflicts and ensuring the correctness of the parsing process.

How to Calculate FIRST Sets

Calculating FIRST sets involves determining the set of terminals that can appear as the first symbol of any string derived from a given non-terminal. Here's a step-by-step method:

  1. For each terminal symbol a, FIRST(a) = {a}.
  2. For each non-terminal A, initialize FIRST(A) to the empty set.
  3. For each production A → α, add FIRST(α) to FIRST(A).
  4. Repeat steps 3 until no more terminals can be added to any FIRST set.

Formal Definition

For a grammar G, the FIRST set of a string of grammar symbols α is defined as:

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

How to Calculate FOLLOW Sets

Calculating FOLLOW sets involves determining the set of terminals that can appear immediately to the right of a given non-terminal in any sentential form. Here's a step-by-step method:

  1. Initialize FOLLOW(S) = {$} where S is the start symbol and $ is the end-of-input marker.
  2. For each production A → αBβ, add FIRST(β) - {ε} to FOLLOW(B).
  3. If FIRST(β) contains ε, add FOLLOW(A) to FOLLOW(B).
  4. Repeat steps 2 and 3 until no more terminals can be added to any FOLLOW set.

Formal Definition

For a grammar G, the FOLLOW set of a non-terminal A is defined as:

FOLLOW(A) = {a | S ⇒* αAaβ}

Example Calculation

Let's consider the following grammar:

S → aB | bA
A → a | bA | ε
B → b | ε

Calculating FIRST Sets

Using the algorithm described above, we can compute the FIRST sets for each non-terminal:

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

Calculating FOLLOW Sets

Using the algorithm described above, we can compute the FOLLOW sets for each non-terminal:

  • FOLLOW(S) = {$}
  • FOLLOW(A) = {$, b}
  • FOLLOW(B) = {$, a}

Note

The example above demonstrates how to compute FIRST and FOLLOW sets for a simple grammar. The actual computation may vary depending on the specific grammar being analyzed.

FAQ

What is the difference between FIRST and FOLLOW sets?

The FIRST set of a string of grammar symbols is the set of terminals that begin the strings derived from those symbols. The FOLLOW set of a non-terminal is the set of terminals that can appear immediately to the right of the non-terminal in any sentential form derived from the start symbol.

How are FIRST and FOLLOW sets used in parsing?

FIRST and FOLLOW sets are used in parsing algorithms like LL(1) parsing to determine which production to use during parsing. They help in resolving parsing conflicts and ensuring the correctness of the parsing process.

Can FIRST and FOLLOW sets be computed for any context-free grammar?

Yes, FIRST and FOLLOW sets can be computed for any context-free grammar. The computation involves iterative application of the rules defined for FIRST and FOLLOW sets.

What is the significance of the ε symbol in FIRST sets?

The ε symbol represents the empty string and indicates that a string of grammar symbols can derive the empty string. It is included in FIRST sets when the string can derive ε.

How can I verify the correctness of my FIRST and FOLLOW sets?

You can verify the correctness of your FIRST and FOLLOW sets by checking that they satisfy the formal definitions and by ensuring that they are consistent with the grammar being analyzed.