Leftmost and Rightmost Derivations in Context-Free Grammars

Understand leftmost and rightmost derivations—fundamental concepts in formal language theory and compiler design. This guide explains how these derivations systematically generate strings from a context-free grammar, clarifying their importance in parsing and validating programming language syntax.

Leftmost and Rightmost Derivations in Context-Free Grammars

Understanding Derivations in CFGs

In formal language theory, a derivation is a sequence of production rules used to generate a string from a grammar's start symbol. This is a fundamental concept in compiler design; it's how compilers determine if an input string is valid according to a programming language's grammar. Two main types of derivations exist: leftmost and rightmost.

Leftmost Derivation

In a leftmost derivation, the leftmost non-terminal symbol in the string is replaced at each step using a production rule from the grammar. The process continues until only terminal symbols (actual characters or tokens in the language) remain. This is done repeatedly until you arrive at the desired string. This provides a systematic way to generate strings from a given grammar.

Example Leftmost Derivation:

Consider this simple grammar:

Grammar Rules

S → S + S | S - S | a | b | c

Let's derive the string "a - b + c":

  1. S → S + S
  2. S → S - S + S
  3. S → a - S + S
  4. S → a - b + S
  5. S → a - b + c

Rightmost Derivation

In a rightmost derivation, the rightmost non-terminal symbol is replaced at each step. Like leftmost derivation, this method continues until the string contains only terminal symbols. It also provides a way to generate valid strings from a grammar, although the steps will be different from a leftmost derivation.

Example Rightmost Derivation:

Using the same grammar as above, let's derive the string "a - b + c" using a rightmost derivation:

  1. S → S - S
  2. S → S - S + S
  3. S → S - S + c
  4. S → S - b + c
  5. S → a - b + c