CFG Derivations

Derivations in the context of Context-Free Grammars (CFGs) refer to the process of generating strings in a language by repeatedly applying the production rules of a grammar. Derivations help illustrate how a string belongs to the language defined by the CFG.

Components of a Derivation

A derivation involves the following steps:

  • Start Symbol: Begin with the start symbol (S) of the grammar.
  • Production Rules: Replace non-terminal symbols in the current string using the grammar’s production rules.
  • Terminal String: Continue applying rules until only terminal symbols remain, forming a valid string in the language.

Derivations can be represented in two ways: Leftmost Derivation and Rightmost Derivation.

Leftmost Derivation

In a leftmost derivation, the leftmost non-terminal symbol in the string is replaced at each step. This process continues until the entire string consists of terminal symbols.

Example: Consider the grammar:

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S → aSb | ε

Derivation for the string aaabb:

Step 1: Start with S
Step 2: Replace S using S → aSb: aSb
Step 3: Replace S again: aaSbb
Step 4: Replace S with ε: aaabb

Thus, the string aaabb is derived using leftmost derivation.

Rightmost Derivation

In a rightmost derivation, the rightmost non-terminal symbol in the string is replaced at each step. This process also continues until only terminal symbols remain.

Example: Using the same grammar:

</>
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S → aSb | ε

Derivation for the string aaabb:

Step 1: Start with S
Step 2: Replace S using S → aSb: aSb
Step 3: Replace the rightmost S: aaSbb
Step 4: Replace S with ε: aaabb

Thus, the string aaabb is derived using rightmost derivation.

Parse Trees

A parse tree (or derivation tree) is a graphical representation of the derivation process. Each internal node represents a non-terminal, and its children represent the production used. The leaves of the tree are terminal symbols or ε.

Example: The parse tree for aaabb is:

Include a diagram showing the hierarchical structure of the grammar’s derivation.

Key Takeaways

  • Derivations are used to generate strings from a CFG by applying production rules repeatedly.
  • Leftmost and rightmost derivations are two systematic ways of applying the rules.
  • Parse trees visually represent the structure of a derivation, aiding in syntax analysis.
  • Derivations are crucial for understanding language syntax and designing parsers in compilers.