Lecture 06

Context-free Language Equivalence Theorem

Theorem A language is context-free if and only if it is generated by context-free gramma.

Proof Sketch CFG\(\rightarrow\)PDA

PDA\(\rightarrow\)CFG

  1. One accepts state
  2. Every transition either pushes a symbol on the stack or pops one off (but not both)
  3. Only accept with an empty stack

Chomsky Normal Form

Definition The context-free gramma \(G\) in Chomsky Normal Form if all its rule are of the form

  1. \(A\rightarrow BC\), for any \(A\in V\), \(B,C\in V-\{S\}\)
  2. \(A\rightarrow a\), for any \(A\in V, a\in\Sigma\)
  3. \(S\rightarrow\epsilon\)

Theorem For every context-free gramma \(G\), there is a context-free gramma \(G'\) in Chomsky Normal Form that generates the same language.

Regular Grammars

Definition The context-free gramma \(G\) is regular if their rules are of the form

Theorem The language \(L\) is regular if and only if it is generated by a regular gramma