DPDA Deterministic PDA Fall 2003 Costas Busch - RPI

Deterministic PDA: DPDA Allowed transitions: (deterministic choices) Fall 2003 Costas Busch - RPI

(deterministic choices) Allowed transitions: (deterministic choices) Fall 2003 Costas Busch - RPI

(non deterministic choices) Not allowed: (non deterministic choices) Fall 2003 Costas Busch - RPI

DPDA example Fall 2003 Costas Busch - RPI

A language is deterministic context-free Definition: A language is deterministic context-free if there exists some DPDA that accepts it Example: The language is deterministic context-free Fall 2003 Costas Busch - RPI

Example of Non-DPDA (PDA) Fall 2003 Costas Busch - RPI

Not allowed in DPDAs Fall 2003 Costas Busch - RPI

PDAs Have More Power than DPDAs Fall 2003 Costas Busch - RPI

Since every DPDA is also a PDA It holds that: Deterministic Context-Free Languages (DPDA) Context-Free Languages PDAs Since every DPDA is also a PDA Fall 2003 Costas Busch - RPI

We will show that there exists a context-free language which is not We will actually show: Deterministic Context-Free Languages (DPDA) Context-Free Languages (PDA) We will show that there exists a context-free language which is not accepted by any DPDA Fall 2003 Costas Busch - RPI

is not deterministic context-free The language is: We will show: is context-free is not deterministic context-free Fall 2003 Costas Busch - RPI

Language is context-free Context-free grammar for : Fall 2003 Costas Busch - RPI

Theorem: The language is not deterministic context-free (there is no DPDA that accepts ) Fall 2003 Costas Busch - RPI

Proof: Assume for contradiction that is deterministic context free Therefore: there is a DPDA that accepts Fall 2003 Costas Busch - RPI

DPDA with accepts accepts Fall 2003 Costas Busch - RPI

Such a path exists due to determinism DPDA with Such a path exists due to determinism Fall 2003 Costas Busch - RPI

Context-free languages Fact 1: The language is not context-free Regular languages Context-free languages (we will prove this at a later class using pumping lemma for context-free languages) Fall 2003 Costas Busch - RPI

Fact 2: The language is not context-free (we can prove this using pumping lemma for context-free languages) Fall 2003 Costas Busch - RPI

We will construct a PDA that accepts: which is a contradiction! Fall 2003 Costas Busch - RPI

DPDA Replace with Modify DPDA Fall 2003 Costas Busch - RPI

Connect the final states of with the final states of A PDA that accepts Connect the final states of with the final states of Fall 2003 Costas Busch - RPI

Since is accepted by a PDA it is context-free Contradiction! (since is not context-free) Fall 2003 Costas Busch - RPI

Is not deterministic context free Therefore: Is not deterministic context free There is no DPDA that accepts it End of Proof Fall 2003 Costas Busch - RPI