NPDAs Accept Context-Free Languages Costas Busch - RPI

Theorem: Context-Free Languages Languages Accepted by (Grammars) NPDAs Costas Busch - RPI

Convert any context-free grammar to a NPDA with: Proof - Step 1: Context-Free Languages (Grammars) Languages Accepted by NPDAs Convert any context-free grammar to a NPDA with: Costas Busch - RPI

Convert any NPDA to a context-free grammar with: Proof - Step 2: Context-Free Languages (Grammars) Languages Accepted by NPDAs Convert any NPDA to a context-free grammar with: Costas Busch - RPI

Converting Context-Free Grammars to NPDAs Proof - step 1 Converting Context-Free Grammars to NPDAs Costas Busch - RPI

We will convert any context-free grammar to an NPDA automaton Such that: Simulates leftmost derivations of Costas Busch - RPI

Simulation of derivation Leftmost derivation Input processed Stack contents leftmost variable Stack Simulation of derivation Input Costas Busch - RPI

Simulation of derivation Leftmost derivation string of terminals Stack Simulation of derivation Input end of input is reached Costas Busch - RPI

What is the equivalent NPDA? An example grammar: What is the equivalent NPDA? Costas Busch - RPI

Grammar: NPDA: Costas Busch - RPI

A leftmost derivation: Grammar: A leftmost derivation: Costas Busch - RPI

Derivation: Input Time 0 Stack Costas Busch - RPI

Derivation: Input Time 0 Stack Costas Busch - RPI

Derivation: Input Time 1 Stack Costas Busch - RPI

Derivation: Input Time 2 Stack Costas Busch - RPI

Derivation: Input Time 3 Stack Costas Busch - RPI

Derivation: Input Time 4 Stack Costas Busch - RPI

Derivation: Input Time 5 Stack Costas Busch - RPI

Derivation: Input Time 6 Stack Costas Busch - RPI

Derivation: Input Time 7 Stack Costas Busch - RPI

Derivation: Input Time 8 Stack Costas Busch - RPI

Derivation: Input Time 9 Stack accept Costas Busch - RPI

In general: Given any grammar We can construct a NPDA With Costas Busch - RPI

Constructing NPDA from grammar : For any production For any terminal Costas Busch - RPI

Grammar generates string if and only if NPDA accepts Costas Busch - RPI

For any context-free language there is a NPDA Therefore: For any context-free language there is a NPDA that accepts the same language Context-Free Languages (Grammars) Languages Accepted by NPDAs Costas Busch - RPI

Converting NPDAs to Context-Free Grammars Proof - step 2 Converting NPDAs to Context-Free Grammars Costas Busch - RPI

a context-free grammar with For any NPDA we will construct a context-free grammar with Costas Busch - RPI

The grammar simulates the machine Intuition: The grammar simulates the machine A derivation in Grammar : terminals variables Input processed Stack contents Current configuration in NPDA Costas Busch - RPI

Some Necessary Modifications Modify (if necessary) the NPDA so that: 1) The stack is never empty 2) It has a single final state and empties the stack when it accepts a string 3) Has transitions in a special form Costas Busch - RPI

the stack is never empty Modify the NPDA so that the stack is never empty Stack OK OK NOT OK Costas Busch - RPI

Introduce the new symbol to denote the bottom of the stack Costas Busch - RPI

At the beginning push into the stack Original NPDA new initial state original initial state Costas Busch - RPI

replace every instance of with In transitions: replace every instance of with Example: Costas Busch - RPI

Convert all transitions so that: if the automaton attempts to pop or replace it will halt Costas Busch - RPI

l , ® $ Convert transitions as follows: halting state Costas Busch - RPI

2) Modify the NPDA so that it empties the stack and has a unique final state Empty the stack NPDA l ® , l ® , l ® , Old final states Costas Busch - RPI

3) modify the NPDA so that transitions have the following forms: Costas Busch - RPI

Convert: Costas Busch - RPI

Convert: symbols Costas Busch - RPI

Convert: symbols Convert recursively Costas Busch - RPI

Example of a NPDA in correct form: Costas Busch - RPI

The Grammar Construction In grammar : Stack symbol Variables: states Terminals: Input symbols of NPDA Costas Busch - RPI

For each transition We add production Costas Busch - RPI

For all possible states in the automaton For each transition We add productions For all possible states in the automaton Costas Busch - RPI

Stack bottom symbol Start Variable: Start state final state Costas Busch - RPI

Example: Grammar production: Costas Busch - RPI

Example: Grammar productions: Costas Busch - RPI

Example: Grammar production: Costas Busch - RPI

Resulting Grammar: Costas Busch - RPI

Costas Busch - RPI

Derivation of string Costas Busch - RPI

the stack doesn’t change below and then is removed from stack In general: if and only if the NPDA goes from to by reading string and the stack doesn’t change below and then is removed from stack Costas Busch - RPI

Therefore: if and only if is accepted by the NPDA Costas Busch - RPI

there is a context-free grammar that accepts the same language Therefore: For any NPDA there is a context-free grammar that accepts the same language Context-Free Languages (Grammars) Languages Accepted by NPDAs Costas Busch - RPI

Deterministic PDA DPDA Costas Busch - RPI

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

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

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

DPDA example Costas Busch - RPI

is deterministic context-free The language is deterministic context-free Costas Busch - RPI

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

Example of Non-DPDA (NPDA) Costas Busch - RPI

Not allowed in DPDAs Costas Busch - RPI

NPDAs Have More Power than DPDAs Costas Busch - RPI

Since every DPDA is also a NPDA It holds that: Deterministic Context-Free Languages (DPDA) Context-Free Languages NPDAs Since every DPDA is also a NPDA 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 (NPDA) We will show that there exists a context-free language which is not accepted by any DPDA Costas Busch - RPI

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

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

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

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

DPDA with accepts accepts Costas Busch - RPI

Such a path exists because of the determinism DPDA with Such a path exists because of the determinism 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) Costas Busch - RPI

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

We will construct a NPDA that accepts: which is a contradiction! Costas Busch - RPI

Replace with Modify Costas Busch - RPI

Connect final states of with final states of The NPDA that accepts Connect final states of with final states of Costas Busch - RPI

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

Not deterministic context free Therefore: Not deterministic context free There is no DPDA that accepts End of Proof Costas Busch - RPI