CS 3240: Languages and Computation Pushdown Automata & CF Grammars NOTE: THESE ARE ONLY PARTIAL SLIDES RELATED TO WEEKS 9 AND 10. PLEASE REFER TO THE TEXTBOOK.

CS 3240: Languages and Computation

Pushdown Automata & CF GrammarsNOTE: THESE ARE ONLY PARTIAL SLIDES RELATED TO WEEKS 9 AND 10 . PLEASE REFER TO THE TEXTBOOK FOR COMPLETE COVERAGE

Project 1

Pushdown Automata

Similar to finite automata, but for CFG’s Finite automata are not adequate for CFG’s

because we cannot keep track of what we’ve doneAt any point, we only know the current state, not

previous states Need memory

PDA’s are finite automata with a stack

FA and PDA Schematics

State control

a a b b

State control

a a b b

xyz

FA

PDA

Stack: Infinite LIFO (last in first out) device

Formal Definition of PDA

A pushdown automaton is a 6-tuple (Q, Σ, , , q0, F), where Q, Σ, , and F are all finite sets, and Q is the set of states Σ is the set of input alphabet is the stack alphabet :Q×Σ× ,P(Q×) is the transition function q0 is the start state and FQ is the set of accept states

How does PDA work?

Start in start state Read input character, perform state transition

and update stack Clone a machine if multiple transitions are

possible At end of input if any machine accepts and the

stack is empty then accept

Example

read 0 & push 0 on stack

read ε & push ε on stack

read ε & push $ on stack

read 1 & pop 0 off stack

read ε & pop $ off stack

Language accepted: {0n1n | n 0}

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()S$

qloop

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

(S)$

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

S)$

(

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

SS)$

(

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

()S)$

(

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

)S)$

((

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

S)$

(()

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

())$

(()

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

))$

(()(

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

)$

(()()

Example

Read (()())

qstart qacceptε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

$

(()())

Example

Read (()())

qstartε, ε S$ ε, $ ε

(,(ε

),)ε

ε,SSS

ε,S(S)

ε,S()

qloop

(()())

qaccept

More Examples

L= {aibjck, i=j or j=k>, and i,j,k>=0}.

L= {wwR | w in {0,1}* }

Differences between PDA’s and NFA’s Transitions read one symbol of the string

and push a string onto or pop a symbol off of the stack

Stack alphabet is not necessarily the same as the alphabet for the languagee.g., $ marks bottom of stack in previous

(0n1n) example

Instantaneous descriptions

Equivalence of pda and cfg

Equivalence of PDA’s and CFG’s

Theorem: A language is context free if and only if some pushdown automaton accepts it by e.s.

Proved in two lemmas – one for the “if” direction and one for the “only if” direction

We will only do the “only if” step – i.e., show that every context-free language has an associated PDA

CFG’s are recognized by PDA’s

Lemma: If a language is context free, then some pushdown automaton accepts it by emp.st.

Proof idea: Construct a PDA following CFG rules

Constructing the PDA

You can read any symbol in when that symbol is at the top of the stack Transitions of the form a,aε

The rules will be pushed onto the stack – when a variable A is on top of the stack and there is a rule Aw, you pop A and push w

You accept only if the input is exhausted and the stack is empty

Idea of PDA construction for AxBz

State control

a b

At

State control

a b

xBzt

Given G build PDA

Deterministic PDA