The document describes Push Down Automata (PDA). It defines PDA as a finite state machine with a stack. A PDA is formally defined as a 7-tuple that includes states, input symbols, stack symbols, transition function, initial state, initial stack symbol, and accepting states. The document provides examples of modeling PDA behavior using instantaneous descriptions that represent the state, remaining input, and stack contents. It also discusses different methods for determining if a PDA accepts an input string, such as checking if the string reduces the stack to empty or reaches an accepting state. Finally, it covers extensions of PDA like non-deterministic and multi-stack models, and how to convert context-free grammars to equivalent

1. Push Down Automata(PDA)

2. Model of PDA
• Push Down Automaton :"Finite state machine" + "a stack"

3. • A pushdown automaton has three components −
• input tape,
• control unit, and
• stack with infinite size.
• The stack head scans the top symbol of the stack.
• A stack does two operations −
• Push − a new symbol is added at the top.
• Pop − the top symbol is read and removed.

4. Def: PDA
• A PDA can be formally described as a 7-tuple
(Q, ∑, Γ, δ, q0, Z0 , F)
• Q-finite number of states
• ∑ -input alphabet
• Γ -stack symbols
• δ -transition function: Q × (∑ ∪ {ε}) × Γ Q × Γ*
• Ex: δ(q0,a, Z0 ) =(q1,aZ0 )
• Q0 initial state (q0 ∈ Q)
• Z0 is the initial stack top symbol (Z0 ∈ Γ)
• F is a set of accepting states (F ∈ Q)

5. Graphical Representation of PDA

7. Instantaneous Description of PDA
• Instantaneous Description (ID) is an informal notation of how
a PDA “computes” a input string and make a decision that
string is accepted or rejected.
• It is denoted by a triple (q, w, γ) where;
• q is the current state
• w is the unread part of the input string or the remaining input
alphabets
• γ is the current contents of the PDA stack

8. Ex 1:Write down the IDs or moves for
input string w = “aabb” of PDA as
M = ({q0, q1}, {a, b}, {a, b, Z0}, δ, q0, Z0,
{q1}), where δ is defined by following rules:
δ(q0, a, Z0) = {(q0, aZ0)} Push
δ(q0, a, a) = {(q0, aa)} Push
δ(q0, b, a) = {(q1, ε)} Pop
δ(q1, b, a) = {(q1, ε)} Pop
Also check string w is accepted by PDA or
not?

9. • Solution: Instantaneous Description for string
w = “aabb”
• (q0, aabb, Z0)
• |- (q0, abb, aZ0)
• |- (q0, bb, aaZ0)
• |- (q1, b, aZ0)
• |- (q1, ε, Z0)
• Finally the input tape is empty or input string
w is completed, PDA stack is empty and PDA
has reached a final state. So the string ‘w’
is accepted.
δ(q0, a, Z0) = {(q0, aZ0)} Push
δ(q0, a, a) = {(q0, aa)} Push
δ(q0, b, a) = {(q1, ε)} Pop
δ(q1, b, a) = {(q1, ε)} Pop

10. Example 2: Write down the Ids for input string w = “aaabb” of the above PDA.
Also check it is accepted by PDA or not?
• (q0, aaabb, Z0)
• |- (q0, aabb, aZ0)
• |- (q0, abb, aaZ0)
• |- (q0, bb, aaaZ0)
• |- (q1, b, aaZ0)
• |- (q1, ε, aZ0)
• |- There is no defined move
• So the pushdown automaton stops at this move and the string is not
accepted because the input tape is empty or input string w is completed
but the PDA stack is not empty. So the string ‘w’ is not accepted.
Note: The above method is also called testing of a string using final state method
δ(q0, a, Z0) = {(q0, aZ0)} Push
δ(q0, a, a) = {(q0, aa)} Push
δ(q0, b, a) = {(q1, ε)} Pop
δ(q1, b, a) = {(q1, ε)} Pop

11. Language Acceptance by PDA
• Method-1: Acceptance by final state
method(Ids Method)
• Method-2: Stack Empty Method

12. Ex: Design a PDA to recognize the language
L={wcwr: w ∈ (a+b)* }

13. Test the string w=aabbcbbaa is accepted by
final state method

14. Test the string w=aabbcbbaa is accepted by
Stack Empty Method

15. step1: a b b a c a b b a
current state=q0 Z0
operation=push(
a)
step2: a b b a c a b b a
a
current state=q0 Z0
operation=push(
b)
step3: a b b a c a b b a b
a
current state=q0 Z0
operation=push(
b)
step 4: a b b a c a b b a b
b
current state=q0 a
operation=push(
a) Z0
step 5: a b b a c a b b a a
b
current state=q0 b
operation= no
push a
no pop Z0
step 6: a b b a c a b b a a
b
current state=q1 b
operation=pop() a
Z0
step 7: a b b a c a b b a
b
current state=q1 b
operation=pop() a
Z0
step 8: a b b a c a b b a
current state=q1 b
operation=pop() a
Z0
step 9: a b b a c a b b a
current state=q1
operation=pop() a
Z0
step 10: a b b a c a b b a
current state=q1
string is empty
stack is empty Z0
Therefore String is accepted

16. Design a PDA which accepts L={anbn: n>=1}.
Check whether the strings I) aaabb II) aabbb and III)aaabbb are
accepted or not using i) Final state method ii) Stack Method

18. step
1: a a a b b
current state=q0 Z0
operation=push(a)
step
2: a a a b b
a
current state=q0 Z0
operation=push(a)
step3: a a a b b a
a
current
state=q0 Z0
operation=pus
h(a)
step 4: a a a b b a
a
current
state=q0 a
operation=pop
() Z0
step 5: a a a b b
current
state=q1 a
operation=
pop() a
Z0
step 6: a a a b b
current
state=q1
Input string:
empty a
Z0
String is rejected
Stack Method

19. • Design a PDA which accepts only odd no of a’s defined
over {a,b}. Check whether the string baababababbbbbaa
is accepted or not using final state and stack methods
• Construct a PDA for the language L={anb2n: n>=1}. Check
whether the string aabbbb is accepted or not by the given
language using i) Final state method ii) Stack Method
• Construct a PDA for the language L={an cb2n: n>=1}. Check
the string aaacbbbbbbb
• Construct a PDA for the language L={a2nbn: n>=1}. Check
the string aaaacbb
• Design a PDA for well formed Parenthesis (),[],{}
• Design PDA for a language L={w/w is in (a+b)* and
na(w)=nb(w) }

20. • Design PDA for a language L={w/w is in
(a+b)* and na(w)>nb(w) }
• Design PDA for a language L={w/w is in
(a+b)* and na(w)<nb(w)}

21. Types of PDA
• DPDA
• Previously constructed PDAs are DPDAs
• NPDA
• Ex1: Design a PDA to recognize the language
L={wwr: w ∈ (a+b)* }
• Ex2: construct PDA for language L containing
all the strings which are palindrome over {a,b}

22. Two stack PDA
• A two stack PDA can be formally described as a 9-tuple
(Q, ∑, Γ, Γ1, δ, q0, Z1 , Z2 , F)
• Q-finite number of states
• ∑ -input alphabet
• Γ –stack1 symbols
• Γ1 –stack2 symbols
• δ -transition function: Q × (∑ ∪ {ε}) × Γ x Γ1  (Q, Γ, Γ1 )
• δ(q0,a, Z1 , Z2 ) =(q1,aZ1 , Z2)
• Q0 initial state (q0 ∈ Q)
• Z1 is the initial stack1 top symbol (Z1∈ Γ)
• Z2 is the initial stack2 top symbol (Z2∈ Γ1)
• F is a set of accepting states (F ∈ Q)
Ex: Design a two stack PDA which accepts L={anbn cn : n>=1}.

23. Construction of PDA from CFG (or) CFG to PDA Conversion
• Step 1 − Convert the productions of the CFG into GNF.
• Step 2 − The PDA will have only one state {q}.
• Step 3 − The start symbol of CFG will be the start symbol in the PDA.
• Step 4 − All non-terminals of the CFG will be the stack symbols of the PDA
and all the terminals of the CFG will be the input symbols of the PDA.
• Step 5 − For each production in the form A → aX make a transition δ (q, a,
A)=(q,X).
• Step 6- For each production in the form A → a make a transition δ (q, a,
A)=(q, ε).

24. Ex: Convert the following CFG in to PDA
SaAA, AaS/bS/a
• The grammar is in GNF
• For SaAA: δ (q, a, S)=(q,AA).
• For AaS : δ (q, a, A)=(q,S).
• For AbS : δ (q, b, A)=(q,S)
• For Aa : δ (q, a, A)=(q, ε).
• The Equivalent PDA:
δ (q, a, S)=(q,AA).
δ (q, a, A)=(q,S).
δ (q, b, A)=(q,S)
δ (q, a, A)=(q, ε)
For A → aX : δ (q, a, A)=(q,X).
For A → a :δ (q, a, A)=(q, ε).

25. • Ia/b
• SaA
• AaABC/bB/a
• Bb
• Cc

26. • SaBB
• BbS/c