1. CS 461 – Sept. 28 Section 2.2 – Pushdown Automata { 0n 1n }
Palindromes
Equal
Next: Converting CFG  PDA

2. { 0n 1n } While reading 0s, push them.
If you read 1, change state. Then, as you read 1s, pop 0s off the stack.
Watch out for bad input!
Unspecified transition  crash (reject)
Now, let’s write this in the form of a table.

3. POP & Go to state “reading 1”
PDA δ for { 0n 1n }
State
Reading 0
Reading 1
Tos
(don’t care)
Empty
Input 1
Action
Push 0
POP & Go to state “reading 1”
Crash
Pop
Notes:
Action depends on input symbol AND what’s on top of stack.
Action includes manipulating stack AND/OR changing state.

4. Example #2 { w # wR } (easier form of palindrome)
Let’s design a PDA that accepts its input by empty stack, as before.
Think about: state, top-of-stack, input, action

5. PDA δ for { w # wR } Before # After # (don’t care) 1 # State Tos Input1
Input #
Action
Push 0
Push 1
Go to “after #”
Pop
Crash

6. Palindrome PDA? Before # After # (don’t care) 1 # State Tos Input1
Input #
Action
Push 0
Push 1
Go to “after #”
Pop
Crash
Changes needed:
Non-deterministically go to “after #” when you push 0 or 1.
Also, non-deterministically don’t push, in case we are dealing with odd-length palindrome!

7. Other examples Think about these “equal” language More 1s than 0s.
How many states do we need? …
More 1s than 0s.
Twice as many 1s as 0s.
Hint: think of the 0s as counting double.

8. Equal PDA State (Just one state) Tos ε 1 Input Action Push 0 Push 11
Input
Action
Push 0
Push 1
Pop