Abstract

Practice running and constructing PDAs.

1. Review the Pushdown Automata section of the Tutorial.

2. Here is the PDA for $0^n1^n$.

   Try executing it (“Input -> Step by Closure”) on these inputs:

   • 0
   • 1
   • 01
   • 10
   • 0011
   • 000111
   • 00110

3. Here is the PDA for $ww^R$.

   Try executing it (“Input -> Step by Closure”) on these inputs:

   • 0
   • 1
   • 00
   • 0101
   • 0110
   • 10101
   • 011001
   • 01100110

4. A palindrome is a string that reads the same backwards as forwards. All strings in our $ww^R$ language are palindromes. But “1001001” is also a palindrome, and is not in $ww^R$.

   Can you modify the $ww^R$ PDA so that it recognizes all palindromes over $\{0, 1\}^*$? (This is a good way to see if you have really embraced thinking about non-determinism.)

   Try it before looking at the answer.

   We just need to allow the “middle” symbol to be consumed without either pushing it onto the stack nor matching against a symbol already on the stack. Of course, we don’t know which input symbol is the middle one, so we pretend that all of them could be. Non-determinism for the win!

   Here is the PDA for palindromes.

   Try running it on the same inputs you used for $ww^R$.