Problem 1: (25 points)
Construct a PDA for each of the following languages:
a) L={a³ⁿb²ⁿ | n > 0}. Construct the PDA without going through grammars.
b) L={a ⁿb^mc^n+m | n > 0, m > 0}. Again, do not go through grammars.
c) The language generated by the grammer in Problem 5 in Homework 3. For that, use the GNF grammar in the solution of that problem to construct the PDA vy following the standard method.

Problem 2: (21 points)
Show that the following languages are not CFL's.

a) {aⁿ⁴ | n > 0}
d) {aⁿ²bⁿ | n > 0}
e) {a^n! | n > 0}.

Problem 3: (24 points)
Using the closure properties of CFL's, show that the following languages are CFL's:
a) L={a ⁿ b ^m | n > 2m}
b) L={a ⁿ b ^m | n != 2m}
c) L={a ⁱ b ^j c ^k | i != j or j != k}
d) L= the complement of {aⁿbⁿcⁿ | n ≥ 0}.

Problem 4: (20 points)
Design Turing machines to recognize the following languages. Your design (in this Problem and in Problem 5) should start with a clear algorithmic description and end with a formal definition of the delta function (with enough comments).
a) {aⁿb^m | n < m}.
b) {aⁿb^n+mc^m | n>=0, m>=0}.

Problem 5: (10 points)
Design a TM that takes as input 0^n10^m and gives output 0 ^|n-m| .