HW-2(2)
Computer Science
Theory of Computing (Csc 520 Course)
2. (12.5 points):
Σ={a,b}, L = {w : w is a palindrome, and #a(w) = #b(w)}. For example, abbaabba ∈ L; aba ∉ L because it has more a’s than b’s; ba ∉ L because it is not the same left-to right as right-to-left. Finish the proof, started below, which shows that L ∉ CFLs.
1. L⋂ = L ⋂ a*b*a*
2. If L ∈ CFLs then L⋂ ∈ CFLs because the CFLs are closed under intersection with an RL.
3. If L⋂ ∉ CFLs then L ∉ CFLs by applying the modus tollens logic rule to step 2.
4. Use the CF pumping theorem to show that L⋂ ∉ CFLs.
4.a Let w = aᴷb²ᴷaᴷ. Then w ∈ L and |w| ≥ k, so that w can be expressed as the
concatenation of the substrings uvxyz, and the CF pumping theorem guarantees
apply.
