CS 4510: Automata and Complexity
Problem Set 2
All problems are worth 10 points.
Problem 1
Prove that the following languages are not regular:
1. {0n1m0n|n≥0}
2. {w|wis not a palindrome}
Problem 2
Consider a new kind of finite automaton called an All-Paths-NFA. An All-Paths-NFA Mis a 5-
tuple (Q, Σ, δ, q0, F ) that accepts x∈Σ∗if every possible computation of Mon xends in a state
from F. Note, in contrast, that an ordinary NFA accepts a string if some computation ends in an
accept state. Prove that All-Paths-NFAs recognize the class of regular languages.
Problem 3
If Ais a language, let A−1
2be the set of all first halves of strings in Aso that
A−1
2={x|for some y, |x|=|y|,and xy ∈A}
Show that if Ais regular, so is A−1
2.
Problem 4
Give context-free grammars that generate the following languages. Also give informal description
of the PDAs accepting these languages. The alphabet is {0,1}.
1. {w|length of wis odd}
2. {w|wcontains more 1s than 0s}
3. {w|wis a palindrome}
Problem 5
For a language A, let SUFFIX(A) denote the set of all suffixes of strings in A, i.e.
SUFFIX(A) = {v|uv ∈Afor some string u}
Show that if Ais a context-free language, so is SUFFIX(A).
Problem 6
Use the pumping lemma to show that the following languages are not context free:
1. {0n1n0n1n|n≥0}
2. {0i1j|i≥1, j ≥1, i =jk for some integer k}
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