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Chapter 0: Q22P (page 1)
Let Show that is a context-free language.
It is shown that C is a context-free language.
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Show that P is closed under homomorphism iff P = NP.
Let CFG be thefollowing grammar.
Give a simple description of in English. Use that description to give a CFG for , the complement of .
Question:Consider the algorithm MINIMIZE, which takes a DFA as input and outputs DFA .
MINIMIZE = 鈥淥n input , where is a DFA:
1.Remove all states of G that are unreachable from the start state.
2. Construct the following undirected graph G whose nodes are the states of .
3. Place an edge in G connecting every accept state with every non accept state. Add additional edges as follows.
4. Repeat until no new edges are added to G :
5. For every pair of distinct states q and r of and every :
6. Add the edge (q,r) to G if is an edge of G .
7. For each state be the collection of edge joins q and r in G }.
8.Form a new DFA where
9. Output ( M')鈥�
a. Show that M and M' are equivalent.
b. Show that M0 is minimal鈥攖hat is, no DFA with fewer states recognizes the same language. You may use the result of Problem 1.52 without proof.
c. Show that MINIMIZE operates in polynomial time.
Let . Show that AMBIGCFG is undecidable. (Hint: Use a reduction from PCP. Given an instance
of the Post Correspondence Problem, construct a CFG Gwith the rules
where a1,...,ak are new terminal symbols. Prove that this reduction works.)
For each of the following languages, give two strings that are members and two strings that are not members鈥攁 total of four strings for each part. Assume the alpha-alphabet in all parts.
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