Let $$L$$ be the language represented by the regular expression $$\sum {^ * 0011\sum {^ * } } $$ where $$\sum { = \left\{ {0,1} \right\}} .$$ What is the minimum number of states in a $$DFA$$ that recognizes $$\overline L $$ (complement of $$L$$)?

Language $${L_1}$$ is polynomial time reducible to language $${L_2}$$ . Language $${L_3}$$ is polynomial time reducible to $${L_2}$$ , which in turn is polynomial time reducible to language $${L_4}$$ . Which of the following is/are true?

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm I}.\,\,\,\,$$ if $$\,\,\,{L_4} \in P,$$ then $$\,\,\,{L_2} \in P$$
$$\,\,\,\,\,\,\,\,\,\,\,\,{\rm I}{\rm I}.\,\,\,\,$$ if $$\,\,\,{L_1} \in P$$ or $$\,\,\,{L_3} \in P,$$ then $$\,\,\,{L_2} \in P$$
$$\,\,\,\,\,\,\,\,\,\,{\rm I}{\rm I}{\rm I}.\,\,\,\,$$ if $$\,\,\,{L_1} \in P,$$ and only $$\,\,\,{L_3} \in P$$
$$\,\,\,\,\,\,\,\,\,\,{\rm I}V.\,\,\,\,$$ if $$\,\,\,{L_4} \in P,$$ then $$\,\,\,{L_1} \in P$$ and $$\,\,\,{L_3} \in P$$

Which of the following languages are context-free? $$$\eqalign{ & {L_1} = \left\{ {{a^m}{b^n}{a^n}{b^m}|m,n \ge 1} \right\} \cr & {L_2} = \left\{ {{a^m}{b^n}{a^m}{b^n}|m,n \ge 1} \right\} \cr & {L_3} = \left\{ {{a^m}{b^n}|m = 2n + 1} \right\} \cr} $$$

In a web server, ten WebPages are stored with the URLs of the form
http://www.yourname.com/$$var$$.html; where, $$var$$ is a different number from $$1$$ to $$10$$ for each Webpage. Suppose, the client stores the Webpage with $$var = 1$$ (say $$W1$$) in local machine, edits and then tests. Rest of the WebPages remains on the web server. $$W1$$ contains several relative URLs of the form “$$var$$.html” referring to the other WebPages. Which one of the following statements needs to be added in $$W1,$$ so that all the relative URLs in $$W1$$ refer to the appropriate WebPages on the web server?