# Undecidability

 Question 1
Consider the following sets:

S1: Set of all recursively enumerable languages over the alphabet {0, 1}.
S2: Set of all syntactically valid C programs.
S3: Set of all languages over the alphabet {0, 1}.
S4: Set of all non-regular languages over the alphabet {0, 1}.

Which of the above sets are uncountable?
 A S1 and S2 B S3 and S4 C S2 and S3 D S1 and S4
GATE CSE 2019      Undecidability
Question 1 Explanation:
 Question 2
Let L(R) be the language represented by regular expression R. Let L(G) be the language generated by a context free grammar G. Let L (M) be the language accepted by a Turning machine M. Which of the following decision problems are undecidable ?

I. Given a regular expression R and a string w, is w$\in$L(R)?
II. Given a context-free grammar G, L(G)=$\phi$?
III. Given a context-free grammar G, is L(G)=$\Sigma$* for some alphabet $\Sigma$?
IV. Given a Turning machine M and a string w, is w$\in$L(M)?
 A I and IV only B II and III only C II, III and IV only D III and IV only
GATE CSE 2017 SET-2      Undecidability
Question 2 Explanation:

 Question 3
Which of the following decision problems are undecidable?
I. Given NFAs N1 and N2, is L(N1)$\cap$L(N2) = $\phi$?
II. Given a CFG G = (N,$\Sigma$,P,S) and a string $x \in\Sigma ^*$, does $x \in$ L(G)?
III. Given CFGs G1 and G2, is L(G1) = L(G2)?
IV. Given a TM M, is L(M) = $\phi$?
 A I and IV only B II and III only C III and IV only D II and IV only
GATE CSE 2016 SET-1      Undecidability
Question 3 Explanation:
 Question 4
Which one of the following problems is undecidable?
 A Deciding if a given context-free grammar is ambiguous B Deciding if a given string is generated by a given context-free grammar C Deciding if the language generated by a given context-free grammar is empty D Deciding if the language generated by a given context-free grammar is finite.
GATE CSE 2014 SET-3      Undecidability
Question 4 Explanation:
 Question 5
Let $\Sigma$ be a finite non-empty alphabet and let $2^{\Sigma^{*}}$ be the power set of $\Sigma^{*}$ . Which one of the following is TRUE?
 A Both $2^{\Sigma^{*}}$ and $\Sigma^{*}$ are countable B $2^{\Sigma^{*}}$ is countable $\Sigma^{*}$ is uncountable C $2^{\Sigma^{*}}$ is uncountable and $\Sigma^{*}$ is countable D Both $2^{\Sigma^{*}}$ and $\Sigma^{*}$ are uncountable
GATE CSE 2014 SET-3      Undecidability
Question 5 Explanation:

There are 5 questions to complete.