1
GATE CSE 2006
+2
-0.6
For $$s \in {\left( {0 + 1} \right)^ * },$$ let $$d(s)$$ denote the decimal value of $$s(e. g.d(101)=5)$$
Let $$L = \left\{ {s \in {{\left( {0 + 1} \right)}^ * }\left| {\,d\left( s \right)\,} \right.} \right.$$ mod $$5=2$$ and $$d(s)$$ mod $$\left. {7 \ne 4} \right\}$$

Which of the following statement is true?

A
$$L$$ is recursively enumerable, but not recursive
B
$$L$$ is recursive, but not context-free
C
$$L$$ is context-free, but not regular
D
$$L$$ is regular
2
GATE CSE 2006
+2
-0.6
Let $${L_1}$$ be a regular language, $${L_2}$$ be a deterministic context-free language and $${L_3}$$ a recursively enumerable, but not recursive, language. Which one of the following statement is false?
A
$${L_1} \cap {L_2}$$ is deterministic $$CFL$$
B
$${L_3} \cap {L_1}$$ is recursive
C
$${L_1} \cup {L_2}$$ is context-free
D
$${L_1} \cap {L_2} \cap {L_3}$$ is recursively enumerable
3
GATE CSE 2005
+2
-0.6
Let $${L_1}$$ be a recursive language, and Let $${L_2}$$ be a recursively enumerable but not a recursive language. Which one of the following is TRUE?
A
$$\overline {{L_1}}$$ is recursive and $$\overline {{L_2}}$$ is recursively enumerable
B
$$\overline {{L_1}}$$ is recursive and $$\overline {{L_2}}$$ is not recursively enumerable
C
$$\overline {{L_1}}$$ and $$\overline {{L_2}}$$ are recursively enumerable
D
$$\overline {{L_1}}$$ is recursively enumerable and $$\overline {{L_2}}$$ is recursive
4
GATE CSE 2003
+2
-0.6
A single tape Turing Machine $$M$$ has two states $${q_0}$$ and $${q_1}$$, of which $${q_0}$$ is the starting state. The tape alphabet of $$M$$ is $$\left\{ {0,\,\,1,\,\,B} \right\}$$ and its input alphabet is $$\left\{ {0,\,\,1} \right\}$$. The symbol $$B$$ is the blank symbol used to indicate end of an input string. The transition function of $$M$$ is described in the following table. The table is interpreted as illustrated below. The entry $$\left( {{q_1},1,\,R} \right)$$ in row $${{q_0}}$$ and column $$1$$ signifies that if $$M$$ is in state $${{q_0}}$$ and reads $$1$$ on the current tape square, then it writes $$1$$ on the same tape square, moves its tape head one position to the right and transitions to state $${{q_1}}$$.

Which of the following statements is true about $$M?$$

A
$$M$$ does not halt on any string in $${\left( {0 + 1} \right)^ + }$$
B
$$M$$ does not halt on any string in $${\left( {00 + 1} \right)^ + }$$
C
$$M$$ halts on all string ending in a $$0$$
D
$$M$$ halts on all string ending in $$a$$
GATE CSE Subjects
Discrete Mathematics
Programming Languages
Theory of Computation
Operating Systems
Digital Logic
Computer Organization
Database Management System
Data Structures
Computer Networks
Algorithms
Compiler Design
Software Engineering
Web Technologies
General Aptitude
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