Computer Science Engineering (CSE) Exam  >  Computer Science Engineering (CSE) Questions  >  Consider the following languages:I. L1 = { &l... Start Learning for Free
Consider the following languages:
I. L
1
= { < M >∣ M takes at least 2016 steps on some input }
II. L
2
= {< M >∣ M takes at least 2016 steps on all inputs }
III. L
3
= {< M >∣ M accepts ϵ}
Where for each Turing machine M, denotes a specific encoding of M.
Which one of the following is true?
  • a)
    L
    1
    is recursive and L
    2
    ,L
    3
    are not recursive
  • b)
    L
    2
    is recursive and L
    1
    ,L
    3
    are not recursive
  • c)
     L
    1
    and L
    2
    are recursive
  • d)
    L
    3
    is not recursive
Correct answer is option 'C,D'. Can you explain this answer?
Most Upvoted Answer
Consider the following languages:I. L1 = { < M > M takes at leas...
Statement I:
L
1
= {< M >∣ M takes at least 2016 steps on some input },
This language is decidable, because counting number of steps can always be decided. We can find TM whether it takes more than 2016 steps on some input string which has length up to 2016. If it processes 2016 steps till end, then will take at least 2016 steps. But if these are not taking 2016 steps, it means these have taken at most 2015 steps which means string is not processed
completely. Also, if Turing machine reaches the halt state before reaching the end of string of input, it can be decided. In all cases, Turing machine M is recursive.
Statement II:
L
2
= {< M >∣ M takes at least 2016 steps on all inputs }
For this, consider M makes less than 2016 steps for some input and we can just give it all possible inputs of length less than 2016 steps and check if it can reach a halt state within 2016 steps. All the possible strings make a finite set. Thus, accepted by Turing machine. So, it is recursive language.
Statement III:
L
3
= {< M >∣ M accepts ϵ},
It is based on the concept of rice theorem i.e. any non - trivial property of the language recognizable by a Turing machine is undecidable. As this language contains ϵ which is non - trivial property of recursive enumerable language and hence undecidable. So, it is not recursive.
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Consider the following languages:I. L1 = { < M > M takes at least 2016 steps on some input }II. L2 = {< M > M takes at least 2016 steps on all inputs }III. L3 = {< M > M accepts }Where for each Turing machine M, denotes a specific encoding of M.Which one of the following is true?a)L1 is recursive and L2,L3 are not recursiveb)L2 is recursive and L1,L3 are not recursivec)L1 and L2 are recursived)L3 is not recursiveCorrect answer is option 'C,D'. Can you explain this answer?
Question Description
Consider the following languages:I. L1 = { < M > M takes at least 2016 steps on some input }II. L2 = {< M > M takes at least 2016 steps on all inputs }III. L3 = {< M > M accepts }Where for each Turing machine M, denotes a specific encoding of M.Which one of the following is true?a)L1 is recursive and L2,L3 are not recursiveb)L2 is recursive and L1,L3 are not recursivec)L1 and L2 are recursived)L3 is not recursiveCorrect answer is option 'C,D'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider the following languages:I. L1 = { < M > M takes at least 2016 steps on some input }II. L2 = {< M > M takes at least 2016 steps on all inputs }III. L3 = {< M > M accepts }Where for each Turing machine M, denotes a specific encoding of M.Which one of the following is true?a)L1 is recursive and L2,L3 are not recursiveb)L2 is recursive and L1,L3 are not recursivec)L1 and L2 are recursived)L3 is not recursiveCorrect answer is option 'C,D'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following languages:I. L1 = { < M > M takes at least 2016 steps on some input }II. L2 = {< M > M takes at least 2016 steps on all inputs }III. L3 = {< M > M accepts }Where for each Turing machine M, denotes a specific encoding of M.Which one of the following is true?a)L1 is recursive and L2,L3 are not recursiveb)L2 is recursive and L1,L3 are not recursivec)L1 and L2 are recursived)L3 is not recursiveCorrect answer is option 'C,D'. Can you explain this answer?.
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