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I need help on this exercise. It is for a theoretical computer science class, so the proof has to be punctual and precise, following a mathematical rigour. Let's denote by M = (Q, Sigma , Gamma , delta , ⊢, , s, a, r) a generic Turing machine. For each language in the following list, determine whether it is decidable or undecidable, providing an appropriate demonstration. A = {M : M has at least 1234 states} B = {M : M executes at least 1234 steps given the input 0} C = {M : M executes at least 1234 steps for some input x} D = {M : M executes at least 1234 steps for every input x} E = {M : there exists an input for which M moves 1234 cells to the right of the start cell of the tape ⊢} F = {M : M accepts the empty string (i.e., accepts if the tape contains only )} G = {M : M accepts some string} H = {M : M accepts a finite set of inputs} I = {M : M accepts exactly the set of prime numbers} J = {M : there exists an input for which M prints a certain symbol gamma in Gamma } K = {M : there exists an input for which M enters a certain state q in Q} Pie = {k : in the decimal expansion of pi there are at least k consecutive zeros}" 

          I need help on this exercise. It is for a theoretical computer science class, so the proof has to be punctual and precise, following a mathematical rigour. Let's denote by M = (Q, Sigma , Gamma , delta , ⊢, , s, a, r) a generic Turing machine. For each language in the following list, determine whether it is decidable or undecidable, providing an appropriate demonstration.
A = {M : M has at least 1234 states}
B = {M : M executes at least 1234 steps given the input 0}
C = {M : M executes at least 1234 steps for some input x}
D = {M : M executes at least 1234 steps for every input x}
E = {M : there exists an input for which M moves 1234 cells to the right of the start cell of the tape ⊢}
F = {M : M accepts the empty string (i.e., accepts if the tape contains only )}
G = {M : M accepts some string}
H = {M : M accepts a finite set of inputs}
I = {M : M accepts exactly the set of prime numbers}
J = {M : there exists an input for which M prints a certain symbol gamma   in  Gamma }
K = {M : there exists an input for which M enters a certain state q  in  Q}
Pie = {k : in the decimal expansion of pi  there are at least k consecutive zeros}"
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Added by Julie G.

Computer Science and Information Technology
Computer Science and Information Technology
Trishna Knowledge Systems 2018 Edition
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I need help on this exercise. It is for a theoretical computer science class, so the proof has to be punctual and precise, following a mathematical rigour. Let's denote by M = (Q, Sigma , Gamma , delta , ⊢, , s, a, r) a generic Turing machine. For each language in the following list, determine whether it is decidable or undecidable, providing an appropriate demonstration. A = {M : M has at least 1234 states} B = {M : M executes at least 1234 steps given the input 0} C = {M : M executes at least 1234 steps for some input x} D = {M : M executes at least 1234 steps for every input x} E = {M : there exists an input for which M moves 1234 cells to the right of the start cell of the tape ⊢} F = {M : M accepts the empty string (i.e., accepts if the tape contains only )} G = {M : M accepts some string} H = {M : M accepts a finite set of inputs} I = {M : M accepts exactly the set of prime numbers} J = {M : there exists an input for which M prints a certain symbol gamma in Gamma } K = {M : there exists an input for which M enters a certain state q in Q} Pie = {k : in the decimal expansion of pi there are at least k consecutive zeros}"
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Transcript

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00:04 We're asked to prove that m and m bar accept the same language in part a.
00:14 So, first, let x be a string accepted by the machine m, and let sx be the state where the string ends in x.
01:06 The string x ends in machine m.
01:11 Now, we know that x is accepted by m, so it follows that sx is a final state of m.
01:26 And by the construction of m bar, it follows that x will end at the state s x, the equivalence class containing s x, so it's the r star equivalence class containing s x in m bar.
02:15 And we have that the r star equivalence class of s x is a final state since all states in the r star equivalence class of s x are final states now this is because s x which is in the equivalence class r star containing x is a final state so it follows all the other states in r star of s x by previous problem are all final or non -final states so it follows that the r -star equivalence class containing sx is a final state in m -bar.
03:56 This implies that m and m -bar accept the same language.
04:30 Now in part b, we're asked to prove that if the machine m has the property f of s 0 x equals s for all states s and some string in i -star, then m -bar, the machine has the minimum number of states of all finite state automaton equivalent to m.
05:00 So to prove this statement, suppose that m has the property f of s not x equals s for all states s in m, and sum string x in i star.
06:38 I'm sorry, this should be written as m as the property that for all states s and s, in m, there is an x in i star such that this statement, f of s not x equals s, is true...
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