***ASKED BEFORE BUT NO ANSWER WAS GIVEN*** ***PLEASE ANSWER CORRECTLY*** WILL LIKE IF ALL CORRECT 1. Let S5 be the set of binary strings of length at least 5, i.e. S5 = {x ∈ {0, 1} ∗ : |x| ≥ 5}. (a) Build (design do not describe) a recogniser for S5 which is not a decider for S5. (b) Build an enumerator for S5. Note that we can enumerate or list all strings in lexicographic order s1 = epsilon , s2 = 0, s3 = 1, s4 = 00, . . . where is the empty string “”. In this case we only consider binary strings but this is valid over any alphabet. Let sn be the nth string enumerated in this way. (c) Is S5 decidable? (d) If you had the option to have access to either a decider or a recogniser when solving membership of a set, which would you choose and why? 2. Consider the set RejTM = { | M is a TM such that M rejects on input w}. (a) Describe an element in the set and one not in the set. You should provide pseudocode for any TM you discuss. (b) Prove RejTM is recognisable. Hint: Similar to the proof of HaltTM in the tutorials. 3. Consider the set Acc010 = { | M is a TM that accepts the string 010}. (a) Describe an element in the set and one not in the set. You should provide pseudocode for any TM you discuss. (b) Prove Acc010 is recognisable. That is, build a recogniser R such that on input of a TM M, R() = acc if and only if M(010) = acc and R rejects or loops if M(010) ≠ acc.
***ASKED BEFORE BUT NO ANSWER WAS GIVEN***
***PLEASE ANSWER CORRECTLY*** WILL LIKE IF ALL CORRECT
1. Let S5 be the set of binary strings of length at least 5, i.e. S5 = {x ∈ {0, 1} ∗ : |x| ≥ 5}.
(a) Build (design do not describe) a recogniser for S5 which is not a decider for S5.
(b) Build an enumerator for S5. Note that we can enumerate or list all strings in lexicographic order s1 = epsilon , s2 = 0, s3 = 1, s4 = 00, . . . where is the empty string “”. In this case we only consider binary strings but this is valid over any alphabet. Let sn be the nth string enumerated in this way.
(c) Is S5 decidable?
(d) If you had the option to have access to either a decider or a recogniser when solving membership of a set, which would you choose and why?
2. Consider the set RejTM = {<M, w> | M is a TM such that M rejects on input w}.
(a) Describe an element in the set and one not in the set. You should provide pseudocode for any TM you discuss.
(b) Prove RejTM is recognisable. Hint: Similar to the proof of HaltTM in the tutorials.
3. Consider the set Acc010 = {<M> | M is a TM that accepts the string 010}.
(a) Describe an element in the set and one not in the set. You should provide pseudocode for any TM you discuss.
(b) Prove Acc010 is recognisable. That is, build a recogniser R such that on input of a TM M, R(<M>) = acc if and only if M(010) = acc and R rejects or loops if M(010) ≠ acc.
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