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aa:lab:04 [2020/10/24 05:20]
claudiu.dorobantu 		aa:lab:04 [2020/11/11 02:06] (current)
claudiu.dorobantu
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 **1.1** Prove that $math[R \subsetneq RE.] **1.1** Prove that $math[R \subsetneq RE.]
  
-**1.2** Show that the following problem is not in $math[R]: $math[f(M) = 1] iff exists $math[w] such that $math[M] accepts $math[w] ​(this is actually the problem complement of establishing if $math[M] accepts the non-trivial problem $math[f(x) = 0]).+**1.2** Show that the following problem is not in $math[R]: $math[f(M) = 1] iff exists $math[w] such that $math[M] accepts $math[w.] 
 +  * This is actually the problem complement of establishing if $math[M] accepts the non-trivial problem $math[f(x) = 0.]
  
 **1.3** Show that the problem in **1.2** is in $math[RE.] **1.3** Show that the problem in **1.2** is in $math[RE.]
  
-**1.4** Show that the following problem is in $math[RE] but not in $math[R]: $math[f(M1, M2, w) = 1] iff $math[M1] and $math[M2] both accept $math[w.]+**1.4** Show that the following problem is in $math[RE] but not in $math[R]: $math[f(M_1, M_2, w) = 1] iff $math[M_1] and $math[M_2] both accept $math[w.]
  
 **1.5** Prove that $math[RE] is countable. **1.5** Prove that $math[RE] is countable.
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 ==== 2. Properties of Turing reductions ==== ==== 2. Properties of Turing reductions ====
  
-**2.1** Answer the following questions and justify your answers: +**2.1** Is $\leq_T$ an equivalence relation over $math[RE]?​ 
-  ​* Is $math[\leq]<​sub>​$math[T]</​sub> ​an equivalence relation over $math[RE]?​ +  * $\leq_T$ is reflexive?​ 
-  * $math[\leq]<​sub>​$math[T]</​sub> ​is reflexive?​ +  * $\leq_T$ is transitive?​ 
-  * $math[\leq]<​sub>​$math[T]</​sub> ​is transitive?​ +  * $\leq_T$ is symmetric? If $\leq_T$ would be symmetric, what would it signify?
-  * $math[\leq]<​sub>​$math[T]</​sub> ​is symmetric? If $math[\leq]<​sub>​$math[T]</​sub> ​would be symmetric, what would it signify?+
  
-**2.2** Suppose $math[f] $math[\leq]<​sub>​$math[T]</​sub> ​$math[f]<​sub>​$math[h]</​sub>​. ​What does that say about $math[f]?+**2.2** Suppose $math[f] $\leq_T$ $math[f_h.] What does that say about $math[f]?
  
-**2.3** Show that $math[k]-$math[Vertex]-$math[Cover] $math[\leq]<​sub>​$math[T]</​sub> ​$math[f]<​sub>​$math[h]</​sub>​.+**2.3** Show that $math[k]-$math[Vertex]-$math[Cover] $\leq_T$ $math[f_h.]
  
 ==== 3. Problems outside RE ==== ==== 3. Problems outside RE ====
  
-**3.1** Show that the following problem is not in $math[R]: $math[f(M1, M2) = 1] iff $math[M1] and $math[M2] accept the same problem $math[g] (solve the same algorithm).+**3.1** Show that the following problem is not in $math[R]: $math[f(M_1, M_2) = 1] iff $math[M_1] and $math[M_2] accept the same problem $math[g] (solve the same algorithm).
  
 **3.2** Show that the following problem is not in $math[R]: $math[f(M) = 1] iff $math[M] accepts all words in $math[\Sigma^*] (does $math[M] accept the problem $math[g(x) = 1]?). **3.2** Show that the following problem is not in $math[R]: $math[f(M) = 1] iff $math[M] accepts all words in $math[\Sigma^*] (does $math[M] accept the problem $math[g(x) = 1]?).