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--- aa:beyondre [2016/10/21 09:25]
pdmatei
+++ aa:beyondre [2016/10/31 14:31] (current)
rstefan
@@ Line 133: / Line 133: @@
 Note that our machine will always terminate if there exists an input for which $math[M] does not halt since the oracle **always terminate**.
-To conclude, in our Universe, $math[f_{all}\in coRE].
+To conclude, in o
+ur Universe, $math[f_{all}\in coRE].
 There is an infinity of classes $math[(co)RE^Y], which contain (co)-recursively enumerable problems, given an oracle from class $math[Y]. Hence, formally, $math[f_{all}\in coRE^{RE}].
@@ Line 164: / Line 165: @@
 Let:
-$math[\Pi_w(\omega) = \mbox{ if } M_x(w) \mbox{halts, then run} M^*(\omega) \mbox{.}]
+$math[\Pi_M(\omega) = \mbox{ if } M_x(M) \mbox{halts, then run} M^*(\omega) \mbox{.}]
-If $math[f_{\Pi_w}] is the problem accepted by $math[\Pi_w], we show that:
+If $math[f_{\Pi_M}] is the problem accepted by $math[\Pi_M], we show that:
-$math[f_{\Pi_w} \in \mathcal{C} \mbox{ iff } M_x(w) \mbox{ halts}]
+$math[f_{\Pi_M} \in \mathcal{C} \mbox{ iff } M_x(M) \mbox{ halts}]
-$math[(\Rightarrow)]. Suppose $math[f_{\Pi_w} \in \mathcal{C}]. Then $math[\Pi_w(\omega)] cannot loop for every input $math[\omega \in \Sigma^*]. If there were so, then $math[f_{\Pi_w}] would be the trivial function always returning $math[0] for any input, which we have assumed is not in $math[\mathcal{C}]. Thus, $math[M_x(w)] halts.
+$math[(\Rightarrow)]. Suppose $math[f_{\Pi_M} \in \mathcal{C}]. Then $math[\Pi_M(\omega)] cannot loop for every input $math[\omega \in \Sigma^*]. If there were so, then $math[f_{\Pi_M}] would be the trivial function always returning $math[0] for any input, which we have assumed is not in $math[\mathcal{C}]. Thus, $math[M_x(M)] halts.
-$math[(\Leftarrow)]. Suppose $math[M_x(w)] halts. Then the behaviour of $math[\Pi_w(\omega)] is precisely that of
+$math[(\Leftarrow)]. Suppose $math[M_x(M)] halts. Then the
-$math[M^*(\omega)]. $math[\Pi_w(\omega)] will return $math[1] whenever $math[M^*(\omega)] will return $math[1] and $math[\Pi_w(\omega) = \perp] whenever $math[M^*(\omega) = \perp]. Since $math[f \in \mathcal{C}], then also $math[f_{\Pi_w} \in \mathcal{C}].
+ behaviour of $math[\Pi_M(\omega)] is precisely that of
+$math[M^*(\omega)]. $math[\Pi_M(\omega)] will return $math[1] whenever $math[M^*(\omega)] will return $math[1] and $math[\Pi_M(\omega) = \perp] whenever $math[M^*(\omega) = \perp]. Since $math[f \in \mathcal{C}], then also $math[f_{\Pi_M} \in \mathcal{C}].
 $end