({{Template:Author|Arjun Suresh|{{arjunweb}} }})
Line 19: Line 19:
 
rankdir=LR;
 
rankdir=LR;
 
size="8,5"
 
size="8,5"
node [shape = doublecircle]; LR_4;
+
node [shape = doublecircle]; LR_3;
 
node [shape = circle];
 
node [shape = circle];
 
LR_0 -> LR_1 [ label = "a" ];
 
LR_0 -> LR_1 [ label = "a" ];
 
LR_1 -> LR_2 [ label = "a" ];
 
LR_1 -> LR_2 [ label = "a" ];
 
LR_2 -> LR_3 [ label = "a" ];
 
LR_2 -> LR_3 [ label = "a" ];
LR_3 -> LR_4 [ label = "a" ];
+
LR_3 -> LR_2 [ label = "a" ];
 
LR_0-> LR_2 [ label = "b" ];
 
LR_0-> LR_2 [ label = "b" ];
 
LR_1 -> LR_0 [ label = "b" ];
 
LR_1 -> LR_0 [ label = "b" ];
 
LR_2 -> LR_2 [ label = "b" ];
 
LR_2 -> LR_2 [ label = "b" ];
 
LR_3 -> LR_2 [ label = "b" ];
 
LR_3 -> LR_2 [ label = "b" ];
LR_4 -> LR_4 [ label = "b" ];
+
}
LR_4 -> LR_0 [ label = "a" ];
+
digraph finite_state_machine {
 +
rankdir=LR;
 +
size="8,5"
 +
node [shape = doublecircle]; LR_0;
 +
node [shape = circle];
 +
LR_1 -> LR_0 [ label = "a" ];
 
 
 
}
 
}

Revision as of 15:01, 24 May 2014

1. If <math>L</math> and <math>L'</math> are both recursively enumerable, then <math>L</math> is recursive. Why?

Solution by Arjun Suresh

Given <math>L</math> is $RE$. So there is a $TM$, which accepts and halts for all words in <math>L</math>. Now, if <math>L'</math> is $RE$, then there is a $TM$, which accepts and halts for all words not in <math>L</math>. So, if a word is given (either from <math>L</math> or not from <math>L</math>), give it to both those $TM$s. If its from $L$, the first $TM$ will halt and we say it belongs to $L$. If its not from $L$, the second one will halt and we say it doesn't belong to <math>L</math>. Thus, <math>L</math> becomes recursive.

2. CYCLE(L) ={xy | yx is in L ,L is regular }

Is this statement true or false ?

Solution by Arjun Suresh

We have a DFA for L, let it be D and have n states. Now we can make a NFA for L' as follows:

For every state of D,

This is a graph with borders and nodes. Maybe there is an Imagemap used so the nodes may be linking to some Pages.



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1. If <math>L</math> and <math>L'</math> are both recursively enumerable, then <math>L</math> is recursive. Why?[edit]

Solution by Arjun Suresh[edit]

Given <math>L</math> is $RE$. So there is a $TM$, which accepts and halts for all words in <math>L</math>. Now, if <math>L'</math> is $RE$, then there is a $TM$, which accepts and halts for all words not in <math>L</math>. So, if a word is given (either from <math>L</math> or not from <math>L</math>), give it to both those $TM$s. If its from $L$, the first $TM$ will halt and we say it belongs to $L$. If its not from $L$, the second one will halt and we say it doesn't belong to <math>L</math>. Thus, <math>L</math> becomes recursive.

2. CYCLE(L) ={xy | yx is in L ,L is regular }[edit]

Is this statement true or false ?

Solution by Arjun Suresh[edit]

We have a DFA for L, let it be D and have n states. Now we can make a NFA for L' as follows:

For every state of D,

This is a graph with borders and nodes. Maybe there is an Imagemap used so the nodes may be linking to some Pages.



blog comments powered by Disqus