Arjun Suresh (talk | contribs) |
Arjun Suresh (talk | contribs) |
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[http://condor.depaul.edu/glancast/444class/docs/nfa2dfa.html Conversion] | [http://condor.depaul.edu/glancast/444class/docs/nfa2dfa.html Conversion] | ||
| − | *'''Q:''' I know that $2^n$ are the maximum states in DFA for an n | + | *'''Q:''' I know that $2^n$ are the maximum states in DFA for an $n$ state NFA. But what about the minimum no of states in a DFA for an $n$ state NFA? |
A: I would say 1 because the $n$ states can be connected through $ε$ moves and hence all of these will be combines to a single start state in DFA. | A: I would say 1 because the $n$ states can be connected through $ε$ moves and hence all of these will be combines to a single start state in DFA. | ||
*'''Q:''' After converting from NFA to DFA and we get $2^n$ states. Now we apply minimization- how many states can we get after this? | *'''Q:''' After converting from NFA to DFA and we get $2^n$ states. Now we apply minimization- how many states can we get after this? | ||
| − | In the worst case this can remain $2^n$ - that is no minimization can be possible. We can see this in the following example | + | A: In the worst case this can remain $2^n$ - that is no minimization can be possible. We can see this in the following example |
| − | + | ===NFA=== | |
<graphviz caption='NFA'> | <graphviz caption='NFA'> | ||
digraph NFA { | digraph NFA { | ||
rankdir=LR; | rankdir=LR; | ||
node [shape = none] ""; | node [shape = none] ""; | ||
| − | node [shape = doublecircle] | + | node [shape = doublecircle] q2; |
node [shape = circle]; | node [shape = circle]; | ||
| − | "" | + | ""-> q1; |
| − | |||
| − | |||
q1->q1 [label="a"]; | q1->q1 [label="a"]; | ||
| − | q1 -> | + | q1->q2 [label="a"]; |
| + | q2 -> q2 [label="b"]; | ||
} | } | ||
| + | </graphviz> | ||
| + | ===DFA=== | ||
<graphviz caption='DFA'> | <graphviz caption='DFA'> | ||
digraph DFA { | digraph DFA { | ||
rankdir=LR; | rankdir=LR; | ||
node [shape = none] ""; | node [shape = none] ""; | ||
| − | node [shape = doublecircle] | + | node [shape = doublecircle] q2, q12; |
node [shape = circle]; | node [shape = circle]; | ||
| − | ""-> | + | ""-> q1; |
| − | + | q1->q12 [label="a"]; | |
| − | + | q12->q12 [label="a"]; | |
| − | + | q12->q2 [label="b"]; | |
| − | q1 -> | + | q2->q2 [label="b"]; |
| + | q2 -> q3 [label="a"]; | ||
| + | q1 -> q3 [label="b"]; | ||
} | } | ||
| + | </graphviz> | ||
A: I would say 1 because the $n$ states can be connected through $ε$ moves and hence all of these will be combines to a single start state in DFA.
A: In the worst case this can remain $2^n$ - that is no minimization can be possible. We can see this in the following example
A: I would say 1 because the $n$ states can be connected through $ε$ moves and hence all of these will be combines to a single start state in DFA.
In the worst case this can remain $2^n$ - that is no minimization can be possible. We can see this in the following example
<graphviz caption='NFA'> digraph NFA { rankdir=LR; node [shape = none] ""; node [shape = doublecircle] q1; node [shape = circle];
""-> q0; q0->q1 [label="a"]; q1->q0 [label="a"]; q1->q1 [label="a"]; q1 -> q0 [label="b"]; }
<graphviz caption='DFA'> digraph DFA { rankdir=LR; node [shape = none] ""; node [shape = doublecircle] q1, q12; node [shape = circle];
""-> q0; q0->q1 [label="a"]; q1->q0 [label="a"]; q1->q1 [label="a"]; q1 -> q0 [label="b"]; }
GATECSE