Mixing
Size of complement of context-free language
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Let $L$ be a context-free language, $bar L$ be its complement and $bar L_n$ be the length $n$ words in $bar L_n$.
What is known about $|bar L_n|$?
Note that it is known that $|L_n|$ is either polynomial (if $L$ is bounded$~$), or grows exponentially.
I wonder if anything similar might be true about $|bar L_n|$.
Warning, $L$ is allowed to be ambiguous!
fl.formal-languages context-free
asked Jan 12 at 12:35
domotorpdomotorp
8,9883080
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add a comment |
$begingroup$
Let $L$ be a context-free language, $bar L$ be its complement and $bar L_n$ be the length $n$ words in $bar L_n$.
What is known about $|bar L_n|$?
Note that it is known that $|L_n|$ is either polynomial (if $L$ is bounded$~$), or grows exponentially.
I wonder if anything similar might be true about $|bar L_n|$.
Warning, $L$ is allowed to be ambiguous!
fl.formal-languages context-free
asked Jan 12 at 12:35
domotorpdomotorp
8,9883080
$endgroup$
add a comment |
$begingroup$
Let $L$ be a context-free language, $bar L$ be its complement and $bar L_n$ be the length $n$ words in $bar L_n$.
What is known about $|bar L_n|$?
Note that it is known that $|L_n|$ is either polynomial (if $L$ is bounded$~$), or grows exponentially.
I wonder if anything similar might be true about $|bar L_n|$.
Warning, $L$ is allowed to be ambiguous!
fl.formal-languages context-free
asked Jan 12 at 12:35
domotorpdomotorp
8,9883080
$endgroup$
Let $L$ be a context-free language, $bar L$ be its complement and $bar L_n$ be the length $n$ words in $bar L_n$.
What is known about $|bar L_n|$?
Note that it is known that $|L_n|$ is either polynomial (if $L$ is bounded$~$), or grows exponentially.
I wonder if anything similar might be true about $|bar L_n|$.
Warning, $L$ is allowed to be ambiguous!
fl.formal-languages context-free
fl.formal-languages context-free
asked Jan 12 at 12:35
domotorpdomotorp
8,9883080
asked Jan 12 at 12:35
domotorpdomotorp
8,9883080
edited Jan 14 at 8:38
domotorp
asked Jan 12 at 12:35
domotorpdomotorp
8,9883080
asked Jan 12 at 12:35
domotorpdomotorp
8,9883080
asked Jan 12 at 12:35
domotorpdomotorp
8,9883080
8,9883080
add a comment |
add a comment |
1 Answer
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From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.
answered Jan 12 at 13:42
Lance FortnowLance Fortnow
7,0363654
$endgroup$
2
$begingroup$
Thanks. I fixed it.
$endgroup$
– Lance Fortnow
Jan 12 at 21:56
add a comment |
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1 Answer
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1 Answer
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$begingroup$
From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.
answered Jan 12 at 13:42
Lance FortnowLance Fortnow
7,0363654
$endgroup$
2
$begingroup$
Thanks. I fixed it.
$endgroup$
– Lance Fortnow
Jan 12 at 21:56
add a comment |
$begingroup$
From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.
answered Jan 12 at 13:42
Lance FortnowLance Fortnow
7,0363654
$endgroup$
2
$begingroup$
Thanks. I fixed it.
$endgroup$
– Lance Fortnow
Jan 12 at 21:56
add a comment |
$begingroup$
From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.
answered Jan 12 at 13:42
Lance FortnowLance Fortnow
7,0363654
$endgroup$
From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.
answered Jan 12 at 13:42
Lance FortnowLance Fortnow
7,0363654
edited Jan 12 at 21:56
answered Jan 12 at 13:42
Lance FortnowLance Fortnow
7,0363654
answered Jan 12 at 13:42
Lance FortnowLance Fortnow
7,0363654
answered Jan 12 at 13:42
Lance FortnowLance Fortnow
7,0363654
7,0363654
2
$begingroup$
Thanks. I fixed it.
$endgroup$
– Lance Fortnow
Jan 12 at 21:56
add a comment |
2
$begingroup$
Thanks. I fixed it.
$endgroup$
– Lance Fortnow
Jan 12 at 21:56
2
2
$begingroup$
Thanks. I fixed it.
$endgroup$
– Lance Fortnow
Jan 12 at 21:56
$begingroup$
Thanks. I fixed it.
$endgroup$
– Lance Fortnow
Jan 12 at 21:56
add a comment |
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