Prove that Dyck language is subset of grammar
last two days I am trying to prove, that Dyck language $L$ over the alphabet ${[, ]}$ is a subset of the language $L(G)$ generated by the grammar with rules ${S rightarrow [S]S | epsilon }$.
Recommended steps by our lector are:
a) Prove that every nonepmty $w in L$ can be written as $w = [u]v$, where $u,v$ $in$ $L$
b) prove with induction where i is number of '[' in $w$ that L $subseteq$ L(G), use a) in induction step
Thanks for any advice.
Edit: I have solved the problem from a, but I do not know, if its
valid proof and how to solve b.
a) Prove that every nonepmty $w in L$ can be written as $w = [u]v$,
where $u,v$ $in$ $L$:
$D = ( {S'}, {[, ]}, { S rightarrow S'S' | [S'] | epsilon }, S' )$ is a grammer defining Dyck language over alphabet ${[,]}$.
- The only way to create $w in L$ where $|w| > 0$ is to apply rule ${S'rightarrow[S'] }$ at least once.
$w_1$: $S' rightarrow [S'] rightarrow^* w_1 $
$w_2$: $S rightarrow S'S' rightarrow^* S^i[S']S^j rightarrow^* w_2 $ where $i, j geq 0$
- In 3rd case, $w = [u]v$, where $v = epsilon$ and $u$ is generated from starting symbol, so $u, v in L$. In 4th case, you have $w = [u]v$ where $u, v$ is generated from starting symbol $S'$ so $u, v in
L$
induction formal-languages formal-grammar
asked Nov 29 '18 at 21:17
TehrynTehryn
62
|
show 2 more comments
last two days I am trying to prove, that Dyck language $L$ over the alphabet ${[, ]}$ is a subset of the language $L(G)$ generated by the grammar with rules ${S rightarrow [S]S | epsilon }$.
Recommended steps by our lector are:
a) Prove that every nonepmty $w in L$ can be written as $w = [u]v$, where $u,v$ $in$ $L$
b) prove with induction where i is number of '[' in $w$ that L $subseteq$ L(G), use a) in induction step
Thanks for any advice.
Edit: I have solved the problem from a, but I do not know, if its
valid proof and how to solve b.
a) Prove that every nonepmty $w in L$ can be written as $w = [u]v$,
where $u,v$ $in$ $L$:
$D = ( {S'}, {[, ]}, { S rightarrow S'S' | [S'] | epsilon }, S' )$ is a grammer defining Dyck language over alphabet ${[,]}$.
- The only way to create $w in L$ where $|w| > 0$ is to apply rule ${S'rightarrow[S'] }$ at least once.
$w_1$: $S' rightarrow [S'] rightarrow^* w_1 $
$w_2$: $S rightarrow S'S' rightarrow^* S^i[S']S^j rightarrow^* w_2 $ where $i, j geq 0$
- In 3rd case, $w = [u]v$, where $v = epsilon$ and $u$ is generated from starting symbol, so $u, v in L$. In 4th case, you have $w = [u]v$ where $u, v$ is generated from starting symbol $S'$ so $u, v in
L$
induction formal-languages formal-grammar
asked Nov 29 '18 at 21:17
TehrynTehryn
62
1
Seems to me the language generated by [ and ] is larger than the language of proper pairings, It includes for example four [ 's and no ]'s. By Dyck language is there a restriction on what strings are in that language?
– coffeemath
Nov 29 '18 at 21:44
2
Dyck language (in our example) is a language, that is generated by grammar with rules S -> SS, S -> epsilon, S-> [S]
– Tehryn
Nov 29 '18 at 22:41
2
Your lector's advice seems pretty good to me. What is giving you trouble following it?
– rici
Nov 30 '18 at 2:56
1
Your definition of the Dyck language is not correct; you only provide the alphabet it uses.
– Peter Leupold
Nov 30 '18 at 8:28
1
@rici I have problem with induction. I was able somehow prove a). Also I was able to prove induction where number of '[' is equal to i where i is from interval <0, |w|>, but was not able to prove it for i+1
– Tehryn
Nov 30 '18 at 9:46
|
show 2 more comments
last two days I am trying to prove, that Dyck language $L$ over the alphabet ${[, ]}$ is a subset of the language $L(G)$ generated by the grammar with rules ${S rightarrow [S]S | epsilon }$.
Recommended steps by our lector are:
a) Prove that every nonepmty $w in L$ can be written as $w = [u]v$, where $u,v$ $in$ $L$
b) prove with induction where i is number of '[' in $w$ that L $subseteq$ L(G), use a) in induction step
Thanks for any advice.
Edit: I have solved the problem from a, but I do not know, if its
valid proof and how to solve b.
a) Prove that every nonepmty $w in L$ can be written as $w = [u]v$,
where $u,v$ $in$ $L$:
$D = ( {S'}, {[, ]}, { S rightarrow S'S' | [S'] | epsilon }, S' )$ is a grammer defining Dyck language over alphabet ${[,]}$.
- The only way to create $w in L$ where $|w| > 0$ is to apply rule ${S'rightarrow[S'] }$ at least once.
$w_1$: $S' rightarrow [S'] rightarrow^* w_1 $
$w_2$: $S rightarrow S'S' rightarrow^* S^i[S']S^j rightarrow^* w_2 $ where $i, j geq 0$
- In 3rd case, $w = [u]v$, where $v = epsilon$ and $u$ is generated from starting symbol, so $u, v in L$. In 4th case, you have $w = [u]v$ where $u, v$ is generated from starting symbol $S'$ so $u, v in
L$
induction formal-languages formal-grammar
asked Nov 29 '18 at 21:17
TehrynTehryn
62
last two days I am trying to prove, that Dyck language $L$ over the alphabet ${[, ]}$ is a subset of the language $L(G)$ generated by the grammar with rules ${S rightarrow [S]S | epsilon }$.
Recommended steps by our lector are:
a) Prove that every nonepmty $w in L$ can be written as $w = [u]v$, where $u,v$ $in$ $L$
b) prove with induction where i is number of '[' in $w$ that L $subseteq$ L(G), use a) in induction step
Thanks for any advice.
Edit: I have solved the problem from a, but I do not know, if its
valid proof and how to solve b.
a) Prove that every nonepmty $w in L$ can be written as $w = [u]v$,
where $u,v$ $in$ $L$:
$D = ( {S'}, {[, ]}, { S rightarrow S'S' | [S'] | epsilon }, S' )$ is a grammer defining Dyck language over alphabet ${[,]}$.
- The only way to create $w in L$ where $|w| > 0$ is to apply rule ${S'rightarrow[S'] }$ at least once.
$w_1$: $S' rightarrow [S'] rightarrow^* w_1 $
$w_2$: $S rightarrow S'S' rightarrow^* S^i[S']S^j rightarrow^* w_2 $ where $i, j geq 0$
- In 3rd case, $w = [u]v$, where $v = epsilon$ and $u$ is generated from starting symbol, so $u, v in L$. In 4th case, you have $w = [u]v$ where $u, v$ is generated from starting symbol $S'$ so $u, v in
L$
induction formal-languages formal-grammar
induction formal-languages formal-grammar
asked Nov 29 '18 at 21:17
TehrynTehryn
62
asked Nov 29 '18 at 21:17
TehrynTehryn
62
edited Dec 5 '18 at 15:44
Tehryn
asked Nov 29 '18 at 21:17
TehrynTehryn
62
asked Nov 29 '18 at 21:17
TehrynTehryn
62
asked Nov 29 '18 at 21:17
TehrynTehryn
62
62
1
Seems to me the language generated by [ and ] is larger than the language of proper pairings, It includes for example four [ 's and no ]'s. By Dyck language is there a restriction on what strings are in that language?
– coffeemath
Nov 29 '18 at 21:44
2
Dyck language (in our example) is a language, that is generated by grammar with rules S -> SS, S -> epsilon, S-> [S]
– Tehryn
Nov 29 '18 at 22:41
2
Your lector's advice seems pretty good to me. What is giving you trouble following it?
– rici
Nov 30 '18 at 2:56
1
Your definition of the Dyck language is not correct; you only provide the alphabet it uses.
– Peter Leupold
Nov 30 '18 at 8:28
1
@rici I have problem with induction. I was able somehow prove a). Also I was able to prove induction where number of '[' is equal to i where i is from interval <0, |w|>, but was not able to prove it for i+1
– Tehryn
Nov 30 '18 at 9:46
|
show 2 more comments
1
Seems to me the language generated by [ and ] is larger than the language of proper pairings, It includes for example four [ 's and no ]'s. By Dyck language is there a restriction on what strings are in that language?
– coffeemath
Nov 29 '18 at 21:44
2
Dyck language (in our example) is a language, that is generated by grammar with rules S -> SS, S -> epsilon, S-> [S]
– Tehryn
Nov 29 '18 at 22:41
2
Your lector's advice seems pretty good to me. What is giving you trouble following it?
– rici
Nov 30 '18 at 2:56
1
Your definition of the Dyck language is not correct; you only provide the alphabet it uses.
– Peter Leupold
Nov 30 '18 at 8:28
1
@rici I have problem with induction. I was able somehow prove a). Also I was able to prove induction where number of '[' is equal to i where i is from interval <0, |w|>, but was not able to prove it for i+1
– Tehryn
Nov 30 '18 at 9:46
1
1
Seems to me the language generated by [ and ] is larger than the language of proper pairings, It includes for example four [ 's and no ]'s. By Dyck language is there a restriction on what strings are in that language?
– coffeemath
Nov 29 '18 at 21:44
Seems to me the language generated by [ and ] is larger than the language of proper pairings, It includes for example four [ 's and no ]'s. By Dyck language is there a restriction on what strings are in that language?
– coffeemath
Nov 29 '18 at 21:44
2
2
Dyck language (in our example) is a language, that is generated by grammar with rules S -> SS, S -> epsilon, S-> [S]
– Tehryn
Nov 29 '18 at 22:41
Dyck language (in our example) is a language, that is generated by grammar with rules S -> SS, S -> epsilon, S-> [S]
– Tehryn
Nov 29 '18 at 22:41
2
2
Your lector's advice seems pretty good to me. What is giving you trouble following it?
– rici
Nov 30 '18 at 2:56
Your lector's advice seems pretty good to me. What is giving you trouble following it?
– rici
Nov 30 '18 at 2:56
1
1
Your definition of the Dyck language is not correct; you only provide the alphabet it uses.
– Peter Leupold
Nov 30 '18 at 8:28
Your definition of the Dyck language is not correct; you only provide the alphabet it uses.
– Peter Leupold
Nov 30 '18 at 8:28
1
1
@rici I have problem with induction. I was able somehow prove a). Also I was able to prove induction where number of '[' is equal to i where i is from interval <0, |w|>, but was not able to prove it for i+1
– Tehryn
Nov 30 '18 at 9:46
@rici I have problem with induction. I was able somehow prove a). Also I was able to prove induction where number of '[' is equal to i where i is from interval <0, |w|>, but was not able to prove it for i+1
– Tehryn
Nov 30 '18 at 9:46
|
show 2 more comments
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1
Seems to me the language generated by [ and ] is larger than the language of proper pairings, It includes for example four [ 's and no ]'s. By Dyck language is there a restriction on what strings are in that language?
– coffeemath
Nov 29 '18 at 21:44
2
Dyck language (in our example) is a language, that is generated by grammar with rules S -> SS, S -> epsilon, S-> [S]
– Tehryn
Nov 29 '18 at 22:41
2
Your lector's advice seems pretty good to me. What is giving you trouble following it?
– rici
Nov 30 '18 at 2:56
1
Your definition of the Dyck language is not correct; you only provide the alphabet it uses.
– Peter Leupold
Nov 30 '18 at 8:28
1
@rici I have problem with induction. I was able somehow prove a). Also I was able to prove induction where number of '[' is equal to i where i is from interval <0, |w|>, but was not able to prove it for i+1
– Tehryn
Nov 30 '18 at 9:46