Are there any computational problems in groups that are harder than P?
$begingroup$
There are several well known classes of groups for which the word problem, conjugacy etc. are solvable in polynomial time (hyperbolic, automatic).
Then there are several classes of groups like asynchronously automatic groups for which it is known that there is an exponential time algorithm to solve the word problem (and whether this can be improved to polynomial is open and conjectured as far as I'm aware).
Going several steps further, there is an algorithm to solve the word problem in one-relator groups in time not bounded by any finite tower of exponentials (and again it is open and conjectured whether this can be improved to P).
On the other, there are algorithms to solve word problems in pathological groups like the Baumslag-Gersten group:
$$G_{(1,2)} = langle a, b | a^{a^b}= a^2 rangle$$
in polynomial time. So even though general algorithms can be very bad, it is unknown whether there are group-specific algorithms for every group in a given class that solve the word problem quickly.
So my question is, are there any groups in which it is known that the word problem (or any other computational problem) is at least exponential, but still solvable? Maybe exp-complete? What are the approaches to proving such a thing?
gr.group-theory computational-complexity geometric-group-theory computational-group-theory word-problem
New contributor
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add a comment |
$begingroup$
There are several well known classes of groups for which the word problem, conjugacy etc. are solvable in polynomial time (hyperbolic, automatic).
Then there are several classes of groups like asynchronously automatic groups for which it is known that there is an exponential time algorithm to solve the word problem (and whether this can be improved to polynomial is open and conjectured as far as I'm aware).
Going several steps further, there is an algorithm to solve the word problem in one-relator groups in time not bounded by any finite tower of exponentials (and again it is open and conjectured whether this can be improved to P).
On the other, there are algorithms to solve word problems in pathological groups like the Baumslag-Gersten group:
$$G_{(1,2)} = langle a, b | a^{a^b}= a^2 rangle$$
in polynomial time. So even though general algorithms can be very bad, it is unknown whether there are group-specific algorithms for every group in a given class that solve the word problem quickly.
So my question is, are there any groups in which it is known that the word problem (or any other computational problem) is at least exponential, but still solvable? Maybe exp-complete? What are the approaches to proving such a thing?
gr.group-theory computational-complexity geometric-group-theory computational-group-theory word-problem
New contributor
$endgroup$
add a comment |
$begingroup$
There are several well known classes of groups for which the word problem, conjugacy etc. are solvable in polynomial time (hyperbolic, automatic).
Then there are several classes of groups like asynchronously automatic groups for which it is known that there is an exponential time algorithm to solve the word problem (and whether this can be improved to polynomial is open and conjectured as far as I'm aware).
Going several steps further, there is an algorithm to solve the word problem in one-relator groups in time not bounded by any finite tower of exponentials (and again it is open and conjectured whether this can be improved to P).
On the other, there are algorithms to solve word problems in pathological groups like the Baumslag-Gersten group:
$$G_{(1,2)} = langle a, b | a^{a^b}= a^2 rangle$$
in polynomial time. So even though general algorithms can be very bad, it is unknown whether there are group-specific algorithms for every group in a given class that solve the word problem quickly.
So my question is, are there any groups in which it is known that the word problem (or any other computational problem) is at least exponential, but still solvable? Maybe exp-complete? What are the approaches to proving such a thing?
gr.group-theory computational-complexity geometric-group-theory computational-group-theory word-problem
New contributor
$endgroup$
There are several well known classes of groups for which the word problem, conjugacy etc. are solvable in polynomial time (hyperbolic, automatic).
Then there are several classes of groups like asynchronously automatic groups for which it is known that there is an exponential time algorithm to solve the word problem (and whether this can be improved to polynomial is open and conjectured as far as I'm aware).
Going several steps further, there is an algorithm to solve the word problem in one-relator groups in time not bounded by any finite tower of exponentials (and again it is open and conjectured whether this can be improved to P).
On the other, there are algorithms to solve word problems in pathological groups like the Baumslag-Gersten group:
$$G_{(1,2)} = langle a, b | a^{a^b}= a^2 rangle$$
in polynomial time. So even though general algorithms can be very bad, it is unknown whether there are group-specific algorithms for every group in a given class that solve the word problem quickly.
So my question is, are there any groups in which it is known that the word problem (or any other computational problem) is at least exponential, but still solvable? Maybe exp-complete? What are the approaches to proving such a thing?
gr.group-theory computational-complexity geometric-group-theory computational-group-theory word-problem
gr.group-theory computational-complexity geometric-group-theory computational-group-theory word-problem
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New contributor
edited 9 hours ago
MSL
New contributor
asked 9 hours ago
MSLMSL
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364
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4 Answers
4
active
oldest
votes
$begingroup$
More or less what Andreas says (in the comments to his answer) is true but one must be careful in the encoding. In
Isoperimetric and Isodiametric Functions of Groups,
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Annals of Mathematics
Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 345-466
and
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Isoperimetric functions of groups and computational complexity of the word problem.
Ann. of Math. (2) 156 (2002), no. 2, 467–518.
groups with NP complete word problem are constructed and other similar results. See in particular, Corollary 1.1 of the first paper listed above.
$endgroup$
1
$begingroup$
This is not a complete answer. It currently appears as the top answer in my view, but it can't be understood by itself.
$endgroup$
– CJ Dennis
4 hours ago
2
$begingroup$
@CJDennis I gave a reference to a pair of papers which are nearly 200 pages that give NP complete word problems and other related results and pointed out a particular Corollary with more detailed information. I obviously cannot give all the details of encoding S-machines etc.
$endgroup$
– Benjamin Steinberg
4 hours ago
$begingroup$
I am rather amused to see that someone has flagged this answer for possible deletion as being "low-quality".
$endgroup$
– Yemon Choi
2 hours ago
add a comment |
$begingroup$
There are finitely presented groups whose word problem is undecidable. See, for example, https://en.wikipedia.org/wiki/Word_problem_for_groups .
$endgroup$
$begingroup$
True, what I really meant was among the groups whose word problem is solvable. Thanks, I'll edit my question!
$endgroup$
– MSL
9 hours ago
6
$begingroup$
@MSL I'd expect that the same method can handle the decidable case. Instead of coding into the presentation a Turing machine whose halting problem is undecidable, code one whose halting problem is decidable but takes a very long time to decide.
$endgroup$
– Andreas Blass
9 hours ago
add a comment |
$begingroup$
An earlier reference for groups with this property is
J. Avenhaus and K. Madlener. Subrekursive Komplexität der Gruppen. I. Gruppen mit vorgeschriebenen Komplexität. Acta Infomat., 9 (1): 87-104, 1977/78.
There is a hierarchy of the recursive functions known as the (difficult to pronounce) Grzegorczyk Hierarchy $E_0 subset E_1 subset E_2 subset cdots$, where (roughly) $E_1$ contains the linearly bounded functions, $E_2$ polynomially bounded functions, and $E_3$ those functions that are bounded by iterated exponentials.
The above paper describes constructions of finitely presented groups $G_n$ for $n ge 3$, in which solving the word problem has time complexity bounded by a function $E_n$ but not by any function in $E_{n-1}$,
$endgroup$
add a comment |
$begingroup$
Classical problem which is believed not to be in P is number factoring, which can be cast as computing a decomposition of a cyclic group into simple ones.
Several problems in permutation groups are known to be as hard as graph isomorphism, thus not believed to be in P, too.
There are also NP-hard problems known for permutation groups, see e.g. https://www.sciencedirect.com/science/article/pii/S0012365X09001289
$endgroup$
$begingroup$
Checking if a cyclic group is simple would seem to be primality testing which is in P.
$endgroup$
– Benjamin Steinberg
2 hours ago
$begingroup$
You are right; I have corrected my answer to reflect this.
$endgroup$
– Dima Pasechnik
2 hours ago
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
More or less what Andreas says (in the comments to his answer) is true but one must be careful in the encoding. In
Isoperimetric and Isodiametric Functions of Groups,
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Annals of Mathematics
Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 345-466
and
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Isoperimetric functions of groups and computational complexity of the word problem.
Ann. of Math. (2) 156 (2002), no. 2, 467–518.
groups with NP complete word problem are constructed and other similar results. See in particular, Corollary 1.1 of the first paper listed above.
$endgroup$
1
$begingroup$
This is not a complete answer. It currently appears as the top answer in my view, but it can't be understood by itself.
$endgroup$
– CJ Dennis
4 hours ago
2
$begingroup$
@CJDennis I gave a reference to a pair of papers which are nearly 200 pages that give NP complete word problems and other related results and pointed out a particular Corollary with more detailed information. I obviously cannot give all the details of encoding S-machines etc.
$endgroup$
– Benjamin Steinberg
4 hours ago
$begingroup$
I am rather amused to see that someone has flagged this answer for possible deletion as being "low-quality".
$endgroup$
– Yemon Choi
2 hours ago
add a comment |
$begingroup$
More or less what Andreas says (in the comments to his answer) is true but one must be careful in the encoding. In
Isoperimetric and Isodiametric Functions of Groups,
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Annals of Mathematics
Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 345-466
and
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Isoperimetric functions of groups and computational complexity of the word problem.
Ann. of Math. (2) 156 (2002), no. 2, 467–518.
groups with NP complete word problem are constructed and other similar results. See in particular, Corollary 1.1 of the first paper listed above.
$endgroup$
1
$begingroup$
This is not a complete answer. It currently appears as the top answer in my view, but it can't be understood by itself.
$endgroup$
– CJ Dennis
4 hours ago
2
$begingroup$
@CJDennis I gave a reference to a pair of papers which are nearly 200 pages that give NP complete word problems and other related results and pointed out a particular Corollary with more detailed information. I obviously cannot give all the details of encoding S-machines etc.
$endgroup$
– Benjamin Steinberg
4 hours ago
$begingroup$
I am rather amused to see that someone has flagged this answer for possible deletion as being "low-quality".
$endgroup$
– Yemon Choi
2 hours ago
add a comment |
$begingroup$
More or less what Andreas says (in the comments to his answer) is true but one must be careful in the encoding. In
Isoperimetric and Isodiametric Functions of Groups,
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Annals of Mathematics
Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 345-466
and
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Isoperimetric functions of groups and computational complexity of the word problem.
Ann. of Math. (2) 156 (2002), no. 2, 467–518.
groups with NP complete word problem are constructed and other similar results. See in particular, Corollary 1.1 of the first paper listed above.
$endgroup$
More or less what Andreas says (in the comments to his answer) is true but one must be careful in the encoding. In
Isoperimetric and Isodiametric Functions of Groups,
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Annals of Mathematics
Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 345-466
and
Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Isoperimetric functions of groups and computational complexity of the word problem.
Ann. of Math. (2) 156 (2002), no. 2, 467–518.
groups with NP complete word problem are constructed and other similar results. See in particular, Corollary 1.1 of the first paper listed above.
edited 3 hours ago
answered 9 hours ago
Benjamin SteinbergBenjamin Steinberg
23.1k265124
23.1k265124
1
$begingroup$
This is not a complete answer. It currently appears as the top answer in my view, but it can't be understood by itself.
$endgroup$
– CJ Dennis
4 hours ago
2
$begingroup$
@CJDennis I gave a reference to a pair of papers which are nearly 200 pages that give NP complete word problems and other related results and pointed out a particular Corollary with more detailed information. I obviously cannot give all the details of encoding S-machines etc.
$endgroup$
– Benjamin Steinberg
4 hours ago
$begingroup$
I am rather amused to see that someone has flagged this answer for possible deletion as being "low-quality".
$endgroup$
– Yemon Choi
2 hours ago
add a comment |
1
$begingroup$
This is not a complete answer. It currently appears as the top answer in my view, but it can't be understood by itself.
$endgroup$
– CJ Dennis
4 hours ago
2
$begingroup$
@CJDennis I gave a reference to a pair of papers which are nearly 200 pages that give NP complete word problems and other related results and pointed out a particular Corollary with more detailed information. I obviously cannot give all the details of encoding S-machines etc.
$endgroup$
– Benjamin Steinberg
4 hours ago
$begingroup$
I am rather amused to see that someone has flagged this answer for possible deletion as being "low-quality".
$endgroup$
– Yemon Choi
2 hours ago
1
1
$begingroup$
This is not a complete answer. It currently appears as the top answer in my view, but it can't be understood by itself.
$endgroup$
– CJ Dennis
4 hours ago
$begingroup$
This is not a complete answer. It currently appears as the top answer in my view, but it can't be understood by itself.
$endgroup$
– CJ Dennis
4 hours ago
2
2
$begingroup$
@CJDennis I gave a reference to a pair of papers which are nearly 200 pages that give NP complete word problems and other related results and pointed out a particular Corollary with more detailed information. I obviously cannot give all the details of encoding S-machines etc.
$endgroup$
– Benjamin Steinberg
4 hours ago
$begingroup$
@CJDennis I gave a reference to a pair of papers which are nearly 200 pages that give NP complete word problems and other related results and pointed out a particular Corollary with more detailed information. I obviously cannot give all the details of encoding S-machines etc.
$endgroup$
– Benjamin Steinberg
4 hours ago
$begingroup$
I am rather amused to see that someone has flagged this answer for possible deletion as being "low-quality".
$endgroup$
– Yemon Choi
2 hours ago
$begingroup$
I am rather amused to see that someone has flagged this answer for possible deletion as being "low-quality".
$endgroup$
– Yemon Choi
2 hours ago
add a comment |
$begingroup$
There are finitely presented groups whose word problem is undecidable. See, for example, https://en.wikipedia.org/wiki/Word_problem_for_groups .
$endgroup$
$begingroup$
True, what I really meant was among the groups whose word problem is solvable. Thanks, I'll edit my question!
$endgroup$
– MSL
9 hours ago
6
$begingroup$
@MSL I'd expect that the same method can handle the decidable case. Instead of coding into the presentation a Turing machine whose halting problem is undecidable, code one whose halting problem is decidable but takes a very long time to decide.
$endgroup$
– Andreas Blass
9 hours ago
add a comment |
$begingroup$
There are finitely presented groups whose word problem is undecidable. See, for example, https://en.wikipedia.org/wiki/Word_problem_for_groups .
$endgroup$
$begingroup$
True, what I really meant was among the groups whose word problem is solvable. Thanks, I'll edit my question!
$endgroup$
– MSL
9 hours ago
6
$begingroup$
@MSL I'd expect that the same method can handle the decidable case. Instead of coding into the presentation a Turing machine whose halting problem is undecidable, code one whose halting problem is decidable but takes a very long time to decide.
$endgroup$
– Andreas Blass
9 hours ago
add a comment |
$begingroup$
There are finitely presented groups whose word problem is undecidable. See, for example, https://en.wikipedia.org/wiki/Word_problem_for_groups .
$endgroup$
There are finitely presented groups whose word problem is undecidable. See, for example, https://en.wikipedia.org/wiki/Word_problem_for_groups .
answered 9 hours ago
Andreas BlassAndreas Blass
57k7135218
57k7135218
$begingroup$
True, what I really meant was among the groups whose word problem is solvable. Thanks, I'll edit my question!
$endgroup$
– MSL
9 hours ago
6
$begingroup$
@MSL I'd expect that the same method can handle the decidable case. Instead of coding into the presentation a Turing machine whose halting problem is undecidable, code one whose halting problem is decidable but takes a very long time to decide.
$endgroup$
– Andreas Blass
9 hours ago
add a comment |
$begingroup$
True, what I really meant was among the groups whose word problem is solvable. Thanks, I'll edit my question!
$endgroup$
– MSL
9 hours ago
6
$begingroup$
@MSL I'd expect that the same method can handle the decidable case. Instead of coding into the presentation a Turing machine whose halting problem is undecidable, code one whose halting problem is decidable but takes a very long time to decide.
$endgroup$
– Andreas Blass
9 hours ago
$begingroup$
True, what I really meant was among the groups whose word problem is solvable. Thanks, I'll edit my question!
$endgroup$
– MSL
9 hours ago
$begingroup$
True, what I really meant was among the groups whose word problem is solvable. Thanks, I'll edit my question!
$endgroup$
– MSL
9 hours ago
6
6
$begingroup$
@MSL I'd expect that the same method can handle the decidable case. Instead of coding into the presentation a Turing machine whose halting problem is undecidable, code one whose halting problem is decidable but takes a very long time to decide.
$endgroup$
– Andreas Blass
9 hours ago
$begingroup$
@MSL I'd expect that the same method can handle the decidable case. Instead of coding into the presentation a Turing machine whose halting problem is undecidable, code one whose halting problem is decidable but takes a very long time to decide.
$endgroup$
– Andreas Blass
9 hours ago
add a comment |
$begingroup$
An earlier reference for groups with this property is
J. Avenhaus and K. Madlener. Subrekursive Komplexität der Gruppen. I. Gruppen mit vorgeschriebenen Komplexität. Acta Infomat., 9 (1): 87-104, 1977/78.
There is a hierarchy of the recursive functions known as the (difficult to pronounce) Grzegorczyk Hierarchy $E_0 subset E_1 subset E_2 subset cdots$, where (roughly) $E_1$ contains the linearly bounded functions, $E_2$ polynomially bounded functions, and $E_3$ those functions that are bounded by iterated exponentials.
The above paper describes constructions of finitely presented groups $G_n$ for $n ge 3$, in which solving the word problem has time complexity bounded by a function $E_n$ but not by any function in $E_{n-1}$,
$endgroup$
add a comment |
$begingroup$
An earlier reference for groups with this property is
J. Avenhaus and K. Madlener. Subrekursive Komplexität der Gruppen. I. Gruppen mit vorgeschriebenen Komplexität. Acta Infomat., 9 (1): 87-104, 1977/78.
There is a hierarchy of the recursive functions known as the (difficult to pronounce) Grzegorczyk Hierarchy $E_0 subset E_1 subset E_2 subset cdots$, where (roughly) $E_1$ contains the linearly bounded functions, $E_2$ polynomially bounded functions, and $E_3$ those functions that are bounded by iterated exponentials.
The above paper describes constructions of finitely presented groups $G_n$ for $n ge 3$, in which solving the word problem has time complexity bounded by a function $E_n$ but not by any function in $E_{n-1}$,
$endgroup$
add a comment |
$begingroup$
An earlier reference for groups with this property is
J. Avenhaus and K. Madlener. Subrekursive Komplexität der Gruppen. I. Gruppen mit vorgeschriebenen Komplexität. Acta Infomat., 9 (1): 87-104, 1977/78.
There is a hierarchy of the recursive functions known as the (difficult to pronounce) Grzegorczyk Hierarchy $E_0 subset E_1 subset E_2 subset cdots$, where (roughly) $E_1$ contains the linearly bounded functions, $E_2$ polynomially bounded functions, and $E_3$ those functions that are bounded by iterated exponentials.
The above paper describes constructions of finitely presented groups $G_n$ for $n ge 3$, in which solving the word problem has time complexity bounded by a function $E_n$ but not by any function in $E_{n-1}$,
$endgroup$
An earlier reference for groups with this property is
J. Avenhaus and K. Madlener. Subrekursive Komplexität der Gruppen. I. Gruppen mit vorgeschriebenen Komplexität. Acta Infomat., 9 (1): 87-104, 1977/78.
There is a hierarchy of the recursive functions known as the (difficult to pronounce) Grzegorczyk Hierarchy $E_0 subset E_1 subset E_2 subset cdots$, where (roughly) $E_1$ contains the linearly bounded functions, $E_2$ polynomially bounded functions, and $E_3$ those functions that are bounded by iterated exponentials.
The above paper describes constructions of finitely presented groups $G_n$ for $n ge 3$, in which solving the word problem has time complexity bounded by a function $E_n$ but not by any function in $E_{n-1}$,
answered 5 hours ago
Derek HoltDerek Holt
26.5k462108
26.5k462108
add a comment |
add a comment |
$begingroup$
Classical problem which is believed not to be in P is number factoring, which can be cast as computing a decomposition of a cyclic group into simple ones.
Several problems in permutation groups are known to be as hard as graph isomorphism, thus not believed to be in P, too.
There are also NP-hard problems known for permutation groups, see e.g. https://www.sciencedirect.com/science/article/pii/S0012365X09001289
$endgroup$
$begingroup$
Checking if a cyclic group is simple would seem to be primality testing which is in P.
$endgroup$
– Benjamin Steinberg
2 hours ago
$begingroup$
You are right; I have corrected my answer to reflect this.
$endgroup$
– Dima Pasechnik
2 hours ago
add a comment |
$begingroup$
Classical problem which is believed not to be in P is number factoring, which can be cast as computing a decomposition of a cyclic group into simple ones.
Several problems in permutation groups are known to be as hard as graph isomorphism, thus not believed to be in P, too.
There are also NP-hard problems known for permutation groups, see e.g. https://www.sciencedirect.com/science/article/pii/S0012365X09001289
$endgroup$
$begingroup$
Checking if a cyclic group is simple would seem to be primality testing which is in P.
$endgroup$
– Benjamin Steinberg
2 hours ago
$begingroup$
You are right; I have corrected my answer to reflect this.
$endgroup$
– Dima Pasechnik
2 hours ago
add a comment |
$begingroup$
Classical problem which is believed not to be in P is number factoring, which can be cast as computing a decomposition of a cyclic group into simple ones.
Several problems in permutation groups are known to be as hard as graph isomorphism, thus not believed to be in P, too.
There are also NP-hard problems known for permutation groups, see e.g. https://www.sciencedirect.com/science/article/pii/S0012365X09001289
$endgroup$
Classical problem which is believed not to be in P is number factoring, which can be cast as computing a decomposition of a cyclic group into simple ones.
Several problems in permutation groups are known to be as hard as graph isomorphism, thus not believed to be in P, too.
There are also NP-hard problems known for permutation groups, see e.g. https://www.sciencedirect.com/science/article/pii/S0012365X09001289
edited 2 hours ago
answered 2 hours ago
Dima PasechnikDima Pasechnik
8,99311851
8,99311851
$begingroup$
Checking if a cyclic group is simple would seem to be primality testing which is in P.
$endgroup$
– Benjamin Steinberg
2 hours ago
$begingroup$
You are right; I have corrected my answer to reflect this.
$endgroup$
– Dima Pasechnik
2 hours ago
add a comment |
$begingroup$
Checking if a cyclic group is simple would seem to be primality testing which is in P.
$endgroup$
– Benjamin Steinberg
2 hours ago
$begingroup$
You are right; I have corrected my answer to reflect this.
$endgroup$
– Dima Pasechnik
2 hours ago
$begingroup$
Checking if a cyclic group is simple would seem to be primality testing which is in P.
$endgroup$
– Benjamin Steinberg
2 hours ago
$begingroup$
Checking if a cyclic group is simple would seem to be primality testing which is in P.
$endgroup$
– Benjamin Steinberg
2 hours ago
$begingroup$
You are right; I have corrected my answer to reflect this.
$endgroup$
– Dima Pasechnik
2 hours ago
$begingroup$
You are right; I have corrected my answer to reflect this.
$endgroup$
– Dima Pasechnik
2 hours ago
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MSL is a new contributor. Be nice, and check out our Code of Conduct.
MSL is a new contributor. Be nice, and check out our Code of Conduct.
MSL is a new contributor. Be nice, and check out our Code of Conduct.
MSL is a new contributor. Be nice, and check out our Code of Conduct.
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