Equivalent forms of the P vs. NP problemCollection of equivalent forms of Riemann HypothesisWhat techniques exist to show that a problem is not NP-complete?Complexity of a variant of the Mandelbrot set decision problem?Proofs that require fundamentally new ways of thinkingWhat could be some potentially useful mathematical databases?Knapsack Problem SpecificsIs this minimization problem NP-Complete ?NP-hardness of a graph partition problem?Surd Partition ProblemCost associated set problem NP-hardNP - hardness of school scheduling problem with a restriction
Equivalent forms of the P vs. NP problem
Collection of equivalent forms of Riemann HypothesisWhat techniques exist to show that a problem is not NP-complete?Complexity of a variant of the Mandelbrot set decision problem?Proofs that require fundamentally new ways of thinkingWhat could be some potentially useful mathematical databases?Knapsack Problem SpecificsIs this minimization problem NP-Complete ?NP-hardness of a graph partition problem?Surd Partition ProblemCost associated set problem NP-hardNP - hardness of school scheduling problem with a restriction
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Many things in math can be formulated quite differently; see the list of statements equivalent to RH here, for example, with RH formulated as a bound on lcm of consecutive integers, as an intergral equality, etc.
I wonder about equivalent formulations of the N vs. NP problem. Formulations that are very much different from the questions such "Is TSP in NP?", formulation that may seem unrelated to complexity theory.
computational-complexity big-list np
asked 7 hours ago
MichaelMichael
1,1502128
$endgroup$
add a comment |
$begingroup$
Many things in math can be formulated quite differently; see the list of statements equivalent to RH here, for example, with RH formulated as a bound on lcm of consecutive integers, as an intergral equality, etc.
I wonder about equivalent formulations of the N vs. NP problem. Formulations that are very much different from the questions such "Is TSP in NP?", formulation that may seem unrelated to complexity theory.
computational-complexity big-list np
asked 7 hours ago
MichaelMichael
1,1502128
$endgroup$
add a comment |
$begingroup$
Many things in math can be formulated quite differently; see the list of statements equivalent to RH here, for example, with RH formulated as a bound on lcm of consecutive integers, as an intergral equality, etc.
I wonder about equivalent formulations of the N vs. NP problem. Formulations that are very much different from the questions such "Is TSP in NP?", formulation that may seem unrelated to complexity theory.
computational-complexity big-list np
asked 7 hours ago
MichaelMichael
1,1502128
$endgroup$
Many things in math can be formulated quite differently; see the list of statements equivalent to RH here, for example, with RH formulated as a bound on lcm of consecutive integers, as an intergral equality, etc.
I wonder about equivalent formulations of the N vs. NP problem. Formulations that are very much different from the questions such "Is TSP in NP?", formulation that may seem unrelated to complexity theory.
computational-complexity big-list np
computational-complexity big-list np
asked 7 hours ago
MichaelMichael
1,1502128
asked 7 hours ago
MichaelMichael
1,1502128
asked 7 hours ago
MichaelMichael
1,1502128
asked 7 hours ago
MichaelMichael
1,1502128
asked 7 hours ago
MichaelMichael
1,1502128
1,1502128
add a comment |
add a comment |
5 Answers
5
active
oldest
votes
$begingroup$
This is not likely what you seek, but $mathopP not= mathopNP$ is a $Pi_2$ sentence,
a sentence of the form $forall n ; exists k ; R(n, k)$,
where $R(n, k)$ is a computable predicate.
Such a sentence represents "a certain relationship among positive integers, which either holds or doesn’t hold."
For example,
$cal T$ is the set of all Turing machines.
$calP$ is the set of all polynomials.
Aaronson, Scott. "$Pmathop =limits^? NP$." In Open problems in mathematics, pp. 1-122. Springer, Cham, 2016. Also,
Electronic Colloquium on Computational Complexity, Report No. 4 (2017).
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
86.9k16241718
$endgroup$
$begingroup$
How is this a different formulation? The sentence literally says "3SAT is not in P".
$endgroup$
– Emil Jeřábek
4 hours ago
add a comment |
$begingroup$
There is the descriptive complexity formulation:
P = NP is equivalent to the statement that every property expressible by a second order existential statement is also expressible in first order logic with a least fixed point operator.
See, e.g., Immerman's survey here: https://people.cs.umass.edu/~immerman/pub/capture.pdf
answered 1 hour ago
Ryan O'DonnellRyan O'Donnell
4,79912138
$endgroup$
add a comment |
$begingroup$
I think "Geometric Complexity Theory" is roughly speaking an attempt to do what you're talking about: formulate P vs. NP in very different language. See https://en.wikipedia.org/wiki/Geometric_complexity_theory. I think that technically it may be dealing with "VP vs. VNP" rather than "P vs. NP" but in spirit it fits your request.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,52212561
$endgroup$
add a comment |
$begingroup$
https://www.cs.toronto.edu/~sacook/homepage/ptime.pdf
The above paper (1991) gives a syntactic method for enumerating all the PTIME functions. P != NP is the proposition that none of the functions in that enumeration recognize 3SAT. That suggests various half-baked proof ideas that I'm sure lots of people have thought of, so they presumably don't work, though who knows.
answered 38 mins ago
nonenone
1
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Check out our Code of Conduct.
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add a comment |
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"This version of Levin's universal search algorithm solves SUBSET-SUM in polynomial time" is equivalent to P=NP.
answered 35 mins ago
nonenone
1
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This is not likely what you seek, but $mathopP not= mathopNP$ is a $Pi_2$ sentence,
a sentence of the form $forall n ; exists k ; R(n, k)$,
where $R(n, k)$ is a computable predicate.
Such a sentence represents "a certain relationship among positive integers, which either holds or doesn’t hold."
For example,
$cal T$ is the set of all Turing machines.
$calP$ is the set of all polynomials.
Aaronson, Scott. "$Pmathop =limits^? NP$." In Open problems in mathematics, pp. 1-122. Springer, Cham, 2016. Also,
Electronic Colloquium on Computational Complexity, Report No. 4 (2017).
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
86.9k16241718
$endgroup$
$begingroup$
How is this a different formulation? The sentence literally says "3SAT is not in P".
$endgroup$
– Emil Jeřábek
4 hours ago
add a comment |
$begingroup$
This is not likely what you seek, but $mathopP not= mathopNP$ is a $Pi_2$ sentence,
a sentence of the form $forall n ; exists k ; R(n, k)$,
where $R(n, k)$ is a computable predicate.
Such a sentence represents "a certain relationship among positive integers, which either holds or doesn’t hold."
For example,
$cal T$ is the set of all Turing machines.
$calP$ is the set of all polynomials.
Aaronson, Scott. "$Pmathop =limits^? NP$." In Open problems in mathematics, pp. 1-122. Springer, Cham, 2016. Also,
Electronic Colloquium on Computational Complexity, Report No. 4 (2017).
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
86.9k16241718
$endgroup$
$begingroup$
How is this a different formulation? The sentence literally says "3SAT is not in P".
$endgroup$
– Emil Jeřábek
4 hours ago
add a comment |
$begingroup$
This is not likely what you seek, but $mathopP not= mathopNP$ is a $Pi_2$ sentence,
a sentence of the form $forall n ; exists k ; R(n, k)$,
where $R(n, k)$ is a computable predicate.
Such a sentence represents "a certain relationship among positive integers, which either holds or doesn’t hold."
For example,
$cal T$ is the set of all Turing machines.
$calP$ is the set of all polynomials.
Aaronson, Scott. "$Pmathop =limits^? NP$." In Open problems in mathematics, pp. 1-122. Springer, Cham, 2016. Also,
Electronic Colloquium on Computational Complexity, Report No. 4 (2017).
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
86.9k16241718
$endgroup$
This is not likely what you seek, but $mathopP not= mathopNP$ is a $Pi_2$ sentence,
a sentence of the form $forall n ; exists k ; R(n, k)$,
where $R(n, k)$ is a computable predicate.
Such a sentence represents "a certain relationship among positive integers, which either holds or doesn’t hold."
For example,
$cal T$ is the set of all Turing machines.
$calP$ is the set of all polynomials.
Aaronson, Scott. "$Pmathop =limits^? NP$." In Open problems in mathematics, pp. 1-122. Springer, Cham, 2016. Also,
Electronic Colloquium on Computational Complexity, Report No. 4 (2017).
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
86.9k16241718
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
86.9k16241718
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
86.9k16241718
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
86.9k16241718
86.9k16241718
$begingroup$
How is this a different formulation? The sentence literally says "3SAT is not in P".
$endgroup$
– Emil Jeřábek
4 hours ago
add a comment |
$begingroup$
How is this a different formulation? The sentence literally says "3SAT is not in P".
$endgroup$
– Emil Jeřábek
4 hours ago
$begingroup$
How is this a different formulation? The sentence literally says "3SAT is not in P".
$endgroup$
– Emil Jeřábek
4 hours ago
$begingroup$
How is this a different formulation? The sentence literally says "3SAT is not in P".
$endgroup$
– Emil Jeřábek
4 hours ago
add a comment |
$begingroup$
There is the descriptive complexity formulation:
P = NP is equivalent to the statement that every property expressible by a second order existential statement is also expressible in first order logic with a least fixed point operator.
See, e.g., Immerman's survey here: https://people.cs.umass.edu/~immerman/pub/capture.pdf
answered 1 hour ago
Ryan O'DonnellRyan O'Donnell
4,79912138
$endgroup$
add a comment |
$begingroup$
There is the descriptive complexity formulation:
P = NP is equivalent to the statement that every property expressible by a second order existential statement is also expressible in first order logic with a least fixed point operator.
See, e.g., Immerman's survey here: https://people.cs.umass.edu/~immerman/pub/capture.pdf
answered 1 hour ago
Ryan O'DonnellRyan O'Donnell
4,79912138
$endgroup$
add a comment |
$begingroup$
There is the descriptive complexity formulation:
P = NP is equivalent to the statement that every property expressible by a second order existential statement is also expressible in first order logic with a least fixed point operator.
See, e.g., Immerman's survey here: https://people.cs.umass.edu/~immerman/pub/capture.pdf
answered 1 hour ago
Ryan O'DonnellRyan O'Donnell
4,79912138
$endgroup$
There is the descriptive complexity formulation:
P = NP is equivalent to the statement that every property expressible by a second order existential statement is also expressible in first order logic with a least fixed point operator.
See, e.g., Immerman's survey here: https://people.cs.umass.edu/~immerman/pub/capture.pdf
answered 1 hour ago
Ryan O'DonnellRyan O'Donnell
4,79912138
answered 1 hour ago
Ryan O'DonnellRyan O'Donnell
4,79912138
answered 1 hour ago
Ryan O'DonnellRyan O'Donnell
4,79912138
answered 1 hour ago
Ryan O'DonnellRyan O'Donnell
4,79912138
4,79912138
add a comment |
add a comment |
$begingroup$
I think "Geometric Complexity Theory" is roughly speaking an attempt to do what you're talking about: formulate P vs. NP in very different language. See https://en.wikipedia.org/wiki/Geometric_complexity_theory. I think that technically it may be dealing with "VP vs. VNP" rather than "P vs. NP" but in spirit it fits your request.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,52212561
$endgroup$
add a comment |
$begingroup$
I think "Geometric Complexity Theory" is roughly speaking an attempt to do what you're talking about: formulate P vs. NP in very different language. See https://en.wikipedia.org/wiki/Geometric_complexity_theory. I think that technically it may be dealing with "VP vs. VNP" rather than "P vs. NP" but in spirit it fits your request.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,52212561
$endgroup$
add a comment |
$begingroup$
I think "Geometric Complexity Theory" is roughly speaking an attempt to do what you're talking about: formulate P vs. NP in very different language. See https://en.wikipedia.org/wiki/Geometric_complexity_theory. I think that technically it may be dealing with "VP vs. VNP" rather than "P vs. NP" but in spirit it fits your request.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,52212561
$endgroup$
I think "Geometric Complexity Theory" is roughly speaking an attempt to do what you're talking about: formulate P vs. NP in very different language. See https://en.wikipedia.org/wiki/Geometric_complexity_theory. I think that technically it may be dealing with "VP vs. VNP" rather than "P vs. NP" but in spirit it fits your request.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,52212561
answered 1 hour ago
Sam HopkinsSam Hopkins
5,52212561
answered 1 hour ago
Sam HopkinsSam Hopkins
5,52212561
answered 1 hour ago
Sam HopkinsSam Hopkins
5,52212561
5,52212561
add a comment |
add a comment |
$begingroup$
https://www.cs.toronto.edu/~sacook/homepage/ptime.pdf
The above paper (1991) gives a syntactic method for enumerating all the PTIME functions. P != NP is the proposition that none of the functions in that enumeration recognize 3SAT. That suggests various half-baked proof ideas that I'm sure lots of people have thought of, so they presumably don't work, though who knows.
answered 38 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
https://www.cs.toronto.edu/~sacook/homepage/ptime.pdf
The above paper (1991) gives a syntactic method for enumerating all the PTIME functions. P != NP is the proposition that none of the functions in that enumeration recognize 3SAT. That suggests various half-baked proof ideas that I'm sure lots of people have thought of, so they presumably don't work, though who knows.
answered 38 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
https://www.cs.toronto.edu/~sacook/homepage/ptime.pdf
The above paper (1991) gives a syntactic method for enumerating all the PTIME functions. P != NP is the proposition that none of the functions in that enumeration recognize 3SAT. That suggests various half-baked proof ideas that I'm sure lots of people have thought of, so they presumably don't work, though who knows.
answered 38 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
https://www.cs.toronto.edu/~sacook/homepage/ptime.pdf
The above paper (1991) gives a syntactic method for enumerating all the PTIME functions. P != NP is the proposition that none of the functions in that enumeration recognize 3SAT. That suggests various half-baked proof ideas that I'm sure lots of people have thought of, so they presumably don't work, though who knows.
answered 38 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 38 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 38 mins ago
nonenone
1
answered 38 mins ago
nonenone
1
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
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Check out our Code of Conduct.
add a comment |
add a comment |
$begingroup$
"This version of Levin's universal search algorithm solves SUBSET-SUM in polynomial time" is equivalent to P=NP.
answered 35 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
"This version of Levin's universal search algorithm solves SUBSET-SUM in polynomial time" is equivalent to P=NP.
answered 35 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
"This version of Levin's universal search algorithm solves SUBSET-SUM in polynomial time" is equivalent to P=NP.
answered 35 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
"This version of Levin's universal search algorithm solves SUBSET-SUM in polynomial time" is equivalent to P=NP.
answered 35 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 35 mins ago
nonenone
1
New contributor
none is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 35 mins ago
nonenone
1
answered 35 mins ago
nonenone
1
1
New contributor
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New contributor
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