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From: sci.math on 6 Jul 2010 16:55

'''P'''. '''NP'''-complete

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<h1 id="firstHeading" class="firstHeading">P versus NP problem</h1>


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<table class="infobox" style="text-align: center;">
<tr>
<th style="background: #ccf; font-size: larger;"><a href="/wiki/
Millennium_Prize_Problems" title="Millennium Prize
Problems">Millennium Prize Problems</a></th>
</tr>
<tr>
<td>P versus NP problem</td>
</tr>
<tr>

<td><a href="/wiki/Hodge_conjecture" title="Hodge conjecture">Hodge
conjecture</a></td>
</tr>
<tr>
<td><a href="/wiki/Poincar%C3%A9_conjecture" title="PoincarÃ©
conjecture">PoincarÃ© conjecture</a> (<a href="/wiki/
Solution_of_the_Poincar%C3%A9_conjecture" title="Solution of the
PoincarÃ© conjecture">solution</a>)</td>
</tr>
<tr>
<td><a href="/wiki/Riemann_hypothesis" title="Riemann
hypothesis">Riemann hypothesis</a></td>
</tr>
<tr>
<td><a href="/wiki/Yang%E2%80%93Mills_existence_and_mass_gap"
title="YangâMills existence and mass gap">YangâMills existence and
mass gap</a></td>

</tr>
<tr>
<td><a href="/wiki/Navier%E2%80%93Stokes_existence_and_smoothness"
title="NavierâStokes existence and smoothness">NavierâStokes existence
and smoothness</a></td>
</tr>
<tr>
<td><a href="/wiki/Birch_and_Swinnerton-Dyer_conjecture" title="Birch
and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</
a></td>
</tr>
</table>
<div class="thumb tright">
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Diagram of complexity classes provided that P â NP. The
existence of problems outside both P and NP-complete in
this case was established by <a href="/wiki/NP-Intermediate" title="NP-
Intermediate">Ladner's theorem</a>.<a href="#cite_note-Ladner-0">[1]</a></div>

</div>
</div>
Update: As it turns out P. NP-complete problems are a
set of problems which any other NP-problem can be reduced to in
polynomial time, but which retain the ability to have their solution
verified in polynomial time. <a href="http://meami.org/gibraltar.htm"
class="external autonumber" rel="nofollow">[1]</a>
The relationship between the <a href="/wiki/Complexity_class"
title="Complexity class">complexity classes</a> <a href="/wiki/
P_(complexity)" title="P (complexity)">P</a> (Polynomial time) and
<a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a>
(Nondeterministic Polynomial time) is an unsolved problem in <a href="/
wiki/Theoretical_computer_science" title="Theoretical computer
science">theoretical computer science</a>, and is considered by many
<a href="/wiki/Theoretical_computer_science" title="Theoretical
computer science">theoretical computer scientists</a> to be the most
important problem in the field.<a href="#cite_note-1">[2]</a> The <a href="/wiki/Clay_Mathematics_Institute"
title="Clay Mathematics Institute">Clay Mathematics Institute</a>,
which is dedicated to increasing and disseminating mathematical
knowledge, has included it in its list of <a href="/wiki/
Millennium_Prize_Problems" title="Millennium Prize
Problems">Millennium Prize Problems</a>; anyone who provides a
satisfactory solution to the problem may be entitled to a million
dollar prize.<a href="#cite_note-
CMI_Millennium_Prize_Problems-2">[3]</a><a href="#cite_note-
Official_Problem_Description-3">[4]</a>

In essence, the question P = NP? asks: if 'yes'-
answers to a <a href="/wiki/Decision_problem" title="Decision
problem">'yes'-or-'no'-question</a> can be verified "quickly"
can the answers themselves also be computed "quickly"? The
theoretical notion of "quick" used here is that of an algorithm that
runs in <a href="/wiki/Polynomial_time" title="Polynomial time"
class="mw-redirect">polynomial time</a>, which usually but not always
corresponds to an algorithm that is fast in practice.

Consider the <a href="/wiki/Subset_sum_problem" title="Subset sum
problem">subset sum problem</a>, an example of a problem which is easy
to verify but whose answer is suspected to be theoretically difficult
to compute. Given a set of <a href="/wiki/Integer"
title="Integer">integers</a>, does some nonempty <a href="/wiki/
Subset" title="Subset">subset</a> of them sum to 0? For instance, does
a subset of the set {â2, â3, 15, 14,
7, â10} add up to 0? The answer "yes, because {â2, â3, â10, 15} add up to zero",
can be quickly verified with three additions. However, finding such a
subset in the first place could take more time. The information needed
to verify a positive answer is also called a certificate. Given
the right certificates, "yes" answers to our problem can be verified
in polynomial time, so this problem is in NP.

An answer to the P = NP question would
determine whether problems like the subset-sum problem that can be
verified in polynomial time can also be solved in polynomial time. If
it turned out that P does not equal NP, it would mean
that some NP problems are harder to compute than to verify:
they could not be solved in polynomial time, but the answer could be
verified in polynomial time.
The restriction to yes/no problems is unimportant; when more
complicated answers are allowed the extension to <a href="/wiki/
Function_problems" title="Function problems" class="mw-
redirect">function problems</a> becomes <a href="/wiki/
FP_(complexity)" title="FP (complexity)">FP</a> = <a href="/
wiki/FNP_(complexity)" title="FNP (complexity)">FNP</a>, which has
been proven to be equivalent to P = NP.<a href="#cite_note-4">[5]</a>

<table id="toc" class="toc">
<tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
</div>
<ul>
<li class="toclevel-1 tocsection-1"><a
href="#Context_of_the_problem">1 Context of the problem</a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Example">1.1 Example</a></
li>

</ul>
</li>
<li class="toclevel-1 tocsection-3"><a href="#NP-complete">2 NP-complete</
a></li>
<li class="toclevel-1 tocsection-4"><a
href="#Still_harder_problems">3 Still harder problems</a></li>
<li class="toclevel-1 tocsection-5"><a href="#Does_P_mean_.22easy.
22.3F">4 Does P
mean "easy"?</a></li>
<li class="toclevel-1 tocsection-6"><a
href="#Reasons_to_believe_P_.E2.89.A0_NP">5 Reasons to believe P â NP</a></li>

<li class="toclevel-1 tocsection-7"><a
href="#Consequences_of_proof">6 Consequences of proof</a></li>
<li class="toclevel-1 tocsection-8"><a
href="#Results_about_difficulty_of_proof">7 Results about difficulty of proof</
a></li>
<li class="toclevel-1 tocsection-9"><a
href="#Logical_characterizations">8
Logical characterizations</a></li>
<li class="toclevel-1 tocsection-10"><a href="#Polynomial-
time_algorithms">9 Polynomial-time algorithms</a></li>
<li class="toclevel-1 tocsection-11"><a href="#Gibraltar_Code">10 Gibraltar Code</a></li>

<li class="toclevel-1 tocsection-12"><a
href="#Formal_definitions_for_P_and_NP">11 Formal definitions for P and NP</
a></li>
<li class="toclevel-1 tocsection-13"><a
href="#Formal_definition_for_NP-completeness">12 Formal definition
for NP-completeness</a></li>
<li class="toclevel-1 tocsection-14"><a href="#See_also">13 See also</a></
li>
<li class="toclevel-1 tocsection-15"><a href="#Notes">14 Notes</a></
li>
<li class="toclevel-1 tocsection-16"><a href="#Further_reading">15 Further reading</a></li>

<li class="toclevel-1 tocsection-17"><a href="#External_links">16 External links</a></li>
</ul>
</td>
</tr>
</table>
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<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=1" title="Edit
section: Context of the problem">edit</a>] Context of the problem</
h2>
The relation between the <a href="/wiki/Complexity_class"
title="Complexity class">complexity classes</a> P and NP
is studied in <a href="/wiki/Computational_complexity_theory"
title="Computational complexity theory">computational complexity
theory</a>, the part of the <a href="/wiki/Theory_of_computation"
title="Theory of computation">theory of computation</a> dealing with
the resources required during computation to solve a given problem.
The most common resources are time (how many steps it takes to solve a
problem) and space (how much memory it takes to solve a problem).

In such analysis, a model of the computer for which time must be
analyzed is required. Typically such models assume that the computer
is <a href="/wiki/Deterministic_computation" title="Deterministic
computation" class="mw-redirect">deterministic</a> (given the
computer's present state and any inputs, there is only one possible
action that the computer might take) and sequential (it
performs actions one after the other).
In this theory, the class <a href="/wiki/P_(complexity)"
title="P (complexity)">P</a> consists of all those <a href="/
wiki/Decision_problem" title="Decision problem">decision problems</a> (defined <a href="#Formal_definitions_for_P_and_NP">below</a>) that
can be solved on a deterministic sequential machine in an amount of
time that is <a href="/wiki/Polynomial" title="Polynomial">polynomial</
a> in the size of the input; the class <a href="/wiki/
NP_(complexity)" title="NP (complexity)">NP</a> consists of all
those decision problems whose positive solutions can be verified in <a
href="/wiki/Polynomial_time" title="Polynomial time" class="mw-
redirect">polynomial time</a> given the right information, or
equivalently, whose solution can be found in polynomial time on a <a
href="/wiki/Non-deterministic_Turing_machine" title="Non-deterministic
Turing machine">non-deterministic</a> machine.<a href="#cite_note-5">[6]</a> Arguably the biggest open question in <a href="/wiki/
Theoretical_computer_science" title="Theoretical computer
science">theoretical computer science</a> concerns the relationship
between those two classes:

<dl>
<dd>Is P equal to NP?</dd>
</dl>
In a 2002 poll of 100 researchers, 61 believed the answer to be no,
9 believed the answer is yes, and 22 were unsure; 8 believed the
question may be independent of the currently accepted axioms and so
impossible to prove or disprove.<a href="#cite_note-poll-6">[7]</a>
<h3>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=2" title="Edit
section: Example">edit</a>] Example</h3>

Let <img class="tex" alt="\mathit{COMPOSITE} = \{x\in N:x=pq \;
\text{for integers}\; p, q > 1 \}" src="http://upload.wikimedia.org/
math/4/4/b/44ba246a84ec1cc45042e9e22ba5ea80.png" /> and <img
class="tex" alt="R = \{(x,y)\in N\times N: 1<y \leq \sqrt x
\; ; \;y\; \text{divides}\; x\}." src="http://
upload.wikimedia.org/math/0/b/4/0b40ea16ba74baa995bec5914cdd41f4.png" /
>
Clearly, the question of whether a given x is a composite is equivalent to the
question of whether x is a member
of COMPOSITE. It can be shown that <img
class="tex" alt="\mathit{COMPOSITE}\in\mathbf{NP}" src="http://
upload.wikimedia.org/math/b/3/0/b307cc622fab34f5958b806f8b6b9c13.png" /
> by verifying that COMPOSITE satisfies
the above definition.

COMPOSITE also happens to be in P.<a href="#cite_note-
Agrawal-7">[8]</a><a href="#cite_note-8">[9]</a>

<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=3" title="Edit
section: NP-complete">edit</a>] NP-complete</h2>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="/wiki/
File:P_np_np-complete_np-hard.svg" class="image"><img alt=""
src="http://upload.wikimedia.org/wikipedia/commons/thumb/a/a0/P_np_np-
complete_np-hard.svg/300px-P_np_np-complete_np-hard.svg.png"
width="300" height="212" class="thumbimage" /></a>
<div class="thumbcaption">
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class="internal" title="Enlarge"><img src="http://bits.wikimedia.org/
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alt="" /></a></div>
<a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagram</a>
for <a href="/wiki/P_(complexity)" title="P (complexity)">P</a>, <a
href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a>, <a href="/
wiki/NP-complete" title="NP-complete">NP-complete</a>, and NP-hard set
of problems</div>

</div>
</div>
To attack the P = NP question the concept of <a
href="/wiki/NP-complete" title="NP-complete">NP-completeness</
a> is very useful. Informally the NP-complete problems are the
"toughest" problems in NP in the sense that they are the ones
most likely not to be in P. NP-complete problems are a
set of problems which any other NP-problem can be reduced to in
polynomial time, but which retain the ability to have their solution
verified in polynomial time. In comparison, <a href="/wiki/NP-hard"
title="NP-hard">NP-hard</a> problems are those which are at
least as hard as NP-complete problems, meaning all NP-
problems can be reduced to them, but not all NP-hard problems
are in NP, meaning not all of them have solutions which can be
verified in polynomial time.

For instance, the decision problem version of the <a href="/wiki/
Travelling_salesman_problem" title="Travelling salesman
problem">travelling salesman problem</a> is NP-complete, so
any instance of any problem in NP can be
transformed mechanically into an instance of the traveling salesman
problem, in polynomial time. The traveling salesman problem is one of
many such NP-complete problems. If any NP-complete
problem is in P, then it would follow that P = NP. Unfortunately, many important problems have been shown to be
NP-complete and as of 2010, not a single fast algorithm for any
of them is known.

Based on the definition alone it's not obvious that NP-
complete problems exist. A trivial and contrived NP-complete
problem can be formulated as: given a description of a Turing machine
M guaranteed to halt in polynomial time, does there exist a polynomial-
size input that M will accept?<a href="#cite_note-Scott-9">[10]</a> It is in NP because (given an
input) it is simple to check whether or not M accepts the input by
simulating M; it is NP-hard because the verifier for any
particular instance of a problem in NP can be encoded as a
polynomial-time machine M that takes the solution to be verified as
input. Then the question of whether the instance is a yes or no
instance is determined by whether a valid input exists.

The first natural problem proven to be NP-complete was the
<a href="/wiki/Boolean_satisfiability_problem" title="Boolean
satisfiability problem">Boolean satisfiability problem</a>. This
result came to be known as <a href="/wiki/Cook%E2%80%93Levin_theorem"
title="CookâLevin theorem">CookâLevin theorem</a>; its proof that
satisfiability is NP-complete contains technical details about Turing
machines as they relate to the definition of NP. However, after
this problem was proved to be NP-complete, <a href="/wiki/
Reduction_(complexity)" title="Reduction (complexity)">proof by
reduction</a> provided a simpler way to show that many other problems
are in this class. Thus, a vast class of seemingly unrelated problems
are all reducible to one another, and are in a sense the "same
problem".
<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=4" title="Edit
section: Still harder problems">edit</a>] Still harder problems</h2>

<div class="rellink boilerplate seealso">See also: <a href="/wiki/
Complexity_class" title="Complexity class">Complexity class</a></div>
Although it is unknown whether P = NP, problems
outside of P are known. A number of succinct problems (problems
which operate not on normal input but on a computational description
of the input) are known to be <a href="/wiki/EXPTIME#EXPTIME-complete"
title="EXPTIME">EXPTIME-complete</a>. Because it can be shown
that P <img class="tex" alt="\subsetneq" src="http://
upload.wikimedia.org/math/d/5/5/d5557175d90cbb263ae71f0dfebbf732.png" /
> <a href="/wiki/EXPTIME" title="EXPTIME">EXPTIME</a>, these
problems are outside P, and so require more than polynomial
time. In fact, by the <a href="/wiki/Time_hierarchy_theorem"
title="Time hierarchy theorem">time hierarchy theorem</a>, they cannot
be solved in significantly less than exponential time. Examples
include finding a perfect strategy for chess (on an NÃN board)<a href="#cite_note-
Fraenkel1981-10">[11]</a> and some
other board games.<a
href="#cite_note-11">[12]</a>

The problem of deciding the truth of a statement in <a href="/wiki/
Presburger_arithmetic" title="Presburger arithmetic">Presburger
arithmetic</a> requires even more time. Fischer and <a href="/wiki/
Michael_O._Rabin" title="Michael O. Rabin">Rabin</a> proved in 1974
that every algorithm which decides the truth of Presburger statements
has a runtime of at least <img class="tex" alt="2^{2^{cn}}"
src="http://upload.wikimedia.org/math/8/9/b/
89b762cb4c6792ffed3450e7351420de.png" /> for some constant c.
Here, n is the length of the Presburger statement. Hence, the
problem is known to need more than exponential run time. Even more
difficult are the <a href="/wiki/List_of_undecidable_problems"
title="List of undecidable problems">undecidable problems</a>, such as
the <a href="/wiki/Halting_problem" title="Halting problem">halting
problem</a>. They cannot be completely solved by any algorithm, in the
sense that for any particular algorithm there is at least one input
for which that algorithm will not produce the right answer; it will
either produce the wrong answer, finish without giving a conclusive
answer, or otherwise run forever without producing any answer at all.

<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=5" title="Edit
section: Does P mean "easy"?">edit</a>] Does P mean "easy"?
</h2>
<div class="thumb tright">
<div class="thumbinner" style="width:312px;"><a href="/wiki/
File:KnapsackEmpComplexity.GIF" class="image"><img alt="" src="http://
upload.wikimedia.org/wikipedia/commons/thumb/1/11/
KnapsackEmpComplexity.GIF/310px-KnapsackEmpComplexity.GIF" width="310"
height="183" class="thumbimage" /></a>
<div class="thumbcaption">
<div class="magnify"><a href="/wiki/File:KnapsackEmpComplexity.GIF"
class="internal" title="Enlarge"><img src="http://bits.wikimedia.org/
skins-1.5/common/images/magnify-clip.png" width="15" height="11"
alt="" /></a></div>
The graph shows time (average of 100 instances in msec using a 933 MHz
Pentium III) vs.problem size for knapsack problems for a state-of-the-
art specialized algorithm. Quadratic fit suggests that empirical
algorithmic complexity for instances with 50â10,000 variables is
O((log n)2).<a href="#cite_note-
Pisinger2003-12">[13]</a></div>

</div>
</div>
All of the above discussion has assumed that P means "easy"
and "not in P" means "hard". This assumption, known as <a
href="/wiki/Cobham%27s_thesis" title="Cobham's thesis">Cobham's
thesis</a>, though a common and reasonably accurate assumption in
complexity theory, is not always true in practice; the size of
constant factors or exponents may have practical importance, or there
may be solutions that work for situations encountered in practice
despite having poor worst-case performance in theory (this is the case
for instance for the simplex algorithm in <a href="/wiki/
Linear_programming" title="Linear programming">linear programming</
a>). Other solutions violate the Turing machine model on which P and NP are defined by introducing concepts like randomness
and quantum computation.

Because of these factors even if a problem is shown to be NP-
complete, and even if P â NP, there may still be
effective approaches to tackling the problem in practice. There are
algorithms for many NP-complete problems, such as the <a href="/wiki/
Knapsack_problem" title="Knapsack problem">knapsack problem</a>, the
<a href="/wiki/Travelling_salesman_problem" title="Travelling salesman
problem">travelling salesman problem</a> and the <a href="/wiki/
Boolean_satisfiability_problem" title="Boolean satisfiability
problem">boolean satisfiability problem</a>, that can solve to
optimality many real-world instances in reasonable time. The empirical
average complexity (time vs. problem size) of such algorithms can be
surprisingly low.
<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=6" title="Edit
section: Reasons to believe P â NP">edit</a>] Reasons to believe P
â NP</h2>

According to a poll<a
href="#cite_note-poll-6">[7]</a> many
computer scientists believe that P â  NP. A key
reason for this belief is that after decades of studying these
problems no one has been able to find a polynomial-time algorithm for
any of more than 3000 important known NP-complete problems (see
<a href="/wiki/List_of_NP-complete_problems" title="List of NP-
complete problems">List of NP-complete problems</a>). These algorithms
were sought long before the concept of NP-completeness was even
defined (<a href="/wiki/Karp%27s_21_NP-complete_problems"
title="Karp's 21 NP-complete problems">Karp's 21 NP-complete problems</
a>, among the first found, were all well-known existing problems at
the time they were shown to be NP-complete). Furthermore, the result
P = NP would imply many other startling results that are
currently believed to be false, such as NP = <a href="/wiki/
Co-NP" title="Co-NP">co-NP</a> and P = <a href="/wiki/
PH_(complexity)" title="PH (complexity)">PH</a>.

It is also intuitively argued that the existence of problems that
are hard to solve but for which the solutions are easy to verify
matches real-world experience.<a href="#cite_note-13">[14]</a>
<blockquote class="templatequote">
<div>If P = NP, then the world would be a profoundly
different place than we usually assume it to be. There would be no
special value in âcreative leaps,â no fundamental gap between solving
a problem and recognizing the solution once itâs found. Everyone who
could appreciate a symphony would be Mozart; everyone who could follow
a step-by-step argument would be Gauss...</div>
<div class="templatequotecite">â <a href="/wiki/Scott_Aaronson"
title="Scott Aaronson">Scott Aaronson</a>, <a href="/wiki/MIT"
title="MIT" class="mw-redirect">MIT</a></div>
</blockquote>
On the other hand, some researchers believe that we are
overconfident in P â NP and should explore proofs of
P = NP as well. For example, in 2002 these statements
were made:<a
href="#cite_note-poll-6">[7]</a>

<blockquote class="templatequote">
<div>The main argument in favor of P â  NP is
the total lack of fundamental progress in the area of exhaustive
search. This is, in my opinion, a very weak argument. The space of
algorithms is very large and we are only at the beginning of its
exploration. [. . .] The resolution of <a href="/wiki/Fermat
%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last
Theorem</a> also shows that very simple questions may be settled only
by very deep theories.</div>
<div class="templatequotecite">â<a href="/wiki/Moshe_Y._Vardi"
title="Moshe Y. Vardi">Moshe Y. Vardi</a>, <a href="/wiki/
Rice_University" title="Rice University">Rice University</a></div>
</blockquote>

<blockquote class="templatequote">
<div>Being attached to a speculation is not a good guide to research
planning. One should always try both directions of every problem.
Prejudice has caused famous mathematicians to fail to solve famous
problems whose solution was opposite to their expectations, even
though they had developed all the methods required.</div>
<div class="templatequotecite">â<a href="/wiki/Anil_Nerode"
title="Anil Nerode">Anil Nerode</a>, <a href="/wiki/
Cornell_University" title="Cornell University">Cornell University</a></
div>
</blockquote>
<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=7" title="Edit
section: Consequences of proof">edit</a>] Consequences of proof</h2>
One of the reasons the problem attracts so much attention is the
consequences of the answer. A proof that P = NP could
have stunning practical consequences, if the proof leads to efficient
methods for solving some of the important problems in NP. It is also
possible that a proof would not lead directly to efficient methods,
perhaps if the proof is non-constructive, or the size of the bounding
polynomial is too big to be efficient in practice. The consequences,
both positive and negative, arise since various NP-complete problems
are fundamental in many fields.

Cryptography, for example, relies on certain problems being
difficult. A constructive and efficient solution to the NP-complete
problem 3-SAT would break many existing cryptosystems such as <a
href="/wiki/Public-key_cryptography" title="Public-key
cryptography">Public-key cryptography</a>, used for economic
transactions over the internet, and <a href="/wiki/Triple_DES"
title="Triple DES">Triple DES</a>, used for transactions between
banks. These would need to be modified or replaced.
On the other hand, there are enormous positive consequences that
would follow from rendering tractable many currently mathematically
intractable problems. For instance, many problems in <a href="/wiki/
Operations_research" title="Operations research">operations research</
a> are NP-complete, such as some types of <a href="/wiki/
Integer_programming" title="Integer programming">integer programming</
a>, and the <a href="/wiki/Travelling_salesman_problem"
title="Travelling salesman problem">travelling salesman problem</a>,
to name two of the most famous examples. Efficient solutions to these
problems would have enormous implications for <a href="/wiki/
Logistics" title="Logistics">logistics</a>. Many other important
problems, such as some problems in <a href="/wiki/
Protein_structure_prediction" title="Protein structure
prediction">protein structure prediction</a> are also NP-
complete;<a
href="#cite_note-Berger-14">[15]</a>
if these problems were efficiently solvable it could spur considerable
advances in biology.

But such changes may pale in significance compared to the
revolution an efficient method for solving NP-complete problems would
cause in mathematics itself. According to <a href="/wiki/Stephen_Cook"
title="Stephen Cook">Stephen Cook</a>,<a
href="#cite_note-Official_Problem_Description-3">[4]</a>
<blockquote class="templatequote">
<div>...it would transform mathematics by allowing a computer to find
a formal proof of any theorem which has a proof of a reasonable
length, since formal proofs can easily be recognized in polynomial
time. Example problems may well include all of the <a href="/wiki/
Clay_Math_Institute#Millennium_Prize_Problems" title="Clay Math
Institute" class="mw-redirect">CMI prize problems</a>.</div>
</blockquote>
Research mathematicians spend their careers trying to prove
theorems, and some proofs have taken decades or even centuries to find
after problems have been stated â for instance, <a href="/wiki/Fermat
%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last
Theorem</a> took over three centuries to prove. A method that is
guaranteed to find proofs to theorems, should one exist of a
"reasonable" size, would essentially end this struggle.

A proof that showed that P â NP, while lacking the
practical computational benefits of a proof that P = NP,
would also represent a very significant advance in computational
complexity theory and provide guidance for future research. It would
allow one to show in a formal way that many common problems cannot be
solved efficiently, so that the attention of researchers can be
focused on partial solutions or solutions to other problems. Due to
widespread belief in P â NP, much of this focusing of
research has already taken place.<a href="#cite_note-15">[16]</a>

<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=8" title="Edit
section: Results about difficulty of proof">edit</a>] Results
about difficulty of proof</h2>
The Clay Mathematics Institute million-dollar prize and a huge
amount of dedicated research with no substantial results suggest that
the problem is difficult. Some of the most fruitful research related
to the P = NP problem has been in showing that existing
proof techniques are not powerful enough to answer the question, thus
suggesting that novel technical approaches are required.
As additional evidence for the difficulty of the problem,
essentially all known proof techniques in <a href="/wiki/
Computational_complexity" title="Computational complexity" class="mw-
redirect">computational complexity</a> theory fall into one of the
following classifications, each of which is known to be insufficient
to prove that P â NP:

<ul>
<li>Relativizing proofs: Imagine a world where every algorithm
is allowed to make queries to some fixed subroutine called an <a
href="/wiki/Oracle_machine" title="Oracle machine">oracle</a>, and the
running time of the oracle is not counted against the running time of
the algorithm. Most proofs (especially classical ones) apply uniformly
in a world with oracles regardless of what the oracle does. These
proofs are called relativizing. In 1975, Baker, Gill, and
Solovay showed that P = NP with respect to some oracles,
while P â NP for other oracles.<a href="#cite_note-16">[17]</a> Since relativizing proofs can only prove statements
that are uniformly true with respect to all possible oracles, this
showed that relativizing techniques cannot resolve P = NP.</li>

<li>Natural proofs: In 1993, <a href="/wiki/Alexander_Razborov"
title="Alexander Razborov">Alexander Razborov</a> and <a href="/wiki/
Steven_Rudich" title="Steven Rudich">Steven Rudich</a> defined a
general class of proof techniques for circuit complexity lower bounds,
called <a href="/wiki/Natural_proof" title="Natural proof">natural
proofs</a>. At the time all previously known circuit lower bounds
were natural, and circuit complexity was considered a very promising
approach for resolving P = NP. However, Razborov and
Rudich showed that in order to prove P â NP using a
natural proof, one necessarily must also prove an even stronger
statement, which is believed to be false. Thus it is unlikely that
natural proofs alone can resolve P = NP.</li>

<li>Algebrizing proofs: After the Baker-Gill-Solovay result,
new non-relativizing proof techniques were successfully used to prove
that <a href="/wiki/IP_(complexity)" title="IP (complexity)">IP</a> =
<a href="/wiki/PSPACE" title="PSPACE">PSPACE</a>. However, in 2008, <a
href="/wiki/Scott_Aaronson" title="Scott Aaronson">Scott Aaronson</a>
and <a href="/wiki/Avi_Wigderson" title="Avi Wigderson">Avi Wigderson</
a> showed that the main technical tool used in the IP =
PSPACE proof, known as arithmetization, was also
insufficient to resolve P = NP.<a href="#cite_note-17">[18]</a></li>

</ul>
These barriers are another reason why NP-complete problems
are useful: if a polynomial-time algorithm can be demonstrated for an
NP-complete problem, this would solve the P = NP
problem in a way which is not excluded by the above results.
These barriers have also led some computer scientists to suggest
that the P versus NP problem may be <a href="/wiki/
Independence_(mathematical_logic)" title="Independence (mathematical
logic)">independent</a> of standard axiom systems like <a href="/wiki/
ZFC" title="ZFC" class="mw-redirect">ZFC</a> (cannot be proved or
disproved within them). The interpretation of an independence result
could be that either no polynomial-time algorithm exists for any NP-
complete problem, but such a proof cannot be constructed in (e.g.)
ZFC, or that polynomial-time algorithms for NP-complete problems may
exist, but it's impossible to prove in ZFC that such algorithms are
correct.<a
href="#cite_note-18">[19]</a> However,
if the problem cannot be decided even with much weaker assumptions
extending the <a href="/wiki/Peano_axioms" title="Peano axioms">Peano
axioms</a> (PA) for integer arithmetic, then there would necessarily
exist nearly-polynomial-time algorithms for every problem in NP.<a href="#cite_note-19">[20]</a> Therefore, if one believes (as most
complexity theorists do) that problems in NP do not have efficient
algorithms, it would follow that such notions of independence cannot
be possible. Additionally, this result implies that proving
independence from PA or ZFC using currently known techniques is no
easier than proving the existence of efficient algorithms for all
problems in NP.

<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=9" title="Edit
section: Logical characterizations">edit</a>] Logical characterizations</h2>
The P = NP problem can be restated in terms of the
expressibility of certain classes of logical statements, as a result
of work in <a href="/wiki/Descriptive_complexity" title="Descriptive
complexity" class="mw-redirect">descriptive complexity</a>. All
languages (of finite structures with a fixed <a href="/wiki/
Signature_(logic)" title="Signature (logic)">signature</a> including a
<a href="/wiki/Linear_order" title="Linear order" class="mw-
redirect">linear order</a> relation) in P can be expressed in
<a href="/wiki/First-order_logic" title="First-order logic">first-
order logic</a> with the addition of a suitable <a href="/wiki/
Least_fixed_point" title="Least fixed point">least fixed point</a>
operator (effectively, this, in combination with the order, allows the
definition of recursive functions); indeed, (as long as the signature
contains at least one predicate or function in addition to the
distinguished order relation [so that the amount of space taken to
store such finite structures is actually polynomial in the number of
elements in the structure]), this precisely characterizes P.
Similarly, NP is the set of languages expressible in
existential <a href="/wiki/Second-order_logic" title="Second-order
logic">second-order logic</a> â that is, second-order logic restricted
to exclude <a href="/wiki/Universal_quantification" title="Universal
quantification">universal quantification</a> over relations,
functions, and subsets. The languages in the <a href="/wiki/
Polynomial_hierarchy" title="Polynomial hierarchy">polynomial
hierarchy</a>, <a href="/wiki/PH_(complexity)" title="PH
(complexity)">PH</a>, correspond to all of <a href="/wiki/Second-
order_logic" title="Second-order logic">second-order logic</a>. Thus,
the question "is P a proper subset of NP" can be
reformulated as "is existential second-order logic able to describe
languages (of finite linearly ordered structures with nontrivial
signature) that first-order logic with least fixed point cannot?". The
word "existential" can even be dropped from the previous
characterization, since P = NP if and only if P =
PH (as the former would establish that NP = co-NP, which in turn would imply that NP = PH). <a href="/
wiki/PSPACE" title="PSPACE">PSPACE</a> = <a href="/wiki/NPSPACE"
title="NPSPACE" class="mw-redirect">NPSPACE</a> as established <a
href="/wiki/Savitch%27s_theorem" title="Savitch's theorem">Savitch's
theorem</a>, this follows directly from the fact that the square of a
polynomial function is still a polynomial function. However, it is
believed, but not proven, a similar relationship may not exist between
the polynomial time complexity classes, <a href="/wiki/P_(complexity)"
title="P (complexity)">P</a> and <a href="/wiki/NP_(complexity)"
title="NP (complexity)">NP</a> so the question is still open.

<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=10" title="Edit
section: Polynomial-time algorithms">edit</a>] Polynomial-time algorithms</h2>
No algorithm for any NP-complete problem is known to run in
polynomial time. However, there are algorithms for NP-complete
problems with the property that if P = NP, then the
algorithm runs in polynomial time (although with enormous constants,
making the algorithm impractical). The following algorithm, due to
Levin, is such an example. It correctly accepts the NP-complete
language <a href="/wiki/Subset_sum_problem" title="Subset sum
problem">SUBSET-SUM</a>, and runs in polynomial time if and only if
P = NP:

<pre>
// Algorithm that accepts the NP-complete language <a href="/wiki/
Subset_sum_problem" title="Subset sum problem">SUBSET-SUM</a>.
//
// This is a polynomial-time algorithm if and only if P=NP.
//
// "Polynomial-time" means it returns "yes" in polynomial time
when
// the answer should be "yes", and runs forever when it is "no".
//
// Input: S = a finite set of integers
// Output: "yes" if any subset of S adds up to 0.
// Runs forever with no output otherwise.
// Note: "Program number P" is the program obtained by
// writing the integer P in binary, then
// considering that string of bits to be a
// program. Every possible program can be
// generated this way, though most do nothing
// because of syntax errors. FOR N = 1...infinity
FOR P = 1...N
Run program number P for N steps with input S
IF the program outputs a list of distinct integers
AND the integers are all in S
AND the integers sum to 0 THEN
OUTPUT "yes" and HALT
</pre>
If, and only if, P = NP, then this is a polynomial-
time algorithm accepting an NP-complete language. "Accepting"
means it gives "yes" answers in polynomial time, but is allowed to run
forever when the answer is "no".

Note that this is enormously impractical, even if P = NP. If the shortest program that can solve SUBSET-SUM in polynomial
time is b bits long, the above algorithm will try 2b
â1 other programs first.
Perhaps we want to "solve" the SUBSET-SUM problem, rather than just
"accept" the SUBSET-SUM language. That means we want the algorithm to
always halt and return a "yes" or "no" answer. If P = NP, then there is an algorithm which does this in polynomial time,
which uses some polynomial-time verification method that there is no
subset sum in the algorithm above. Another algorithm that is obtained
by replacing the IF statement in the above algorithm with this:

<pre>
IF the program outputs a complete math proof
AND each step of the proof is legal
AND the conclusion is that S does (or does not) have a
subset summing to 0
THEN
OUTPUT "yes" (or "no") and HALT
</pre>
<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=11" title="Edit
section: Gibraltar Code">edit</a>] Gibraltar Code</h2>
These are the codes for the sets P.NP.
<code>Copyright Â© 2010 www.meami.org </code>
<code>The information on this page may not be reproduced or
republished on another web page or web site unless done so by M. M.
Musatov (with the exception of Wikipedia). M. M. Musatov, Meami.org,
released this code for private non-profit non-commercial use only -
All Other Rights Reserved.</code>
<pre>

<code> // 
// #include <cassert> 
// #include <climits> 
</code>
</pre>
<pre>
<code> // #include <cstdlib> 

// #include <cstdio> 
// 
// using namespace std; 
// 
// typedef struct node 
// { 

</code>
</pre>
<pre>
<code> // int nElem; 
// struct node *pNextNode; 
// }Node; 
// 
// 

// int pushElem(Node **argpRoot, int argnElem) 
// { 
</code>
</pre>
<pre>
<code> // Node *pNewNode; 
// pNewNode = (Node *)malloc(sizeof(Node)); 
// if(!pNewNode) 

// { 
// fprintf(stderr,"\n\t ERR: Memory allocation
failure
</code>
</pre>
<code>for Node \n"); </code>
<pre>
<code> // return -1; 
</code>
</pre>
<pre>
<code> // } 

// 
// pNewNode->nElem = argnElem; 
// pNewNode->pNextNode = NULL; 
// if(*argpRoot==NULL) 
// { 

</code>
</pre>
<pre>
<code> // *argpRoot = pNewNode; 
// } 
// else 
// { 
// pNewNode->pNextNode = *argpRoot; 

// *argpRoot = pNewNode; 
</code>
</pre>
<pre>
<code> // } 
// return 1; 
// } 
// 

// 
// int popElem(Node **argpRoot) 
// { 
</code>
</pre>
<pre>
<code> // assert(*argpRoot!=NULL); 
// int nRetElem; 

// Node *pDeleteNode; 
// pDeleteNode = *argpRoot; 
// *argpRoot = (*argpRoot)->pNextNode; 
// nRetElem = pDeleteNode->nElem; 
</code>
</pre>
<pre>

<code> // free(pDeleteNode); 
// return nRetElem; 
// } 
// 
// void deleteList(Node **argpRoot) 
// { 

</code>
</pre>
<pre>
<code> // while(*argpRoot) 
// { 
// popElem(argpRoot); 
// } 
// } 

// 
// void printElems(Node *argpRoot) 
</code>
</pre>
<pre>
<code> // { 
// //assert(argpRoot != NULL); 
// if(argpRoot!=NULL) 

// { 
// Node *pTempNode = argpRoot; 
// while(pTempNode->pNextNode) 
</code>
</pre>
<pre>
<code> // { 
// fprintf(stdout,"%d->",pTempNode-
>nElem); 

// pTempNode = pTempNode->pNextNode; 
// } 
// fprintf(stdout,"%d",pTempNode->nElem); 
// } 
</code>
</pre>
<pre>

<code> // } 
// 
// int findElemFromListEnd(Node *argpRoot,int nTargetPos,int
</code>
</pre>
<ul>
<li><code>argpnElem) </code></li>
</ul>
<pre>
<code> // { 

// /* 
// assert(argpRoot!=NULL); 
</code>
</pre>
<pre>
<code> // assert(nTargetPos <= INT_MAX &&
nTargetPos > 0); 

// assert(argpnElem!=NULL); 
// */ 
// 
// if(argpRoot == NULL) 
</code>
</pre>
<pre>
<code> // { 

// fprintf(stderr, "\n\t ERR: list is empty
\n"); 
// return -1; 
// } 
// 
// if((nTargetPos > INT_MAX) || (nTargetPos <=
0)) 

</code>
</pre>
<pre>
<code> // { 
// fprintf(stderr, "\n\t ERR: target position
should be
</code>
</pre>
<code><=INT_MAX and non-zero positive value\n"); </code>
<pre>
<code> // return -1; 
// } 

// 
// if(argpnElem == NULL) 
</code>
</pre>
<pre>
<code> // { 
// fprintf(stderr, "\n\t ERR: no memory allocated
to
</code>
</pre>
<code>store the element at target position in the input list
\n"); </code>

<pre>
<code> // return -1; 
// } 
// 
// Node *pFwdNode,*pLagNode; 
</code>
</pre>
<pre>
<code> // int nCurrentPos = 1; 

// 
// pFwdNode = argpRoot; 
// pLagNode = NULL; 
// 
// while(pFwdNode) 
// { 

</code>
</pre>
<pre>
<code> // if(nCurrentPos == nTargetPos) 
// { 
// pLagNode = argpRoot; 
// break; 
// } 

// pFwdNode = pFwdNode->pNextNode; 
</code>
</pre>
<pre>
<code> // nCurrentPos++; 
// } 
// 
// if(!pLagNode) 

// { 
// fprintf(stderr, "\n\t ERR: target position
specified
</code>
</pre>
<code>is non-existent for the current list\n"); </code>
<pre>
<code> // *argpnElem = -1; 
// return -1; 
// } 

// 
// while(pFwdNode->pNextNode) 
// { 
</code>
</pre>
<pre>
<code> // pLagNode = pLagNode->pNextNode; 

// pFwdNode = pFwdNode->pNextNode; 
// } 
// *argpnElem = pLagNode->nElem; 
// return 1; 
// } 

</code>
</pre>
<pre>
<code> // int main() 
// { 
// Node *pRoot = NULL; 
// int nNumElems = 0; 
// int nCurElem; 

// unsigned int unTestCaseId; 
</code>
</pre>
<pre>
<code> // int nTargetPos; 
// int i; 
// 
// while(!feof(stdin)) 

// { 
// fscanf(stdin,"---\n"); 
</code>
</pre>
<pre>
<code> // fprintf(stdout,"---\n"); 
// fscanf(stdin,"NumOfElems :%d
\n",&nNumElems); 
// fprintf(stdout,"NumOfElems :%d
\n",nNumElems); 

// fflush(stdout); 
// for(i=0;i<nNumElems;i++) 
// { 
</code>
</pre>
<pre>
<code> //
fscanf(stdin,"%d,",&nCurElem); 

// pushElem(&pRoot,nCurElem); 
// } 
// printElems(pRoot); 
// fflush(stdout); 
</code>
</pre>
<pre>
<code> // fscanf(stdin,"\nTarget Position :%d
\n",&nTargetPos);</

</code>
</pre>
<pre>
<code> // fprintf(stdout,"\nTarget Position :%d
\n",nTargetPos);</
</code>
</pre>
<pre>
<code> // fflush(stdout); 
//
if(findElemFromListEnd(pRoot,nTargetPos,&nCurElem)<0)</

</code>
</pre>
<pre>
<code> // { 
</code>
</pre>
<pre>
<code> // fprintf(stdout,"ERROR\n"); 
// 
// } 

// else 
// { 
// fprintf(stdout,"Element:%d
\n",nCurElem); 
</code>
</pre>
<pre>
<code> // } 
// fscanf(stdin,"---\n"); 

// fprintf(stdout,"---\n"); 
// fflush(stdout); 
// 
// deleteList(&pRoot); 
// } 
</code>

</pre>
<pre>
<code> // return 0; 
// } </code><a href="http://meami.org/gibraltar.htm"
class="external autonumber" rel="nofollow">[2]</a>
</pre>
<h2>[<a href="/w/index.php?
title=P_versus_NP_problem&action=edit&section=12" title="Edit
section: Formal definitions for P and NP">edit</a>] Formal
definitions for P and NP</h2>
Conceptually a decision problem is a problem that takes as
input some <a href="/wiki/String_(computer_science)" title="String
(computer science)">string</a>, and outputs "yes" or "no". If there is
an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> (say a <a
href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>,
or a <a href="/wiki/Computer_programming" title="Computer
programming">computer program</a> with unbounded memory) which is able
to produce the correct answer for any input string of length n in at most <img class="tex" alt="c
\cdot n^k" src="http://upload.wikimedia.org/math/1/6/c/
16c462a08b39445092909372ffc5355b.png" /> steps, where k and c are constants independent of the input string, then we say that
the problem can be solved in polynomial time and we place it in
the class P. Formally, P is defined as the set of all
languages which can be decided by a deterministic polynomial-time
Turing machine. That is,

P = {L:L = L(M) for some deterministic polynomial-time Turing machine
M}
where <img class="tex" alt="L(M) = \{ w\in\Sigma^{*}: M
\text{ accepts } w \}" src="http://upload.wikimedia.org/math/
2/9/9/29923cd12c8e3f483b7307217f2faddb.png" />

and a deterministic polynomial-time Turing machine is a
deterministic Turing machine M
which satisfies the following two conditions:
<ol>
<li>M halts on all input w;
and</li>
<li>there exists <img class="tex" alt="k \in N" src="http://
upload.wikimedia.org/math/0/6/d/06dfb1cc3c34f77feeb5622736ac1559.png" /
> such that <img class="tex" alt="T_{M}(n)\in\; " src="http://
upload.wikimedia.org/math/c/c/4/cc451be9c433fe9d9b1665e02a1451ea.png" /
><a href="/wiki/Big_O_notation#Formal_definition" title="Big O
notation">O</a>(nk),</li>

</ol>
<dl>
<dd>
<dl>
<dd>where <img class=

|
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