Diagram of complexity classes provided that P ≠ NP. The existence of problems outside both P and NP-complete in this case was established by Ladner.^[1]

The relationship between the complexity classes P and NP is an unsolved question in theoretical computer science. It is considered to be the most important problem in the field - the Clay Mathematics Institute has offered a $1 million US prize for the first correct proof. Image File history File links Complexity_classes. ... Image File history File links Complexity_classes. ... In computational complexity theory, a complexity class is a set of problems of related complexity. ... In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ... In computational complexity theory, NP (Non-deterministic Polynomial time) is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine. ... Computer science (informally, CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ... The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts. ...

In essence, the P = NP question asks: if 'yes'-answers to a 'yes'-or-'no'-question can be verified quickly, can the answers themselves also be computed quickly? In this context, "quickly" means "in polynomial time". In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ... In computational complexity theory, polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ...

Consider, for instance, the subset-sum problem, an example of a problem which is easy to verify, but whose answer is believed (but not proven) to be difficult to compute. Given a set of integers, does some nonempty subset of them sum to 0? For instance, does a subset of the set {−2, −3, 15, 14, 7, −10} add up to 0? The answer is YES, though it may take a while to find a subset that does, depending on its size. On the other hand, if someone claims that the answer is "YES, because {−2, −3, −10, 15} add up to zero", then we can quickly check that with a few additions. Verifying that the subset adds up to zero is much faster than finding the subset in the first place. The information needed to verify a positive answer is also called a certificate. So we conclude that given the right certificates, positive answers to our problem can be verified quickly (in polynomial time) and that's why this problem is in NP. The subset sum problem is an important problem in complexity theory and cryptography. ... The integers are commonly denoted by the above symbol. ... “Superset” redirects here. ...

An answer to the P = NP question would determine whether problems like SUBSET-SUM are as easy to compute as to verify. If it turned out P does not equal NP, it would mean that some NP problems are substantially harder to compute than to verify. The subset sum problem is an important problem in complexity theory and cryptography. ...

The restriction to YES/NO problems doesn't really make a difference; even if we allow more complicated answers, the resulting problem (whether FP = FNP) is equivalent. In computational complexity theory, the complexity class FP is the set of function problems which can be solved by a deterministic Turing machine in polynomial time; it is the function problem version of the decision problem class P. Roughly speaking, it is the class of functions that can be efficiently... In computational complexity theory, the complexity class FNP is the function problem extension of the decision problem class NP. The name is a bit of a misnomer, since technically it is a class of binary relations, not functions, as the following formal definition explains: A binary relation P(x,y...

The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. ... The Hodge conjecture is a major unsolved problem of algebraic geometry. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... There is also the Riemann hypothesis for curves over finite fields. ... The Clay Mathematics Institute has offered the prize of 1 million dollars for each of 7 great problems in mathematics. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function L(E, s) at s = 1. ...

Context of the problem

The relation between the complexity classes P and NP is studied in computational complexity theory, the part of the theory of computation 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 computational complexity theory, a complexity class is a set of problems of related complexity. ... As a branch of the theory of computation in computer science, computational complexity theory investigates the problems related to the amounts of resources required for the execution of algorithms (e. ... The theory of computation is the branch of computer science that deals with whether and how efficiently problems can be solved on a computer. ...

In such analysis, a model of the computer for which time must be analyzed is required. Typically, such models assume that the computer is deterministic (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). These assumptions reflect the behavior of all practical computers yet devised, even including machines featuring parallel computing. We dont have an article called Deterministic computation Start this article Search for Deterministic computation in. ... Parallel computing is the simultaneous execution of the same task (split up and specially adapted) on multiple processors in order to obtain results faster. ...

In this theory, the class P consists of all those decision problems that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; the class NP consists of all those decision problems whose positive solutions can be verified in polynomial time given the right information, or equivalently, whose solution can be found in polynomial time on a non-deterministic machine. Arguably, the biggest open question in theoretical computer science concerns the relationship between those two classes: In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ... In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In computational complexity theory, NP (Non-deterministic Polynomial time) is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine. ... In computational complexity theory, polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ... In theoretical computer science, an ordinary (deterministic) Turing machine (DTM) has a transition rule that specifies for a given current state of the head and computer (s,q) a single instruction (s, q, d), where s is the symbol to be written by the head, q is the subsequent state... The theory of computation is the branch of computer science that deals with whether and how efficiently problems can be solved on a computer. ...

Is P equal to NP?

In a 2002 poll of 100 researchers, 61 believed the answer is no, 9 believed the answer is yes, 22 were unsure, and 8 believed the question may be independent of the currently accepted axioms, and so impossible to prove or disprove.^[2]

Formal definitions for P and NP

Conceptually, a decision problem is a problem that takes as input some string and requires as output either YES or NO. If there is an algorithm (say a Turing machine, or a Lisp or Pascal program with unbounded memory) which is able to produce the correct answer for any input string of length n in at most steps, where k and c are some 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, In computer programming and formal language theory, (and other branches of mathematics), a string is an ordered sequence of symbols. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... For the test of artificial intelligence, see Turing test. ... Lisp is a family of computer programming languages with a long history and a distinctive fully-parenthesized syntax. ... Pascal is a structured imperative computer programming language, developed in 1970 by Niklaus Wirth as a language particularly suitable for structured programming. ...

P = { L : L=L(M) for some deterministic polynomial-time Turing machine M }

where L(M) = { w: M accepts w }

and a deterministic polynomial-time Turing machine is a deterministic Turing machine M which satisfies the following two conditions:

(i) M halts on all input w

(ii) There exists kN such that T_M(n)O(n^k) For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

where T_M(n) = max{ t_M(w) : w, = n }

and t_M(w) = number of steps M takes to halt on input w.

NP can be defined similarly using nondeterministic Turing machines (the traditional way). However, a modern approach to define NP is to use the concept of certificate and verifier. Formally, NP is defined as the set of languages that satifies the following conditions. Let L be a language over a finite alphabet, Σ. If there exist a binary relation and a positive integer k such that for all , the following conditions hold:

(i) such that and O For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

(ii) The language L_R= over is decidable

then, the Turing machine that decides L_R (Let's call it V) is called the Verifier for L and y is called the Certificate of membership of x in L.

Finally, L is in NP if, and only if, V runs in polynomial time.

Example.

Let COMPOSITE =

R =

Clearly, the question of whether a given x is a composite is equivalent to the question of whether x is a member of COMPOSITE. In fact, one can easily show that by verifying that COMPOSITE satifies the above definition.

(Note. It should be mentioned that COMPOSITE was recently shown to be also in P ^[3])

NP-complete

To attack the P = NP question, the concept of NP-completeness 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-hard problems are those to which any problem in NP can be reduced in polynomial time. NP-complete problems are those NP-hard problems which are in NP. For instance, the decision problem version of the traveling salesman problem 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. So, if the traveling salesman problem turned out to be in P, then P = NP. 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 not a single fast algorithm for any of them is known. In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use... In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing... The traveling salesman problem (TSP), is a problem in discrete or combinatorial optimization. ...

It is relatively obvious that the class NP-complete is non-empty because a trivial NP and NP-hard decision problem called DUH can be formulated: given a description of a Turing machine M guaranteed to halt in polynomial time, DUH is the question of whether there exists a polynomial-size input that M will accept.^[4] However, since DUH is contrived and of primarily theoretical interest, it came as a breakthrough to discover that numerous existing, highly practical problems were also NP-complete.

The first natural problem proven to be NP-complete was the Boolean satisfiability problem. This result was proven by Stephen Cook in 1971, and came to be known as Cook's theorem. Cook's proof that satisfiability is NP-complete is very complicated. However, after this problem was proved to be NP-complete, proof by reduction has 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" – a profound and unexpected result. 3SAT redirects here. ... Stephen A. Cook is a noted computer scientist. ... It has been suggested that Proof that Boolean satisfiability problem is NP-complete be merged into this article or section. ... In computability theory and computational complexity theory, a reduction is a transformation of one problem into another problem. ...

Formal definition for NP-completeness

Although there are many equivalent ways of describing NP-completeness, in the context of the P vs NP question, it is best to define NP-complete problems in terms of NP problems.

Let be a language over a finite alphabet .

is NP-complete if, and only if, the following two conditions are satisfied:

(i) NP

(ii) Any NP is polynomial time reducible to (written as )

where if, and only if, the following two conditions are satisfied:

(a) There exists such that ,

(b) There exists a polynomial time Turing machine which halts with on its tape on any input .

Still harder problems

Although it is unknown whether P = NP, problems outside of P are known. A number of succinct problems, that is, problems which operate not on normal input but on a computational description of the input, are known to be EXPTIME-complete. Because it can be shown that P EXPTIME, these problems are outside P, and so require more than polynomial time. In fact, by the time hierarchy theorem, they cannot be solved in significantly less than exponential time. In computational complexity theory, the complexity class EXPTIME (sometimes called EXP) is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n. ... In computational complexity theory, the complexity class EXPTIME (sometimes called EXP) is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n. ... In computational complexity theory, the time hierarchy theorems are important statements that ensure the existence of certain hard problems which cannot be solved in a given amount of time. ...

The problem of deciding the truth of a statement in Presburger arithmetic is even harder. Fischer and Rabin proved in 1974 that every algorithm which decides the truth of Presburger statements has a runtime of at least 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 undecidable problems, such as the halting problem. They cannot be solved in general given any finite amount of time. Presburger arithmetic is the first-order theory of the natural numbers with addition. ... Michael Oser Rabin (born 1931 in Breslau, Germany, today in Poland) is a noted computer scientist and a recipient of the Turing Award, the most prestigious award in the field. ... Year 1974 (MCMLXXIV) was a common year starting on Tuesday (link will display full calendar) of the 1974 Gregorian calendar. ... In computability theory, an undecidable problem is a problem whose language is not a recursively enumerable set. ... In computability theory the halting problem is a decision problem which can be stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ...

Is P really practical?

Main article: Cobham's thesis

All of the above discussion has assumed that P means "easy" and "not in P" means "hard". While this is a common and reasonably accurate assumption in complexity theory, it is not always true in practice, for several reasons: Alan Cobham wrote a paper in 1964 entitled “The intrinsic computational difficulty of functions. ...

• It ignores constant factors. A problem that takes time 10¹⁰⁰n is in P (it is linear time), but is completely impractical. A problem that takes time (1+10^-100)ⁿ is not in P (it is exponential time), but is very practical for even quite large values of n.
• It ignores the size of the exponents. A problem with time n¹⁰⁰ is in P, yet impractical. Problems have been proven to exist in P that require arbitrarily large exponents (see time hierarchy theorem). A problem with time 2^n/100 is not in P, yet is practical for n up into the low thousands.
• It only considers worst-case times. There might be a problem that arises in the real world such that most of the time, it can be solved in time n, but on very rare occasions you'll see an instance of the problem that takes time 2ⁿ. This problem might have an average time that is polynomial, but the worst case is exponential, so the problem would not be in P. The simplex algorithm is an example of a practical worst-case exponential algorithm.
• It only considers deterministic solutions. Imagine a problem that you can solve quickly if you accept a tiny error probability, but a guaranteed correct answer is much harder to get. The problem would not belong to P even though in practice it can be solved quickly. This is in fact a common approach to attack problems in NP not known to be in P (see RP, BPP). Even if P = BPP, as many researchers believe, it is often considerably easier to find probabilistic algorithms.
• New computing models such as quantum computers may be able to quickly solve some problems not known to be in P; though quantum algorithms have not to date solved any NP-hard problem in polynomial time. However, the definition of P and NP are in terms of classical computing models like Turing machines. Therefore, even if a quantum computer algorithm were discovered to efficiently solve an NP-hard problem, we would only have a way of physically solving difficult problems quickly, not a proof that the mathematical classes P and NP are equal.
• Advances in CPU and other technology may make exponential-time algorithms efficient for practical ranges of problem sizes.

In computational complexity theory, the time hierarchy theorems are important statements that ensure the existence of certain hard problems which cannot be solved in a given amount of time. ... In mathematical optimization theory, the simplex algorithm, created by the American mathematician George Dantzig in 1947, is a popular technique for numerical solution of the linear programming problem. ... In complexity theory, RP (randomized polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists with these properties: It always runs in polynomial time in the input size If the correct answer is NO, it always returns NO If the correct answer is YES, then... This article is about the complexity class. ... A randomized algorithm is an algorithm which is allowed to flip a truly random coin. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ...

Why do many computer scientists think P ≠ NP?

Most computer scientists believe that P≠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 NP-hard problem. Moreover, these algorithms were sought long before the concept of NP-completeness was even known (Karp's 21 NP-complete problems, among the first found, were all well-known existing problems). Furthermore, the result P = NP would imply many other startling results that are currently believed to be false, such as NP = co-NP and P = PH. One of the most important results of computational complexity theory was Stephen Cooks 1971 paper that demonstrated the first NP-complete problem, the boolean satisfiability problem. ... In computational complexity theory, co-NP is a complexity class. ... In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy: PH is contained in the complexity classes PPP (the class of problems that are decidable by a polynomial time Turing machine with an access to PP oracle) and PSPACE. PH has...

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.

On the other hand, some researchers believe that we are overconfident in P ≠ NP and should explore proofs of P = NP as well. For example, in 2002 these statements were made:^[2]

Consequences of proof

One of the reasons the problem attracts so much attention is the consequences of the answer.

A proof of P = NP could have stunning practical consequences, if the proof leads to efficient methods for solving some of the important problems in NP. Various NP-complete problems are fundamental in many fields. There are enormous positive consequences that would follow from rendering tractable many currently mathematically intractable problems. For instance, many problems in operations research are NP-complete, such as some types of integer programming, and the travelling salesman problem, to name two of the most famous examples. Efficient solutions to these problems would have enormous implications for logistics. But it is by no means the only field that would be profoundly changed. To take one of very many examples, important problems in Protein structure prediction are NP-complete;^[5] if these problems were solvable efficiently it could spur considerable advances in biology. Operations Research or Operational Research (OR) is an interdisciplinary branch of mathematics which uses methods like mathematical modeling, statistics, and algorithms to arrive at optimal or good decisions in complex problems which are concerned with optimizing the maxima (profit, faster assembly line, greater crop yield, higher bandwidth, etc) or minima... In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ... If a salesman starts at point A, and if the distances between every pair of points are known, what is the shortest route which visits all points and returns to point A? The traveling salesman problem (TSP) is a problem in discrete or combinatorial optimization. ... Look up Logistics in Wiktionary, the free dictionary. ... Protein structure prediction is one of the most significant technologies pursued by computational structural biology and theoretical chemistry. ...

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 Stephen Cook,^[6]

...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 CMI prize problems.

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 - for instance, Fermat's Last Theorem 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. The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ...

A proof that showed that P ≠ NP, while lacking the practical computational benefits of a proof that P = NP, would also represent a massive 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 ≠ NP, much of this focusing of research has already taken place.

Results about difficulty of proof

A million-dollar prize and a huge amount of dedicated research with no substantial results are enough to show the problem is difficult. There have also been some formal results demonstrating why the problem might be difficult to solve.

One of the most frequently-cited is a result involving oracles. Imagine you have a magical machine called an oracle that can solve only one problem, such as determining if a given number is prime, but can solve it in constant time. Our new question is now, if we're allowed to use this oracle as much as we want, are there problems we can verify in polynomial time that we cannot solve in polynomial time? It turns out that, depending on the problem that the oracle solves, with certain oracles one has P = NP, while for other oracles one has P ≠ NP. The practical consequence of this is that any proof which can be modified to account for the existence of these oracles cannot solve the problem. Unfortunately, most known methods and nearly all classical methods can be modified in such a way (we say they are relativizing). In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. ...

Furthermore, a 1993 result by Alexander Razborov and Steven Rudich showed that, given a certain credible assumption, proofs that are "natural" in a certain sense cannot solve the P = NP problem (see natural proof). This demonstrated that some of the most seemingly-promising methods of the time were also unlikely to succeed. As more theorems of this kind are proved, a potential proof of the theorem has more and more traps to avoid. Alexander Razborov is a Russian mathematician who won the Nevanlinna Prize in 1990 for his work in theoretical aspects of computer science. ... Steven Rudich is a professor in the Carnegie Mellon School of Computer Science. ... In computational complexity theory, natural proof is a notion introduced by Alexander Razborov and Steven Rudich, to classify the set of proofs that will fail to prove separations between complexity classes. ...

This is actually 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.

Polynomial-time algorithms

No one knows whether polynomial-time algorithms exist for NP-complete languages. But if such algorithms do exist, we already know some of them! For example, the following algorithm (due to Levin) correctly accepts an NP-complete language, but no one knows how long it takes in general. This is a polynomial-time algorithm if and only if P = NP.

 // Algorithm that accepts the NP-complete language SUBSET-SUM. // // 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. // Otherwise, it runs forever with no output. // Note: "Program number P" is the program you get 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 

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". The subset sum problem is an important problem in complexity theory and cryptography. ...

Perhaps we want to "solve" the SUBSET-SUM problem, rather than just "accept" the SUBSET-SUM language. That means we want it to always halt and return a "YES" or "NO" answer. Does any algorithm exist that can provably do this in polynomial time? No one knows. But if such algorithms do exist, then we already know some of them (by an argument of Schnorr)! Just replace the IF statement in the above algorithm with this:

 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" if that were proved) and HALT 

Logical characterizations

The P = NP problem can be restated in terms of the expressibility of certain classes of logical statements. All languages in P can be expressed in first-order logic with the addition of a least fixed point operator and an order relation (effectively, this allows the definition of recursive functions). Similarly, NP is the set of languages expressible in existential second-order logic — that is, second-order logic restricted to exclude universal quantification over relations, functions, and subsets. The languages in the polynomial hierarchy, PH, correspond to all of second-order logic. Thus, the question "is P a proper subset of NP" can be reformulated as "is existential second-order logic able to describe languages that first-order logic with least fixed point cannot?" First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ... In mathematics, the least fixed point in order theory of a function is the fixed point which is less than or equal to all other fixed points, according to some partial order. ... In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ... In predicate logic, universal quantification is an attempt to formalize the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In computational complexity theory, the polynomial hierarchy is a hierarchy of complexity classes that generalize the classes P, NP and co-NP to oracle machines. ... In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy: PH is contained in the complexity classes PPP (the class of problems that are decidable by a polynomial time Turing machine with an access to PP oracle) and PSPACE. PH has... In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ...

Humor and cultural references

The Princeton University computer science building has the question "P = NP?" encoded in binary ASCII in its brickwork on the top floor of the west side. If it is proven that P = NP, the bricks can easily be changed to encode "P = NP!". If P does not equal NP, it can be changed to "P<NP!". ^[7] Princeton University is a private coeducational research university located in Princeton, New Jersey. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... Image:ASCII fullsvg There are 95 printable ASCII characters, numbered 32 to 126. ...

In the science fiction story Antibodies by Charles Stross (which appears in his collection "Toast"), the discovery that P = NP quickly leads to the emergence of Artificial Intelligence bent on enslaving humanity. Charles David George Charlie Stross (born Leeds, October 18, 1964) is a writer based in Edinburgh, Scotland. ... AI redirects here. ...

In the second episode of the CBS show NUMB3RS, Charlie, a mathematician, works with his brother, an FBI agent, to predict which bank a group of seemingly non-violent robbers will hit next. When the FBI's attempt to arrest the criminals ends in bloodshed, Charlie tries to deal with his emotional reaction by attempting to solve P = NP. (The show used the popular computer game Minesweeper to help explain what he was working on.) Numb3rs (also capitalized as NUMB3RS and pronounced as Numbers) is an American television show produced by brothers Ridley Scott and Tony Scott. ... The game begins when the user clicks on a blank square. ...

In the episode "Put Your Head on My Shoulder" from the FOX cartoon series Futurama, a bookcase behind the two characters Fry and Amy contains two books, one labeled P and the other labeled NP. A similar reference appeared in the 1995 The Simpsons episode "Treehouse of Horror VI" when Homer enters a three-dimensional universe where several equations are hanging in mid-air, including P=NP. The inclusion of both of these references was due to David X. Cohen's affection for mathematical jokes (Cohen was a writer on The Simpsons and an executive producer on Futurama).^[8] Put Your Head on My Shoulder is episode seven in season two of Futurama. ... This article is about the animal. ... This article is about the television series. ... Simpsons redirects here. ... Treehouse of Horror VI is the sixth episode of The Simpsons seventh season, as well as the sixth Halloween episode. ... David X. Cohen (born 1966), born David Samuel Cohen, is an American television writer. ... This equation uses mathematical symbols to write Sex is fun. A mathematical joke is a form of humor which relies on aspects of mathematics or a stereotype of mathematicians to derive humor. ...

See also

• P (complexity)
• NP (complexity)
• NP-complete
• Game complexity
• Cobham's thesis
• List of open problems in computer science
• Unsolved problems in mathematics

In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ... In computational complexity theory, NP (Non-deterministic Polynomial time) is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine. ... In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use... Combinatorial game theory has several ways of measuring game complexity. ... Alan Cobham wrote a paper in 1964 entitled “The intrinsic computational difficulty of functions. ... This is a list of unsolved problems in computer science. ... This article lists some unsolved problems in mathematics. ...

References

A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 239th day of the year (240th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 239th day of the year (240th in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 96th day of the year (97th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 271st day of the year (272nd in leap years) in the Gregorian calendar. ...

Further reading

• A. S. Fraenkel and D. Lichtenstein, Computing a perfect strategy for n*n chess requires time exponential in n, Proc. 8th Int. Coll. Automata, Languages, and Programming, Springer LNCS 115 (1981) 278-293 and J. Comb. Th. A 31 (1981) 199-214.
• E. Berlekamp and D. Wolfe, Mathematical Go: Chilling Gets the Last Point, A. K. Peters, 1994. D. Wolfe, Go endgames are hard, MSRI Combinatorial Game Theory Research Worksh., 2000.
• Neil Immerman. Languages Which Capture Complexity Classes. 15th ACM STOC Symposium, pp.347-354. 1983.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001). "Chapter 34: NP-Completeness", Introduction to Algorithms, Second Edition, MIT Press and McGraw-Hill, pp.966–1021. ISBN 0-262-03293-7. 
• Christos Papadimitriou (1993). "Chapter 14: On P vs. NP", Computational Complexity, 1st edition, Addison Wesley, pp.329–356. ISBN 0-201-53082-1. 

Neil Immerman is one of the key developers of an active research program called Descriptive Complexity, an approach he is currently applying to research in model checking, database theory, and computational complexity theory. ... Thomas H. Cormen is the co-author of Introduction to Algorithms, along with Charles Leiserson, Ron Rivest, and Cliff Stein. ... Charles E. Leiserson is a computer scientist, specializing in the theory of parallel computing and distributed computing, and particularly practical applications thereof; as part of this effort, he developed the Cilk multithreaded language. ... Professor Ron Rivest Professor Ronald Linn Rivest (born 1947, Schenectady, New York) is a cryptographer, and is the Viterbi Professor of Computer Science at MITs Department of Electrical Engineering and Computer Science. ... Clifford Stein is a computer scientist, currently working as a professor at Columbia University in New York, NY. He earned his BSE from Princeton University in 1987, a MS from Massachusetts Institute of Technology in 1989, and a PhD from Massachusetts Institute of Technology in 1992. ... Cover of the second edition Introduction to Algorithms is a book by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. ... Christos Papadimitriou is a Professor in the Computer Science Division at the University of California, Berkeley, USA. He studied at the National Technical University of Athens (BS in Electrical Engineering, 1972) and at Princeton University (MS in Electrical Engineering, 1974 and PhD in Electrical Engineering and Computer Science, 1976). ...

External links

• The Clay Math Institute Millennium Prize Problems
• The Clay Math Institute Official Problem Description (pdf)
• Ian Stewart on Minesweeper as NP-complete at The Clay Math Institute
• Gerhard J. Woeginger. The P-versus-NP page. A list of links to a number of purported solutions to the problem. Some of these links state that P equals NP, some of them state the opposite. It is probable that all these alleged solutions are incorrect.
• Computational Complexity of Games and Puzzles
• Scott Aaronson's Complexity Zoo: P, NP
• Qeden, a wiki that aims to solve the Millennium Prize Problems
• Scott Aaronson's Shtetl Optimized blog: Reasons to believe, a list of justifications for the belief that P ≠ NP