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 computer science, computational complexity theory is the branch of the theory of computation that studies the resources required during computation to solve a given problem. ... In computational complexity theory, a complexity class is a set of problems of related complexity. ... In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ... The Turing machine is an abstract machine introduced in 1936 by Alan Turing to give a mathematically precise definition of algorithm or mechanical procedure. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... 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. ...

P is known to contain many natural problems, including linear programming, calculating the greatest common divisor, and finding a maximum matching. In 2002, it was shown that the problem of determining if a number is prime is in P. The related class of function problems is FP. In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf) of two integers which are not both zero is the largest integer that divides both numbers. ... This article is about mathematical matchings. ... In mathematics, a prime number (or a prime) is a natural number greater than one whose only positive divisors are one and itself. ... In computational complexity theory, a function problem is a problem other than a decision problem, that is, a problem requiring a more complex answer than just YES or NO. Notable examples include the traveling salesman problem which asks for the route taken by the salesman, and the integer factorization problem... 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...

P is often taken to be the class of computational problems which are "efficiently solvable" or "tractable", although there are potentially larger classes that are also considered tractable such as RP and BPP. Also, there exist problems in P which are intractable in practical terms; for example, some require at least n^1000000 operations. See even harder problems of complexity classes for further discussion. This article relates to the theory of computation. ... This article is about the complexity class. ... Diagram of complexity classes provided that P ≠ NP. If P = NP, then all three classes are equal. ...

Relationships to other classes

A generalization of P is NP, which is the class of languages decidable in polynomial time on a non-deterministic Turing machine. We then trivially have P is a subset of NP. One of the largest open problems currently in theoretical computer science is whether P = NP; see complexity classes P and NP. 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 mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... 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... An open problem is a problem that can be formally stated and for which a solution is known to exist but which has not yet been solved. ... Diagram of complexity classes provided that P ≠ NP. If P = NP, then all three classes are equal. ...

P is also known to be at least as large as L, the class of problems decidable in a logarithmic amount of memory space. A decider using O(log n) space cannot use more than 2^{O(log n)}=n^O(1) time, because this is the total number of possible configurations; thus, L is a subset of P. Another important problem is whether L = P. We do know that P = ALOGSPACE, the set of problems solvable in logarithmic memory by alternating Turing machines. P is also known to be no larger than PSPACE, the class of problems decidable in polynomial space. Again, whether P = PSPACE is an open problem. To summarize: In computational complexity theory, L is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a logarithmic amount of memory space. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... The terms storage (U.K.) or memory (U.S.) refer to the parts of a digital computer that retain physical state (data) for some interval of time, possibly even after electrical power to the computer is turned off. ... In computational complexity theory, an alternating Turing machine is a non-deterministic Turing machine with a rule for accepting computations that generalizes the rules used in the definition of the complexity classes NP and co-NP. The definition of NP uses the existential mode of computation: if any choice leads... In complexity theory the class PSPACE, which equals NPSPACE by Savitchs theorem, is the set of decision problems that can be solved by a deterministic or nondeterministic Turing machine using a polynomial amount of memory and unlimited time. ...

Here, EXPTIME is the class of problems solvable in exponential time. Of all the classes shown above, only two strict containments are known: 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. ...

• P is strictly contained in EXPTIME. Consequently, all EXPTIME-hard problems lie outside P, and at least one of the containments to the right of P above is strict (in fact, it is widely believed that all three are strict).
• L is strictly contained in PSPACE.

The most difficult problems in P are P-complete problems. In complexity theory, the complexity class P-complete is a set of decision problems and is useful in the analysis of which problems can be efficiently solved on parallel computers. ...

Another generalization of P is P/poly, or Nonuniform Polynomial-Time. If a problem is in P/poly, then it can be solved in deterministic polynomial time provided that an advice string is given that depends only on the length of the input. Unlike for NP, however, the polynomial-time machine need not detect fraudulent advice strings; it is not a verifier. P/poly is a large class containing nearly all practical algorithms, including all of BPP. If it contains NP, then the polynomial hierarchy collapses to the second level. On the other hand, it also contains some impractical algorithms, including some undecidable problems such as the unary version of any undecidable problem. Advice is a concept in complexity theory. ... This article is about the complexity class. ... 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 logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ...

Properties

Polynomial-time algorithms are closed under composition. Intuitively, this says that if I write a function which is polynomial-time assuming that function calls are constant-time, and if those called functions themselves require polynomial time, then the entire algorithm takes polynomial time. One consequence of this is that P is low for itself. This is also one of the main reasons that P is considered to be a machine-independent class; any machine "feature", such as random access, which can be simulated in polynomial time can simply be composed with the main polynomial-time algorithm to reduce it to a polynomial-time algorithm on a more basic machine. In computational complexity theory, it is said that a complexity class B is low for a complexity class A if AB = A; that is, A with an oracle for B is equal to A. Such a statement implies that an abstract machine which solves problems in A achieves no additional... In computer science, random access is the ability to access a random element of a group in equal time. ...

References

• C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994. ISBN 0201530821.
• Complexity Zoo: P, P/poly
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 34.1: Polynomial time, pp.971–979.
• Michael Sipser (1997). Introduction to the Theory of Computation, PWS Publishing. ISBN 0-534-94728-X. Section 7.2: The Class P, pp.234–241.

Famous coauthor of Introduction to Algorithms, along with Charles Leiserson, Ron Rivest, and Cliff Stein. ... Charles 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. ... Introduction to Algorithms is a book by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. ... Michael Sipser is a Professor of Applied Mathematics in the Theory of Computation Group at MIT. His research area is complexity theory. ...