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In
computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ...
, a complexity class is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of
computational problem In theoretical computer science, a computational problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring :"Given a positive integer ''n'', find a nontrivial prime factor of ''n''." is a computati ...
s "of related resource-based
complexity Complexity characterizes the behavior of a system or model whose components interact in multiple ways and follow local rules, leading to non-linearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to c ...
". The two most commonly analyzed resources are
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
and
memory Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembe ...
. In general, a complexity class is defined in terms of a type of computational problem, a
model of computation In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how ...
, and a bounded resource like
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
or
memory Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembe ...
. In particular, most complexity classes consist of
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
s that are solvable with a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
, and are differentiated by their time or space (memory) requirements. For instance, the class P is the set of decision problems solvable by a deterministic Turing machine in
polynomial time In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...
. There are, however, many complexity classes defined in terms of other types of problems (e.g. counting problems and
function problem In computational complexity theory, a function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem. For function problems, the ou ...
s) and using other models of computation (e.g.
probabilistic Turing machine In theoretical computer science, a probabilistic Turing machine is a non-deterministic Turing machine that chooses between the available transitions at each point according to some probability distribution. As a consequence, a probabilistic Tur ...
s, interactive proof systems,
Boolean circuit In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible inpu ...
s, and quantum computers). The study of the relationships between complexity classes is a major area of research in
theoretical computer science Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Associati ...
. There are often general hierarchies of complexity classes; for example, it is known that a number of fundamental time and space complexity classes relate to each other in the following way: LNLPNP
PSPACE In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. Formal definition If we denote by SPACE(''f''(''n'')), the set of all problems that can ...
EXPTIME In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2''p''(''n'')) time, w ...
NEXPTIME In computational complexity theory, the complexity class NEXPTIME (sometimes called NEXP) is the set of decision problems that can be solved by a non-deterministic Turing machine using time 2^. In terms of NTIME, :\mathsf = \bigcup_ \mathsf(2^) ...
EXPSPACE Where ⊆ denotes the
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
relation. However, many relationships are not yet known; for example, one of the most famous
open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is kno ...
s in computer science concerns whether P equals NP. The relationships between classes often answer questions about the fundamental nature of computation. The P versus NP problem, for instance, is directly related to questions of whether nondeterminism adds any computational power to computers and whether problems having solutions that can be quickly checked for correctness can also be quickly solved.


Background

Complexity classes are sets of related
computational problem In theoretical computer science, a computational problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring :"Given a positive integer ''n'', find a nontrivial prime factor of ''n''." is a computati ...
s. They are defined in terms of the computational difficulty of solving the problems contained within them with respect to particular computational resources like time or memory. More formally, the definition of a complexity class consists of three things: a type of computational problem, a model of computation, and a bounded computational resource. In particular, most complexity classes consist of
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
s that can be solved by a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
with bounded
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
or
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
resources. For example, the complexity class P is defined as the set of
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
s that can be solved by a deterministic Turing machine in
polynomial time In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...
.


Computational problems

Intuitively, a
computational problem In theoretical computer science, a computational problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring :"Given a positive integer ''n'', find a nontrivial prime factor of ''n''." is a computati ...
is just a question that can be solved by an
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
. For example, "is the
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
n
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
?" is a computational problem. A computational problem is mathematically represented as the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of answers to the problem. In the primality example, the problem (call it \texttt) is represented by the set of all natural numbers that are prime: \texttt = \. In the theory of computation, these answers are represented as strings; for example, in the primality example the natural numbers could be represented as strings of bits that represent
binary number A binary number is a number expressed in the Radix, base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may ...
s. For this reason, computational problems are often synonymously referred to as languages, since strings of bits represent
formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...
s (a concept borrowed from
linguistics Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds ...
); for example, saying that the \texttt problem is in the complexity class P is equivalent to saying that the language \texttt is in P.


Decision problems

The most commonly analyzed problems in theoretical computer science are
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
s—the kinds of problems that can be posed as
yes–no question In linguistics, a yes–no question, also known as a binary question, a polar question, or a general question, is a closed-ended question whose expected answer is one of two choices, one that provides an affirmative answer to the question versus ...
s. The primality example above, for instance, is an example of a decision problem as it can be represented by the yes–no question "is the
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
n
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
". In terms of the theory of computation, a decision problem is represented as the set of input strings that a computer running a correct
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
would answer "yes" to. In the primality example, \texttt is the set of strings representing natural numbers that, when input into a computer running an algorithm that correctly tests for primality, the algorithm answers "yes, this number is prime". This "yes-no" format is often equivalently stated as "accept-reject"; that is, an algorithm "accepts" an input string if the answer to the decision problem is "yes" and "rejects" if the answer is "no". While some problems cannot easily be expressed as decision problems, they nonetheless encompass a broad range of computational problems. Other types of problems that certain complexity classes are defined in terms of include: *
Function problem In computational complexity theory, a function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem. For function problems, the ou ...
s (e.g. FP) * Counting problems (e.g. #P) *
Optimization problem In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goo ...
s *
Promise problem In computational complexity theory, a promise problem is a generalization of a decision problem where the input is promised to belong to a particular subset of all possible inputs. Unlike decision problems, the ''yes'' instances (the inputs for whi ...
s (see section "Other types of problems")


Computational models

To make concrete the notion of a "computer", in theoretical computer science problems are analyzed in the context of a
computational model A computational model uses computer programs to simulate and study complex systems using an algorithmic or mechanistic approach and is widely used in a diverse range of fields spanning from physics, engineering, chemistry and biology to economics ...
. Computational models make exact the notions of computational resources like "time" and "memory". In
computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ...
, complexity classes deal with the ''inherent'' resource requirements of problems and not the resource requirements that depend upon how a physical computer is constructed. For example, in the real world different computers may require different amounts of time and memory to solve the same problem because of the way that they have been engineered. By providing an abstract mathematical representations of computers, computational models abstract away superfluous complexities of the real world (like differences in processor speed) that obstruct an understanding of fundamental principles. The most commonly used computational model is the
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
. While other models exist and many complexity classes are defined in terms of them (see section "Other models of computation"), the Turing machine is used to define most basic complexity classes. With the Turing machine, instead of using standard units of time like the second (which make it impossible to disentangle running time from the speed of physical hardware) and standard units of memory like
byte The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable un ...
s, the notion of time is abstracted as the number of elementary steps that a Turing machine takes to solve a problem and the notion of memory is abstracted as the number of cells that are used on the machine's tape. These are explained in greater detail below. It is also possible to use the Blum axioms to define complexity classes without referring to a concrete
computational model A computational model uses computer programs to simulate and study complex systems using an algorithmic or mechanistic approach and is widely used in a diverse range of fields spanning from physics, engineering, chemistry and biology to economics ...
, but this approach is less frequently used in complexity theory.


Deterministic Turing machines

A Turing machine is a mathematical model of a general computing machine. It is the most commonly used model in complexity theory, owing in large part to the fact that it is believed to be as powerful as any other model of computation and is easy to analyze mathematically. Importantly, it is believed that if there exists an algorithm that solves a particular problem then there also exists a Turing machine that solves that same problem (this is known as the
Church–Turing thesis In Computability theory (computation), computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) ...
); this means that it is believed that ''every'' algorithm can be represented as a Turing machine. Mechanically, a Turing machine (TM) manipulates symbols (generally restricted to the bits 0 and 1 to provide an intuitive connection to real-life computers) contained on an infinitely long strip of tape. The TM can read and write, one at a time, using a tape head. Operation is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; if the symbol seen is 1, change into state 17; in state 17, if the symbol seen is 0, write a 1 and change to state 6". The Turing machine starts with only the input string on its tape and blanks everywhere else. The TM accepts the input if it enters a designated accept state and rejects the input if it enters a reject state. The deterministic Turing machine (DTM) is the most basic type of Turing machine. It uses a fixed set of rules to determine its future actions (which is why it is called "
deterministic Determinism is the metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping mo ...
"). A computational problem can then be defined in terms of a Turing machine as the set of input strings that a particular Turing machine accepts. For example, the primality problem \texttt from above is the set of strings (representing natural numbers) that a Turing machine running an algorithm that correctly tests for primality accepts. A Turing machine is said to recognize a language (recall that "problem" and "language" are largely synonymous in computability and complexity theory) if it accepts all inputs that are in the language and is said to decide a language if it additionally rejects all inputs that are not in the language (certain inputs may cause a Turing machine to run forever, so decidability places the additional constraint over recognizability that the Turing machine must halt on all inputs). A Turing machine that "solves" a problem is generally meant to mean one that decides the language. Turing machines enable intuitive notions of "time" and "space". The
time complexity In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations ...
of a TM on a particular input is the number of elementary steps that the Turing machine takes to reach either an accept or reject state. The
space complexity The space complexity of an algorithm or a data structure is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it exec ...
is the number of cells on its tape that it uses to reach either an accept or reject state.


Nondeterministic Turing machines

The deterministic Turing machine (DTM) is a variant of the nondeterministic Turing machine (NTM). Intuitively, an NTM is just a regular Turing machine that has the added capability of being able to explore multiple possible future actions from a given state, and "choosing" a branch that accepts (if any accept). That is, while a DTM must follow only one branch of computation, an NTM can be imagined as a computation tree, branching into many possible computational pathways at each step (see image). If at least one branch of the tree halts with an "accept" condition, then the NTM accepts the input. In this way, an NTM can be thought of as simultaneously exploring all computational possibilities in parallel and selecting an accepting branch. NTMs are not meant to be physically realizable models, they are simply theoretically interesting abstract machines that give rise to a number of interesting complexity classes (which often do have physically realizable equivalent definitions). The
time complexity In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations ...
of an NTM is the maximum number of steps that the NTM uses on ''any'' branch of its computation. Similarly, the
space complexity The space complexity of an algorithm or a data structure is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it exec ...
of an NTM is the maximum number of cells that the NTM uses on any branch of its computation. DTMs can be viewed as a special case of NTMs that do not make use of the power of nondeterminism. Hence, every computation that can be carried out by a DTM can also be carried out by an equivalent NTM. It is also possible to simulate any NTM using a DTM (the DTM will simply compute every possible computational branch one-by-one). Hence, the two are equivalent in terms of computability. However, simulating an NTM with a DTM often requires greater time and/or memory resources; as will be seen, how significant this slowdown is for certain classes of computational problems is an important question in computational complexity theory.


Resource bounds

Complexity classes group computational problems by their resource requirements. To do this, computational problems are differentiated by
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less ...
s on the maximum amount of resources that the most efficient algorithm takes to solve them. More specifically, complexity classes are concerned with the ''rate of growth'' in the resources required to solve particular computational problems as the input size increases. For example, the amount of time it takes to solve problems in the complexity class P grows at a
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
rate as the input size increases, which is comparatively slow compared to problems in the exponential complexity class
EXPTIME In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2''p''(''n'')) time, w ...
(or more accurately, for problems in EXPTIME that are outside of P, since \mathsf\subseteq\mathsf). Note that the study of complexity classes is intended primarily to understand the ''inherent'' complexity required to solve computational problems. Complexity theorists are thus generally concerned with finding the smallest complexity class that a problem falls into and are therefore concerned with identifying which class a computational problem falls into using the ''most efficient'' algorithm. There may be an algorithm, for instance, that solves a particular problem in exponential time, but if the most efficient algorithm for solving this problem runs in polynomial time then the inherent time complexity of that problem is better described as polynomial.


Time bounds

The
time complexity In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations ...
of an algorithm with respect to the Turing machine model is the number of steps it takes for a Turing machine to run an algorithm on a given input size. Formally, the time complexity for an algorithm implemented with a Turing machine M is defined as the function t_M: \mathbb \to \mathbb, where t_M(n) is the maximum number of steps that M takes on any input of length n. In computational complexity theory, theoretical computer scientists are concerned less with particular runtime values and more with the general class of functions that the time complexity function falls into. For instance, is the time complexity function a
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
? A
logarithmic function Logarithmic can refer to: * Logarithm, a transcendental function in mathematics * Logarithmic scale, the use of the logarithmic function to describe measurements * Logarithmic spiral, * Logarithmic growth * Logarithmic distribution, a discrete p ...
? An exponential function? Or another kind of function?


Space bounds

The
space complexity The space complexity of an algorithm or a data structure is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it exec ...
of an algorithm with respect to the Turing machine model is the number of cells on the Turing machine's tape that are required to run an algorithm on a given input size. Formally, the space complexity of an algorithm implemented with a Turing machine M is defined as the function s_M: \mathbb \to \mathbb, where s_M(n) is the maximum number of cells that M uses on any input of length n.


Basic complexity classes


Basic definitions

Complexity classes are often defined using granular sets of complexity classes called DTIME and NTIME (for time complexity) and DSPACE and NSPACE (for space complexity). Using
big O notation Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
, they are defined as follows: * The time complexity class \mathsf(t(n)) is the set of all problems that are decided by an O(t(n)) time deterministic Turing machine. * The time complexity class \mathsf(t(n)) is the set of all problems that are decided by an O(t(n)) time nondeterministic Turing machine. * The space complexity class \mathsf(s(n)) is the set of all problems that are decided by an O(s(n)) space deterministic Turing machine. * The space complexity class \mathsf(s(n)) is the set of all problems that are decided by an O(s(n)) space nondeterministic Turing machine.


Time complexity classes


P and NP

P is the class of problems that are solvable by a deterministic Turing machine in
polynomial time In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...
and NP is the class of problems that are solvable by a
nondeterministic Turing machine In theoretical computer science, a nondeterministic Turing machine (NTM) is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations. That is, an NTM's next state is ''not'' comp ...
in polynomial time. Or more formally, : \mathsf = \bigcup_ \mathsf(n^k) : \mathsf = \bigcup_ \mathsf(n^k) P is often said to be the class of problems that can be solved "quickly" or "efficiently" by a deterministic computer, since the
time complexity In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations ...
of solving a problem in P increases relatively slowly with the input size. An important characteristic of the class NP is that it can be equivalently defined as the class of problems whose solutions are ''verifiable'' by a deterministic Turing machine in polynomial time. That is, a language is in NP if there exists a ''deterministic'' polynomial time Turing machine, referred to as the verifier, that takes as input a string w ''and'' a polynomial-size certificate string c, and accepts w if w is in the language and rejects w if w is not in the language. Intuitively, the certificate acts as a
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...
that the input w is in the language. Formally: : NP is the class of languages L for which there exists a polynomial-time deterministic Turing machine M and a polynomial p such that for all w \in \^*, w is in L ''if and only if'' there exists some c \in \^ such that M(w,c) accepts. This equivalence between the nondeterministic definition and the verifier definition highlights a fundamental connection between nondeterminism and solution verifiability. Furthermore, it also provides a useful method for proving that a language is in NP—simply identify a suitable certificate and show that it can be verified in polynomial time.


=The P versus NP problem

= While there might seem to be an obvious difference between the class of problems that are efficiently solvable and the class of problems whose solutions are merely efficiently checkable, P and NP are actually at the center of one of the most famous unsolved problems in computer science: the P versus NP problem. While it is known that \mathsf \subseteq \mathsf (intuitively, deterministic Turing machines are just a subclass of nondeterministic Turing machines that don't make use of their nondeterminism; or under the verifier definition, P is the class of problems whose polynomial time verifiers need only receive the empty string as their certificate), it is not known whether NP is strictly larger than P. If P=NP, then it follows that nondeterminism provides ''no additional computational power'' over determinism with regards to the ability to quickly find a solution to a problem; that is, being able to explore ''all possible branches'' of computation provides ''at most'' a polynomial speedup over being able to explore only a single branch. Furthermore, it would follow that if there exists a proof for a problem instance and that proof can be quickly be checked for correctness (that is, if the problem is in NP), then there also exists an algorithm that can quickly ''construct'' that proof (that is, the problem is in P). However, the overwhelming majority of computer scientists believe that \mathsf\neq\mathsf, and most cryptographic schemes employed today rely on the assumption that \mathsf\neq\mathsf.


EXPTIME and NEXPTIME

EXPTIME (sometimes shortened to EXP) is the class of decision problems solvable by a deterministic Turing machine in exponential time and NEXPTIME (sometimes shortened to NEXP) is the class of decision problems solvable by a nondeterministic Turing machine in exponential time. Or more formally, : \mathsf = \bigcup_ \mathsf(2^) : \mathsf = \bigcup_ \mathsf(2^) EXPTIME is a strict superset of P and NEXPTIME is a strict superset of NP. It is further the case that EXPTIME\subseteqNEXPTIME. It is not known whether this is proper, but if P=NP then EXPTIME must equal NEXPTIME.


Space complexity classes


L and NL

While it is possible to define logarithmic time complexity classes, these are extremely narrow classes as sublinear times do not even enable a Turing machine to read the entire input (because \log n < n ). However, there are a meaningful number of problems that can be solved in logarithmic space. The definitions of these classes require a two-tape Turing machine so that it is possible for the machine to store the entire input (it can be shown that in terms of
computability Computability is the ability to solve a problem by an effective procedure. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is c ...
the two-tape Turing machine is equivalent to the single-tape Turing machine). In the two-tape Turing machine model, one tape is the input tape, which is read-only. The other is the work tape, which allows both reading and writing and is the tape on which the Turing machine performs computations. The space complexity of the Turing machine is measured as the number of cells that are used on the work tape. L (sometimes lengthened to LOGSPACE) is then defined as the class of problems solvable in logarithmic space on a deterministic Turing machine and NL (sometimes lengthened to NLOGSPACE) is the class of problems solvable in logarithmic space on a nondeterministic Turing machine. Or more formally, :\mathsf = \mathsf(\log n) :\mathsf = \mathsf(\log n) It is known that \mathsf\subseteq\mathsf\subseteq\mathsf. However, it is not known whether any of these relationships is proper.


PSPACE and NPSPACE

The complexity classes PSPACE and NPSPACE are the space analogues to P and NP. That is, PSPACE is the class of problems solvable in polynomial space by a deterministic Turing machine and NPSPACE is the class of problems solvable in polynomial space by a nondeterministic Turing machine. More formally, :\mathsf = \bigcup_ \mathsf(n^k) :\mathsf = \bigcup_ \mathsf(n^k) While it is not known whether P=NP, Savitch's theorem famously showed that PSPACE=NPSPACE. It is also known that \mathsf \subseteq \mathsf, which follows intuitively from the fact that, since writing to a cell on a Turing machine's tape is defined as taking one unit of time, a Turing machine operating in polynomial time can only write to polynomially many cells. It is suspected that P is strictly smaller than PSPACE, but this has not been proven.


EXPSPACE and NEXPSPACE

The complexity classes EXPSPACE and NEXPSPACE are the space analogues to
EXPTIME In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2''p''(''n'')) time, w ...
and
NEXPTIME In computational complexity theory, the complexity class NEXPTIME (sometimes called NEXP) is the set of decision problems that can be solved by a non-deterministic Turing machine using time 2^. In terms of NTIME, :\mathsf = \bigcup_ \mathsf(2^) ...
. That is, EXPSPACE is the class of problems solvable in exponential space by a deterministic Turing machine and NEXPSPACE is the class of problems solvable in exponential space by a nondeterministic Turing machine. Or more formally, :\mathsf = \bigcup_ \mathsf(2^) :\mathsf = \bigcup_ \mathsf(2^) Savitch's theorem showed that EXPSPACE=NEXPSPACE. This class is extremely broad: it is known to be a strict superset of PSPACE, NP, and P, and is believed to be a strict superset of EXPTIME.


Properties of complexity classes


Closure

Complexity classes have a variety of closure properties. For example, decision classes may be closed under
negation In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...
,
disjunction In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is ...
, conjunction, or even under all Boolean operations. Moreover, they might also be closed under a variety of quantification schemes. P, for instance, is closed under all Boolean operations, and under quantification over polynomially sized domains. Closure properties can be helpful in separating classes—one possible route to separating two complexity classes is to find some closure property possessed by one class but not by the other. Each class X that is not closed under negation has a complement class co-X, which consists of the complements of the languages contained in X (i.e. \textsf = \). co-NP, for instance, is one important complement complexity class, and sits at the center of the unsolved problem over whether co-NP=NP. Closure properties are one of the key reasons many complexity classes are defined in the way that they are. Take, for example, a problem that can be solved in O(n) time (that is, in linear time) and one that can be solved in, at best, O(n^) time. Both of these problems are in P, yet the runtime of the second grows considerably faster than the runtime of the first as the input size increases. One might ask whether it would be better to define the class of "efficiently solvable" problems using some smaller polynomial bound, like O(n^3), rather than all polynomials, which allows for such large discrepancies. It turns out, however, that the set of all polynomials is the smallest class of functions containing the linear functions that is also closed under addition, multiplication, and composition (for instance, O(n^3) \circ O(n^2) = O(n^6), which is a polynomial but O(n^6)>O(n^3)). Since we would like composing one efficient algorithm with another efficient algorithm to still be considered efficient, the polynomials are the smallest class that ensures composition of "efficient algorithms". (Note that the definition of P is also useful because, empirically, almost all problems in P that are practically useful do in fact have low order polynomial runtimes, and almost all problems outside of P that are practically useful do not have any known algorithms with small exponential runtimes, i.e. with O(c^n) runtimes where is close to 1.)


Reductions

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem, i.e. a reduction takes inputs from one problem and transforms them into inputs of another problem. For instance, you can reduce ordinary base-10 addition x+y to base-2 addition by transforming x and y to their base-2 notation (e.g. 5+7 becomes 101+111). Formally, a problem X reduces to a problem Y if there exists a function f such that for every x \in \Sigma^* , x \in X ''if and only if'' f(x) \in Y. Generally, reductions are used to capture the notion of a problem being at least as difficult as another problem. Thus we are generally interested in using a polynomial-time reduction, since any problem X that can be efficiently reduced to another problem Y is no more difficult than Y. Formally, a problem X is polynomial-time reducible to a problem Y if there exists a ''polynomial-time'' computable function p such that for all x \in \Sigma^*, x \in X ''if and only if'' p(x) \in Y. Note that reductions can be defined in many different ways. Common reductions are
Cook reduction In computability theory, a Turing reduction from a decision problem A to a decision problem B is an oracle machine that decides problem A given an oracle for B (Rogers 1967, Soare 1987) in finitely many steps. It can be understood as an algorithm ...
s, Karp reductions and Levin reductions, and can vary based on resource bounds, such as
polynomial-time reduction In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming ...
s and
log-space reduction In computational complexity theory, a log-space reduction is a reduction (complexity), reduction computable by a deterministic Turing machine using logarithmic space. Conceptually, this means it can keep a constant number of Pointer (computer progr ...
s.


Hardness

Reductions motivate the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be polynomial-time reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C with at most polynomial slowdown. Of particular importance, the set of problems that are hard for NP is called the set of
NP-hard In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assumi ...
problems.


Completeness

If a problem X is hard for C and is also in C, then X is said to be complete for C. This means that X is the hardest problem in C (since there could be many problems that are equally hard, more precisely X is as hard as the hardest problems in C). Of particular importance is the class of NP-complete problems—the most difficult problems in NP. Because all problems in NP can be polynomial-time reduced to NP-complete problems, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.


Relationships between complexity classes


Savitch's theorem

Savitch's theorem establishes the relationship between deterministic and nondetermistic space resources. It shows that if a nondeterministic Turing machine can solve a problem using f(n) space, then a deterministic Turing machine can solve the same problem in f(n)^2 space, i.e. in the square of the space. Formally, Savitch's theorem states that for any f(n) > n , :\mathsf\left(f\left(n\right)\right) \subseteq \mathsf\left(f\left(n\right)^2\right). Important corollaries of Savitch's theorem are that PSPACE = NPSPACE (since the square of a polynomial is still a polynomial) and EXPSPACE = NEXPSPACE (since the square of an exponential is still an exponential). These relationships answer fundamental questions about the power of nondeterminism compared to determinism. Specifically, Savitch's theorem shows that any problem that a nondeterministic Turing machine can solve in polynomial space, a deterministic Turing machine can also solve in polynomial space. Similarly, any problem that a nondeterministic Turing machine can solve in exponential space, a deterministic Turing machine can also solve in exponential space.


Hierarchy theorems

By definition of DTIME, it follows that \mathsf(n^) is contained in \mathsf(n^) if k_1 \leq k_2 , since O(n^) \subseteq O(n^) if k_1 \leq k_2. However, this definition gives no indication of whether this inclusion is strict. For time and space requirements, the conditions under which the inclusion is strict are given by the time and space hierarchy theorems, respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. The hierarchy theorems enable one to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved. The time hierarchy theorem states that :\mathsf\big(f(n) \big) \subsetneq \mathsf \big(f(n) \sdot \log^(f(n)) \big). The space hierarchy theorem states that :\mathsf\big(f(n)\big) \subsetneq \mathsf \big(f(n) \sdot \log(f(n)) \big). The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem establishes that P is strictly contained in EXPTIME, and the space hierarchy theorem establishes that L is strictly contained in PSPACE.


Other models of computation

While deterministic and non-deterministic
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
s are the most commonly used models of computation, many complexity classes are defined in terms of other computational models. In particular, * A number of classes are defined using
probabilistic Turing machine In theoretical computer science, a probabilistic Turing machine is a non-deterministic Turing machine that chooses between the available transitions at each point according to some probability distribution. As a consequence, a probabilistic Tur ...
s, including the classes BPP, PP, RP, and ZPP * A number of classes are defined using interactive proof systems, including the classes IP, MA, and AM * A number of classes are defined using
Boolean circuit In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible inpu ...
s, including the classes P/poly and its subclasses NC and AC * A number of classes are defined using quantum Turing machines, including the classes BQP and QMA These are explained in greater detail below.


Randomized computation

A number of important complexity classes are defined using the
probabilistic Turing machine In theoretical computer science, a probabilistic Turing machine is a non-deterministic Turing machine that chooses between the available transitions at each point according to some probability distribution. As a consequence, a probabilistic Tur ...
, a variant of the
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
that can toss random coins. These classes help to better describe the complexity of
randomized algorithm A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performan ...
s. A probabilistic Turing machine is similar to a deterministic Turing machine, except rather than following a single transition function (a set of rules for how to proceed at each step of the computation) it probabilistically selects between multiple transition functions at each step. The standard definition of a probabilistic Turing machine specifies two transition functions, so that the selection of transition function at each step resembles a coin flip. The randomness introduced at each step of the computation introduces the potential for error; that is, strings that the Turing machine is meant to accept may on some occasions be rejected and strings that the Turing machine is meant to reject may on some occasions be accepted. As a result, the complexity classes based on the probabilistic Turing machine are defined in large part around the amount of error that is allowed. Formally, they are defined using an error probability \epsilon. A probabilistic Turing machine M is said to recognize a language L with error probability \epsilon if: # a string w in L implies that \text \text w\geq 1 - \epsilon # a string w not in L implies that \text \text w\geq 1 - \epsilon


Important complexity classes

The fundamental randomized time complexity classes are ZPP, RP, co-RP, BPP, and PP. The strictest class is ZPP (zero-error probabilistic polynomial time), the class of problems solvable in polynomial time by a probabilistic Turing machine with error probability 0. Intuitively, this is the strictest class of probabilistic problems because it demands ''no error whatsoever''. A slightly looser class is RP (randomized polynomial time), which maintains no error for strings not in the language but allows bounded error for strings in the language. More formally, a language is in RP if there is a probabilistic polynomial-time Turing machine M such that if a string is not in the language then M always rejects and if a string is in the language then M accepts with a probability at least 1/2. The class co-RP is similarly defined except the roles are flipped: error is not allowed for strings in the language but is allowed for strings not in the language. Taken together, the classes RP and co-RP encompass all of the problems that can be solved by probabilistic Turing machines with one-sided error. Loosening the error requirements further to allow for two-sided error yields the class BPP (bounded-error probabilistic polynomial time), the class of problems solvable in polynomial time by a probabilistic Turing machine with error probability less than 1/3 (for both strings in the language and not in the language). BPP is the most practically relevant of the probabilistic complexity classes—problems in BPP have efficient
randomized algorithm A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performan ...
s that can be run quickly on real computers. BPP is also at the center of the important unsolved problem in computer science over whether P=BPP, which if true would mean that randomness does not increase the computational power of computers, i.e. any probabilistic Turing machine could be simulated by a deterministic Turing machine with at most polynomial slowdown. The broadest class of efficiently-solvable probabilistic problems is PP (probabilistic polynomial time), the set of languages solvable by a probabilistic Turing machine in polynomial time with an error probability of less than 1/2 for all strings. ZPP, RP and co-RP are all subsets of BPP, which in turn is a subset of PP. The reason for this is intuitive: the classes allowing zero error and only one-sided error are all contained within the class that allows two-sided error, and PP simply relaxes the error probability of BPP. ZPP relates to RP and co-RP in the following way: \textsf=\textsf\cap\textsf. That is, ZPP consists exactly of those problems that are in both RP and co-RP. Intuitively, this follows from the fact that RP and co-RP allow only one-sided error: co-RP does not allow error for strings in the language and RP does not allow error for strings not in the language. Hence, if a problem is in both RP and co-RP, then there must be no error for strings both in ''and'' not in the language (i.e. no error whatsoever), which is exactly the definition of ZPP. Important randomized space complexity classes include BPL, RL, and RLP.


Interactive proof systems

A number of complexity classes are defined using interactive proof systems. Interactive proofs generalize the proofs definition of the complexity class NP and yield insights into
cryptography Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), ...
,
approximation algorithm In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned sol ...
s, and
formal verification In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of a system with respect to a certain formal specification or property, using formal methods of mathematics. Formal ver ...
. Interactive proof systems are
abstract machine In computer science, an abstract machine is a theoretical model that allows for a detailed and precise analysis of how a computer system functions. It is similar to a mathematical function in that it receives inputs and produces outputs based on p ...
s that model computation as the exchange of messages between two parties: a prover P and a verifier V. The parties interact by exchanging messages, and an input string is accepted by the system if the verifier decides to accept the input on the basis of the messages it has received from the prover. The prover P has unlimited computational power while the verifier has bounded computational power (the standard definition of interactive proof systems defines the verifier to be polynomially-time bounded). The prover, however, is untrustworthy (this prevents all languages from being trivially recognized by the proof system by having the computationally unbounded prover solve for whether a string is in a language and then sending a trustworthy "YES" or "NO" to the verifier), so the verifier must conduct an "interrogation" of the prover by "asking it" successive rounds of questions, accepting only if it develops a high degree of confidence that the string is in the language.


Important complexity classes

The class NP is a simple proof system in which the verifier is restricted to being a deterministic polynomial-time
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
and the procedure is restricted to one round (that is, the prover sends only a single, full proof—typically referred to as the certificate—to the verifier). Put another way, in the definition of the class NP (the set of decision problems for which the problem instances, when the answer is "YES", have proofs verifiable in polynomial time by a deterministic Turing machine) is a proof system in which the proof is constructed by an unmentioned prover and the deterministic Turing machine is the verifier. For this reason, NP can also be called dIP (deterministic interactive proof), though it is rarely referred to as such. It turns out that NP captures the full power of interactive proof systems with deterministic (polynomial-time) verifiers because it can be shown that for any proof system with a deterministic verifier it is never necessary to need more than a single round of messaging between the prover and the verifier. Interactive proof systems that provide greater computational power over standard complexity classes thus require ''probabilistic'' verifiers, which means that the verifier's questions to the prover are computed using
probabilistic algorithm A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performan ...
s. As noted in the section above on randomized computation, probabilistic algorithms introduce error into the system, so complexity classes based on probabilistic proof systems are defined in terms of an error probability \epsilon. The most general complexity class arising out of this characterization is the class IP (interactive polynomial time), which is the class of all problems solvable by an interactive proof system (P,V), where V is probabilistic polynomial-time and the proof system satisfies two properties: for a language L \in \mathsf # (Completeness) a string w in L implies \Pr \textw \text P\ge \tfrac # (Soundness) a string w not in L implies \Pr \textw \text P\le \tfrac An important feature of IP is that it equals
PSPACE In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. Formal definition If we denote by SPACE(''f''(''n'')), the set of all problems that can ...
. In other words, any problem that can be solved by a polynomial-time interactive proof system can also be solved by a deterministic Turing machine with polynomial space resources, and vice versa. A modification of the protocol for IP produces another important complexity class: AM (Arthur–Merlin protocol). In the definition of interactive proof systems used by IP, the prover was not able to see the coins utilized by the verifier in its probabilistic computation—it was only able to see the messages that the verifier produced with these coins. For this reason, the coins are called ''private random coins''. The interactive proof system can be constrained so that the coins used by the verifier are ''public random coins''; that is, the prover is able to see the coins. Formally, AM is defined as the class of languages with an interactive proof in which the verifier sends a random string to the prover, the prover responds with a message, and the verifier either accepts or rejects by applying a deterministic polynomial-time function to the message from the prover. AM can be generalized to AM 'k'' where ''k'' is the number of messages exchanged (so in the generalized form the standard AM defined above is AM . However, it is the case that for all k \geq 2, AM 'k''AM It is also the case that \mathsf subseteq\mathsf /math>. Other complexity classes defined using interactive proof systems include MIP (multiprover interactive polynomial time) and QIP (quantum interactive polynomial time).


Boolean circuits

An alternative model of computation to the
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
is the
Boolean circuit In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible inpu ...
, a simplified model of the
digital circuit In theoretical computer science, a circuit is a model of computation in which input values proceed through a sequence of gates, each of which computes a function. Circuits of this kind provide a generalization of Boolean circuits and a mathematica ...
s used in modern
computer A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic set ...
s. Not only does this model provide an intuitive connection between computation in theory and computation in practice, but it is also a natural model for non-uniform computation (computation in which different input sizes within the same problem use different algorithms). Formally, a Boolean circuit C is a
directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one ...
in which edges represent wires (which carry the bit values 0 and 1), the input bits are represented by source vertices (vertices with no incoming edges), and all non-source vertices represent
logic gate A logic gate is a device that performs a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic gate, one that has, for ...
s (generally the AND, OR, and
NOT gate Not or NOT may also refer to: Language * Not, the general declarative form of "no", indicating a negation of a related statement that usually precedes * ... Not!, a grammatical construction used as a contradiction, popularized in the early 1990 ...
s). One logic gate is designated the output gate, and represents the end of the computation. The input/output behavior of a circuit C with n input variables is represented by the
Boolean function In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth functi ...
f_C:\^n \to \; for example, on input bits x_1,x_2,...,x_n, the output bit b of the circuit is represented mathematically as b = f_C(x_1,x_2,...,x_n). The circuit C is said to ''compute'' the Boolean function f_C. Any particular circuit has a fixed number of input vertices, so it can only act on inputs of that size.
Languages Language is a structured system of communication that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed forms, and may also be conveyed through writing. Human language is ch ...
(the formal representations of
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
s), however, contain strings of differing lengths, so languages cannot be fully captured by a single circuit (this contrasts with the Turing machine model, in which a language is fully described by a single Turing machine that can act on any input size). A language is thus represented by a circuit family. A circuit family is an infinite list of circuits (C_0,C_1,C_2,...), where C_n is a circuit with n input variables. A circuit family is said to decide a language L if, for every string w, w is in the language L if and only if C_n(w)=1, where n is the length of w. In other words, a string w of size n is in the language represented by the circuit family (C_0,C_1,C_2,...) if the circuit C_n (the circuit with the same number of input vertices as the number of bits in w) evaluates to 1 when w is its input. While complexity classes defined using Turing machines are described in terms of
time complexity In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations ...
, circuit complexity classes are defined in terms of circuit size — the number of vertices in the circuit. The size complexity of a circuit family (C_0,C_1,C_2,...) is the function f:\mathbb \to \mathbb, where f(n) is the circuit size of C_n. The familiar function classes follow naturally from this; for example, a polynomial-size circuit family is one such that the function f is a
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
.


Important complexity classes

The complexity class P/poly is the set of languages that are decidable by polynomial-size circuit families. It turns out that there is a natural connection between circuit complexity and time complexity. Intuitively, a language with small time complexity (that is, requires relatively few sequential operations on a Turing machine), also has a small circuit complexity (that is, requires relatively few Boolean operations). Formally, it can be shown that if a language is in \mathsf(t(n)), where t is a function t:\mathbb \to \mathbb, then it has circuit complexity O(t^2(n)). It follows directly from this fact that \mathsf\subset\textsf. In other words, any problem that can be solved in polynomial time by a deterministic Turing machine can also be solved by a polynomial-size circuit family. It is further the case that the inclusion is proper, i.e. \textsf\subsetneq \textsf (for example, there are some
undecidable problem In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an ...
s that are in P/poly). P/poly has a number of properties that make it highly useful in the study of the relationships between complexity classes. In particular, it is helpful in investigating problems related to P versus NP. For example, if there is any language in NP that is not in P/poly, then \mathsf\neq\mathsf. P/poly is also helpful in investigating properties of the
polynomial hierarchy In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. ...
. For example, if NP ⊆ P/poly, then PH collapses to \Sigma_2^. A full description of the relations between P/poly and other complexity classes is available at " Importance of P/poly". P/poly is also helpful in the general study of the properties of
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
s, as the class can be equivalently defined as the class of languages recognized by a polynomial-time Turing machine with a polynomial-bounded advice function. Two subclasses of P/poly that have interesting properties in their own right are NC and AC. These classes are defined not only in terms of their circuit size but also in terms of their depth. The depth of a circuit is the length of the longest directed path from an input node to the output node. The class NC is the set of languages that can be solved by circuit families that are restricted not only to having polynomial-size but also to having polylogarithmic depth. The class AC is defined similarly to NC, however gates are allowed to have unbounded fan-in (that is, the AND and OR gates can be applied to more than two bits). NC is a notable class because it can be equivalently defined as the class of languages that have efficient
parallel algorithm In computer science, a parallel algorithm, as opposed to a traditional serial algorithm, is an algorithm which can do multiple operations in a given time. It has been a tradition of computer science to describe serial algorithms in abstract mach ...
s.


Quantum computation

The classes BQP and QMA, which are of key importance in
quantum information science Quantum information science is a field that combines the principles of quantum mechanics with information theory to study the processing, analysis, and transmission of information. It covers both theoretical and experimental aspects of quantum phys ...
, are defined using quantum Turing machines.


Other types of problems

While most complexity classes studied by computer scientists are sets of
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
s, there are also a number of complexity classes defined in terms of other types of problems. In particular, there are complexity classes consisting of counting problems,
function problem In computational complexity theory, a function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem. For function problems, the ou ...
s, and
promise problem In computational complexity theory, a promise problem is a generalization of a decision problem where the input is promised to belong to a particular subset of all possible inputs. Unlike decision problems, the ''yes'' instances (the inputs for whi ...
s. These are explained in greater detail below.


Counting problems

A counting problem asks not only ''whether'' a solution exists (as with a
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
), but asks ''how many'' solutions exist. For example, the decision problem \texttt asks ''whether'' a particular graph G has a simple cycle (the answer is a simple yes/no); the corresponding counting problem \#\texttt (pronounced "sharp cycle") asks ''how many'' simple cycles G has. The output to a counting problem is thus a number, in contrast to the output for a decision problem, which is a simple yes/no (or accept/reject, 0/1, or other equivalent scheme). Thus, whereas decision problems are represented mathematically as
formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...
s, counting problems are represented mathematically as functions: a counting problem is formalized as the function f:\^* \to \mathbb such that for every input w \in \^*, f(w) is the number of solutions. For example, in the \#\texttt problem, the input is a graph G \in \^* (a graph represented as a string of bits) and f(G) is the number of simple cycles in G. Counting problems arise in a number of fields, including
statistical estimation Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value ...
,
statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, network design, and
economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...
.


Important complexity classes

#P (pronounced "sharp P") is an important class of counting problems that can be thought of as the counting version of NP. The connection to NP arises from the fact that the number of solutions to a problem equals the number of accepting branches in a
nondeterministic Turing machine In theoretical computer science, a nondeterministic Turing machine (NTM) is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations. That is, an NTM's next state is ''not'' comp ...
's computation tree. #P is thus formally defined as follows: : #P is the set of all functions f:\^* \to \mathbb such that there is a polynomial time nondeterministic Turing machine M such that for all w \in \^*, f(w) equals the number of accepting branches in M's computation tree on w. And just as NP can be defined both in terms of nondeterminism and in terms of a verifier (i.e. as an interactive proof system), so too can #P be equivalently defined in terms of a verifier. Recall that a decision problem is in NP if there exists a polynomial-time checkable certificate to a given problem instance—that is, NP asks whether there exists a proof of membership (a certificate) for the input that can be checked for correctness in polynomial time. The class #P asks ''how many'' such certificates exist. In this context, #P is defined as follows: : #P is the set of functions f: \^* \to \mathbb such that there exists a polynomial p: \mathbb \to \mathbb and a polynomial-time Turing machine V (the verifier), such that for every w \in \^*, f(w)=\Big, \big\\Big, . In other words, f(w) equals the size of the set containing all of the polynomial-size certificates for w.


Function problems

Counting problems are a subset of a broader class of problems called function problems. A function problem is a type of problem in which the values of a function f:A \to B are computed. Formally, a function problem f is defined as a relation R over strings of an arbitrary
alphabet An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from a ...
\Sigma: : R \subseteq \Sigma^* \times \Sigma^* An algorithm solves f if for every input x such that there exists a y satisfying (x, y) \in R, the algorithm produces one such y. This is just another way of saying that f is a function and the algorithm solves f(x) for all x \in \Sigma^*.


Important complexity classes

An important function complexity class is FP, the class of efficiently solvable functions. More specifically, FP is the set of function problems that can be solved by a deterministic Turing machine in
polynomial time In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...
. FP can be thought of as the function problem equivalent of P. Importantly, FP provides some insight into both counting problems and P versus NP. If #P=FP, then the functions that determine the number of certificates for problems in NP are efficiently solvable. And since computing the number of certificates is at least as hard as determining whether a certificate exists, it must follow that if #P=FP then P=NP (it is not known whether this holds in the reverse, i.e. whether P=NP implies #P=FP). Just as FP is the function problem equivalent of P, FNP is the function problem equivalent of NP. Importantly, FP=FNP if and only if P=NP.


Promise problems

Promise problems are a generalization of decision problems in which the input to a problem is guaranteed ("promised") to be from a particular subset of all possible inputs. Recall that with a decision problem L \subseteq \^*, an algorithm M for L must act (correctly) on ''every'' w \in \^*. A promise problem loosens the input requirement on M by restricting the input to some subset of \^*. Specifically, a promise problem is defined as a pair of non-intersecting sets (\Pi_,\Pi_), where: * \Pi_ \subseteq \^* is the set of all inputs that are accepted. * \Pi_ \subseteq \^* is the set of all inputs that are rejected. The input to an algorithm M for a promise problem (\Pi_,\Pi_) is thus \Pi_ \cup \Pi_, which is called the promise. Strings in \Pi_ \cup \Pi_ are said to ''satisfy the promise''. By definition, \Pi_ and \Pi_ must be disjoint, i.e. \Pi_ \cap \Pi_ = \emptyset. Within this formulation, it can be seen that decision problems are just the subset of promise problems with the trivial promise \Pi_ \cup \Pi_ = \^*. With decision problems it is thus simpler to simply define the problem as only \Pi_ (with \Pi_ implicitly being \^* / \Pi_), which throughout this page is denoted L to emphasize that \Pi_=L is a
formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...
. Promise problems make for a more natural formulation of many computational problems. For instance, a computational problem could be something like "given a
planar graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...
, determine whether or not..." This is often stated as a decision problem, where it is assumed that there is some translation schema that takes ''every'' string s \in \^* to a planar graph. However, it is more straightforward to define this as a promise problem in which the input is promised to be a planar graph.


Relation to complexity classes

Promise problems provide an alternate definition for standard complexity classes of decision problems. P, for instance, can be defined as a promise problem: : P is the class of promise problems that are solvable in deterministic polynomial time. That is, the promise problem (\Pi_,\Pi_) is in P if there exists a polynomial-time algorithm M such that: :* For every x \in \Pi_, M(x)=1 :* For ever x \in \Pi_, M(x)=0 Classes of decision problems—that is, classes of problems defined as formal languages—thus translate naturally to promise problems, where a language L in the class is simply L= \Pi_ and \Pi_ is implicitly \^* / \Pi_. Formulating many basic complexity classes like P as promise problems provides little additional insight into their nature. However, there are some complexity classes for which formulating them as promise problems have been useful to computer scientists. Promise problems have, for instance, played a key role in the study of SZK (statistical zero-knowledge).


Summary of relationships between complexity classes

The following table shows some of the classes of problems that are considered in complexity theory. If class X is a strict
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of Y, then X is shown below Y with a dark line connecting them. If X is a subset, but it is unknown whether they are equal sets, then the line is lighter and dotted. Technically, the breakdown into decidable and undecidable pertains more to the study of
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ex ...
, but is useful for putting the complexity classes in perspective.


See also

* List of complexity classes


Notes


References


Bibliography

* * * * * * * * * * * *


Further reading


The Complexity Zoo
: A huge list of complexity classes, a reference for experts. * Includes a diagram showing the hierarchy of complexity classes and how they fit together. * Michael Garey, and David S. Johnson: ''Computers and Intractability: A Guide to the Theory of NP-Completeness.'' New York: W. H. Freeman & Co., 1979. The standard reference on NP-Complete problems - an important category of problems whose solutions appear to require an impractically long time to compute. {{ComplexityClasses * Measures of complexity Theoretical computer science