770:
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).
762:
5927:
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
685:
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 "
5454:. 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.
6586:
6204:
31:
7220:
9075:
661:
469:
221:. 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
6658:
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.
789:
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.
6684:(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,
3754:, 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.
615:
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.
2887:
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
6657:
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
5161:
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.
769:
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
684:
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
788:
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
672:
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
6699:
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
5926:
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
8642:
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
126:
The study of the relationships between complexity classes is a major area of research in theoretical computer science. 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
7972:), 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
7077:
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.
3952:, 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,
631:), 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
11945:
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
893:
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.
888:
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
5137:
5606:
5760:
635:, 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.
1055:
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
3916:, 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
554:, 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".
7379:
1086:
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
3408:, 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
4090:). 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
6254:(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
2413:
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
3685:
3570:
2752:
2643:
912:
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
3334:
3232:
10126:
6362:(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).
1866:
557:
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:
6384:, 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.
4160:. 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
1782:
7065:, 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
3089:
380:
11374:
8836:. 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.
11306:
3828:
8833:
6477:
6637:. 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
11202:
11158:
4385:
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
11660:
11118:
10939:
8868:
879:
11048:
10994:
3406:
3039:
10341:
2971:
13572:
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.
7188:
2379:
5412:
5020:
1148:
974:
10700:
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
5467:
11936:
11797:
11741:
11482:
8937:
2502:
2461:
4030:
9927:
8751:
8546:
7908:
5622:
9891:
9657:
9346:
7736:
5300:
5241:
2326:
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,
9284:
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).
6193:
6100:
11982:
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
8993:
1612:
1514:
8884:
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
8695:
8510:
8365:
8045:
7029:
6927:
4914:
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
1416:
1321:
11855:
11535:
10743:
805:
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
7801:
2269:
11882:
11428:
11401:
11260:
11233:
4088:
2381:(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,
9447:
9258:
7226:
6824:
692:
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
724:
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
9203:
6393:(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.
6337:
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
714:
548:
454:
424:
323:
11595:
10831:
10548:
9992:
9722:
9492:
9391:
4541:
4317:
2162:
5452:
5340:
6267:(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
13467:
12780:
12745:
12527:
12475:
12466:
12459:
12436:
12414:
12391:
12382:
12375:
12295:
12286:
12279:
12256:
12247:
12240:
12211:
12202:
12195:
12157:
12133:
12109:
10890:
9046:
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
8796:
3910:
2877:
12589:
12565:
12558:
12536:
12520:
12513:
12491:
12484:
12452:
12445:
12407:
12400:
12368:
12361:
12311:
12304:
12272:
12265:
12233:
12188:
12181:
12085:
12061:
12022:
12013:
10442:
8214:
2793:
6730:
6005:
5945:
5012:
4604:
4380:
10229:
4977:
4136:
3950:
1184:
1010:
616:
By providing an abstract mathematical representations of computers, computational models abstract away superfluous complexities of the real world (like differences in
12664:
10292:
7498:
7478:
7445:
7412:
7122:
1650:
1552:
1454:
1359:
12690:
7575:
4567:
4343:
3576:
2304:
8602:
8392:
8072:
7959:
7546:
7522:
6771:
6712:, probabilistic algorithms introduce error into the system, so complexity classes based on probabilistic proof systems are defined in terms of an error probability
3464:
10515:
10155:
9751:
9525:
9420:
8575:
4941:
3874:
2649:
5766:
The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem establishes that
5162:
Similarly, any problem that a nondeterministic Turing machine can solve in exponential space, a deterministic Turing machine can also solve in exponential space.
4184:
2543:
10612:
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
6415:. 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
12710:
11822:
11684:
11502:
11072:
10851:
10783:
10763:
10482:
10462:
10404:
10384:
10364:
10269:
10249:
10175:
9947:
9791:
9771:
9677:
9545:
9282:
9227:
8715:
8622:
8432:
8412:
8294:
8274:
8254:
8234:
8172:
8152:
8132:
8112:
8092:
7928:
7821:
7663:
7643:
7599:
6969:
6949:
6867:
6847:
6791:
6655:
6635:
6615:
6329:
6309:
6289:
6142:
6122:
6049:
6029:
5985:
5965:
4855:
4831:
4801:
4773:
4731:
4711:
4687:
4659:
4508:
4484:
4464:
4444:
4424:
4404:
4284:
4264:
4244:
4224:
4204:
2202:
2182:
2117:
2097:
2077:
2050:
2026:
2006:
1986:
1966:
1946:
1919:
1224:
1204:
1105:
1050:
1030:
931:
517:
292:
3847:
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
3238:
3139:
11544:
Promise problems make for a more natural formulation of many computational problems. For instance, a computational problem could be something like "given a
9997:
7574:. 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
1788:
8643:
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
1707:
205:
adds any computational power to computers and whether problems having solutions that can be quickly checked for correctness can also be quickly solved.
9139:, there are also a number of complexity classes defined in terms of other types of problems. In particular, there are complexity classes consisting of
801:
on the maximum amount of resources that the most efficient algorithm takes to solve them. More specifically, complexity classes are concerned with the
13452:
13406:
11804:
Classes of decision problems—that is, classes of problems defined as formal languages—thus translate naturally to promise problems, where a language
2847:
time complexity classes, these are extremely narrow classes as sublinear times do not even enable a Turing machine to read the entire input (because
736:
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.
13258:
8441:, circuit complexity classes are defined in terms of circuit size — the number of vertices in the circuit. The size complexity of a circuit family
13231:
3047:
673:
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
328:
11597:
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.
11548:, determine whether or not..." This is often stated as a decision problem, where it is assumed that there is some translation schema that takes
9821:
asks whether there exists a proof of membership (a certificate) for the input that can be checked for correctness in polynomial time. The class
11314:
765:
A comparison of deterministic and nondeterministic computation. If any branch on the nondeterministic computation accepts then the NTM accepts.
11265:
9011:, as the class can be equivalently defined as the class of languages recognized by a polynomial-time Turing machine with a polynomial-bounded
5794:
are the most commonly used models of computation, many complexity classes are defined in terms of other computational models. In particular,
3772:
743:
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
13313:
7609:
values 0 and 1), the input bits are represented by source vertices (vertices with no incoming edges), and all non-source vertices represent
728:
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
13597:
6540:
2385:
is the class of problems whose polynomial time verifiers need only receive the empty string as their certificate), it is not known whether
13549:
8802:
6438:
2879:). However, there are a meaningful number of problems that can be solved in logarithmic space. The definitions of these classes require a
522:". In terms of the theory of computation, a decision problem is represented as the set of input strings that a computer running a correct
797:
Complexity classes group computational problems by their resource requirements. To do this, computational problems are differentiated by
496:. 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
8604:. The familiar function classes follow naturally from this; for example, a polynomial-size circuit family is one such that the function
13514:
17:
13362:
13999:
13176:
11167:
11123:
7625:). One logic gate is designated the output gate, and represents the end of the computation. The input/output behavior of a circuit
13267:
13204:
11619:
11077:
10898:
2310:
This equivalence between the nondeterministic definition and the verifier definition highlights a fundamental connection between
13486:
11311:
Within this formulation, it can be seen that decision problems are just the subset of promise problems with the trivial promise
8839:
832:
13534:
12423:
11001:
10947:
6676:
and the procedure is restricted to one round (that is, the prover sends only a single, full proof—typically referred to as the
5804:
3362:
2977:
1874:
is often said to be the class of problems that can be solved "quickly" or "efficiently" by a deterministic computer, since the
10300:
2912:
14009:
13356:
13212:
7135:
9054:, however gates are allowed to have unbounded fan-in (that is, the AND and OR gates can be applied to more than two bits).
5132:{\displaystyle {\mathsf {NSPACE}}\left(f\left(n\right)\right)\subseteq {\mathsf {DSPACE}}\left(f\left(n\right)^{2}\right).}
2347:
394:. For this reason, computational problems are often synonymously referred to as languages, since strings of bits represent
5601:{\displaystyle {\mathsf {DTIME}}{\big (}f(n){\big )}\subsetneq {\mathsf {DTIME}}{\big (}f(n)\cdot \log ^{2}(f(n)){\big )}}
3836:, for instance, is one important complement complexity class, and sits at the center of the unsolved problem over whether
716:
from above is the set of strings (representing natural numbers) that a Turing machine running an algorithm that correctly
2537:) is the class of decision problems solvable by a nondeterministic Turing machine in exponential time. Or more formally,
5345:
1110:
936:
550:
is the set of strings representing natural numbers that, when input into a computer running an algorithm that correctly
13446:
7044:. In other words, any problem that can be solved by a polynomial-time interactive proof system can also be solved by a
5755:{\displaystyle {\mathsf {DSPACE}}{\big (}f(n){\big )}\subsetneq {\mathsf {DSPACE}}{\big (}f(n)\cdot \log(f(n)){\big )}}
815:
rate as the input size increases, which is comparatively slow compared to problems in the exponential complexity class
603:
To make concrete the notion of a "computer", in theoretical computer science problems are analyzed in the context of a
11887:
11748:
11692:
11433:
8905:
2470:
2429:
13959:
13497:
13480:
13282:
3955:
2405:
over determinism with regards to the ability to quickly find a solution to a problem; that is, being able to explore
9896:
9583:(pronounced "sharp P") is an important class of counting problems that can be thought of as the counting version of
8720:
8515:
7826:
13590:
9843:
9609:
9298:
11616:
is the class of promise problems that are solvable in deterministic polynomial time. That is, the promise problem
7671:
5246:
5187:
14004:
12033:
9592:
6147:
6054:
2906:) is the class of problems solvable in logarithmic space on a nondeterministic Turing machine. Or more formally,
1698:
756:
608:
39:
8964:
3458:
is the class of problems solvable in exponential space by a nondeterministic Turing machine. Or more formally,
2898:) is then defined as the class of problems solvable in logarithmic space on a deterministic Turing machine and
11958:
The following table shows some of the classes of problems that are considered in complexity theory. If class
4098:
that are practically useful do in fact have low order polynomial runtimes, and almost all problems outside of
1560:
1462:
9160:
9140:
8646:
8444:
8299:
7979:
6974:
6872:
1367:
1272:
574:
100:
11827:
11507:
10703:
9591:
arises from the fact that the number of solutions to a problem equals the number of accepting branches in a
7374:{\displaystyle f_{C}(x_{1},x_{2},x_{3})=\neg (x_{1}\wedge x_{2})\wedge ((x_{2}\wedge x_{3})\vee \neg x_{3})}
2314:
and solution verifiability. Furthermore, it also provides a useful method for proving that a language is in
193:. The relationships between classes often answer questions about the fundamental nature of computation. The
13429:
12048:
10569:
7741:
7045:
5911:
5799:
3133:
is the class of problems solvable in polynomial space by a nondeterministic Turing machine. More formally,
2210:
1686:
248:
108:
13391:
11860:
11406:
11379:
11238:
11211:
6741:(interactive polynomial time), which is the class of all problems solvable by an interactive proof system
4035:
2888:
space complexity of the Turing machine is measured as the number of cells that are used on the work tape.
13994:
13583:
13243:
10272:
9425:
9236:
9118:
6796:
9058:
is a notable class because it can be equivalently defined as the class of languages that have efficient
7964:
Any particular circuit has a fixed number of input vertices, so it can only act on inputs of that size.
4102:
that are practically useful do not have any known algorithms with small exponential runtimes, i.e. with
761:
13975:
13782:
13223:
12618:
11605:
Promise problems provide an alternate definition for standard complexity classes of decision problems.
9184:
9086:
7195:
2529:) is the class of decision problems solvable by a deterministic Turing machine in exponential time and
2311:
1235:
733:
695:
529:
435:
405:
304:
202:
11555:
10791:
10520:
9952:
9682:
9452:
9351:
6672:
is a simple proof system in which the verifier is restricted to being a deterministic polynomial-time
4513:
4289:
2122:
607:. Computational models make exact the notions of computational resources like "time" and "memory". In
12222:
6700:
proof systems that provide greater computational power over standard complexity classes thus require
6560:
6208:
5417:
5305:
4622:
383:
13964:
13402:
9814:
9802:
6677:
6553:
6207:
The relationships between the fundamental probabilistic complexity classes. BQP is a probabilistic
5829:
2883:
so that it is possible for the machine to store the entire input (it can be shown that in terms of
2880:
1925:
674:
112:
55:
12750:
12715:
4737:
with at most polynomial slowdown. Of particular importance, the set of problems that are hard for
13300:
10856:
8756:
6705:
6575:
5613:
5175:
3879:
2850:
627:. While other models exist and many complexity classes are defined in terms of them (see section
10409:
8177:
2778:
2334:
are actually at the center of one of the most famous unsolved problems in computer science: the
99:. There are, however, many complexity classes defined in terms of other types of problems (e.g.
13918:
13509:
13308:
10485:
10197:
9551:
9292:
9123:
9034:. These classes are defined not only in terms of their circuit size but also in terms of their
7602:
7578:(computation in which different input sizes within the same problem use different algorithms).
6793:
is probabilistic polynomial-time and the proof system satisfies two properties: for a language
6715:
6709:
5990:
5930:
5904:
5877:
5458:
5171:
4982:
4574:
4350:
4151:
3680:{\displaystyle {\mathsf {NEXPSPACE}}=\bigcup _{k\in \mathbb {N} }{\mathsf {NSPACE}}(2^{n^{k}})}
2464:
386:; for example, in the primality example the natural numbers could be represented as strings of
89:, and are differentiated by their time or space (memory) requirements. For instance, the class
10202:
4946:
4105:
3919:
3565:{\displaystyle {\mathsf {EXPSPACE}}=\bigcup _{k\in \mathbb {N} }{\mathsf {DSPACE}}(2^{n^{k}})}
1153:
979:
179:
relation). However, many relationships are not yet known; for example, one of the most famous
13913:
13908:
12640:
10277:
7483:
7450:
7417:
7384:
7101:
7069:. The interactive proof system can be constrained so that the coins used by the verifier are
4805:
3731:
3454:
is the class of problems solvable in exponential space by a deterministic Turing machine and
2747:{\displaystyle {\mathsf {NEXPTIME}}=\bigcup _{k\in \mathbb {N} }{\mathsf {NTIME}}(2^{n^{k}})}
2318:—simply identify a suitable certificate and show that it can be verified in polynomial time.
1617:
1519:
1421:
1326:
785:
of an NTM is the maximum number of cells that the NTM uses on any branch of its computation.
617:
264:
218:
51:
13501:
13192:
12669:
6380:
4546:
4322:
3129:
is the class of problems solvable in polynomial space by a deterministic Turing machine and
2638:{\displaystyle {\mathsf {EXPTIME}}=\bigcup _{k\in \mathbb {N} }{\mathsf {DTIME}}(2^{n^{k}})}
2274:
297:?" is a computational problem. A computational problem is mathematically represented as the
13903:
13553:
13254:
12576:
11983:
8944:
8580:
8370:
8050:
7937:
7531:
7507:
6744:
6371:
5920:
4909:
3690:
3348:
1065:
1061:
747:
is the number of cells on its tape that it uses to reach either an accept or reject state.
586:
70:
10491:
10131:
9727:
9501:
9396:
8551:
4917:
3850:
1901:
polynomial time Turing machine, referred to as the verifier, that takes as input a string
8:
13558:
Includes a diagram showing the hierarchy of complexity classes and how they fit together.
13335:
13328:
9555:
9012:
8871:
6579:
6378:
is also at the center of the important unsolved problem in computer science over whether
4626:
4163:
3743:
3329:{\displaystyle {\mathsf {NPSPACE}}=\bigcup _{k\in \mathbb {N} }{\mathsf {NSPACE}}(n^{k})}
3100:
643:
604:
493:
69:
In general, a complexity class is defined in terms of a type of computational problem, a
4979:
space, i.e. in the square of the space. Formally, Savitch's theorem states that for any
3227:{\displaystyle {\mathsf {PSPACE}}=\bigcup _{k\in \mathbb {N} }{\mathsf {DSPACE}}(n^{k})}
13382:
12695:
12072:
11807:
11669:
11487:
11057:
10836:
10768:
10748:
10467:
10447:
10389:
10369:
10349:
10254:
10234:
10160:
9932:
9776:
9756:
9662:
9530:
9267:
9212:
9059:
8700:
8607:
8417:
8397:
8279:
8259:
8239:
8219:
8157:
8137:
8117:
8097:
8077:
7913:
7806:
7648:
7628:
7584:
7214:
6954:
6934:
6852:
6832:
6776:
6640:
6620:
6600:
6345:
encompass all of the problems that can be solved by probabilistic Turing machines with
6314:
6294:
6274:
6127:
6107:
6034:
6014:
5970:
5950:
4840:
4816:
4786:
4758:
4716:
4696:
4672:
4644:
4493:
4469:
4449:
4429:
4409:
4389:
4269:
4249:
4229:
4209:
4189:
3747:
2844:
2187:
2167:
2102:
2082:
2062:
2035:
2029:
2011:
1991:
1971:
1951:
1931:
1904:
1209:
1189:
1090:
1035:
1015:
916:
502:
277:
13288:
9801:
can be defined both in terms of nondeterminism and in terms of a verifier (i.e. as an
6704:
verifiers, which means that the verifier's questions to the prover are computed using
1245:
Complexity classes are often defined using granular sets of complexity classes called
13923:
13476:
13442:
13352:
13278:
13208:
13200:
13193:
10121:{\displaystyle f(w)={\Big |}{\big \{}c\in \{0,1\}^{p(|w|)}:V(w,c)=1{\big \}}{\Big |}}
9809:
be equivalently defined in terms of a verifier. Recall that a decision problem is in
6507:
does not allow error for strings not in the language. Hence, if a problem is in both
6366:
is the most practically relevant of the probabilistic complexity classes—problems in
551:
298:
214:
47:
13531:
9000:
6735:
The most general complexity class arising out of this characterization is the class
5919:
that can toss random coins. These classes help to better describe the complexity of
1889:
is that it can be equivalently defined as the class of problems whose solutions are
13887:
13777:
13709:
13699:
13565:
13378:
13344:
11998:
10657:
10187:
9174:
9144:
9136:
7969:
7666:
7201:
6594:
6528:
6519:
not in the language (i.e. no error whatsoever), which is exactly the definition of
6358:
6250:
6235:
6217:
5822:
2823:
1861:{\displaystyle {\mathsf {NP}}=\bigcup _{k\in \mathbb {N} }{\mathsf {NTIME}}(n^{k})}
1083:
1077:
782:
744:
562:
489:
473:
244:
234:
222:
120:
104:
82:
78:
63:
10157:
equals the size of the set containing all of the polynomial-size certificates for
4609:
Note that reductions can be defined in many different ways. Common reductions are
1777:{\displaystyle {\mathsf {P}}=\bigcup _{k\in \mathbb {N} }{\mathsf {DTIME}}(n^{k})}
884:
Note that the study of complexity classes is intended primarily to understand the
34:
A representation of the relationships between several important complexity classes
13882:
13872:
13829:
13819:
13812:
13802:
13792:
13750:
13745:
13740:
13724:
13704:
13682:
13677:
13672:
13660:
13655:
13650:
13645:
13538:
13343:. Lecture Notes in Computer Science. Vol. 3895. Springer. pp. 254–290.
12600:
12545:
12348:
11538:
11376:. With decision problems it is thus simpler to simply define the problem as only
10692:
10663:
10573:
10560:
9288:
9148:
9030:
9024:
8949:
8438:
8437:
While complexity classes defined using Turing machines are described in terms of
7965:
7567:
7562:
7061:(Arthur–Merlin protocol). In the definition of interactive proof systems used by
7057:
6737:
6668:
6597:
that model computation as the exchange of messages between two parties: a prover
6585:
6566:
6534:
6389:
6353:
6346:
6263:
6241:
6223:
5870:
5864:
5853:
5846:
5840:
5834:
5816:
5810:
4618:
3750:. Moreover, they might also be closed under a variety of quantification schemes.
3121:
2838:
1875:
1690:
1677:
1662:
909:
903:
774:
740:
591:
567:
428:
395:
252:
230:
147:
135:
116:
96:
74:
59:
13141:
6203:
1893:
by a deterministic Turing machine in polynomial time. That is, a language is in
13877:
13765:
13687:
13640:
13463:
13324:
13168:
13164:
12500:
10608:, then the functions that determine the number of certificates for problems in
10582:
9559:
9008:
8799:
7557:
6673:
5916:
5791:
4614:
4610:
3115:
2834:
1673:
1262:
807:
717:
655:
624:
497:
272:
239:
226:
141:
128:
91:
86:
13117:
10895:
Specifically, a promise problem is defined as a pair of non-intersecting sets
7219:
6564:. Interactive proofs generalize the proofs definition of the complexity class
95:
is the set of decision problems solvable by a deterministic Turing machine in
13988:
13561:
13545:
13438:
13184:
9039:
8394:(the circuit with the same number of input vertices as the number of bits in
7090:
is the number of messages exchanged (so in the generalized form the standard
2884:
729:
686:
391:
13129:
6692:(deterministic interactive proof), though it is rarely referred to as such.
3084:{\displaystyle {\mathsf {L}}\subseteq {\mathsf {NL}}\subseteq {\mathsf {P}}}
30:
11545:
9230:
6571:
519:
375:{\displaystyle {\texttt {PRIME}}=\{n\in \mathbb {N} |n{\text{ is prime}}\}}
294:
180:
4837:(since there could be many problems that are equally hard, more precisely
2426:). However, the overwhelming majority of computer scientists believe that
301:
of answers to the problem. In the primality example, the problem (call it
13755:
13665:
13425:
13337:
Theoretical
Computer Science. Lecture Notes in Computer Science, vol 3895
13090:
11369:{\displaystyle \Pi _{\text{ACCEPT}}\cup \Pi _{\text{REJECT}}=\{0,1\}^{*}}
10591:
9135:
While most complexity classes studied by computer scientists are sets of
8885:
7193:
Other complexity classes defined using interactive proof systems include
4943:
space, then a deterministic Turing machine can solve the same problem in
4865:
4226:
to their base-2 notation (e.g. 5+7 becomes 101+111). Formally, a problem
3739:
3091:. However, it is not known whether any of these relationships is proper.
2335:
798:
639:
399:
184:
13348:
11301:{\displaystyle \Pi _{\text{ACCEPT}}\cap \Pi _{\text{REJECT}}=\emptyset }
9074:
4887:-complete problem that can be solved in polynomial time would mean that
3701:. This class is extremely broad: it is known to be a strict superset of
664:
An illustration of a Turing machine. The "B" indicates the blank symbol.
488:
The most commonly analyzed problems in theoretical computer science are
13867:
13692:
13570:
Computers and
Intractability: A Guide to the Theory of NP-Completeness.
13377:. Handbook of Theoretical Computer Science. Elsevier. pp. 67–161.
13239:
13188:
8625:
7610:
3876:
time (that is, in linear time) and one that can be solved in, at best,
3823:{\displaystyle {\textsf {co-X}}=\{L|{\overline {L}}\in {\mathsf {X}}\}}
1057:
812:
11953:
2079:
for which there exists a polynomial-time deterministic Turing machine
13862:
13575:
10192:
Counting problems are a subset of a broader class of problems called
9659:
such that there is a polynomial time nondeterministic Turing machine
9563:
6387:
The broadest class of efficiently-solvable probabilistic problems is
523:
268:
13334:. In Goldreich, Oded; Rosenberg, Arnold L.; Selman, Alen L. (eds.).
9233:(the answer is a simple yes/no); the corresponding counting problem
8828:{\displaystyle {\mathsf {\color {Blue}P}}\subset {\textsf {P/poly}}}
6472:{\displaystyle {\textsf {ZPP}}={\textsf {RP}}\cap {\textsf {co-RP}}}
739:
Turing machines enable intuitive notions of "time" and "space". The
660:
13857:
13852:
13797:
13620:
12120:
12096:
11986:, but is useful for putting the complexity classes in perspective.
10196:. A function problem is a type of problem in which the values of a
7622:
7614:
7571:
7549:
7501:
6680:—to the verifier). Put another way, in the definition of the class
3735:
3446:
3425:
2517:
468:
325:) is represented by the set of all natural numbers that are prime:
201:
problem, for instance, is directly related to questions of whether
171:
165:
10564:, the class of efficiently solvable functions. More specifically,
9287:
Thus, whereas decision problems are represented mathematically as
3765:, which consists of the complements of the languages contained in
646:, but this approach is less frequently used in complexity theory.
13847:
12144:
10628:(it is not known whether this holds in the reverse, i.e. whether
7618:
7525:
4743:
3440:
2513:
817:
620:
speed) that obstruct an understanding of fundamental principles.
579:
382:. In the theory of computation, these answers are represented as
159:
13954:
13949:
13824:
13807:
13431:
Automata, Computability and
Complexity: Theory and Applications
12168:
11963:
9575:
8638:
8094:
input variables. A circuit family is said to decide a language
7040:
5909:
A number of important complexity classes are defined using the
5858:
2028:
is not in the language. Intuitively, the certificate acts as a
176:
153:
5158:(since the square of an exponential is still an exponential).
3734:
properties. For example, decision classes may be closed under
777:
of an NTM is the maximum number of steps that the NTM uses on
13944:
13939:
13760:
13541:: A huge list of complexity classes, a reference for experts.
13373:
Johnson, David S. (1990). "A Catalog of
Complexity Classes".
13042:
12526:
12474:
12465:
12458:
12435:
12413:
12390:
12381:
12374:
12320:
12294:
12285:
12278:
12255:
12246:
12239:
12210:
12201:
12194:
12156:
12132:
12108:
11197:{\displaystyle \Pi _{\text{ACCEPT}}\cup \Pi _{\text{REJECT}}}
11153:{\displaystyle \Pi _{\text{ACCEPT}}\cup \Pi _{\text{REJECT}}}
6333:
6229:
5150:(since the square of a polynomial is still a polynomial) and
3832:
642:
to define complexity classes without referring to a concrete
13080:
13078:
13076:
13074:
12588:
12564:
12557:
12535:
12519:
12512:
12490:
12483:
12451:
12444:
12406:
12399:
12367:
12360:
12310:
12303:
12271:
12264:
12232:
12187:
12180:
12084:
12060:
12021:
12012:
4094:
is also useful because, empirically, almost all problems in
13772:
13630:
12596:
12572:
12541:
12496:
12419:
12344:
12329:
12316:
12218:
12164:
12140:
12116:
12092:
12068:
12044:
12029:
11994:
11988:
11655:{\displaystyle (\Pi _{\text{ACCEPT}},\Pi _{\text{REJECT}})}
11113:{\displaystyle (\Pi _{\text{ACCEPT}},\Pi _{\text{REJECT}})}
10934:{\displaystyle (\Pi _{\text{ACCEPT}},\Pi _{\text{REJECT}})}
6271:
if there is a probabilistic polynomial-time Turing machine
4898:
4156:
Many complexity classes are defined using the concept of a
632:
9007:
is also helpful in the general study of the properties of
8863:{\displaystyle {\textsf {P}}\subsetneq {\textsf {P/poly}}}
7073:; that is, the prover is able to see the coins. Formally,
874:{\displaystyle {\mathsf {P}}\subseteq {\mathsf {EXPTIME}}}
13787:
13719:
13714:
13635:
13625:
13071:
13030:
12994:
12970:
12333:
11043:{\displaystyle \Pi _{\text{REJECT}}\subseteq \{0,1\}^{*}}
10989:{\displaystyle \Pi _{\text{ACCEPT}}\subseteq \{0,1\}^{*}}
9550:
Counting problems arise in a number of fields, including
9495:
9113:
9107:
7606:
5888:
5882:
3761:
that is not closed under negation has a complement class
3401:{\displaystyle {\mathsf {P}}\subseteq {\mathsf {PSPACE}}}
3034:{\displaystyle {\mathsf {NL}}={\mathsf {NSPACE}}(\log n)}
387:
12924:
12922:
12909:
12907:
12820:
10580:
can be thought of as the function problem equivalent of
10568:
is the set of function problems that can be solved by a
10336:{\displaystyle R\subseteq \Sigma ^{*}\times \Sigma ^{*}}
9022:
that have interesting properties in their own right are
6211:
class and is described in the quantum computing section.
2966:{\displaystyle {\mathsf {L}}={\mathsf {DSPACE}}(\log n)}
2418:), then there also exists an algorithm that can quickly
13107:
13105:
12808:
7223:
Example
Boolean circuit computing the Boolean function
6589:
General representation of an interactive proof protocol
6311:
always rejects and if a string is in the language then
6215:
The fundamental randomized time complexity classes are
10590:
provides some insight into both counting problems and
9291:, counting problems are represented mathematically as
9038:. The depth of a circuit is the length of the longest
7183:{\displaystyle {\mathsf {AM}}\subseteq {\mathsf {IP}}}
7010:
6908:
6526:
Important randomized space complexity classes include
6352:
Loosening the error requirements further to allow for
13228:
Computer
Science 522: Computational Complexity Theory
12982:
12946:
12934:
12919:
12904:
12892:
12844:
12753:
12718:
12698:
12672:
12643:
11890:
11863:
11830:
11810:
11751:
11695:
11672:
11622:
11609:, for instance, can be defined as a promise problem:
11558:
11510:
11490:
11436:
11409:
11382:
11317:
11268:
11241:
11214:
11170:
11126:
11080:
11060:
11004:
10950:
10901:
10859:
10839:
10833:. A promise problem loosens the input requirement on
10794:
10771:
10751:
10706:
10523:
10494:
10470:
10450:
10412:
10392:
10372:
10352:
10303:
10280:
10257:
10237:
10205:
10163:
10134:
10000:
9955:
9935:
9899:
9846:
9779:
9759:
9730:
9685:
9665:
9612:
9533:
9504:
9455:
9428:
9399:
9354:
9301:
9270:
9239:
9215:
9187:
8967:
8908:
8842:
8805:
8759:
8723:
8703:
8649:
8610:
8583:
8554:
8518:
8447:
8420:
8400:
8373:
8302:
8296:
is in the language represented by the circuit family
8282:
8262:
8242:
8222:
8180:
8160:
8140:
8120:
8100:
8080:
8053:
7982:
7940:
7916:
7829:
7809:
7744:
7674:
7651:
7631:
7587:
7534:
7510:
7486:
7453:
7420:
7387:
7229:
7138:
7104:
6977:
6957:
6937:
6875:
6855:
6835:
6799:
6779:
6747:
6718:
6643:
6623:
6603:
6503:
does not allow error for strings in the language and
6441:
6317:
6297:
6277:
6150:
6130:
6110:
6057:
6037:
6017:
5993:
5973:
5953:
5933:
5625:
5470:
5420:
5348:
5308:
5249:
5190:
5023:
4985:
4949:
4920:
4843:
4819:
4789:
4761:
4719:
4699:
4675:
4647:
4577:
4549:
4516:
4496:
4472:
4452:
4432:
4412:
4392:
4353:
4325:
4292:
4272:
4252:
4232:
4212:
4192:
4166:
4108:
4038:
3958:
3922:
3882:
3853:
3775:
3579:
3467:
3365:
3241:
3142:
3050:
2980:
2915:
2853:
2781:
2652:
2546:
2473:
2432:
2374:{\displaystyle {\mathsf {P}}\subseteq {\mathsf {NP}}}
2350:
2277:
2213:
2190:
2170:
2125:
2105:
2085:
2065:
2038:
2014:
1994:
1974:
1954:
1934:
1907:
1791:
1710:
1620:
1563:
1522:
1465:
1424:
1370:
1329:
1275:
1212:
1192:
1156:
1113:
1093:
1038:
1018:
982:
939:
919:
835:
698:
532:
505:
438:
408:
331:
307:
280:
13273:. In Hemaspaandra, Lane A.; Selman, Alan L. (eds.).
13102:
13061:
13059:
13057:
12880:
12856:
12832:
6483:
consists exactly of those problems that are in both
5142:
Important corollaries of
Savitch's theorem are that
81:. In particular, most complexity classes consist of
13006:
11954:
Summary of relationships between complexity classes
9295:: a counting problem is formalized as the function
9181:solutions exist. For example, the decision problem
8943:is also helpful in investigating properties of the
7976:. A circuit family is an infinite list of circuits
6331:accepts with a probability at least 1/2. The class
5774:, and the space hierarchy theorem establishes that
4637:Reductions motivate the concept of a problem being
4406:that can be efficiently reduced to another problem
3720:
750:
12868:
12796:
12774:
12739:
12712:, allows a Turning machine to read inputs of size
12704:
12684:
12658:
11930:
11876:
11849:
11816:
11791:
11735:
11678:
11654:
11589:
11529:
11496:
11476:
11422:
11395:
11368:
11300:
11254:
11227:
11196:
11152:
11112:
11066:
11042:
10988:
10933:
10884:
10845:
10825:
10777:
10757:
10737:
10542:
10509:
10476:
10456:
10436:
10398:
10378:
10358:
10335:
10286:
10263:
10243:
10223:
10169:
10149:
10120:
9986:
9941:
9921:
9885:
9785:
9765:
9745:
9716:
9671:
9651:
9539:
9519:
9486:
9441:
9414:
9385:
9340:
9276:
9252:
9221:
9197:
8987:
8931:
8862:
8827:
8790:
8745:
8709:
8689:
8616:
8596:
8569:
8540:
8504:
8426:
8406:
8386:
8359:
8288:
8268:
8248:
8228:
8208:
8166:
8146:
8126:
8106:
8086:
8066:
8039:
7953:
7922:
7902:
7815:
7795:
7730:
7657:
7637:
7593:
7540:
7516:
7492:
7472:
7439:
7406:
7373:
7182:
7116:
7023:
6963:
6943:
6921:
6861:
6841:
6818:
6785:
6765:
6724:
6649:
6629:
6609:
6515:, then there must be no error for strings both in
6471:
6323:
6303:
6291:such that if a string is not in the language then
6283:
6187:
6136:
6116:
6094:
6043:
6023:
5999:
5979:
5959:
5939:
5754:
5600:
5446:
5407:{\displaystyle O(n^{k_{1}})\subseteq O(n^{k_{2}})}
5406:
5334:
5294:
5235:
5131:
5006:
4971:
4935:
4849:
4825:
4795:
4767:
4725:
4705:
4681:
4653:
4598:
4561:
4535:
4502:
4478:
4458:
4438:
4418:
4398:
4374:
4337:
4311:
4278:
4258:
4238:
4218:
4198:
4178:
4130:
4082:
4024:
3944:
3904:
3868:
3822:
3679:
3564:
3400:
3328:
3226:
3083:
3033:
2965:
2871:
2787:
2746:
2637:
2496:
2455:
2373:
2298:
2263:
2196:
2176:
2156:
2111:
2091:
2071:
2044:
2020:
2000:
1980:
1960:
1940:
1913:
1860:
1776:
1644:
1614:is the set of all problems that are decided by an
1606:
1546:
1516:is the set of all problems that are decided by an
1508:
1448:
1418:is the set of all problems that are decided by an
1410:
1353:
1323:is the set of all problems that are decided by an
1315:
1218:
1198:
1178:
1143:{\displaystyle s_{M}:\mathbb {N} \to \mathbb {N} }
1142:
1099:
1044:
1024:
1004:
969:{\displaystyle t_{M}:\mathbb {N} \to \mathbb {N} }
968:
925:
873:
708:
681:algorithm can be represented as a Turing machine.
623:The most commonly used computational model is the
542:
511:
448:
418:
374:
317:
286:
27:Set of problems in computational complexity theory
13054:
13018:
11600:
10553:
10113:
10018:
9569:
9042:from an input node to the output node. The class
8631:
7048:with polynomial space resources, and vice versa.
6661:
6558:A number of complexity classes are defined using
4621:, and can vary based on resource bounds, such as
2798:. It is not known whether this is proper, but if
1882:increases relatively slowly with the input size.
526:would answer "yes" to. In the primality example,
13986:
13475:(2nd ed.). USA: Thomson Course Technology.
13173:Electronic Colloquim on Computational Complexity
11931:{\displaystyle \{0,1\}^{*}/\Pi _{\text{ACCEPT}}}
11792:{\displaystyle x\in \Pi _{\text{REJECT}},M(x)=0}
11736:{\displaystyle x\in \Pi _{\text{ACCEPT}},M(x)=1}
11477:{\displaystyle \{0,1\}^{*}/\Pi _{\text{ACCEPT}}}
9422:is the number of solutions. For example, in the
8932:{\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}}
7823:of the circuit is represented mathematically as
7605:in which edges represent wires (which carry the
6978:
6876:
2497:{\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}}
2456:{\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}}
1697:is the class of problems that are solvable by a
1685:is the class of problems that are solvable by a
649:
628:
58:". The two most commonly analyzed resources are
13224:"Complexity classes having to do with counting"
12958:
11941:Formulating many basic complexity classes like
9151:. These are explained in greater detail below.
6491:. Intuitively, this follows from the fact that
6198:
4025:{\displaystyle O(n^{3})\circ O(n^{2})=O(n^{6})}
2401:, then it follows that nondeterminism provides
13502:"Lecture 22: Quantum computational complexity"
13251:Computer Science 522: Computational Complexity
9922:{\displaystyle p:\mathbb {N} \to \mathbb {N} }
8999:and other complexity classes is available at "
8995:. A full description of the relations between
8746:{\displaystyle t:\mathbb {N} \to \mathbb {N} }
8541:{\displaystyle f:\mathbb {N} \to \mathbb {N} }
7903:{\displaystyle b=f_{C}(x_{1},x_{2},...,x_{n})}
7199:(multiprover interactive polynomial time) and
5785:
107:) and using other models of computation (e.g.
13591:
13399:CSE431: Introduction to Theory of Computation
10106:
10025:
9886:{\displaystyle f:\{0,1\}^{*}\to \mathbb {N} }
9652:{\displaystyle f:\{0,1\}^{*}\to \mathbb {N} }
9341:{\displaystyle f:\{0,1\}^{*}\to \mathbb {N} }
7055:produces another important complexity class:
5895:These are explained in greater detail below.
5747:
5701:
5669:
5650:
5593:
5540:
5511:
5492:
3713:, and is believed to be a strict superset of
237:resources. For example, the complexity class
12747:, there will invariably reach a point where
11904:
11891:
11666:if there exists a polynomial-time algorithm
11578:
11565:
11450:
11437:
11357:
11344:
11031:
11018:
10977:
10964:
10873:
10860:
10814:
10801:
10726:
10713:
10049:
10036:
9975:
9962:
9866:
9853:
9813:if there exists a polynomial-time checkable
9705:
9692:
9632:
9619:
9475:
9462:
9374:
9361:
9321:
9308:
7731:{\displaystyle f_{C}:\{0,1\}^{n}\to \{0,1\}}
7725:
7713:
7701:
7688:
6547:
5295:{\displaystyle {\mathsf {DTIME}}(n^{k_{2}})}
5236:{\displaystyle {\mathsf {DTIME}}(n^{k_{1}})}
3817:
3786:
2233:
2220:
2145:
2132:
492:—the kinds of problems that can be posed as
369:
342:
267:is just a question that can be solved by an
13195:Computational Complexity: A Modern Approach
11050:is the set of all inputs that are rejected.
10996:is the set of all inputs that are accepted.
10853:by restricting the input to some subset of
10231:are computed. Formally, a function problem
9753:equals the number of accepting branches in
8894:. For example, if there is any language in
7556:An alternative model of computation to the
6374:that can be run quickly on real computers.
6188:{\displaystyle {\text{Pr}}\geq 1-\epsilon }
6095:{\displaystyle {\text{Pr}}\geq 1-\epsilon }
2817:
2467:employed today rely on the assumption that
1229:
13598:
13584:
13183:
13084:
13048:
13036:
13000:
12976:
12814:
10558:An important function complexity class is
10464:. This is just another way of saying that
9829:such certificates exist. In this context,
9130:
8798:. It follows directly from this fact that
5790:While deterministic and non-deterministic
4466:is polynomial-time reducible to a problem
2321:
1656:
781:branch of its computation. Similarly, the
432:is equivalent to saying that the language
13544:
13469:Introduction to the Theory of Computation
13323:
13147:
13135:
13123:
11484:), which throughout this page is denoted
9915:
9907:
9879:
9645:
9334:
8988:{\displaystyle \Sigma _{2}^{\mathsf {P}}}
8855:
8845:
8820:
8739:
8731:
8534:
8526:
6464:
6454:
6444:
5898:
4864:Of particular importance is the class of
3778:
3626:
3511:
3419:
3282:
3180:
2696:
2587:
1885:An important characteristic of the class
1817:
1733:
1136:
1128:
962:
954:
352:
258:
13207:from the original on February 23, 2022.
13163:
12952:
12940:
12928:
12886:
12862:
12850:
9434:
9245:
9190:
7218:
7098:). However, it is the case that for all
6584:
6419:simply relaxes the error probability of
6202:
4899:Relationships between complexity classes
4871:problems—the most difficult problems in
1607:{\displaystyle {\mathsf {NSPACE}}(s(n))}
1509:{\displaystyle {\mathsf {DSPACE}}(s(n))}
760:
701:
659:
535:
467:
441:
411:
334:
310:
29:
13496:
13372:
13298:
13265:
13111:
13012:
12874:
12802:
8690:{\displaystyle {\mathsf {DTIME}}(t(n))}
8505:{\displaystyle (C_{0},C_{1},C_{2},...)}
8360:{\displaystyle (C_{0},C_{1},C_{2},...)}
8040:{\displaystyle (C_{0},C_{1},C_{2},...)}
7205:(quantum interactive polynomial time).
7024:{\displaystyle \Pr\leq {\tfrac {1}{3}}}
6922:{\displaystyle \Pr\geq {\tfrac {2}{3}}}
2507:
2422:that proof (that is, the problem is in
1411:{\displaystyle {\mathsf {NTIME}}(t(n))}
1316:{\displaystyle {\mathsf {DTIME}}(t(n))}
677:); this means that it is believed that
598:
594:(see section "Other types of problems")
14:
13987:
13605:
13462:
13458:from the original on January 21, 2022.
13412:from the original on November 29, 2021
13319:from the original on November 2, 2021.
13301:"Guest Column: The Third P =? NP Poll"
13150:, p. 266 (11–12 in provided pdf).
12988:
12913:
12898:
12838:
12826:
11978:with a dark line connecting them. If
11850:{\displaystyle L=\Pi _{\text{ACCEPT}}}
11530:{\displaystyle \Pi _{\text{ACCEPT}}=L}
10738:{\displaystyle L\subseteq \{0,1\}^{*}}
10661:is the function problem equivalent of
10651:is the function problem equivalent of
9065:
8979:
8924:
8921:
8911:
8809:
8664:
8661:
8658:
8655:
8652:
7665:input variables is represented by the
7166:
7163:
7144:
7141:
6811:
6808:
5876:A number of classes are defined using
5852:A number of classes are defined using
5828:A number of classes are defined using
5798:A number of classes are defined using
5694:
5691:
5688:
5685:
5682:
5679:
5643:
5640:
5637:
5634:
5631:
5628:
5533:
5530:
5527:
5524:
5521:
5485:
5482:
5479:
5476:
5473:
5264:
5261:
5258:
5255:
5252:
5205:
5202:
5199:
5196:
5193:
5090:
5087:
5084:
5081:
5078:
5075:
5041:
5038:
5035:
5032:
5029:
5026:
4857:is as hard as the hardest problems in
3812:
3649:
3646:
3643:
3640:
3637:
3634:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3534:
3531:
3528:
3525:
3522:
3519:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3393:
3390:
3387:
3384:
3381:
3378:
3368:
3305:
3302:
3299:
3296:
3293:
3290:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3203:
3200:
3197:
3194:
3191:
3188:
3160:
3157:
3154:
3151:
3148:
3145:
3076:
3066:
3063:
3053:
3011:
3008:
3005:
3002:
2999:
2996:
2986:
2983:
2943:
2940:
2937:
2934:
2931:
2928:
2918:
2716:
2713:
2710:
2707:
2704:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2607:
2604:
2601:
2598:
2595:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2489:
2486:
2476:
2448:
2445:
2435:
2366:
2363:
2353:
1837:
1834:
1831:
1828:
1825:
1797:
1794:
1753:
1750:
1747:
1744:
1741:
1713:
1701:in polynomial time. Or more formally,
1652:space nondeterministic Turing machine.
1581:
1578:
1575:
1572:
1569:
1566:
1483:
1480:
1477:
1474:
1471:
1468:
1385:
1382:
1379:
1376:
1373:
1290:
1287:
1284:
1281:
1278:
866:
863:
860:
857:
854:
851:
848:
838:
732:places the additional constraint over
13579:
13238:
13221:
13065:
13024:
9929:and a polynomial-time Turing machine
9817:to a given problem instance—that is,
9599:is thus formally defined as follows:
8808:
7796:{\displaystyle x_{1},x_{2},...,x_{n}}
5165:
3730:Complexity classes have a variety of
3094:
2264:{\displaystyle c\in \{0,1\}^{p(|w|)}}
1456:time nondeterministic Turing machine.
821:(or more accurately, for problems in
720:accepts. A Turing machine is said to
484:(alternatively, 1 or 0) on any input.
183:in computer science concerns whether
13424:
13126:, p. 255 (2–3 in provided pdf).
13099:, p. 689 (510 in provided PDF).
13096:
11877:{\displaystyle \Pi _{\text{REJECT}}}
11423:{\displaystyle \Pi _{\text{REJECT}}}
11396:{\displaystyle \Pi _{\text{ACCEPT}}}
11255:{\displaystyle \Pi _{\text{REJECT}}}
11228:{\displaystyle \Pi _{\text{ACCEPT}}}
10181:
9949:(the verifier), such that for every
9893:such that there exists a polynomial
9494:(a graph represented as a string of
9154:
9069:
4903:
4083:{\displaystyle O(n^{6})>O(n^{3})}
3912:time. Both of these problems are in
1240:
1186:is the maximum number of cells that
1012:is the maximum number of steps that
463:
13520:from the original on June 18, 2022.
13389:
13261:from the original on April 3, 2021.
13179:from the original on June 17, 2020.
12964:
10686:
9442:{\displaystyle \#{\texttt {CYCLE}}}
9253:{\displaystyle \#{\texttt {CYCLE}}}
7208:
7051:A modification of the protocol for
6819:{\displaystyle L\in {\mathsf {IP}}}
6708:. As noted in the section above on
4186:to base-2 addition by transforming
1554:space deterministic Turing machine.
611:, complexity classes deal with the
426:problem is in the complexity class
24:
13525:
13383:10.1016/b978-0-444-88071-0.50007-2
13275:Complexity Theory Retrospective II
13234:from the original on May 21, 2022.
13138:, p. 257 (4 in provided pdf).
11919:
11865:
11838:
11759:
11703:
11640:
11627:
11512:
11465:
11411:
11384:
11332:
11319:
11295:
11283:
11270:
11243:
11216:
11185:
11172:
11141:
11128:
11098:
11085:
11006:
10952:
10919:
10906:
10531:
10444:, the algorithm produces one such
10324:
10311:
10281:
9527:is the number of simple cycles in
9429:
9240:
8969:
7535:
7355:
7285:
6997: after interacting with
6895: after interacting with
4879:can be polynomial-time reduced to
4733:allows us to solve any problem in
4669:can be polynomial-time reduced to
4641:for a complexity class. A problem
4524:
4300:
1361:time deterministic Turing machine.
792:
25:
14021:
13960:Probabilistically checkable proof
13550:"Computational Complexity Theory"
13368:from the original on May 6, 2021.
13175:. Weizmann Institute of Science.
9198:{\displaystyle {\texttt {CYCLE}}}
9117:, which are of key importance in
8753:, then it has circuit complexity
5947:. A probabilistic Turing machine
2403:no additional computational power
709:{\displaystyle {\texttt {PRIME}}}
543:{\displaystyle {\texttt {PRIME}}}
449:{\displaystyle {\texttt {PRIME}}}
419:{\displaystyle {\texttt {PRIME}}}
318:{\displaystyle {\texttt {PRIME}}}
12587:
12563:
12556:
12534:
12525:
12518:
12511:
12489:
12482:
12473:
12464:
12457:
12450:
12443:
12434:
12412:
12405:
12398:
12389:
12380:
12373:
12366:
12359:
12309:
12302:
12293:
12284:
12277:
12270:
12263:
12254:
12245:
12238:
12231:
12209:
12200:
12193:
12186:
12179:
12155:
12131:
12107:
12083:
12059:
12020:
12011:
11590:{\displaystyle s\in \{0,1\}^{*}}
10826:{\displaystyle w\in \{0,1\}^{*}}
10543:{\displaystyle x\in \Sigma ^{*}}
9987:{\displaystyle w\in \{0,1\}^{*}}
9717:{\displaystyle w\in \{0,1\}^{*}}
9487:{\displaystyle G\in \{0,1\}^{*}}
9386:{\displaystyle w\in \{0,1\}^{*}}
9260:(pronounced "sharp cycle") asks
9073:
5967:is said to recognize a language
4661:is hard for a class of problems
4536:{\displaystyle x\in \Sigma ^{*}}
4312:{\displaystyle x\in \Sigma ^{*}}
3721:Properties of complexity classes
3416:, but this has not been proven.
2344:problem. While it is known that
2157:{\displaystyle w\in \{0,1\}^{*}}
1068:? Or another kind of function?
751:Nondeterministic Turing machines
402:); for example, saying that the
14000:Computational complexity theory
13329:"On Promise Problems: A Survey"
13157:
12637:While a logarithmic runtime of
12034:Type 0 (Recursively enumerable)
9593:nondeterministic Turing machine
7968:(the formal representations of
6411:, which in turn is a subset of
5447:{\displaystyle k_{1}\leq k_{2}}
5335:{\displaystyle k_{1}\leq k_{2}}
4883:-complete problems, finding an
4750:
2843:While it is possible to define
1988:is in the language and rejects
1699:nondeterministic Turing machine
1265:, they are defined as follows:
1071:
757:Nondeterministic Turing machine
638:It is also possible to use the
609:computational complexity theory
476:has only two possible outputs,
40:computational complexity theory
13390:Lee, James R. (May 22, 2014).
13222:Arora, Sanjeev (Spring 2003).
13199:. Cambridge University Press.
12631:
11950:(statistical zero-knowledge).
11780:
11774:
11724:
11718:
11649:
11623:
11601:Relation to complexity classes
11107:
11081:
10928:
10902:
10504:
10498:
10425:
10413:
10215:
10144:
10138:
10095:
10083:
10072:
10068:
10060:
10056:
10010:
10004:
9911:
9875:
9740:
9734:
9641:
9514:
9508:
9449:problem, the input is a graph
9409:
9403:
9330:
8785:
8782:
8776:
8763:
8735:
8684:
8681:
8675:
8669:
8564:
8558:
8530:
8499:
8448:
8354:
8303:
8197:
8191:
8034:
7983:
7897:
7846:
7710:
7368:
7349:
7323:
7320:
7314:
7288:
7279:
7240:
7177:
7171:
7155:
7149:
7003:
6981:
6901:
6879:
6760:
6748:
6593:Interactive proof systems are
6170:
6156:
6077:
6063:
5742:
5739:
5733:
5727:
5715:
5709:
5664:
5658:
5588:
5585:
5579:
5573:
5554:
5548:
5506:
5500:
5401:
5381:
5372:
5352:
5289:
5269:
5230:
5210:
4995:
4989:
4960:
4953:
4930:
4924:
4587:
4581:
4363:
4357:
4125:
4112:
4077:
4064:
4055:
4042:
4019:
4006:
3997:
3984:
3975:
3962:
3939:
3926:
3899:
3886:
3863:
3857:
3793:
3674:
3654:
3559:
3539:
3339:While it is not known whether
3323:
3310:
3221:
3208:
3028:
3016:
2960:
2948:
2772:. It is further the case that
2741:
2721:
2632:
2612:
2293:
2281:
2256:
2252:
2244:
2240:
2052:is in the language. Formally:
1855:
1842:
1771:
1758:
1639:
1636:
1630:
1624:
1601:
1598:
1592:
1586:
1541:
1538:
1532:
1526:
1503:
1500:
1494:
1488:
1443:
1440:
1434:
1428:
1405:
1402:
1396:
1390:
1348:
1345:
1339:
1333:
1310:
1307:
1301:
1295:
1261:(for space complexity). Using
1173:
1167:
1132:
999:
993:
958:
897:
357:
73:, and a bounded resource like
13:
1:
13277:. Springer. pp. 81–106.
12789:
10271:over strings of an arbitrary
9173:a solution exists (as with a
9161:Counting problem (complexity)
8870:(for example, there are some
7738:; for example, on input bits
5800:probabilistic Turing machines
4145:
1032:takes on any input of length
650:Deterministic Turing machines
629:"Other models of computation"
208:
109:probabilistic Turing machines
14010:Theoretical computer science
13299:Gasarch, William I. (2019).
12775:{\displaystyle n>c\log n}
12740:{\displaystyle n<c\log n}
10570:deterministic Turing machine
10554:Important complexity classes
9606:is the set of all functions
9570:Important complexity classes
8632:Important complexity classes
7581:Formally, a Boolean circuit
7566:, a simplified model of the
7046:deterministic Turing machine
6662:Important complexity classes
6499:allow only one-sided error:
6199:Important complexity classes
5912:probabilistic Turing machine
4032:, which is a polynomial but
3802:
1687:deterministic Turing machine
1206:uses on any input of length
249:deterministic Turing machine
7:
12612:
10885:{\displaystyle \{0,1\}^{*}}
9119:quantum information science
8791:{\displaystyle O(t^{2}(n))}
8256:. In other words, a string
7132:. It is also the case that
6261:A slightly looser class is
5786:Other models of computation
4632:
4266:if there exists a function
3905:{\displaystyle O(n^{1000})}
3438:are the space analogues to
3113:are the space analogues to
2872:{\displaystyle \log n<n}
2828:
1667:
1557:The space complexity class
1459:The space complexity class
1107:is defined as the function
933:is defined as the function
175:(where ⊆ denotes the
54:"of related resource-based
10:
14026:
13976:List of complexity classes
12619:List of complexity classes
12223:Type 1 (Context Sensitive)
11054:The input to an algorithm
10690:
10437:{\displaystyle (x,y)\in R}
10185:
9573:
9348:such that for every input
9158:
8209:{\displaystyle C_{n}(w)=1}
7212:
6551:
5902:
5169:
4907:
4875:. Because all problems in
4833:is the hardest problem in
4623:polynomial-time reductions
4426:is no more difficult than
4149:
3725:
3423:
3098:
2832:
2821:
2788:{\displaystyle \subseteq }
2511:
2059:is the class of languages
1671:
1660:
1364:The time complexity class
1269:The time complexity class
1253:(for time complexity) and
1236:List of complexity classes
1233:
1075:
901:
754:
653:
13973:
13932:
13896:
13840:
13733:
13613:
13375:Algorithms and Complexity
12692:multiplied by a constant
12595:
12586:
12571:
12562:
12540:
12495:
12343:
12163:
12154:
12139:
12130:
12115:
12106:
12091:
12082:
12067:
12058:
12043:
12028:
12017:
12008:
11993:
11991:
10488:and the algorithm solves
10386:such that there exists a
10251:is defined as a relation
6725:{\displaystyle \epsilon }
6561:interactive proof systems
6548:Interactive proof systems
6000:{\displaystyle \epsilon }
5940:{\displaystyle \epsilon }
5830:interactive proof systems
5778:is strictly contained in
5770:is strictly contained in
5007:{\displaystyle f(n)>n}
4713:, since an algorithm for
4599:{\displaystyle p(x)\in Y}
4375:{\displaystyle f(x)\in Y}
3412:is strictly smaller than
2902:(sometimes lengthened to
2894:(sometimes lengthened to
398:(a concept borrowed from
243:is defined as the set of
113:interactive proof systems
85:that are solvable with a
18:Class (complexity theory)
13965:Interactive proof system
13403:University of Washington
13244:"Complexity of counting"
13230:. Princeton University.
12624:
12431:
12185:
10785:must act (correctly) on
10224:{\displaystyle f:A\to B}
9840:is the set of functions
9803:interactive proof system
9050:is defined similarly to
7034:An important feature of
6829:(Completeness) a string
6706:probabilistic algorithms
6576:approximation algorithms
6570:and yield insights into
6554:Interactive proof system
5880:, including the classes
5856:, including the classes
5832:, including the classes
5802:, including the classes
4972:{\displaystyle f(n)^{2}}
4131:{\displaystyle O(c^{n})}
3945:{\displaystyle O(n^{3})}
3359:. It is also known that
2818:Space complexity classes
2768:is a strict superset of
2760:is a strict superset of
2533:(sometimes shortened to
2525:(sometimes shortened to
2409:of computation provides
2389:is strictly larger than
1878:of solving a problem in
1230:Basic complexity classes
1179:{\displaystyle s_{M}(n)}
1005:{\displaystyle t_{M}(n)}
247:that can be solved by a
225:that can be solved by a
13266:Fortnow, Lance (1997).
12659:{\displaystyle c\log n}
11824:in the class is simply
11262:must be disjoint, i.e.
10287:{\displaystyle \Sigma }
9833:is defined as follows:
9773:'s computation tree on
9131:Other types of problems
9124:quantum Turing machines
8577:is the circuit size of
7576:non-uniform computation
7493:{\displaystyle \wedge }
7473:{\displaystyle x_{3}=0}
7440:{\displaystyle x_{2}=1}
7407:{\displaystyle x_{1}=0}
7117:{\displaystyle k\geq 2}
6248:The strictest class is
5987:with error probability
5878:quantum Turing machines
5614:space hierarchy theorem
5176:Space hierarchy theorem
3430:The complexity classes
3105:The complexity classes
2881:two-tape Turing machine
2322:The P versus NP problem
1657:Time complexity classes
1645:{\displaystyle O(s(n))}
1547:{\displaystyle O(s(n))}
1449:{\displaystyle O(t(n))}
1354:{\displaystyle O(t(n))}
271:. For example, "is the
213:Complexity classes are
14005:Measures of complexity
13919:Arithmetical hierarchy
13510:University of Waterloo
13309:University of Maryland
13085:Arora & Barak 2009
13049:Arora & Barak 2009
13037:Arora & Barak 2009
13001:Arora & Barak 2009
12977:Arora & Barak 2009
12815:Arora & Barak 2009
12776:
12741:
12706:
12686:
12685:{\displaystyle \log n}
12660:
11932:
11878:
11851:
11818:
11793:
11737:
11680:
11656:
11591:
11531:
11498:
11478:
11424:
11397:
11370:
11302:
11256:
11229:
11198:
11160:, which is called the
11154:
11114:
11074:for a promise problem
11068:
11044:
10990:
10935:
10886:
10847:
10827:
10779:
10759:
10739:
10544:
10511:
10478:
10458:
10438:
10400:
10380:
10360:
10337:
10288:
10265:
10245:
10225:
10171:
10151:
10122:
9988:
9943:
9923:
9887:
9787:
9767:
9747:
9718:
9673:
9653:
9552:statistical estimation
9541:
9521:
9488:
9443:
9416:
9387:
9342:
9278:
9254:
9223:
9199:
8989:
8933:
8864:
8829:
8792:
8747:
8711:
8691:
8618:
8598:
8571:
8542:
8506:
8428:
8414:) evaluates to 1 when
8408:
8388:
8361:
8290:
8270:
8250:
8230:
8210:
8168:
8148:
8128:
8108:
8088:
8068:
8041:
7955:
7924:
7904:
7817:
7797:
7732:
7659:
7639:
7603:directed acyclic graph
7595:
7553:
7542:
7518:
7494:
7474:
7441:
7408:
7375:
7184:
7118:
7082:can be generalized to
7025:
6965:
6945:
6923:
6863:
6843:
6820:
6787:
6767:
6726:
6710:randomized computation
6651:
6631:
6611:
6590:
6473:
6435:in the following way:
6325:
6305:
6285:
6212:
6189:
6138:
6118:
6096:
6045:
6025:
6001:
5981:
5961:
5941:
5905:Randomized computation
5899:Randomized computation
5756:
5602:
5459:time hierarchy theorem
5448:
5408:
5336:
5296:
5237:
5172:Time hierarchy theorem
5133:
5008:
4973:
4937:
4851:
4827:
4797:
4769:
4727:
4707:
4683:
4655:
4600:
4563:
4562:{\displaystyle x\in X}
4537:
4504:
4480:
4460:
4446:. Formally, a problem
4440:
4420:
4400:
4376:
4339:
4338:{\displaystyle x\in X}
4313:
4280:
4260:
4240:
4220:
4200:
4180:
4152:Reduction (complexity)
4132:
4084:
4026:
3946:
3906:
3870:
3824:
3681:
3566:
3420:EXPSPACE and NEXPSPACE
3402:
3330:
3228:
3085:
3035:
2967:
2873:
2789:
2748:
2639:
2498:
2457:
2375:
2300:
2299:{\displaystyle M(w,c)}
2265:
2198:
2178:
2158:
2113:
2093:
2073:
2046:
2022:
2002:
1982:
1962:
1942:
1915:
1862:
1778:
1646:
1608:
1548:
1510:
1450:
1412:
1355:
1317:
1220:
1200:
1180:
1144:
1101:
1046:
1026:
1006:
970:
927:
875:
766:
710:
665:
544:
513:
485:
450:
420:
376:
319:
288:
259:Computational problems
219:computational problems
52:computational problems
35:
13914:Grzegorczyk hierarchy
13909:Exponential hierarchy
13841:Considered infeasible
13268:"Counting Complexity"
12777:
12742:
12707:
12687:
12661:
12577:Type 2 (Context Free)
11933:
11879:
11852:
11819:
11794:
11738:
11681:
11657:
11592:
11532:
11499:
11479:
11425:
11398:
11371:
11303:
11257:
11230:
11199:
11155:
11115:
11069:
11045:
10991:
10936:
10887:
10848:
10828:
10780:
10760:
10740:
10545:
10512:
10479:
10459:
10439:
10401:
10381:
10361:
10338:
10289:
10266:
10246:
10226:
10172:
10152:
10123:
9989:
9944:
9924:
9888:
9788:
9768:
9748:
9719:
9674:
9654:
9595:'s computation tree.
9542:
9522:
9489:
9444:
9417:
9388:
9343:
9279:
9255:
9224:
9200:
8990:
8934:
8865:
8830:
8793:
8748:
8712:
8692:
8636:The complexity class
8619:
8599:
8597:{\displaystyle C_{n}}
8572:
8543:
8507:
8429:
8409:
8389:
8387:{\displaystyle C_{n}}
8362:
8291:
8271:
8251:
8231:
8211:
8169:
8149:
8129:
8114:if, for every string
8109:
8089:
8069:
8067:{\displaystyle C_{n}}
8042:
7956:
7954:{\displaystyle f_{C}}
7934:the Boolean function
7925:
7905:
7818:
7798:
7733:
7660:
7640:
7596:
7543:
7541:{\displaystyle \neg }
7519:
7517:{\displaystyle \vee }
7495:
7475:
7442:
7409:
7381:, with example input
7376:
7222:
7185:
7119:
7026:
6966:
6946:
6931:(Soundness) a string
6924:
6864:
6844:
6821:
6788:
6768:
6766:{\displaystyle (P,V)}
6727:
6652:
6632:
6612:
6588:
6474:
6372:randomized algorithms
6326:
6306:
6286:
6206:
6190:
6139:
6119:
6097:
6046:
6026:
6002:
5982:
5962:
5942:
5921:randomized algorithms
5757:
5603:
5449:
5409:
5337:
5297:
5238:
5134:
5009:
4974:
4938:
4852:
4828:
4798:
4770:
4741:is called the set of
4728:
4708:
4689:. Thus no problem in
4684:
4656:
4601:
4564:
4538:
4505:
4481:
4461:
4441:
4421:
4401:
4377:
4340:
4314:
4281:
4261:
4246:reduces to a problem
4241:
4221:
4201:
4181:
4133:
4085:
4027:
3947:
3907:
3871:
3825:
3682:
3567:
3403:
3351:famously showed that
3331:
3229:
3086:
3036:
2968:
2874:
2790:
2749:
2640:
2499:
2465:cryptographic schemes
2458:
2407:all possible branches
2376:
2301:
2266:
2199:
2179:
2159:
2114:
2094:
2074:
2047:
2023:
2003:
1983:
1963:
1943:
1916:
1863:
1779:
1647:
1609:
1549:
1511:
1451:
1413:
1356:
1318:
1221:
1201:
1181:
1145:
1102:
1047:
1027:
1007:
971:
928:
876:
764:
711:
663:
587:Optimization problems
545:
514:
471:
451:
421:
377:
320:
289:
265:computational problem
33:
13904:Polynomial hierarchy
13734:Suspected infeasible
13492:on February 7, 2022.
13255:Princeton University
12751:
12716:
12696:
12670:
12641:
11984:computability theory
11888:
11861:
11828:
11808:
11749:
11693:
11670:
11620:
11556:
11508:
11488:
11434:
11407:
11380:
11315:
11266:
11239:
11212:
11168:
11124:
11078:
11058:
11002:
10948:
10899:
10857:
10837:
10792:
10769:
10749:
10704:
10521:
10510:{\displaystyle f(x)}
10492:
10468:
10448:
10410:
10390:
10370:
10350:
10346:An algorithm solves
10301:
10278:
10255:
10235:
10203:
10161:
10150:{\displaystyle f(w)}
10132:
9998:
9953:
9933:
9897:
9844:
9777:
9757:
9746:{\displaystyle f(w)}
9728:
9683:
9663:
9610:
9587:. The connection to
9531:
9520:{\displaystyle f(G)}
9502:
9453:
9426:
9415:{\displaystyle f(w)}
9397:
9352:
9299:
9268:
9237:
9213:
9185:
9121:, are defined using
9001:Importance of P/poly
8965:
8945:polynomial hierarchy
8906:
8872:undecidable problems
8840:
8803:
8757:
8721:
8701:
8647:
8608:
8581:
8570:{\displaystyle f(n)}
8552:
8516:
8445:
8418:
8398:
8371:
8300:
8280:
8260:
8240:
8220:
8178:
8158:
8138:
8118:
8098:
8078:
8051:
7980:
7938:
7914:
7827:
7807:
7742:
7672:
7649:
7629:
7585:
7532:
7508:
7484:
7451:
7418:
7385:
7227:
7136:
7102:
7067:private random coins
6975:
6955:
6935:
6873:
6853:
6833:
6797:
6777:
6745:
6716:
6641:
6621:
6601:
6439:
6315:
6295:
6275:
6148:
6128:
6108:
6055:
6035:
6015:
5991:
5971:
5951:
5931:
5623:
5468:
5418:
5346:
5306:
5247:
5188:
5021:
4983:
4947:
4936:{\displaystyle f(n)}
4918:
4841:
4817:
4787:
4759:
4717:
4697:
4673:
4665:if every problem in
4645:
4627:log-space reductions
4575:
4547:
4514:
4494:
4490:computable function
4470:
4450:
4430:
4410:
4390:
4351:
4323:
4290:
4286:such that for every
4270:
4250:
4230:
4210:
4190:
4164:
4106:
4036:
3956:
3920:
3880:
3869:{\displaystyle O(n)}
3851:
3773:
3746:, or even under all
3577:
3465:
3363:
3239:
3140:
3048:
2978:
2913:
2851:
2779:
2650:
2544:
2508:EXPTIME and NEXPTIME
2471:
2430:
2348:
2275:
2211:
2188:
2168:
2123:
2103:
2083:
2063:
2036:
2012:
1992:
1972:
1952:
1932:
1905:
1789:
1708:
1618:
1561:
1520:
1463:
1422:
1368:
1327:
1273:
1210:
1190:
1154:
1111:
1091:
1066:exponential function
1062:logarithmic function
1036:
1016:
980:
937:
917:
833:
825:that are outside of
696:
675:Church–Turing thesis
599:Computational models
530:
503:
436:
406:
329:
305:
278:
71:model of computation
13933:Families of classes
13614:Considered feasible
13349:10.1007/11685654_12
11206:satisfy the promise
10366:if for every input
9556:statistical physics
9209:a particular graph
9066:Quantum computation
9060:parallel algorithms
8984:
8154:is in the language
7071:public random coins
6989: accepts
6887: accepts
6688:can also be called
6580:formal verification
6407:are all subsets of
6256:no error whatsoever
6164: rejects
6071: accepts
5915:, a variant of the
5862:and its subclasses
4179:{\displaystyle x+y}
3101:PSPACE (complexity)
718:tests for primality
644:computational model
605:computational model
552:tests for primality
13995:Complexity classes
13607:Complexity classes
13537:2019-08-27 at the
13532:The Complexity Zoo
13500:(April 11, 2006).
13167:(8 January 2017).
13051:, p. 341–342.
12829:, p. 48, 150.
12772:
12737:
12702:
12682:
12656:
11928:
11874:
11847:
11814:
11789:
11733:
11676:
11652:
11587:
11527:
11504:to emphasize that
11494:
11474:
11420:
11393:
11366:
11298:
11252:
11225:
11194:
11150:
11110:
11064:
11040:
10986:
10931:
10882:
10843:
10823:
10775:
10755:
10735:
10540:
10507:
10474:
10454:
10434:
10396:
10376:
10356:
10333:
10284:
10261:
10241:
10221:
10167:
10147:
10128:. In other words,
10118:
9984:
9939:
9919:
9883:
9783:
9763:
9743:
9714:
9679:such that for all
9669:
9649:
9537:
9517:
9484:
9439:
9412:
9383:
9338:
9274:
9250:
9219:
9195:
9085:. You can help by
9018:Two subclasses of
8985:
8968:
8947:. For example, if
8929:
8860:
8825:
8812:
8788:
8743:
8707:
8687:
8614:
8594:
8567:
8538:
8502:
8424:
8404:
8384:
8357:
8286:
8266:
8246:
8226:
8206:
8164:
8144:
8124:
8104:
8084:
8074:is a circuit with
8064:
8037:
7951:
7920:
7900:
7813:
7793:
7728:
7655:
7635:
7591:
7554:
7538:
7514:
7490:
7470:
7437:
7404:
7371:
7215:Circuit complexity
7180:
7114:
7038:is that it equals
7021:
7019:
6961:
6941:
6919:
6917:
6859:
6839:
6816:
6783:
6763:
6722:
6695:It turns out that
6647:
6627:
6607:
6591:
6469:
6321:
6301:
6281:
6213:
6209:quantum complexity
6185:
6134:
6114:
6092:
6041:
6021:
5997:
5977:
5957:
5937:
5752:
5598:
5444:
5404:
5332:
5292:
5233:
5184:, it follows that
5166:Hierarchy theorems
5129:
5004:
4969:
4933:
4847:
4823:
4813:. This means that
4793:
4765:
4723:
4703:
4679:
4651:
4596:
4559:
4533:
4510:such that for all
4500:
4486:if there exists a
4476:
4456:
4436:
4416:
4396:
4372:
4335:
4309:
4276:
4256:
4236:
4216:
4196:
4176:
4128:
4080:
4022:
3942:
3902:
3866:
3820:
3748:Boolean operations
3677:
3631:
3562:
3516:
3398:
3349:Savitch's theorem
3326:
3287:
3224:
3185:
3095:PSPACE and NPSPACE
3081:
3031:
2963:
2869:
2785:
2744:
2701:
2635:
2592:
2494:
2453:
2371:
2296:
2261:
2207:there exists some
2194:
2174:
2154:
2119:such that for all
2109:
2089:
2069:
2042:
2018:
1998:
1978:
1958:
1938:
1924:a polynomial-size
1911:
1897:if there exists a
1858:
1822:
1774:
1738:
1642:
1604:
1544:
1506:
1446:
1408:
1351:
1313:
1216:
1196:
1176:
1140:
1097:
1042:
1022:
1002:
966:
923:
871:
767:
706:
666:
540:
509:
486:
446:
416:
372:
315:
284:
36:
13982:
13981:
13924:Boolean hierarchy
13897:Class hierarchies
13358:978-3-540-32881-0
13294:on June 18, 2022.
13214:978-0-521-42426-4
12705:{\displaystyle c}
12610:
12609:
12606:
12605:
12582:
12581:
12551:
12550:
12506:
12505:
12429:
12428:
12354:
12353:
12339:
12338:
12326:
12325:
12228:
12227:
12174:
12173:
12150:
12149:
12126:
12125:
12102:
12101:
12078:
12077:
12054:
12053:
12039:
12038:
12004:
12003:
11925:
11871:
11844:
11817:{\displaystyle L}
11765:
11709:
11679:{\displaystyle M}
11646:
11633:
11518:
11497:{\displaystyle L}
11471:
11430:implicitly being
11417:
11390:
11338:
11325:
11289:
11276:
11249:
11222:
11208:. By definition,
11191:
11178:
11147:
11134:
11104:
11091:
11067:{\displaystyle M}
11012:
10958:
10925:
10912:
10846:{\displaystyle M}
10778:{\displaystyle L}
10758:{\displaystyle M}
10477:{\displaystyle f}
10457:{\displaystyle y}
10399:{\displaystyle y}
10379:{\displaystyle x}
10359:{\displaystyle f}
10264:{\displaystyle R}
10244:{\displaystyle f}
10194:function problems
10182:Function problems
10170:{\displaystyle w}
9942:{\displaystyle V}
9786:{\displaystyle w}
9766:{\displaystyle M}
9672:{\displaystyle M}
9540:{\displaystyle G}
9436:
9277:{\displaystyle G}
9247:
9222:{\displaystyle G}
9192:
9155:Counting problems
9145:function problems
9141:counting problems
9137:decision problems
9103:
9102:
8857:
8847:
8822:
8710:{\displaystyle t}
8617:{\displaystyle f}
8427:{\displaystyle w}
8407:{\displaystyle w}
8289:{\displaystyle n}
8269:{\displaystyle w}
8249:{\displaystyle w}
8236:is the length of
8229:{\displaystyle n}
8167:{\displaystyle L}
8147:{\displaystyle w}
8127:{\displaystyle w}
8107:{\displaystyle L}
8087:{\displaystyle n}
7970:decision problems
7923:{\displaystyle C}
7816:{\displaystyle b}
7803:, the output bit
7658:{\displaystyle n}
7638:{\displaystyle C}
7594:{\displaystyle C}
7094:defined above is
7018:
6998:
6990:
6964:{\displaystyle L}
6944:{\displaystyle w}
6916:
6896:
6888:
6862:{\displaystyle L}
6842:{\displaystyle w}
6786:{\displaystyle V}
6650:{\displaystyle P}
6630:{\displaystyle V}
6610:{\displaystyle P}
6595:abstract machines
6466:
6456:
6446:
6356:yields the class
6324:{\displaystyle M}
6304:{\displaystyle M}
6284:{\displaystyle M}
6165:
6154:
6137:{\displaystyle L}
6117:{\displaystyle w}
6072:
6061:
6044:{\displaystyle L}
6024:{\displaystyle w}
5980:{\displaystyle L}
5960:{\displaystyle M}
5180:By definition of
4910:Savitch's theorem
4904:Savitch's theorem
4850:{\displaystyle X}
4826:{\displaystyle X}
4796:{\displaystyle X}
4768:{\displaystyle X}
4726:{\displaystyle X}
4706:{\displaystyle X}
4682:{\displaystyle X}
4654:{\displaystyle X}
4503:{\displaystyle p}
4479:{\displaystyle Y}
4459:{\displaystyle X}
4439:{\displaystyle Y}
4419:{\displaystyle Y}
4399:{\displaystyle X}
4279:{\displaystyle f}
4259:{\displaystyle Y}
4239:{\displaystyle X}
4219:{\displaystyle y}
4199:{\displaystyle x}
3805:
3780:
3691:Savitch's theorem
3614:
3499:
3270:
3168:
3044:It is known that
2684:
2575:
2197:{\displaystyle L}
2177:{\displaystyle w}
2112:{\displaystyle p}
2099:and a polynomial
2092:{\displaystyle M}
2072:{\displaystyle L}
2045:{\displaystyle w}
2021:{\displaystyle w}
2001:{\displaystyle w}
1981:{\displaystyle w}
1961:{\displaystyle w}
1941:{\displaystyle c}
1914:{\displaystyle w}
1805:
1721:
1241:Basic definitions
1219:{\displaystyle n}
1199:{\displaystyle M}
1100:{\displaystyle M}
1045:{\displaystyle n}
1025:{\displaystyle M}
926:{\displaystyle M}
703:
575:Counting problems
563:Function problems
537:
512:{\displaystyle n}
490:decision problems
464:Decision problems
443:
413:
367:
336:
312:
287:{\displaystyle n}
245:decision problems
223:decision problems
121:quantum computers
105:function problems
101:counting problems
83:decision problems
16:(Redirected from
14017:
13600:
13593:
13586:
13577:
13576:
13566:David S. Johnson
13557:
13552:. Archived from
13521:
13519:
13506:
13493:
13491:
13485:. Archived from
13474:
13459:
13457:
13436:
13421:
13419:
13417:
13411:
13396:
13386:
13369:
13367:
13342:
13333:
13320:
13318:
13305:
13295:
13293:
13287:. Archived from
13272:
13262:
13248:
13235:
13218:
13198:
13180:
13151:
13145:
13139:
13133:
13127:
13121:
13115:
13109:
13100:
13094:
13088:
13082:
13069:
13063:
13052:
13046:
13040:
13034:
13028:
13022:
13016:
13010:
13004:
12998:
12992:
12986:
12980:
12974:
12968:
12962:
12956:
12950:
12944:
12938:
12932:
12926:
12917:
12911:
12902:
12896:
12890:
12884:
12878:
12872:
12866:
12860:
12854:
12848:
12842:
12836:
12830:
12824:
12818:
12812:
12806:
12800:
12783:
12781:
12779:
12778:
12773:
12746:
12744:
12743:
12738:
12711:
12709:
12708:
12703:
12691:
12689:
12688:
12683:
12665:
12663:
12662:
12657:
12635:
12601:Type 3 (Regular)
12597:
12591:
12573:
12567:
12560:
12542:
12538:
12529:
12522:
12515:
12497:
12493:
12486:
12477:
12468:
12461:
12454:
12447:
12438:
12420:
12416:
12409:
12402:
12393:
12384:
12377:
12370:
12363:
12345:
12330:
12317:
12313:
12306:
12297:
12288:
12281:
12274:
12267:
12258:
12249:
12242:
12235:
12219:
12213:
12204:
12197:
12190:
12183:
12165:
12159:
12141:
12135:
12117:
12111:
12093:
12087:
12069:
12063:
12045:
12030:
12024:
12015:
11999:Decision Problem
11995:
11989:
11937:
11935:
11934:
11929:
11927:
11926:
11923:
11917:
11912:
11911:
11883:
11881:
11880:
11875:
11873:
11872:
11869:
11856:
11854:
11853:
11848:
11846:
11845:
11842:
11823:
11821:
11820:
11815:
11798:
11796:
11795:
11790:
11767:
11766:
11763:
11742:
11740:
11739:
11734:
11711:
11710:
11707:
11685:
11683:
11682:
11677:
11661:
11659:
11658:
11653:
11648:
11647:
11644:
11635:
11634:
11631:
11596:
11594:
11593:
11588:
11586:
11585:
11536:
11534:
11533:
11528:
11520:
11519:
11516:
11503:
11501:
11500:
11495:
11483:
11481:
11480:
11475:
11473:
11472:
11469:
11463:
11458:
11457:
11429:
11427:
11426:
11421:
11419:
11418:
11415:
11402:
11400:
11399:
11394:
11392:
11391:
11388:
11375:
11373:
11372:
11367:
11365:
11364:
11340:
11339:
11336:
11327:
11326:
11323:
11307:
11305:
11304:
11299:
11291:
11290:
11287:
11278:
11277:
11274:
11261:
11259:
11258:
11253:
11251:
11250:
11247:
11234:
11232:
11231:
11226:
11224:
11223:
11220:
11203:
11201:
11200:
11195:
11193:
11192:
11189:
11180:
11179:
11176:
11159:
11157:
11156:
11151:
11149:
11148:
11145:
11136:
11135:
11132:
11119:
11117:
11116:
11111:
11106:
11105:
11102:
11093:
11092:
11089:
11073:
11071:
11070:
11065:
11049:
11047:
11046:
11041:
11039:
11038:
11014:
11013:
11010:
10995:
10993:
10992:
10987:
10985:
10984:
10960:
10959:
10956:
10940:
10938:
10937:
10932:
10927:
10926:
10923:
10914:
10913:
10910:
10891:
10889:
10888:
10883:
10881:
10880:
10852:
10850:
10849:
10844:
10832:
10830:
10829:
10824:
10822:
10821:
10784:
10782:
10781:
10776:
10764:
10762:
10761:
10756:
10744:
10742:
10741:
10736:
10734:
10733:
10698:Promise problems
10687:Promise problems
10549:
10547:
10546:
10541:
10539:
10538:
10516:
10514:
10513:
10508:
10483:
10481:
10480:
10475:
10463:
10461:
10460:
10455:
10443:
10441:
10440:
10435:
10405:
10403:
10402:
10397:
10385:
10383:
10382:
10377:
10365:
10363:
10362:
10357:
10342:
10340:
10339:
10334:
10332:
10331:
10319:
10318:
10293:
10291:
10290:
10285:
10270:
10268:
10267:
10262:
10250:
10248:
10247:
10242:
10230:
10228:
10227:
10222:
10188:Function problem
10176:
10174:
10173:
10168:
10156:
10154:
10153:
10148:
10127:
10125:
10124:
10119:
10117:
10116:
10110:
10109:
10076:
10075:
10071:
10063:
10029:
10028:
10022:
10021:
9993:
9991:
9990:
9985:
9983:
9982:
9948:
9946:
9945:
9940:
9928:
9926:
9925:
9920:
9918:
9910:
9892:
9890:
9889:
9884:
9882:
9874:
9873:
9792:
9790:
9789:
9784:
9772:
9770:
9769:
9764:
9752:
9750:
9749:
9744:
9723:
9721:
9720:
9715:
9713:
9712:
9678:
9676:
9675:
9670:
9658:
9656:
9655:
9650:
9648:
9640:
9639:
9546:
9544:
9543:
9538:
9526:
9524:
9523:
9518:
9493:
9491:
9490:
9485:
9483:
9482:
9448:
9446:
9445:
9440:
9438:
9437:
9421:
9419:
9418:
9413:
9392:
9390:
9389:
9384:
9382:
9381:
9347:
9345:
9344:
9339:
9337:
9329:
9328:
9289:formal languages
9283:
9281:
9280:
9275:
9259:
9257:
9256:
9251:
9249:
9248:
9228:
9226:
9225:
9220:
9204:
9202:
9201:
9196:
9194:
9193:
9175:decision problem
9167:counting problem
9149:promise problems
9098:
9095:
9077:
9070:
8994:
8992:
8991:
8986:
8983:
8982:
8976:
8938:
8936:
8935:
8930:
8928:
8927:
8915:
8914:
8869:
8867:
8866:
8861:
8859:
8858:
8849:
8848:
8834:
8832:
8831:
8826:
8824:
8823:
8814:
8813:
8797:
8795:
8794:
8789:
8775:
8774:
8752:
8750:
8749:
8744:
8742:
8734:
8716:
8714:
8713:
8708:
8696:
8694:
8693:
8688:
8668:
8667:
8623:
8621:
8620:
8615:
8603:
8601:
8600:
8595:
8593:
8592:
8576:
8574:
8573:
8568:
8547:
8545:
8544:
8539:
8537:
8529:
8512:is the function
8511:
8509:
8508:
8503:
8486:
8485:
8473:
8472:
8460:
8459:
8433:
8431:
8430:
8425:
8413:
8411:
8410:
8405:
8393:
8391:
8390:
8385:
8383:
8382:
8366:
8364:
8363:
8358:
8341:
8340:
8328:
8327:
8315:
8314:
8295:
8293:
8292:
8287:
8275:
8273:
8272:
8267:
8255:
8253:
8252:
8247:
8235:
8233:
8232:
8227:
8215:
8213:
8212:
8207:
8190:
8189:
8173:
8171:
8170:
8165:
8153:
8151:
8150:
8145:
8133:
8131:
8130:
8125:
8113:
8111:
8110:
8105:
8093:
8091:
8090:
8085:
8073:
8071:
8070:
8065:
8063:
8062:
8046:
8044:
8043:
8038:
8021:
8020:
8008:
8007:
7995:
7994:
7960:
7958:
7957:
7952:
7950:
7949:
7929:
7927:
7926:
7921:
7909:
7907:
7906:
7901:
7896:
7895:
7871:
7870:
7858:
7857:
7845:
7844:
7822:
7820:
7819:
7814:
7802:
7800:
7799:
7794:
7792:
7791:
7767:
7766:
7754:
7753:
7737:
7735:
7734:
7729:
7709:
7708:
7684:
7683:
7667:Boolean function
7664:
7662:
7661:
7656:
7644:
7642:
7641:
7636:
7600:
7598:
7597:
7592:
7568:digital circuits
7547:
7545:
7544:
7539:
7523:
7521:
7520:
7515:
7499:
7497:
7496:
7491:
7479:
7477:
7476:
7471:
7463:
7462:
7446:
7444:
7443:
7438:
7430:
7429:
7413:
7411:
7410:
7405:
7397:
7396:
7380:
7378:
7377:
7372:
7367:
7366:
7348:
7347:
7335:
7334:
7313:
7312:
7300:
7299:
7278:
7277:
7265:
7264:
7252:
7251:
7239:
7238:
7209:Boolean circuits
7189:
7187:
7186:
7181:
7170:
7169:
7148:
7147:
7123:
7121:
7120:
7115:
7030:
7028:
7027:
7022:
7020:
7011:
6999:
6996:
6991:
6988:
6970:
6968:
6967:
6962:
6950:
6948:
6947:
6942:
6928:
6926:
6925:
6920:
6918:
6909:
6897:
6894:
6889:
6886:
6868:
6866:
6865:
6860:
6848:
6846:
6845:
6840:
6825:
6823:
6822:
6817:
6815:
6814:
6792:
6790:
6789:
6784:
6772:
6770:
6769:
6764:
6731:
6729:
6728:
6723:
6656:
6654:
6653:
6648:
6636:
6634:
6633:
6628:
6616:
6614:
6613:
6608:
6478:
6476:
6475:
6470:
6468:
6467:
6458:
6457:
6448:
6447:
6330:
6328:
6327:
6322:
6310:
6308:
6307:
6302:
6290:
6288:
6287:
6282:
6194:
6192:
6191:
6186:
6166:
6163:
6155:
6152:
6143:
6141:
6140:
6135:
6123:
6121:
6120:
6115:
6101:
6099:
6098:
6093:
6073:
6070:
6062:
6059:
6050:
6048:
6047:
6042:
6030:
6028:
6027:
6022:
6006:
6004:
6003:
5998:
5986:
5984:
5983:
5978:
5966:
5964:
5963:
5958:
5946:
5944:
5943:
5938:
5854:Boolean circuits
5761:
5759:
5758:
5753:
5751:
5750:
5705:
5704:
5698:
5697:
5673:
5672:
5654:
5653:
5647:
5646:
5607:
5605:
5604:
5599:
5597:
5596:
5569:
5568:
5544:
5543:
5537:
5536:
5515:
5514:
5496:
5495:
5489:
5488:
5453:
5451:
5450:
5445:
5443:
5442:
5430:
5429:
5413:
5411:
5410:
5405:
5400:
5399:
5398:
5397:
5371:
5370:
5369:
5368:
5341:
5339:
5338:
5333:
5331:
5330:
5318:
5317:
5301:
5299:
5298:
5293:
5288:
5287:
5286:
5285:
5268:
5267:
5243:is contained in
5242:
5240:
5239:
5234:
5229:
5228:
5227:
5226:
5209:
5208:
5138:
5136:
5135:
5130:
5125:
5121:
5120:
5119:
5114:
5094:
5093:
5069:
5065:
5064:
5045:
5044:
5013:
5011:
5010:
5005:
4978:
4976:
4975:
4970:
4968:
4967:
4942:
4940:
4939:
4934:
4856:
4854:
4853:
4848:
4832:
4830:
4829:
4824:
4802:
4800:
4799:
4794:
4774:
4772:
4771:
4766:
4732:
4730:
4729:
4724:
4712:
4710:
4709:
4704:
4688:
4686:
4685:
4680:
4660:
4658:
4657:
4652:
4619:Levin reductions
4605:
4603:
4602:
4597:
4568:
4566:
4565:
4560:
4542:
4540:
4539:
4534:
4532:
4531:
4509:
4507:
4506:
4501:
4485:
4483:
4482:
4477:
4465:
4463:
4462:
4457:
4445:
4443:
4442:
4437:
4425:
4423:
4422:
4417:
4405:
4403:
4402:
4397:
4381:
4379:
4378:
4373:
4344:
4342:
4341:
4336:
4318:
4316:
4315:
4310:
4308:
4307:
4285:
4283:
4282:
4277:
4265:
4263:
4262:
4257:
4245:
4243:
4242:
4237:
4225:
4223:
4222:
4217:
4205:
4203:
4202:
4197:
4185:
4183:
4182:
4177:
4142:is close to 1.)
4141:
4137:
4135:
4134:
4129:
4124:
4123:
4089:
4087:
4086:
4081:
4076:
4075:
4054:
4053:
4031:
4029:
4028:
4023:
4018:
4017:
3996:
3995:
3974:
3973:
3951:
3949:
3948:
3943:
3938:
3937:
3911:
3909:
3908:
3903:
3898:
3897:
3875:
3873:
3872:
3867:
3829:
3827:
3826:
3821:
3816:
3815:
3806:
3798:
3796:
3782:
3781:
3686:
3684:
3683:
3678:
3673:
3672:
3671:
3670:
3653:
3652:
3630:
3629:
3610:
3609:
3571:
3569:
3568:
3563:
3558:
3557:
3556:
3555:
3538:
3537:
3515:
3514:
3495:
3494:
3407:
3405:
3404:
3399:
3397:
3396:
3372:
3371:
3335:
3333:
3332:
3327:
3322:
3321:
3309:
3308:
3286:
3285:
3266:
3265:
3233:
3231:
3230:
3225:
3220:
3219:
3207:
3206:
3184:
3183:
3164:
3163:
3090:
3088:
3087:
3082:
3080:
3079:
3070:
3069:
3057:
3056:
3040:
3038:
3037:
3032:
3015:
3014:
2990:
2989:
2972:
2970:
2969:
2964:
2947:
2946:
2922:
2921:
2878:
2876:
2875:
2870:
2824:Space complexity
2794:
2792:
2791:
2786:
2753:
2751:
2750:
2745:
2740:
2739:
2738:
2737:
2720:
2719:
2700:
2699:
2680:
2679:
2644:
2642:
2641:
2636:
2631:
2630:
2629:
2628:
2611:
2610:
2591:
2590:
2571:
2570:
2503:
2501:
2500:
2495:
2493:
2492:
2480:
2479:
2462:
2460:
2459:
2454:
2452:
2451:
2439:
2438:
2380:
2378:
2377:
2372:
2370:
2369:
2357:
2356:
2305:
2303:
2302:
2297:
2270:
2268:
2267:
2262:
2260:
2259:
2255:
2247:
2203:
2201:
2200:
2195:
2183:
2181:
2180:
2175:
2163:
2161:
2160:
2155:
2153:
2152:
2118:
2116:
2115:
2110:
2098:
2096:
2095:
2090:
2078:
2076:
2075:
2070:
2051:
2049:
2048:
2043:
2027:
2025:
2024:
2019:
2007:
2005:
2004:
1999:
1987:
1985:
1984:
1979:
1967:
1965:
1964:
1959:
1947:
1945:
1944:
1939:
1920:
1918:
1917:
1912:
1867:
1865:
1864:
1859:
1854:
1853:
1841:
1840:
1821:
1820:
1801:
1800:
1783:
1781:
1780:
1775:
1770:
1769:
1757:
1756:
1737:
1736:
1717:
1716:
1651:
1649:
1648:
1643:
1613:
1611:
1610:
1605:
1585:
1584:
1553:
1551:
1550:
1545:
1515:
1513:
1512:
1507:
1487:
1486:
1455:
1453:
1452:
1447:
1417:
1415:
1414:
1409:
1389:
1388:
1360:
1358:
1357:
1352:
1322:
1320:
1319:
1314:
1294:
1293:
1225:
1223:
1222:
1217:
1205:
1203:
1202:
1197:
1185:
1183:
1182:
1177:
1166:
1165:
1149:
1147:
1146:
1141:
1139:
1131:
1123:
1122:
1106:
1104:
1103:
1098:
1084:space complexity
1078:Space complexity
1051:
1049:
1048:
1043:
1031:
1029:
1028:
1023:
1011:
1009:
1008:
1003:
992:
991:
975:
973:
972:
967:
965:
957:
949:
948:
932:
930:
929:
924:
880:
878:
877:
872:
870:
869:
842:
841:
783:space complexity
745:space complexity
715:
713:
712:
707:
705:
704:
592:Promise problems
549:
547:
546:
541:
539:
538:
518:
516:
515:
510:
494:yes–no questions
474:decision problem
455:
453:
452:
447:
445:
444:
425:
423:
422:
417:
415:
414:
396:formal languages
381:
379:
378:
373:
368:
365:
360:
355:
338:
337:
324:
322:
321:
316:
314:
313:
293:
291:
290:
285:
117:Boolean circuits
44:complexity class
21:
14025:
14024:
14020:
14019:
14018:
14016:
14015:
14014:
13985:
13984:
13983:
13978:
13969:
13928:
13892:
13836:
13830:PSPACE-complete
13729:
13609:
13604:
13539:Wayback Machine
13528:
13526:Further reading
13517:
13504:
13489:
13483:
13472:
13464:Sipser, Michael
13455:
13449:
13434:
13415:
13413:
13409:
13394:
13365:
13359:
13340:
13331:
13325:Goldreich, Oded
13316:
13303:
13291:
13285:
13270:
13246:
13242:(Spring 2006).
13215:
13165:Aaronson, Scott
13160:
13155:
13154:
13146:
13142:
13134:
13130:
13122:
13118:
13110:
13103:
13095:
13091:
13083:
13072:
13064:
13055:
13047:
13043:
13035:
13031:
13023:
13019:
13011:
13007:
12999:
12995:
12987:
12983:
12975:
12971:
12963:
12959:
12951:
12947:
12939:
12935:
12927:
12920:
12912:
12905:
12897:
12893:
12885:
12881:
12873:
12869:
12861:
12857:
12849:
12845:
12837:
12833:
12825:
12821:
12813:
12809:
12801:
12797:
12792:
12787:
12786:
12752:
12749:
12748:
12717:
12714:
12713:
12697:
12694:
12693:
12671:
12668:
12667:
12642:
12639:
12638:
12636:
12632:
12627:
12615:
11974:is shown below
11956:
11922:
11918:
11913:
11907:
11903:
11889:
11886:
11885:
11868:
11864:
11862:
11859:
11858:
11841:
11837:
11829:
11826:
11825:
11809:
11806:
11805:
11762:
11758:
11750:
11747:
11746:
11706:
11702:
11694:
11691:
11690:
11671:
11668:
11667:
11643:
11639:
11630:
11626:
11621:
11618:
11617:
11603:
11581:
11577:
11557:
11554:
11553:
11539:formal language
11515:
11511:
11509:
11506:
11505:
11489:
11486:
11485:
11468:
11464:
11459:
11453:
11449:
11435:
11432:
11431:
11414:
11410:
11408:
11405:
11404:
11387:
11383:
11381:
11378:
11377:
11360:
11356:
11335:
11331:
11322:
11318:
11316:
11313:
11312:
11286:
11282:
11273:
11269:
11267:
11264:
11263:
11246:
11242:
11240:
11237:
11236:
11219:
11215:
11213:
11210:
11209:
11188:
11184:
11175:
11171:
11169:
11166:
11165:
11144:
11140:
11131:
11127:
11125:
11122:
11121:
11101:
11097:
11088:
11084:
11079:
11076:
11075:
11059:
11056:
11055:
11034:
11030:
11009:
11005:
11003:
11000:
10999:
10980:
10976:
10955:
10951:
10949:
10946:
10945:
10922:
10918:
10909:
10905:
10900:
10897:
10896:
10876:
10872:
10858:
10855:
10854:
10838:
10835:
10834:
10817:
10813:
10793:
10790:
10789:
10770:
10767:
10766:
10750:
10747:
10746:
10745:, an algorithm
10729:
10725:
10705:
10702:
10701:
10695:
10693:Promise problem
10689:
10675:if and only if
10667:. Importantly,
10586:. Importantly,
10574:polynomial time
10556:
10534:
10530:
10522:
10519:
10518:
10493:
10490:
10489:
10469:
10466:
10465:
10449:
10446:
10445:
10411:
10408:
10407:
10391:
10388:
10387:
10371:
10368:
10367:
10351:
10348:
10347:
10327:
10323:
10314:
10310:
10302:
10299:
10298:
10279:
10276:
10275:
10256:
10253:
10252:
10236:
10233:
10232:
10204:
10201:
10200:
10190:
10184:
10162:
10159:
10158:
10133:
10130:
10129:
10112:
10111:
10105:
10104:
10067:
10059:
10052:
10048:
10024:
10023:
10017:
10016:
9999:
9996:
9995:
9978:
9974:
9954:
9951:
9950:
9934:
9931:
9930:
9914:
9906:
9898:
9895:
9894:
9878:
9869:
9865:
9845:
9842:
9841:
9778:
9775:
9774:
9758:
9755:
9754:
9729:
9726:
9725:
9708:
9704:
9684:
9681:
9680:
9664:
9661:
9660:
9644:
9635:
9631:
9611:
9608:
9607:
9578:
9572:
9532:
9529:
9528:
9503:
9500:
9499:
9478:
9474:
9454:
9451:
9450:
9433:
9432:
9427:
9424:
9423:
9398:
9395:
9394:
9377:
9373:
9353:
9350:
9349:
9333:
9324:
9320:
9300:
9297:
9296:
9269:
9266:
9265:
9244:
9243:
9238:
9235:
9234:
9214:
9211:
9210:
9189:
9188:
9186:
9183:
9182:
9163:
9157:
9133:
9099:
9093:
9090:
9083:needs expansion
9068:
9013:advice function
9009:Turing machines
8978:
8977:
8972:
8966:
8963:
8962:
8920:
8919:
8910:
8909:
8907:
8904:
8903:
8898:that is not in
8854:
8853:
8844:
8843:
8841:
8838:
8837:
8819:
8818:
8807:
8806:
8804:
8801:
8800:
8770:
8766:
8758:
8755:
8754:
8738:
8730:
8722:
8719:
8718:
8702:
8699:
8698:
8651:
8650:
8648:
8645:
8644:
8634:
8609:
8606:
8605:
8588:
8584:
8582:
8579:
8578:
8553:
8550:
8549:
8533:
8525:
8517:
8514:
8513:
8481:
8477:
8468:
8464:
8455:
8451:
8446:
8443:
8442:
8439:time complexity
8419:
8416:
8415:
8399:
8396:
8395:
8378:
8374:
8372:
8369:
8368:
8367:if the circuit
8336:
8332:
8323:
8319:
8310:
8306:
8301:
8298:
8297:
8281:
8278:
8277:
8261:
8258:
8257:
8241:
8238:
8237:
8221:
8218:
8217:
8185:
8181:
8179:
8176:
8175:
8174:if and only if
8159:
8156:
8155:
8139:
8136:
8135:
8119:
8116:
8115:
8099:
8096:
8095:
8079:
8076:
8075:
8058:
8054:
8052:
8049:
8048:
8016:
8012:
8003:
7999:
7990:
7986:
7981:
7978:
7977:
7945:
7941:
7939:
7936:
7935:
7915:
7912:
7911:
7891:
7887:
7866:
7862:
7853:
7849:
7840:
7836:
7828:
7825:
7824:
7808:
7805:
7804:
7787:
7783:
7762:
7758:
7749:
7745:
7743:
7740:
7739:
7704:
7700:
7679:
7675:
7673:
7670:
7669:
7650:
7647:
7646:
7630:
7627:
7626:
7613:(generally the
7586:
7583:
7582:
7570:used in modern
7563:Boolean circuit
7533:
7530:
7529:
7509:
7506:
7505:
7485:
7482:
7481:
7458:
7454:
7452:
7449:
7448:
7425:
7421:
7419:
7416:
7415:
7392:
7388:
7386:
7383:
7382:
7362:
7358:
7343:
7339:
7330:
7326:
7308:
7304:
7295:
7291:
7273:
7269:
7260:
7256:
7247:
7243:
7234:
7230:
7228:
7225:
7224:
7217:
7211:
7162:
7161:
7140:
7139:
7137:
7134:
7133:
7103:
7100:
7099:
7009:
6995:
6987:
6976:
6973:
6972:
6956:
6953:
6952:
6936:
6933:
6932:
6907:
6893:
6885:
6874:
6871:
6870:
6854:
6851:
6850:
6834:
6831:
6830:
6807:
6806:
6798:
6795:
6794:
6778:
6775:
6774:
6746:
6743:
6742:
6717:
6714:
6713:
6664:
6642:
6639:
6638:
6622:
6619:
6618:
6617:and a verifier
6602:
6599:
6598:
6556:
6550:
6463:
6462:
6453:
6452:
6443:
6442:
6440:
6437:
6436:
6370:have efficient
6354:two-sided error
6347:one-sided error
6316:
6313:
6312:
6296:
6293:
6292:
6276:
6273:
6272:
6201:
6162:
6151:
6149:
6146:
6145:
6129:
6126:
6125:
6109:
6106:
6105:
6069:
6058:
6056:
6053:
6052:
6036:
6033:
6032:
6016:
6013:
6012:
5992:
5989:
5988:
5972:
5969:
5968:
5952:
5949:
5948:
5932:
5929:
5928:
5907:
5901:
5792:Turing machines
5788:
5746:
5745:
5700:
5699:
5678:
5677:
5668:
5667:
5649:
5648:
5627:
5626:
5624:
5621:
5620:
5592:
5591:
5564:
5560:
5539:
5538:
5520:
5519:
5510:
5509:
5491:
5490:
5472:
5471:
5469:
5466:
5465:
5438:
5434:
5425:
5421:
5419:
5416:
5415:
5393:
5389:
5388:
5384:
5364:
5360:
5359:
5355:
5347:
5344:
5343:
5326:
5322:
5313:
5309:
5307:
5304:
5303:
5281:
5277:
5276:
5272:
5251:
5250:
5248:
5245:
5244:
5222:
5218:
5217:
5213:
5192:
5191:
5189:
5186:
5185:
5178:
5170:Main articles:
5168:
5115:
5104:
5103:
5099:
5095:
5074:
5073:
5054:
5050:
5046:
5025:
5024:
5022:
5019:
5018:
4984:
4981:
4980:
4963:
4959:
4948:
4945:
4944:
4919:
4916:
4915:
4912:
4906:
4901:
4842:
4839:
4838:
4818:
4815:
4814:
4788:
4785:
4784:
4779:and is also in
4760:
4757:
4756:
4753:
4718:
4715:
4714:
4698:
4695:
4694:
4693:is harder than
4674:
4671:
4670:
4646:
4643:
4642:
4635:
4615:Karp reductions
4611:Cook reductions
4576:
4573:
4572:
4548:
4545:
4544:
4527:
4523:
4515:
4512:
4511:
4495:
4492:
4491:
4488:polynomial-time
4471:
4468:
4467:
4451:
4448:
4447:
4431:
4428:
4427:
4411:
4408:
4407:
4391:
4388:
4387:
4352:
4349:
4348:
4324:
4321:
4320:
4303:
4299:
4291:
4288:
4287:
4271:
4268:
4267:
4251:
4248:
4247:
4231:
4228:
4227:
4211:
4208:
4207:
4191:
4188:
4187:
4165:
4162:
4161:
4154:
4148:
4139:
4138:runtimes where
4119:
4115:
4107:
4104:
4103:
4071:
4067:
4049:
4045:
4037:
4034:
4033:
4013:
4009:
3991:
3987:
3969:
3965:
3957:
3954:
3953:
3933:
3929:
3921:
3918:
3917:
3893:
3889:
3881:
3878:
3877:
3852:
3849:
3848:
3811:
3810:
3797:
3792:
3777:
3776:
3774:
3771:
3770:
3728:
3723:
3666:
3662:
3661:
3657:
3633:
3632:
3625:
3618:
3581:
3580:
3578:
3575:
3574:
3551:
3547:
3546:
3542:
3518:
3517:
3510:
3503:
3469:
3468:
3466:
3463:
3462:
3428:
3422:
3377:
3376:
3367:
3366:
3364:
3361:
3360:
3317:
3313:
3289:
3288:
3281:
3274:
3243:
3242:
3240:
3237:
3236:
3215:
3211:
3187:
3186:
3179:
3172:
3144:
3143:
3141:
3138:
3137:
3103:
3097:
3075:
3074:
3062:
3061:
3052:
3051:
3049:
3046:
3045:
2995:
2994:
2982:
2981:
2979:
2976:
2975:
2927:
2926:
2917:
2916:
2914:
2911:
2910:
2852:
2849:
2848:
2841:
2839:NL (complexity)
2833:Main articles:
2831:
2826:
2820:
2780:
2777:
2776:
2733:
2729:
2728:
2724:
2703:
2702:
2695:
2688:
2654:
2653:
2651:
2648:
2647:
2624:
2620:
2619:
2615:
2594:
2593:
2586:
2579:
2548:
2547:
2545:
2542:
2541:
2520:
2512:Main articles:
2510:
2485:
2484:
2475:
2474:
2472:
2469:
2468:
2444:
2443:
2434:
2433:
2431:
2428:
2427:
2362:
2361:
2352:
2351:
2349:
2346:
2345:
2324:
2276:
2273:
2272:
2251:
2243:
2236:
2232:
2212:
2209:
2208:
2189:
2186:
2185:
2169:
2166:
2165:
2148:
2144:
2124:
2121:
2120:
2104:
2101:
2100:
2084:
2081:
2080:
2064:
2061:
2060:
2037:
2034:
2033:
2032:that the input
2013:
2010:
2009:
1993:
1990:
1989:
1973:
1970:
1969:
1953:
1950:
1949:
1933:
1930:
1929:
1906:
1903:
1902:
1876:time complexity
1849:
1845:
1824:
1823:
1816:
1809:
1793:
1792:
1790:
1787:
1786:
1765:
1761:
1740:
1739:
1732:
1725:
1712:
1711:
1709:
1706:
1705:
1691:polynomial time
1680:
1678:NP (complexity)
1672:Main articles:
1670:
1665:
1663:Time complexity
1659:
1619:
1616:
1615:
1565:
1564:
1562:
1559:
1558:
1521:
1518:
1517:
1467:
1466:
1464:
1461:
1460:
1423:
1420:
1419:
1372:
1371:
1369:
1366:
1365:
1328:
1325:
1324:
1277:
1276:
1274:
1271:
1270:
1243:
1238:
1232:
1211:
1208:
1207:
1191:
1188:
1187:
1161:
1157:
1155:
1152:
1151:
1135:
1127:
1118:
1114:
1112:
1109:
1108:
1092:
1089:
1088:
1080:
1074:
1037:
1034:
1033:
1017:
1014:
1013:
987:
983:
981:
978:
977:
961:
953:
944:
940:
938:
935:
934:
918:
915:
914:
910:time complexity
906:
904:Time complexity
900:
847:
846:
837:
836:
834:
831:
830:
795:
793:Resource bounds
775:time complexity
759:
753:
741:time complexity
734:recognizability
700:
699:
697:
694:
693:
658:
652:
601:
534:
533:
531:
528:
527:
504:
501:
500:
466:
440:
439:
437:
434:
433:
410:
409:
407:
404:
403:
390:that represent
364:
356:
351:
333:
332:
330:
327:
326:
309:
308:
306:
303:
302:
279:
276:
275:
263:Intuitively, a
261:
253:polynomial time
211:
127:following way:
97:polynomial time
28:
23:
22:
15:
12:
11:
5:
14023:
14013:
14012:
14007:
14002:
13997:
13980:
13979:
13974:
13971:
13970:
13968:
13967:
13962:
13957:
13952:
13947:
13942:
13936:
13934:
13930:
13929:
13927:
13926:
13921:
13916:
13911:
13906:
13900:
13898:
13894:
13893:
13891:
13890:
13885:
13880:
13875:
13870:
13865:
13860:
13855:
13850:
13844:
13842:
13838:
13837:
13835:
13834:
13833:
13832:
13822:
13817:
13816:
13815:
13805:
13800:
13795:
13790:
13785:
13780:
13775:
13770:
13769:
13768:
13766:co-NP-complete
13763:
13758:
13753:
13743:
13737:
13735:
13731:
13730:
13728:
13727:
13722:
13717:
13712:
13707:
13702:
13697:
13696:
13695:
13685:
13680:
13675:
13670:
13669:
13668:
13658:
13653:
13648:
13643:
13638:
13633:
13628:
13623:
13617:
13615:
13611:
13610:
13603:
13602:
13595:
13588:
13580:
13574:
13573:
13559:
13556:on 2016-04-16.
13542:
13527:
13524:
13523:
13522:
13494:
13481:
13460:
13448:978-0132288064
13447:
13422:
13387:
13370:
13357:
13321:
13296:
13283:
13263:
13236:
13219:
13213:
13185:Arora, Sanjeev
13181:
13159:
13156:
13153:
13152:
13148:Goldreich 2006
13140:
13136:Goldreich 2006
13128:
13124:Goldreich 2006
13116:
13101:
13089:
13087:, p. 344.
13070:
13053:
13041:
13039:, p. 342.
13029:
13017:
13005:
13003:, p. 286.
12993:
12991:, p. 355.
12981:
12979:, p. 144.
12969:
12957:
12945:
12933:
12918:
12916:, p. 321.
12903:
12901:, p. 320.
12891:
12879:
12867:
12855:
12843:
12841:, p. 255.
12831:
12819:
12807:
12803:Johnson (1990)
12794:
12793:
12791:
12788:
12785:
12784:
12771:
12768:
12765:
12762:
12759:
12756:
12736:
12733:
12730:
12727:
12724:
12721:
12701:
12681:
12678:
12675:
12655:
12652:
12649:
12646:
12629:
12628:
12626:
12623:
12622:
12621:
12614:
12611:
12608:
12607:
12604:
12603:
12593:
12592:
12584:
12583:
12580:
12579:
12569:
12568:
12561:
12553:
12552:
12549:
12548:
12539:
12531:
12530:
12523:
12516:
12508:
12507:
12504:
12503:
12494:
12487:
12479:
12478:
12471:
12469:
12462:
12455:
12448:
12440:
12439:
12432:
12430:
12427:
12426:
12417:
12410:
12403:
12395:
12394:
12387:
12385:
12378:
12371:
12364:
12356:
12355:
12352:
12351:
12342:
12340:
12337:
12336:
12327:
12324:
12323:
12314:
12307:
12299:
12298:
12291:
12289:
12282:
12275:
12268:
12260:
12259:
12252:
12250:
12243:
12236:
12229:
12226:
12225:
12215:
12214:
12207:
12205:
12198:
12191:
12184:
12176:
12175:
12172:
12171:
12161:
12160:
12152:
12151:
12148:
12147:
12137:
12136:
12128:
12127:
12124:
12123:
12113:
12112:
12104:
12103:
12100:
12099:
12089:
12088:
12080:
12079:
12076:
12075:
12065:
12064:
12056:
12055:
12052:
12051:
12042:
12040:
12037:
12036:
12026:
12025:
12018:
12016:
12009:
12006:
12005:
12002:
12001:
11992:
11955:
11952:
11921:
11916:
11910:
11906:
11902:
11899:
11896:
11893:
11884:is implicitly
11867:
11840:
11836:
11833:
11813:
11802:
11801:
11800:
11799:
11788:
11785:
11782:
11779:
11776:
11773:
11770:
11761:
11757:
11754:
11743:
11732:
11729:
11726:
11723:
11720:
11717:
11714:
11705:
11701:
11698:
11675:
11651:
11642:
11638:
11629:
11625:
11602:
11599:
11584:
11580:
11576:
11573:
11570:
11567:
11564:
11561:
11526:
11523:
11514:
11493:
11467:
11462:
11456:
11452:
11448:
11445:
11442:
11439:
11413:
11386:
11363:
11359:
11355:
11352:
11349:
11346:
11343:
11334:
11330:
11321:
11297:
11294:
11285:
11281:
11272:
11245:
11218:
11187:
11183:
11174:
11143:
11139:
11130:
11109:
11100:
11096:
11087:
11083:
11063:
11052:
11051:
11037:
11033:
11029:
11026:
11023:
11020:
11017:
11008:
10997:
10983:
10979:
10975:
10972:
10969:
10966:
10963:
10954:
10930:
10921:
10917:
10908:
10904:
10879:
10875:
10871:
10868:
10865:
10862:
10842:
10820:
10816:
10812:
10809:
10806:
10803:
10800:
10797:
10774:
10754:
10732:
10728:
10724:
10721:
10718:
10715:
10712:
10709:
10691:Main article:
10688:
10685:
10555:
10552:
10537:
10533:
10529:
10526:
10506:
10503:
10500:
10497:
10473:
10453:
10433:
10430:
10427:
10424:
10421:
10418:
10415:
10395:
10375:
10355:
10344:
10343:
10330:
10326:
10322:
10317:
10313:
10309:
10306:
10283:
10260:
10240:
10220:
10217:
10214:
10211:
10208:
10186:Main article:
10183:
10180:
10179:
10178:
10166:
10146:
10143:
10140:
10137:
10115:
10108:
10103:
10100:
10097:
10094:
10091:
10088:
10085:
10082:
10079:
10074:
10070:
10066:
10062:
10058:
10055:
10051:
10047:
10044:
10041:
10038:
10035:
10032:
10027:
10020:
10015:
10012:
10009:
10006:
10003:
9981:
9977:
9973:
9970:
9967:
9964:
9961:
9958:
9938:
9917:
9913:
9909:
9905:
9902:
9881:
9877:
9872:
9868:
9864:
9861:
9858:
9855:
9852:
9849:
9805:), so too can
9795:
9794:
9782:
9762:
9742:
9739:
9736:
9733:
9711:
9707:
9703:
9700:
9697:
9694:
9691:
9688:
9668:
9647:
9643:
9638:
9634:
9630:
9627:
9624:
9621:
9618:
9615:
9574:Main article:
9571:
9568:
9560:network design
9536:
9516:
9513:
9510:
9507:
9481:
9477:
9473:
9470:
9467:
9464:
9461:
9458:
9431:
9411:
9408:
9405:
9402:
9380:
9376:
9372:
9369:
9366:
9363:
9360:
9357:
9336:
9332:
9327:
9323:
9319:
9316:
9313:
9310:
9307:
9304:
9273:
9264:simple cycles
9242:
9218:
9169:asks not only
9159:Main article:
9156:
9153:
9132:
9129:
9101:
9100:
9080:
9078:
9067:
9064:
8981:
8975:
8971:
8926:
8923:
8918:
8913:
8852:
8817:
8811:
8787:
8784:
8781:
8778:
8773:
8769:
8765:
8762:
8741:
8737:
8733:
8729:
8726:
8717:is a function
8706:
8686:
8683:
8680:
8677:
8674:
8671:
8666:
8663:
8660:
8657:
8654:
8633:
8630:
8613:
8591:
8587:
8566:
8563:
8560:
8557:
8536:
8532:
8528:
8524:
8521:
8501:
8498:
8495:
8492:
8489:
8484:
8480:
8476:
8471:
8467:
8463:
8458:
8454:
8450:
8434:is its input.
8423:
8403:
8381:
8377:
8356:
8353:
8350:
8347:
8344:
8339:
8335:
8331:
8326:
8322:
8318:
8313:
8309:
8305:
8285:
8265:
8245:
8225:
8205:
8202:
8199:
8196:
8193:
8188:
8184:
8163:
8143:
8123:
8103:
8083:
8061:
8057:
8036:
8033:
8030:
8027:
8024:
8019:
8015:
8011:
8006:
8002:
7998:
7993:
7989:
7985:
7974:circuit family
7948:
7944:
7919:
7910:. The circuit
7899:
7894:
7890:
7886:
7883:
7880:
7877:
7874:
7869:
7865:
7861:
7856:
7852:
7848:
7843:
7839:
7835:
7832:
7812:
7790:
7786:
7782:
7779:
7776:
7773:
7770:
7765:
7761:
7757:
7752:
7748:
7727:
7724:
7721:
7718:
7715:
7712:
7707:
7703:
7699:
7696:
7693:
7690:
7687:
7682:
7678:
7654:
7634:
7590:
7558:Turing machine
7537:
7513:
7489:
7469:
7466:
7461:
7457:
7436:
7433:
7428:
7424:
7403:
7400:
7395:
7391:
7370:
7365:
7361:
7357:
7354:
7351:
7346:
7342:
7338:
7333:
7329:
7325:
7322:
7319:
7316:
7311:
7307:
7303:
7298:
7294:
7290:
7287:
7284:
7281:
7276:
7272:
7268:
7263:
7259:
7255:
7250:
7246:
7242:
7237:
7233:
7213:Main article:
7210:
7207:
7179:
7176:
7173:
7168:
7165:
7160:
7157:
7154:
7151:
7146:
7143:
7113:
7110:
7107:
7032:
7031:
7017:
7014:
7008:
7005:
7002:
6994:
6986:
6983:
6980:
6960:
6940:
6929:
6915:
6912:
6906:
6903:
6900:
6892:
6884:
6881:
6878:
6858:
6838:
6813:
6810:
6805:
6802:
6782:
6762:
6759:
6756:
6753:
6750:
6721:
6674:Turing machine
6663:
6660:
6646:
6626:
6606:
6552:Main article:
6549:
6546:
6461:
6451:
6320:
6300:
6280:
6200:
6197:
6196:
6195:
6184:
6181:
6178:
6175:
6172:
6169:
6161:
6158:
6133:
6113:
6102:
6091:
6088:
6085:
6082:
6079:
6076:
6068:
6065:
6040:
6020:
5996:
5976:
5956:
5936:
5917:Turing machine
5903:Main article:
5900:
5897:
5893:
5892:
5874:
5850:
5826:
5787:
5784:
5764:
5763:
5749:
5744:
5741:
5738:
5735:
5732:
5729:
5726:
5723:
5720:
5717:
5714:
5711:
5708:
5703:
5696:
5693:
5690:
5687:
5684:
5681:
5676:
5671:
5666:
5663:
5660:
5657:
5652:
5645:
5642:
5639:
5636:
5633:
5630:
5610:
5609:
5595:
5590:
5587:
5584:
5581:
5578:
5575:
5572:
5567:
5563:
5559:
5556:
5553:
5550:
5547:
5542:
5535:
5532:
5529:
5526:
5523:
5518:
5513:
5508:
5505:
5502:
5499:
5494:
5487:
5484:
5481:
5478:
5475:
5441:
5437:
5433:
5428:
5424:
5403:
5396:
5392:
5387:
5383:
5380:
5377:
5374:
5367:
5363:
5358:
5354:
5351:
5329:
5325:
5321:
5316:
5312:
5291:
5284:
5280:
5275:
5271:
5266:
5263:
5260:
5257:
5254:
5232:
5225:
5221:
5216:
5212:
5207:
5204:
5201:
5198:
5195:
5167:
5164:
5140:
5139:
5128:
5124:
5118:
5113:
5110:
5107:
5102:
5098:
5092:
5089:
5086:
5083:
5080:
5077:
5072:
5068:
5063:
5060:
5057:
5053:
5049:
5043:
5040:
5037:
5034:
5031:
5028:
5003:
5000:
4997:
4994:
4991:
4988:
4966:
4962:
4958:
4955:
4952:
4932:
4929:
4926:
4923:
4908:Main article:
4905:
4902:
4900:
4897:
4846:
4822:
4803:is said to be
4792:
4764:
4752:
4749:
4722:
4702:
4678:
4650:
4634:
4631:
4595:
4592:
4589:
4586:
4583:
4580:
4570:if and only if
4558:
4555:
4552:
4530:
4526:
4522:
4519:
4499:
4475:
4455:
4435:
4415:
4395:
4371:
4368:
4365:
4362:
4359:
4356:
4346:if and only if
4334:
4331:
4328:
4306:
4302:
4298:
4295:
4275:
4255:
4235:
4215:
4195:
4175:
4172:
4169:
4147:
4144:
4127:
4122:
4118:
4114:
4111:
4079:
4074:
4070:
4066:
4063:
4060:
4057:
4052:
4048:
4044:
4041:
4021:
4016:
4012:
4008:
4005:
4002:
3999:
3994:
3990:
3986:
3983:
3980:
3977:
3972:
3968:
3964:
3961:
3941:
3936:
3932:
3928:
3925:
3901:
3896:
3892:
3888:
3885:
3865:
3862:
3859:
3856:
3819:
3814:
3809:
3804:
3801:
3795:
3791:
3788:
3785:
3727:
3724:
3722:
3719:
3688:
3687:
3676:
3669:
3665:
3660:
3656:
3651:
3648:
3645:
3642:
3639:
3636:
3628:
3624:
3621:
3617:
3613:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3572:
3561:
3554:
3550:
3545:
3541:
3536:
3533:
3530:
3527:
3524:
3521:
3513:
3509:
3506:
3502:
3498:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3424:Main article:
3421:
3418:
3395:
3392:
3389:
3386:
3383:
3380:
3375:
3370:
3337:
3336:
3325:
3320:
3316:
3312:
3307:
3304:
3301:
3298:
3295:
3292:
3284:
3280:
3277:
3273:
3269:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3234:
3223:
3218:
3214:
3210:
3205:
3202:
3199:
3196:
3193:
3190:
3182:
3178:
3175:
3171:
3167:
3162:
3159:
3156:
3153:
3150:
3147:
3099:Main article:
3096:
3093:
3078:
3073:
3068:
3065:
3060:
3055:
3042:
3041:
3030:
3027:
3024:
3021:
3018:
3013:
3010:
3007:
3004:
3001:
2998:
2993:
2988:
2985:
2973:
2962:
2959:
2956:
2953:
2950:
2945:
2942:
2939:
2936:
2933:
2930:
2925:
2920:
2868:
2865:
2862:
2859:
2856:
2835:L (complexity)
2830:
2827:
2822:Main article:
2819:
2816:
2784:
2755:
2754:
2743:
2736:
2732:
2727:
2723:
2718:
2715:
2712:
2709:
2706:
2698:
2694:
2691:
2687:
2683:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2645:
2634:
2627:
2623:
2618:
2614:
2609:
2606:
2603:
2600:
2597:
2589:
2585:
2582:
2578:
2574:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2509:
2506:
2491:
2488:
2483:
2478:
2450:
2447:
2442:
2437:
2368:
2365:
2360:
2355:
2323:
2320:
2312:nondeterminism
2308:
2307:
2295:
2292:
2289:
2286:
2283:
2280:
2258:
2254:
2250:
2246:
2242:
2239:
2235:
2231:
2228:
2225:
2222:
2219:
2216:
2205:if and only if
2193:
2173:
2151:
2147:
2143:
2140:
2137:
2134:
2131:
2128:
2108:
2088:
2068:
2041:
2017:
1997:
1977:
1957:
1948:, and accepts
1937:
1910:
1869:
1868:
1857:
1852:
1848:
1844:
1839:
1836:
1833:
1830:
1827:
1819:
1815:
1812:
1808:
1804:
1799:
1796:
1784:
1773:
1768:
1764:
1760:
1755:
1752:
1749:
1746:
1743:
1735:
1731:
1728:
1724:
1720:
1715:
1674:P (complexity)
1669:
1666:
1661:Main article:
1658:
1655:
1654:
1653:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1603:
1600:
1597:
1594:
1591:
1588:
1583:
1580:
1577:
1574:
1571:
1568:
1555:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1505:
1502:
1499:
1496:
1493:
1490:
1485:
1482:
1479:
1476:
1473:
1470:
1457:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1407:
1404:
1401:
1398:
1395:
1392:
1387:
1384:
1381:
1378:
1375:
1362:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1312:
1309:
1306:
1303:
1300:
1297:
1292:
1289:
1286:
1283:
1280:
1263:big O notation
1242:
1239:
1231:
1228:
1215:
1195:
1175:
1172:
1169:
1164:
1160:
1138:
1134:
1130:
1126:
1121:
1117:
1096:
1076:Main article:
1073:
1070:
1041:
1021:
1001:
998:
995:
990:
986:
964:
960:
956:
952:
947:
943:
922:
902:Main article:
899:
896:
891:most efficient
868:
865:
862:
859:
856:
853:
850:
845:
840:
803:rate of growth
794:
791:
755:Main article:
752:
749:
670:Turing machine
656:Turing machine
654:Main article:
651:
648:
625:Turing machine
600:
597:
596:
595:
589:
584:
572:
508:
498:natural number
465:
462:
392:binary numbers
371:
366: is prime
363:
359:
354:
350:
347:
344:
341:
283:
273:natural number
260:
257:
227:Turing machine
210:
207:
203:nondeterminism
87:Turing machine
26:
9:
6:
4:
3:
2:
14022:
14011:
14008:
14006:
14003:
14001:
13998:
13996:
13993:
13992:
13990:
13977:
13972:
13966:
13963:
13961:
13958:
13956:
13953:
13951:
13948:
13946:
13943:
13941:
13938:
13937:
13935:
13931:
13925:
13922:
13920:
13917:
13915:
13912:
13910:
13907:
13905:
13902:
13901:
13899:
13895:
13889:
13886:
13884:
13881:
13879:
13876:
13874:
13871:
13869:
13866:
13864:
13861:
13859:
13856:
13854:
13851:
13849:
13846:
13845:
13843:
13839:
13831:
13828:
13827:
13826:
13823:
13821:
13818:
13814:
13811:
13810:
13809:
13806:
13804:
13801:
13799:
13796:
13794:
13791:
13789:
13786:
13784:
13781:
13779:
13776:
13774:
13771:
13767:
13764:
13762:
13759:
13757:
13754:
13752:
13749:
13748:
13747:
13744:
13742:
13739:
13738:
13736:
13732:
13726:
13723:
13721:
13718:
13716:
13713:
13711:
13708:
13706:
13703:
13701:
13698:
13694:
13691:
13690:
13689:
13686:
13684:
13681:
13679:
13676:
13674:
13671:
13667:
13664:
13663:
13662:
13659:
13657:
13654:
13652:
13649:
13647:
13644:
13642:
13639:
13637:
13634:
13632:
13629:
13627:
13624:
13622:
13619:
13618:
13616:
13612:
13608:
13601:
13596:
13594:
13589:
13587:
13582:
13581:
13578:
13571:
13567:
13563:
13562:Michael Garey
13560:
13555:
13551:
13547:
13546:Neil Immerman
13543:
13540:
13536:
13533:
13530:
13529:
13516:
13512:
13511:
13503:
13499:
13498:Watrous, John
13495:
13488:
13484:
13482:0-534-95097-3
13478:
13471:
13470:
13465:
13461:
13454:
13450:
13444:
13440:
13439:Prentice Hall
13433:
13432:
13427:
13423:
13408:
13404:
13400:
13393:
13388:
13384:
13380:
13376:
13371:
13364:
13360:
13354:
13350:
13346:
13339:
13338:
13330:
13326:
13322:
13315:
13311:
13310:
13302:
13297:
13290:
13286:
13284:9780387949734
13280:
13276:
13269:
13264:
13260:
13256:
13252:
13245:
13241:
13237:
13233:
13229:
13225:
13220:
13216:
13210:
13206:
13202:
13197:
13196:
13190:
13186:
13182:
13178:
13174:
13170:
13166:
13162:
13161:
13149:
13144:
13137:
13132:
13125:
13120:
13113:
13108:
13106:
13098:
13093:
13086:
13081:
13079:
13077:
13075:
13067:
13062:
13060:
13058:
13050:
13045:
13038:
13033:
13026:
13021:
13014:
13009:
13002:
12997:
12990:
12985:
12978:
12973:
12966:
12961:
12954:
12953:Aaronson 2017
12949:
12942:
12941:Aaronson 2017
12937:
12930:
12929:Aaronson 2017
12925:
12923:
12915:
12910:
12908:
12900:
12895:
12888:
12887:Aaronson 2017
12883:
12876:
12871:
12864:
12863:Aaronson 2017
12859:
12853:, p. 12.
12852:
12851:Aaronson 2017
12847:
12840:
12835:
12828:
12823:
12817:, p. 28.
12816:
12811:
12804:
12799:
12795:
12769:
12766:
12763:
12760:
12757:
12754:
12734:
12731:
12728:
12725:
12722:
12719:
12699:
12679:
12676:
12673:
12653:
12650:
12647:
12644:
12634:
12630:
12620:
12617:
12616:
12602:
12599:
12598:
12594:
12590:
12585:
12578:
12575:
12574:
12570:
12566:
12559:
12555:
12554:
12547:
12544:
12543:
12537:
12533:
12532:
12528:
12524:
12521:
12517:
12514:
12510:
12509:
12502:
12499:
12498:
12492:
12488:
12485:
12481:
12480:
12476:
12472:
12470:
12467:
12463:
12460:
12456:
12453:
12449:
12446:
12442:
12441:
12437:
12433:
12425:
12422:
12421:
12418:
12415:
12411:
12408:
12404:
12401:
12397:
12396:
12392:
12388:
12386:
12383:
12379:
12376:
12372:
12369:
12365:
12362:
12358:
12357:
12350:
12347:
12346:
12341:
12335:
12332:
12331:
12328:
12322:
12319:
12318:
12315:
12312:
12308:
12305:
12301:
12300:
12296:
12292:
12290:
12287:
12283:
12280:
12276:
12273:
12269:
12266:
12262:
12261:
12257:
12253:
12251:
12248:
12244:
12241:
12237:
12234:
12230:
12224:
12221:
12220:
12217:
12216:
12212:
12208:
12206:
12203:
12199:
12196:
12192:
12189:
12182:
12178:
12177:
12170:
12167:
12166:
12162:
12158:
12153:
12146:
12143:
12142:
12138:
12134:
12129:
12122:
12119:
12118:
12114:
12110:
12105:
12098:
12095:
12094:
12090:
12086:
12081:
12074:
12071:
12070:
12066:
12062:
12057:
12050:
12047:
12046:
12041:
12035:
12032:
12031:
12027:
12023:
12019:
12014:
12010:
12007:
12000:
11997:
11996:
11990:
11987:
11985:
11981:
11977:
11973:
11969:
11965:
11961:
11951:
11949:
11944:
11939:
11914:
11908:
11900:
11897:
11894:
11834:
11831:
11811:
11786:
11783:
11777:
11771:
11768:
11755:
11752:
11744:
11730:
11727:
11721:
11715:
11712:
11699:
11696:
11688:
11687:
11673:
11665:
11636:
11615:
11612:
11611:
11610:
11608:
11598:
11582:
11574:
11571:
11568:
11562:
11559:
11551:
11547:
11542:
11540:
11524:
11521:
11491:
11460:
11454:
11446:
11443:
11440:
11361:
11353:
11350:
11347:
11341:
11328:
11309:
11292:
11279:
11207:
11181:
11164:. Strings in
11163:
11137:
11094:
11061:
11035:
11027:
11024:
11021:
11015:
10998:
10981:
10973:
10970:
10967:
10961:
10944:
10943:
10942:
10915:
10893:
10877:
10869:
10866:
10863:
10840:
10818:
10810:
10807:
10804:
10798:
10795:
10788:
10772:
10752:
10730:
10722:
10719:
10716:
10710:
10707:
10699:
10694:
10684:
10682:
10678:
10674:
10670:
10666:
10665:
10660:
10659:
10654:
10650:
10645:
10643:
10639:
10635:
10631:
10627:
10623:
10619:
10615:
10611:
10607:
10603:
10599:
10598:
10594:
10589:
10585:
10584:
10579:
10575:
10571:
10567:
10563:
10562:
10551:
10535:
10527:
10524:
10501:
10495:
10487:
10471:
10451:
10431:
10428:
10422:
10419:
10416:
10393:
10373:
10353:
10328:
10320:
10315:
10307:
10304:
10297:
10296:
10295:
10274:
10258:
10238:
10218:
10212:
10209:
10206:
10199:
10195:
10189:
10164:
10141:
10135:
10101:
10098:
10092:
10089:
10086:
10080:
10077:
10064:
10053:
10045:
10042:
10039:
10033:
10030:
10013:
10007:
10001:
9979:
9971:
9968:
9965:
9959:
9956:
9936:
9903:
9900:
9870:
9862:
9859:
9856:
9850:
9847:
9839:
9836:
9835:
9834:
9832:
9828:
9824:
9820:
9816:
9812:
9808:
9804:
9800:
9780:
9760:
9737:
9731:
9709:
9701:
9698:
9695:
9689:
9686:
9666:
9636:
9628:
9625:
9622:
9616:
9613:
9605:
9602:
9601:
9600:
9598:
9594:
9590:
9586:
9582:
9577:
9567:
9565:
9561:
9557:
9553:
9548:
9534:
9511:
9505:
9497:
9479:
9471:
9468:
9465:
9459:
9456:
9406:
9400:
9378:
9370:
9367:
9364:
9358:
9355:
9325:
9317:
9314:
9311:
9305:
9302:
9294:
9290:
9285:
9271:
9263:
9232:
9216:
9208:
9180:
9176:
9172:
9168:
9162:
9152:
9150:
9146:
9142:
9138:
9128:
9126:
9125:
9120:
9116:
9115:
9110:
9109:
9097:
9088:
9084:
9081:This section
9079:
9076:
9072:
9071:
9063:
9061:
9057:
9053:
9049:
9045:
9041:
9040:directed path
9037:
9033:
9032:
9027:
9026:
9021:
9016:
9014:
9010:
9006:
9002:
8998:
8973:
8961:collapses to
8960:
8956:
8952:
8951:
8946:
8942:
8916:
8901:
8897:
8893:
8892:
8888:
8883:
8879:
8877:
8873:
8850:
8835:
8815:
8779:
8771:
8767:
8760:
8727:
8724:
8704:
8678:
8672:
8641:
8640:
8629:
8627:
8611:
8589:
8585:
8561:
8555:
8522:
8519:
8496:
8493:
8490:
8487:
8482:
8478:
8474:
8469:
8465:
8461:
8456:
8452:
8440:
8435:
8421:
8401:
8379:
8375:
8351:
8348:
8345:
8342:
8337:
8333:
8329:
8324:
8320:
8316:
8311:
8307:
8283:
8263:
8243:
8223:
8203:
8200:
8194:
8186:
8182:
8161:
8141:
8121:
8101:
8081:
8059:
8055:
8031:
8028:
8025:
8022:
8017:
8013:
8009:
8004:
8000:
7996:
7991:
7987:
7975:
7971:
7967:
7962:
7946:
7942:
7933:
7917:
7892:
7888:
7884:
7881:
7878:
7875:
7872:
7867:
7863:
7859:
7854:
7850:
7841:
7837:
7833:
7830:
7810:
7788:
7784:
7780:
7777:
7774:
7771:
7768:
7763:
7759:
7755:
7750:
7746:
7722:
7719:
7716:
7705:
7697:
7694:
7691:
7685:
7680:
7676:
7668:
7652:
7632:
7624:
7620:
7616:
7612:
7608:
7604:
7588:
7579:
7577:
7573:
7569:
7565:
7564:
7559:
7551:
7527:
7511:
7503:
7487:
7467:
7464:
7459:
7455:
7434:
7431:
7426:
7422:
7401:
7398:
7393:
7389:
7363:
7359:
7352:
7344:
7340:
7336:
7331:
7327:
7317:
7309:
7305:
7301:
7296:
7292:
7282:
7274:
7270:
7266:
7261:
7257:
7253:
7248:
7244:
7235:
7231:
7221:
7216:
7206:
7204:
7203:
7198:
7197:
7191:
7174:
7158:
7152:
7131:
7127:
7111:
7108:
7105:
7097:
7093:
7089:
7085:
7081:
7076:
7072:
7068:
7064:
7060:
7059:
7054:
7049:
7047:
7043:
7042:
7037:
7015:
7012:
7006:
7000:
6992:
6984:
6958:
6938:
6930:
6913:
6910:
6904:
6898:
6890:
6882:
6856:
6836:
6828:
6827:
6826:
6803:
6800:
6780:
6757:
6754:
6751:
6740:
6739:
6733:
6719:
6711:
6707:
6703:
6702:probabilistic
6698:
6693:
6691:
6687:
6683:
6679:
6675:
6671:
6670:
6659:
6644:
6624:
6604:
6596:
6587:
6583:
6581:
6577:
6573:
6569:
6568:
6563:
6562:
6555:
6545:
6543:
6542:
6537:
6536:
6531:
6530:
6524:
6522:
6518:
6514:
6510:
6506:
6502:
6498:
6494:
6490:
6486:
6482:
6459:
6449:
6434:
6430:
6426:
6422:
6418:
6414:
6410:
6406:
6402:
6398:
6394:
6392:
6391:
6385:
6383:
6382:
6377:
6373:
6369:
6365:
6361:
6360:
6355:
6350:
6348:
6344:
6340:
6336:
6335:
6318:
6298:
6278:
6270:
6266:
6265:
6259:
6257:
6253:
6252:
6246:
6244:
6243:
6238:
6237:
6232:
6231:
6226:
6225:
6220:
6219:
6210:
6205:
6182:
6179:
6176:
6173:
6167:
6159:
6144:implies that
6131:
6111:
6103:
6089:
6086:
6083:
6080:
6074:
6066:
6051:implies that
6038:
6018:
6010:
6009:
6008:
5994:
5974:
5954:
5934:
5924:
5922:
5918:
5914:
5913:
5906:
5896:
5891:
5890:
5885:
5884:
5879:
5875:
5873:
5872:
5867:
5866:
5861:
5860:
5855:
5851:
5849:
5848:
5843:
5842:
5837:
5836:
5831:
5827:
5825:
5824:
5819:
5818:
5813:
5812:
5807:
5806:
5801:
5797:
5796:
5795:
5793:
5783:
5781:
5777:
5773:
5769:
5736:
5730:
5724:
5721:
5718:
5712:
5706:
5674:
5661:
5655:
5619:
5618:
5617:
5615:
5582:
5576:
5570:
5565:
5561:
5557:
5551:
5545:
5516:
5503:
5497:
5464:
5463:
5462:
5460:
5455:
5439:
5435:
5431:
5426:
5422:
5394:
5390:
5385:
5378:
5375:
5365:
5361:
5356:
5349:
5327:
5323:
5319:
5314:
5310:
5282:
5278:
5273:
5223:
5219:
5214:
5183:
5177:
5173:
5163:
5159:
5157:
5153:
5149:
5145:
5126:
5122:
5116:
5111:
5108:
5105:
5100:
5096:
5070:
5066:
5061:
5058:
5055:
5051:
5047:
5017:
5016:
5015:
5001:
4998:
4992:
4986:
4964:
4956:
4950:
4927:
4921:
4911:
4896:
4894:
4891: =
4890:
4886:
4882:
4878:
4874:
4870:
4868:
4862:
4860:
4844:
4836:
4820:
4812:
4808:
4807:
4790:
4782:
4778:
4762:
4755:If a problem
4748:
4746:
4745:
4740:
4736:
4720:
4700:
4692:
4676:
4668:
4664:
4648:
4640:
4630:
4628:
4624:
4620:
4616:
4612:
4607:
4593:
4590:
4584:
4578:
4571:
4556:
4553:
4550:
4528:
4520:
4517:
4497:
4489:
4473:
4453:
4433:
4413:
4393:
4383:
4369:
4366:
4360:
4354:
4347:
4332:
4329:
4326:
4304:
4296:
4293:
4273:
4253:
4233:
4213:
4193:
4173:
4170:
4167:
4159:
4153:
4143:
4120:
4116:
4109:
4101:
4097:
4093:
4072:
4068:
4061:
4058:
4050:
4046:
4039:
4014:
4010:
4003:
4000:
3992:
3988:
3981:
3978:
3970:
3966:
3959:
3934:
3930:
3923:
3915:
3894:
3890:
3883:
3860:
3854:
3845:
3843:
3839:
3835:
3834:
3807:
3799:
3789:
3783:
3768:
3764:
3760:
3755:
3753:
3749:
3745:
3741:
3737:
3733:
3718:
3716:
3712:
3708:
3704:
3700:
3696:
3692:
3667:
3663:
3658:
3622:
3619:
3615:
3611:
3573:
3552:
3548:
3543:
3507:
3504:
3500:
3496:
3461:
3460:
3459:
3457:
3453:
3449:
3448:
3443:
3442:
3437:
3433:
3427:
3417:
3415:
3411:
3373:
3358:
3354:
3350:
3346:
3342:
3318:
3314:
3278:
3275:
3271:
3267:
3235:
3216:
3212:
3176:
3173:
3169:
3165:
3136:
3135:
3134:
3132:
3128:
3124:
3123:
3118:
3117:
3112:
3108:
3102:
3092:
3071:
3058:
3025:
3022:
3019:
2991:
2974:
2957:
2954:
2951:
2923:
2909:
2908:
2907:
2905:
2901:
2897:
2893:
2889:
2886:
2885:computability
2882:
2866:
2863:
2860:
2857:
2854:
2846:
2840:
2836:
2825:
2815:
2813:
2809:
2805:
2801:
2797:
2782:
2775:
2771:
2767:
2763:
2759:
2734:
2730:
2725:
2692:
2689:
2685:
2681:
2646:
2625:
2621:
2616:
2583:
2580:
2576:
2572:
2540:
2539:
2538:
2536:
2532:
2528:
2524:
2519:
2515:
2505:
2481:
2466:
2440:
2425:
2421:
2417:
2412:
2408:
2404:
2400:
2396:
2392:
2388:
2384:
2358:
2343:
2342:
2338:
2333:
2329:
2319:
2317:
2313:
2290:
2287:
2284:
2278:
2248:
2237:
2229:
2226:
2223:
2217:
2214:
2206:
2191:
2171:
2149:
2141:
2138:
2135:
2129:
2126:
2106:
2086:
2066:
2058:
2055:
2054:
2053:
2039:
2031:
2015:
1995:
1975:
1955:
1935:
1927:
1923:
1908:
1900:
1899:deterministic
1896:
1892:
1888:
1883:
1881:
1877:
1873:
1850:
1846:
1813:
1810:
1806:
1802:
1785:
1766:
1762:
1729:
1726:
1722:
1718:
1704:
1703:
1702:
1700:
1696:
1692:
1688:
1684:
1679:
1675:
1664:
1633:
1627:
1621:
1595:
1589:
1556:
1535:
1529:
1523:
1497:
1491:
1458:
1437:
1431:
1425:
1399:
1393:
1363:
1342:
1336:
1330:
1304:
1298:
1268:
1267:
1266:
1264:
1260:
1256:
1252:
1248:
1237:
1227:
1213:
1193:
1170:
1162:
1158:
1124:
1119:
1115:
1094:
1085:
1079:
1069:
1067:
1063:
1059:
1053:
1039:
1019:
996:
988:
984:
950:
945:
941:
920:
911:
905:
895:
892:
887:
882:
843:
828:
824:
820:
819:
814:
810:
809:
804:
800:
790:
786:
784:
780:
776:
771:
763:
758:
748:
746:
742:
737:
735:
731:
727:
723:
719:
690:
688:
687:deterministic
682:
680:
676:
671:
662:
657:
647:
645:
641:
636:
634:
630:
626:
621:
619:
614:
610:
606:
593:
590:
588:
585:
582:
581:
576:
573:
570:
569:
564:
561:
560:
559:
555:
553:
525:
521:
506:
499:
495:
491:
483:
479:
475:
470:
461:
459:
431:
430:
401:
397:
393:
389:
385:
361:
348:
345:
339:
300:
296:
281:
274:
270:
266:
256:
254:
250:
246:
242:
241:
236:
232:
229:with bounded
228:
224:
220:
216:
206:
204:
200:
196:
192:
191:
187:
182:
181:open problems
178:
174:
173:
168:
167:
162:
161:
156:
155:
150:
149:
144:
143:
138:
137:
132:
131:
124:
122:
118:
114:
110:
106:
102:
98:
94:
93:
88:
84:
80:
76:
72:
67:
65:
61:
57:
53:
49:
45:
41:
32:
19:
13606:
13569:
13554:the original
13508:
13487:the original
13468:
13430:
13426:Rich, Elaine
13414:. Retrieved
13398:
13392:"Lecture 16"
13374:
13336:
13307:
13289:the original
13274:
13250:
13227:
13194:
13172:
13158:Bibliography
13143:
13131:
13119:
13114:, p. 1.
13112:Watrous 2006
13092:
13044:
13032:
13020:
13013:Fortnow 1997
13008:
12996:
12984:
12972:
12960:
12955:, p. 6.
12948:
12943:, p. 5.
12936:
12931:, p. 7.
12894:
12889:, p. 4.
12882:
12875:Gasarch 2019
12870:
12865:, p. 3.
12858:
12846:
12834:
12822:
12810:
12798:
12633:
11979:
11975:
11971:
11967:
11962:is a strict
11959:
11957:
11947:
11942:
11940:
11803:
11663:
11613:
11606:
11604:
11549:
11546:planar graph
11543:
11310:
11205:
11204:are said to
11161:
11053:
10894:
10786:
10697:
10696:
10680:
10676:
10672:
10668:
10662:
10656:
10652:
10648:
10646:
10641:
10637:
10633:
10629:
10625:
10621:
10617:
10613:
10609:
10605:
10601:
10596:
10592:
10587:
10581:
10577:
10565:
10559:
10557:
10345:
10193:
10191:
9837:
9830:
9826:
9822:
9818:
9810:
9806:
9798:
9797:And just as
9796:
9603:
9596:
9588:
9584:
9580:
9579:
9549:
9286:
9261:
9231:simple cycle
9206:
9178:
9177:), but asks
9170:
9166:
9164:
9134:
9122:
9112:
9106:
9105:The classes
9104:
9091:
9087:adding to it
9082:
9055:
9051:
9047:
9043:
9035:
9029:
9023:
9019:
9017:
9004:
8996:
8958:
8954:
8948:
8940:
8899:
8895:
8890:
8886:
8881:
8880:
8875:
8874:that are in
8637:
8635:
8436:
7973:
7963:
7931:
7580:
7561:
7555:
7200:
7194:
7192:
7129:
7125:
7095:
7091:
7087:
7083:
7079:
7074:
7070:
7066:
7062:
7056:
7052:
7050:
7039:
7035:
7033:
6736:
6734:
6701:
6696:
6694:
6689:
6685:
6681:
6667:
6665:
6592:
6572:cryptography
6565:
6559:
6557:
6539:
6533:
6527:
6525:
6520:
6516:
6512:
6508:
6504:
6500:
6496:
6492:
6488:
6484:
6480:
6432:
6428:
6424:
6420:
6416:
6412:
6408:
6404:
6400:
6396:
6395:
6388:
6386:
6379:
6375:
6367:
6363:
6357:
6351:
6342:
6338:
6332:
6268:
6262:
6260:
6255:
6249:
6247:
6240:
6234:
6228:
6222:
6216:
6214:
5925:
5910:
5908:
5894:
5887:
5881:
5869:
5863:
5857:
5845:
5839:
5833:
5821:
5815:
5809:
5803:
5789:
5779:
5775:
5771:
5767:
5765:
5616:states that
5611:
5461:states that
5456:
5181:
5179:
5160:
5155:
5151:
5147:
5143:
5141:
4913:
4892:
4888:
4884:
4880:
4876:
4872:
4866:
4863:
4858:
4834:
4810:
4804:
4780:
4776:
4775:is hard for
4754:
4751:Completeness
4742:
4738:
4734:
4690:
4666:
4662:
4638:
4636:
4608:
4569:
4487:
4384:
4345:
4157:
4155:
4099:
4095:
4091:
3913:
3846:
3841:
3837:
3831:
3766:
3762:
3758:
3756:
3751:
3729:
3714:
3710:
3706:
3702:
3698:
3694:
3693:showed that
3689:
3455:
3451:
3445:
3439:
3435:
3431:
3429:
3413:
3409:
3356:
3352:
3344:
3340:
3338:
3130:
3126:
3120:
3114:
3110:
3106:
3104:
3043:
2903:
2899:
2895:
2891:
2890:
2842:
2811:
2807:
2803:
2799:
2795:
2773:
2769:
2765:
2761:
2757:
2756:
2534:
2530:
2526:
2522:
2521:
2423:
2419:
2415:
2410:
2406:
2402:
2398:
2394:
2390:
2386:
2382:
2340:
2336:
2331:
2327:
2325:
2315:
2309:
2204:
2056:
1921:
1898:
1894:
1890:
1886:
1884:
1879:
1871:
1870:
1694:
1682:
1681:
1258:
1254:
1250:
1246:
1244:
1081:
1072:Space bounds
1054:
907:
890:
885:
883:
826:
822:
816:
806:
802:
799:upper bounds
796:
787:
778:
772:
768:
738:
730:decidability
725:
721:
691:
683:
678:
669:
667:
637:
622:
612:
602:
578:
566:
556:
487:
481:
477:
457:
427:
262:
238:
212:
198:
194:
189:
185:
170:
164:
158:
152:
146:
140:
134:
129:
125:
90:
68:
43:
37:
13813:#P-complete
13751:NP-complete
13666:NL-complete
13240:Barak, Boaz
13189:Barak, Boaz
12989:Sipser 2006
12914:Sipser 2006
12899:Sipser 2006
12839:Sipser 2006
12827:Sipser 2006
12049:Undecidable
11686:such that:
10406:satisfying
9815:certificate
7930:is said to
7611:logic gates
6678:certificate
6479:. That is,
6427:relates to
3757:Each class
3744:conjunction
3740:disjunction
3450:. That is,
3125:. That is,
2845:logarithmic
2810:must equal
2463:, and most
1926:certificate
898:Time bounds
811:grows at a
640:Blum axioms
400:linguistics
217:of related
13989:Categories
13868:ELEMENTARY
13693:P-complete
13416:October 5,
13066:Barak 2006
13025:Arora 2003
12790:References
11689:For every
9094:April 2017
8626:polynomial
7548:nodes are
7528:, and the
7524:nodes are
7500:nodes are
6666:The class
4747:problems.
4150:See also:
4146:Reductions
2271:such that
1891:verifiable
1234:See also:
1058:polynomial
813:polynomial
209:Background
56:complexity
13863:2-EXPTIME
13097:Rich 2008
12767:
12732:
12677:
12651:
12073:Decidable
11920:Π
11909:∗
11866:Π
11839:Π
11760:Π
11756:∈
11745:For ever
11704:Π
11700:∈
11641:Π
11628:Π
11583:∗
11563:∈
11513:Π
11466:Π
11455:∗
11412:Π
11385:Π
11362:∗
11333:Π
11329:∪
11320:Π
11296:∅
11284:Π
11280:∩
11271:Π
11244:Π
11217:Π
11186:Π
11182:∪
11173:Π
11142:Π
11138:∪
11129:Π
11099:Π
11086:Π
11036:∗
11016:⊆
11007:Π
10982:∗
10962:⊆
10953:Π
10941:, where:
10920:Π
10907:Π
10878:∗
10819:∗
10799:∈
10731:∗
10711:⊆
10536:∗
10532:Σ
10528:∈
10429:∈
10329:∗
10325:Σ
10321:×
10316:∗
10312:Σ
10308:⊆
10282:Σ
10216:→
10034:∈
9980:∗
9960:∈
9912:→
9876:→
9871:∗
9710:∗
9690:∈
9642:→
9637:∗
9564:economics
9480:∗
9460:∈
9430:#
9379:∗
9359:∈
9331:→
9326:∗
9293:functions
9241:#
8970:Σ
8917:≠
8851:⊊
8816:⊂
8736:→
8531:→
7966:Languages
7711:→
7623:NOT gates
7572:computers
7550:NOT gates
7536:¬
7512:∨
7502:AND gates
7488:∧
7356:¬
7353:∨
7337:∧
7318:∧
7302:∧
7286:¬
7159:⊆
7109:≥
7007:≤
6905:≥
6804:∈
6720:ϵ
6460:∩
6183:ϵ
6180:−
6174:≥
6104:a string
6090:ϵ
6087:−
6081:≥
6011:a string
5995:ϵ
5935:ϵ
5725:
5719:⋅
5675:⊊
5571:
5558:⋅
5517:⊊
5432:≤
5376:⊆
5320:≤
5156:NEXPSPACE
5071:⊆
4869:-complete
4591:∈
4554:∈
4529:∗
4525:Σ
4521:∈
4367:∈
4330:∈
4305:∗
4301:Σ
4297:∈
4158:reduction
3979:∘
3808:∈
3803:¯
3699:NEXPSPACE
3623:∈
3616:⋃
3508:∈
3501:⋃
3456:NEXPSPACE
3436:NEXPSPACE
3374:⊆
3279:∈
3272:⋃
3177:∈
3170:⋃
3072:⊆
3059:⊆
3023:
2955:
2904:NLOGSPACE
2858:
2783:⊆
2693:∈
2686:⋃
2584:∈
2577:⋃
2482:≠
2441:≠
2420:construct
2359:⊆
2218:∈
2150:∗
2130:∈
1814:∈
1807:⋃
1730:∈
1723:⋃
1133:→
959:→
844:⊆
722:recognize
618:processor
524:algorithm
349:∈
269:algorithm
13858:EXPSPACE
13853:NEXPTIME
13621:DLOGTIME
13535:Archived
13515:Archived
13466:(2006).
13453:Archived
13428:(2008).
13407:Archived
13363:Archived
13327:(2006).
13314:Archived
13259:Archived
13232:Archived
13205:Archived
13191:(2009).
13177:Archived
12965:Lee 2014
12613:See also
12121:NEXPTIME
12097:EXPSPACE
11120:is thus
10647:Just as
10636:implies
10517:for all
10486:function
10273:alphabet
10198:function
9827:how many
9262:how many
9179:how many
8697:, where
8548:, where
8276:of size
8216:, where
8047:, where
7526:OR gates
7086:, where
6971:implies
6869:implies
6773:, where
5342:, since
5152:EXPSPACE
4806:complete
4633:Hardness
3736:negation
3695:EXPSPACE
3452:EXPSPACE
3447:NEXPTIME
3432:EXPSPACE
3426:EXPSPACE
2896:LOGSPACE
2829:L and NL
2812:NEXPTIME
2796:NEXPTIME
2766:NEXPTIME
2531:NEXPTIME
2518:NEXPTIME
2306:accepts.
1668:P and NP
1150:, where
976:, where
886:inherent
829:, since
613:inherent
172:EXPSPACE
166:NEXPTIME
13848:EXPTIME
13756:NP-hard
13169:"P=?NP"
12666:, i.e.
12145:EXPTIME
11970:, then
11552:string
11162:promise
10595:versus
9207:whether
9171:whether
8957:, then
8902:, then
8889:versus
7932:compute
7560:is the
6951:not in
6124:not in
5772:EXPTIME
5148:NPSPACE
4783:, then
4744:NP-hard
3732:closure
3726:Closure
3715:EXPTIME
3441:EXPTIME
3357:NPSPACE
3131:NPSPACE
3111:NPSPACE
2808:EXPTIME
2774:EXPTIME
2758:EXPTIME
2523:EXPTIME
2514:EXPTIME
2411:at most
2339:versus
1928:string
823:EXPTIME
818:EXPTIME
384:strings
197:versus
188:equals
163:⊆
160:EXPTIME
157:⊆
151:⊆
145:⊆
139:⊆
13955:NSPACE
13950:DSPACE
13825:PSPACE
13564:, and
13479:
13445:
13355:
13281:
13211:
12169:PSPACE
11964:subset
11924:ACCEPT
11870:REJECT
11843:ACCEPT
11764:REJECT
11708:ACCEPT
11662:is in
11645:REJECT
11632:ACCEPT
11517:ACCEPT
11470:ACCEPT
11416:REJECT
11403:(with
11389:ACCEPT
11337:REJECT
11324:ACCEPT
11288:REJECT
11275:ACCEPT
11248:REJECT
11221:ACCEPT
11190:REJECT
11177:ACCEPT
11146:REJECT
11133:ACCEPT
11103:REJECT
11090:ACCEPT
11011:REJECT
10957:ACCEPT
10924:REJECT
10911:ACCEPT
9562:, and
9498:) and
9229:has a
9147:, and
9020:P/poly
9005:P/poly
8997:P/poly
8955:P/poly
8941:P/poly
8900:P/poly
8882:P/poly
8876:P/poly
8856:P/poly
8821:P/poly
8639:P/poly
7621:, and
7504:, the
7480:. The
7447:, and
7041:PSPACE
6578:, and
6538:, and
6239:, and
5859:P/poly
5844:, and
5820:, and
5780:PSPACE
5144:PSPACE
3769:(i.e.
3709:, and
3703:PSPACE
3414:PSPACE
3353:PSPACE
3127:PSPACE
3107:PSPACE
2184:is in
1259:NSPACE
1255:DSPACE
726:decide
577:(e.g.
565:(e.g.
456:is in
177:subset
154:PSPACE
119:, and
79:memory
64:memory
13945:NTIME
13940:DTIME
13761:co-NP
13518:(PDF)
13505:(PDF)
13490:(PDF)
13473:(PDF)
13456:(PDF)
13435:(PDF)
13410:(PDF)
13395:(PDF)
13366:(PDF)
13341:(PDF)
13332:(PDF)
13317:(PDF)
13304:(PDF)
13292:(PDF)
13271:(PDF)
13247:(PDF)
13201:Draft
12625:Notes
12321:co-NP
11550:every
11537:is a
10787:every
10644:).
10620:then
10600:. If
10484:is a
9825:asks
9435:CYCLE
9246:CYCLE
9205:asks
9191:CYCLE
9036:depth
8624:is a
7645:with
7601:is a
6513:co-RP
6501:co-RP
6497:co-RP
6489:co-RP
6465:co-RP
6433:co-RP
6405:co-RP
6381:P=BPP
6343:co-RP
6334:co-RP
6230:co-RP
5182:DTIME
3838:co-NP
3833:co-NP
2806:then
2393:. If
2030:proof
1251:NTIME
1247:DTIME
1064:? An
702:PRIME
679:every
633:bytes
536:PRIME
520:prime
442:PRIME
412:PRIME
335:PRIME
311:PRIME
295:prime
235:space
46:is a
13773:TFNP
13477:ISBN
13443:ISBN
13418:2022
13353:ISBN
13279:ISBN
13209:ISBN
12758:>
12723:<
11857:and
11235:and
10765:for
9496:bits
9111:and
9028:and
6511:and
6495:and
6487:and
6431:and
6403:and
6341:and
6007:if:
5886:and
5868:and
5612:The
5457:The
5174:and
4999:>
4861:).
4809:for
4639:hard
4625:and
4617:and
4206:and
4059:>
3895:1000
3779:co-X
3763:co-X
3444:and
3434:and
3119:and
3109:and
2864:<
2837:and
2764:and
2535:NEXP
2516:and
2330:and
1693:and
1676:and
1257:and
1249:and
1082:The
1060:? A
908:The
773:The
689:").
388:bits
231:time
215:sets
103:and
75:time
62:and
60:time
42:, a
13888:ALL
13788:QMA
13778:FNP
13720:APX
13715:BQP
13710:BPP
13700:ZPP
13631:ACC
13379:doi
13345:doi
12764:log
12729:log
12674:log
12648:log
12424:BPP
12334:BQP
11966:of
11948:SZK
11308:.
10892:.
10673:FNP
10658:FNP
10572:in
9547:.
9114:QMA
9108:BQP
9089:.
9003:".
8878:).
7615:AND
7607:bit
7202:QIP
7196:MIP
6849:in
6732:.
6690:dIP
6541:RLP
6529:BPL
6523:.
6521:ZPP
6517:and
6481:ZPP
6445:ZPP
6425:ZPP
6421:BPP
6409:BPP
6397:ZPP
6376:BPP
6368:BPP
6364:BPP
6359:BPP
6349:.
6258:.
6251:ZPP
6236:BPP
6218:ZPP
6031:in
5889:QMA
5883:BQP
5823:ZPP
5805:BPP
5722:log
5562:log
5414:if
5302:if
4629:.
3830:).
3020:log
2952:log
2855:log
2527:EXP
2008:if
1968:if
1922:and
1689:in
881:).
779:any
480:or
478:yes
299:set
251:in
233:or
123:).
77:or
66:.
50:of
48:set
38:In
13991::
13883:RE
13873:PR
13820:IP
13808:#P
13803:PP
13798:⊕P
13793:PH
13783:AM
13746:NP
13741:UP
13725:FP
13705:RP
13683:CC
13678:SC
13673:NC
13661:NL
13656:FL
13651:RL
13646:SL
13636:TC
13626:AC
13568::
13548:.
13513:.
13507:.
13451:.
13441:.
13437:.
13405:.
13401:.
13397:.
13361:.
13351:.
13312:.
13306:.
13257:.
13253:.
13249:.
13226:.
13203:.
13187:;
13171:.
13104:^
13073:^
13056:^
12921:^
12906:^
12546:NC
12349:NP
11938:.
11541:.
10683:.
10681:NP
10669:FP
10664:NP
10655:,
10649:FP
10642:FP
10638:#P
10634:NP
10626:NP
10618:FP
10614:#P
10610:NP
10606:FP
10602:#P
10597:NP
10588:FP
10578:FP
10576:.
10566:FP
10561:FP
10550:.
10294::
9994:,
9838:#P
9831:#P
9823:#P
9819:NP
9811:NP
9807:#P
9799:NP
9724:,
9604:#P
9597:#P
9589:NP
9585:NP
9581:#P
9576:♯P
9566:.
9558:,
9554:,
9393:,
9165:A
9143:,
9127:.
9062:.
9056:NC
9052:NC
9048:AC
9044:NC
9031:AC
9025:NC
9015:.
8959:PH
8953:⊆
8950:NP
8939:.
8896:NP
8891:NP
8628:.
8134:,
7961:.
7619:OR
7617:,
7414:,
7190:.
7130:AM
7126:AM
7124:,
7096:AM
7092:AM
7084:AM
7080:AM
7075:AM
7063:IP
7058:AM
7053:IP
7036:IP
6979:Pr
6877:Pr
6738:IP
6697:NP
6686:NP
6682:NP
6669:NP
6582:.
6574:,
6567:NP
6544:.
6535:RL
6532:,
6509:RP
6505:RP
6493:RP
6485:RP
6455:RP
6429:RP
6423:.
6417:PP
6413:PP
6401:RP
6399:,
6390:PP
6339:RP
6269:RP
6264:RP
6245:.
6242:PP
6233:,
6227:,
6224:RP
6221:,
6153:Pr
6060:Pr
5923:.
5871:AC
5865:NC
5847:AM
5841:MA
5838:,
5835:IP
5817:RP
5814:,
5811:PP
5808:,
5782:.
5154:=
5146:=
5014:,
4895:.
4893:NP
4885:NP
4881:NP
4877:NP
4873:NP
4867:NP
4739:NP
4613:,
4606:.
4543:,
4382:.
4319:,
3844:.
3842:NP
3742:,
3738:,
3717:.
3707:NP
3705:,
3347:,
3345:NP
3122:NP
2900:NL
2814:.
2804:NP
2770:NP
2504:.
2416:NP
2399:NP
2387:NP
2341:NP
2332:NP
2316:NP
2164:,
2057:NP
1895:NP
1887:NP
1695:NP
1226:.
1052:.
668:A
580:#P
568:FP
482:no
472:A
460:.
458:NP
429:NP
255:.
199:NP
190:NP
148:NP
136:NL
115:,
111:,
13878:R
13688:P
13641:L
13599:e
13592:t
13585:v
13420:.
13385:.
13381::
13347::
13217:.
13068:.
13027:.
13015:.
12967:.
12877:.
12805:.
12782:.
12770:n
12761:c
12755:n
12735:n
12726:c
12720:n
12700:c
12680:n
12654:n
12645:c
12501:P
11980:X
11976:Y
11972:X
11968:Y
11960:X
11943:P
11915:/
11905:}
11901:1
11898:,
11895:0
11892:{
11835:=
11832:L
11812:L
11787:0
11784:=
11781:)
11778:x
11775:(
11772:M
11769:,
11753:x
11731:1
11728:=
11725:)
11722:x
11719:(
11716:M
11713:,
11697:x
11674:M
11664:P
11650:)
11637:,
11624:(
11614:P
11607:P
11579:}
11575:1
11572:,
11569:0
11566:{
11560:s
11525:L
11522:=
11492:L
11461:/
11451:}
11447:1
11444:,
11441:0
11438:{
11358:}
11354:1
11351:,
11348:0
11345:{
11342:=
11293:=
11108:)
11095:,
11082:(
11062:M
11032:}
11028:1
11025:,
11022:0
11019:{
10978:}
10974:1
10971:,
10968:0
10965:{
10929:)
10916:,
10903:(
10874:}
10870:1
10867:,
10864:0
10861:{
10841:M
10815:}
10811:1
10808:,
10805:0
10802:{
10796:w
10773:L
10753:M
10727:}
10723:1
10720:,
10717:0
10714:{
10708:L
10679:=
10677:P
10671:=
10653:P
10640:=
10632:=
10630:P
10624:=
10622:P
10616:=
10604:=
10593:P
10583:P
10525:x
10505:)
10502:x
10499:(
10496:f
10472:f
10452:y
10432:R
10426:)
10423:y
10420:,
10417:x
10414:(
10394:y
10374:x
10354:f
10305:R
10259:R
10239:f
10219:B
10213:A
10210::
10207:f
10177:.
10165:w
10145:)
10142:w
10139:(
10136:f
10114:|
10107:}
10102:1
10099:=
10096:)
10093:c
10090:,
10087:w
10084:(
10081:V
10078::
10073:)
10069:|
10065:w
10061:|
10057:(
10054:p
10050:}
10046:1
10043:,
10040:0
10037:{
10031:c
10026:{
10019:|
10014:=
10011:)
10008:w
10005:(
10002:f
9976:}
9972:1
9969:,
9966:0
9963:{
9957:w
9937:V
9916:N
9908:N
9904::
9901:p
9880:N
9867:}
9863:1
9860:,
9857:0
9854:{
9851::
9848:f
9793:.
9781:w
9761:M
9741:)
9738:w
9735:(
9732:f
9706:}
9702:1
9699:,
9696:0
9693:{
9687:w
9667:M
9646:N
9633:}
9629:1
9626:,
9623:0
9620:{
9617::
9614:f
9535:G
9515:)
9512:G
9509:(
9506:f
9476:}
9472:1
9469:,
9466:0
9463:{
9457:G
9410:)
9407:w
9404:(
9401:f
9375:}
9371:1
9368:,
9365:0
9362:{
9356:w
9335:N
9322:}
9318:1
9315:,
9312:0
9309:{
9306::
9303:f
9272:G
9217:G
9096:)
9092:(
8980:P
8974:2
8925:P
8922:N
8912:P
8887:P
8846:P
8810:P
8786:)
8783:)
8780:n
8777:(
8772:2
8768:t
8764:(
8761:O
8740:N
8732:N
8728::
8725:t
8705:t
8685:)
8682:)
8679:n
8676:(
8673:t
8670:(
8665:E
8662:M
8659:I
8656:T
8653:D
8612:f
8590:n
8586:C
8565:)
8562:n
8559:(
8556:f
8535:N
8527:N
8523::
8520:f
8500:)
8497:.
8494:.
8491:.
8488:,
8483:2
8479:C
8475:,
8470:1
8466:C
8462:,
8457:0
8453:C
8449:(
8422:w
8402:w
8380:n
8376:C
8355:)
8352:.
8349:.
8346:.
8343:,
8338:2
8334:C
8330:,
8325:1
8321:C
8317:,
8312:0
8308:C
8304:(
8284:n
8264:w
8244:w
8224:n
8204:1
8201:=
8198:)
8195:w
8192:(
8187:n
8183:C
8162:L
8142:w
8122:w
8102:L
8082:n
8060:n
8056:C
8035:)
8032:.
8029:.
8026:.
8023:,
8018:2
8014:C
8010:,
8005:1
8001:C
7997:,
7992:0
7988:C
7984:(
7947:C
7943:f
7918:C
7898:)
7893:n
7889:x
7885:,
7882:.
7879:.
7876:.
7873:,
7868:2
7864:x
7860:,
7855:1
7851:x
7847:(
7842:C
7838:f
7834:=
7831:b
7811:b
7789:n
7785:x
7781:,
7778:.
7775:.
7772:.
7769:,
7764:2
7760:x
7756:,
7751:1
7747:x
7726:}
7723:1
7720:,
7717:0
7714:{
7706:n
7702:}
7698:1
7695:,
7692:0
7689:{
7686::
7681:C
7677:f
7653:n
7633:C
7589:C
7552:.
7468:0
7465:=
7460:3
7456:x
7435:1
7432:=
7427:2
7423:x
7402:0
7399:=
7394:1
7390:x
7369:)
7364:3
7360:x
7350:)
7345:3
7341:x
7332:2
7328:x
7324:(
7321:(
7315:)
7310:2
7306:x
7297:1
7293:x
7289:(
7283:=
7280:)
7275:3
7271:x
7267:,
7262:2
7258:x
7254:,
7249:1
7245:x
7241:(
7236:C
7232:f
7178:]
7175:k
7172:[
7167:P
7164:I
7156:]
7153:k
7150:[
7145:M
7142:A
7128:=
7112:2
7106:k
7088:k
7016:3
7013:1
7004:]
7001:P
6993:w
6985:V
6982:[
6959:L
6939:w
6914:3
6911:2
6902:]
6899:P
6891:w
6883:V
6880:[
6857:L
6837:w
6812:P
6809:I
6801:L
6781:V
6761:)
6758:V
6755:,
6752:P
6749:(
6645:P
6625:V
6605:P
6450:=
6319:M
6299:M
6279:M
6177:1
6171:]
6168:w
6160:M
6157:[
6132:L
6112:w
6084:1
6078:]
6075:w
6067:M
6064:[
6039:L
6019:w
5975:L
5955:M
5776:L
5768:P
5762:.
5748:)
5743:)
5740:)
5737:n
5734:(
5731:f
5728:(
5716:)
5713:n
5710:(
5707:f
5702:(
5695:E
5692:C
5689:A
5686:P
5683:S
5680:D
5670:)
5665:)
5662:n
5659:(
5656:f
5651:(
5644:E
5641:C
5638:A
5635:P
5632:S
5629:D
5608:.
5594:)
5589:)
5586:)
5583:n
5580:(
5577:f
5574:(
5566:2
5555:)
5552:n
5549:(
5546:f
5541:(
5534:E
5531:M
5528:I
5525:T
5522:D
5512:)
5507:)
5504:n
5501:(
5498:f
5493:(
5486:E
5483:M
5480:I
5477:T
5474:D
5440:2
5436:k
5427:1
5423:k
5402:)
5395:2
5391:k
5386:n
5382:(
5379:O
5373:)
5366:1
5362:k
5357:n
5353:(
5350:O
5328:2
5324:k
5315:1
5311:k
5290:)
5283:2
5279:k
5274:n
5270:(
5265:E
5262:M
5259:I
5256:T
5253:D
5231:)
5224:1
5220:k
5215:n
5211:(
5206:E
5203:M
5200:I
5197:T
5194:D
5127:.
5123:)
5117:2
5112:)
5109:n
5106:(
5101:f
5097:(
5091:E
5088:C
5085:A
5082:P
5079:S
5076:D
5067:)
5062:)
5059:n
5056:(
5052:f
5048:(
5042:E
5039:C
5036:A
5033:P
5030:S
5027:N
5002:n
4996:)
4993:n
4990:(
4987:f
4965:2
4961:)
4957:n
4954:(
4951:f
4931:)
4928:n
4925:(
4922:f
4889:P
4859:C
4845:X
4835:C
4821:X
4811:C
4791:X
4781:C
4777:C
4763:X
4735:C
4721:X
4701:X
4691:C
4677:X
4667:C
4663:C
4649:X
4594:Y
4588:)
4585:x
4582:(
4579:p
4557:X
4551:x
4518:x
4498:p
4474:Y
4454:X
4434:Y
4414:Y
4394:X
4370:Y
4364:)
4361:x
4358:(
4355:f
4333:X
4327:x
4294:x
4274:f
4254:Y
4234:X
4214:y
4194:x
4174:y
4171:+
4168:x
4140:c
4126:)
4121:n
4117:c
4113:(
4110:O
4100:P
4096:P
4092:P
4078:)
4073:3
4069:n
4065:(
4062:O
4056:)
4051:6
4047:n
4043:(
4040:O
4020:)
4015:6
4011:n
4007:(
4004:O
4001:=
3998:)
3993:2
3989:n
3985:(
3982:O
3976:)
3971:3
3967:n
3963:(
3960:O
3940:)
3935:3
3931:n
3927:(
3924:O
3914:P
3900:)
3891:n
3887:(
3884:O
3864:)
3861:n
3858:(
3855:O
3840:=
3818:}
3813:X
3800:L
3794:|
3790:L
3787:{
3784:=
3767:X
3759:X
3752:P
3711:P
3697:=
3675:)
3668:k
3664:n
3659:2
3655:(
3650:E
3647:C
3644:A
3641:P
3638:S
3635:N
3627:N
3620:k
3612:=
3607:E
3604:C
3601:A
3598:P
3595:S
3592:P
3589:X
3586:E
3583:N
3560:)
3553:k
3549:n
3544:2
3540:(
3535:E
3532:C
3529:A
3526:P
3523:S
3520:D
3512:N
3505:k
3497:=
3492:E
3489:C
3486:A
3483:P
3480:S
3477:P
3474:X
3471:E
3410:P
3394:E
3391:C
3388:A
3385:P
3382:S
3379:P
3369:P
3355:=
3343:=
3341:P
3324:)
3319:k
3315:n
3311:(
3306:E
3303:C
3300:A
3297:P
3294:S
3291:N
3283:N
3276:k
3268:=
3263:E
3260:C
3257:A
3254:P
3251:S
3248:P
3245:N
3222:)
3217:k
3213:n
3209:(
3204:E
3201:C
3198:A
3195:P
3192:S
3189:D
3181:N
3174:k
3166:=
3161:E
3158:C
3155:A
3152:P
3149:S
3146:P
3116:P
3077:P
3067:L
3064:N
3054:L
3029:)
3026:n
3017:(
3012:E
3009:C
3006:A
3003:P
3000:S
2997:N
2992:=
2987:L
2984:N
2961:)
2958:n
2949:(
2944:E
2941:C
2938:A
2935:P
2932:S
2929:D
2924:=
2919:L
2892:L
2867:n
2861:n
2802:=
2800:P
2762:P
2742:)
2735:k
2731:n
2726:2
2722:(
2717:E
2714:M
2711:I
2708:T
2705:N
2697:N
2690:k
2682:=
2677:E
2674:M
2671:I
2668:T
2665:P
2662:X
2659:E
2656:N
2633:)
2626:k
2622:n
2617:2
2613:(
2608:E
2605:M
2602:I
2599:T
2596:D
2588:N
2581:k
2573:=
2568:E
2565:M
2562:I
2559:T
2556:P
2553:X
2550:E
2490:P
2487:N
2477:P
2449:P
2446:N
2436:P
2424:P
2397:=
2395:P
2391:P
2383:P
2367:P
2364:N
2354:P
2337:P
2328:P
2294:)
2291:c
2288:,
2285:w
2282:(
2279:M
2257:)
2253:|
2249:w
2245:|
2241:(
2238:p
2234:}
2230:1
2227:,
2224:0
2221:{
2215:c
2192:L
2172:w
2146:}
2142:1
2139:,
2136:0
2133:{
2127:w
2107:p
2087:M
2067:L
2040:w
2016:w
1996:w
1976:w
1956:w
1936:c
1909:w
1880:P
1872:P
1856:)
1851:k
1847:n
1843:(
1838:E
1835:M
1832:I
1829:T
1826:N
1818:N
1811:k
1803:=
1798:P
1795:N
1772:)
1767:k
1763:n
1759:(
1754:E
1751:M
1748:I
1745:T
1742:D
1734:N
1727:k
1719:=
1714:P
1683:P
1640:)
1637:)
1634:n
1631:(
1628:s
1625:(
1622:O
1602:)
1599:)
1596:n
1593:(
1590:s
1587:(
1582:E
1579:C
1576:A
1573:P
1570:S
1567:N
1542:)
1539:)
1536:n
1533:(
1530:s
1527:(
1524:O
1504:)
1501:)
1498:n
1495:(
1492:s
1489:(
1484:E
1481:C
1478:A
1475:P
1472:S
1469:D
1444:)
1441:)
1438:n
1435:(
1432:t
1429:(
1426:O
1406:)
1403:)
1400:n
1397:(
1394:t
1391:(
1386:E
1383:M
1380:I
1377:T
1374:N
1349:)
1346:)
1343:n
1340:(
1337:t
1334:(
1331:O
1311:)
1308:)
1305:n
1302:(
1299:t
1296:(
1291:E
1288:M
1285:I
1282:T
1279:D
1214:n
1194:M
1174:)
1171:n
1168:(
1163:M
1159:s
1137:N
1129:N
1125::
1120:M
1116:s
1095:M
1040:n
1020:M
1000:)
997:n
994:(
989:M
985:t
963:N
955:N
951::
946:M
942:t
921:M
867:E
864:M
861:I
858:T
855:P
852:X
849:E
839:P
827:P
808:P
583:)
571:)
507:n
370:}
362:n
358:|
353:N
346:n
343:{
340:=
282:n
240:P
195:P
186:P
169:⊆
142:P
133:⊆
130:L
92:P
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
