Incomputational complexity theory, thecomplexity classEXPTIME (sometimes calledEXP orDEXPTIME) is theset of alldecision problems that are solvable by adeterministic Turing machine inexponential time, i.e., inO(2^p(n)) time, wherep(n) is a polynomial function ofn.

EXPTIME is one intuitive class in anexponential hierarchy of complexity classes with increasingly more complex oracles or quantifier alternations. For example, the class2-EXPTIME is defined similarly to EXPTIME but with adoubly exponential time bound. This can be generalized to higher and higher time bounds.

EXPTIME can also be reformulated as the space class APSPACE, the set of all problems that can be solved by analternating Turing machine in polynomial space.

EXPTIME relates to the other basic time and space complexity classes in the following way:P ⊆NP ⊆PSPACE ⊆ EXPTIME ⊆NEXPTIME ⊆EXPSPACE. Furthermore, by thetime hierarchy theorem and thespace hierarchy theorem, it is known that P ⊊ EXPTIME, NP ⊊ NEXPTIME and PSPACE ⊊ EXPSPACE.

In terms ofDTIME,

${\displaystyle \mathsf{E}\mathsf{X}\mathsf{P}\mathsf{T}\mathsf{I}\mathsf{M}\mathsf{E}=\bigcup _{k\in \mathbb{N}}\mathsf{D}\mathsf{T}\mathsf{I}\mathsf{M}\mathsf{E}\left(2^{n^k}\right).}$

It is known that

and also, by thetime hierarchy theorem and thespace hierarchy theorem, that

In the above expressions, the symbol ⊆ means "is a subset of", and the symbol ⊊ means "is a strict subset of".

so at least one of the first three inclusions and at least one of the last three inclusions must be proper, but it is not known which ones are. It is also known that ifP = NP, then EXPTIME =NEXPTIME, the class of problems solvable in exponential time by anondeterministic Turing machine.^[1] More precisely,E ≠NE if and only if there existsparse languages inNP that are not inP.^[2]

EXPTIME can be reformulated as the space class APSPACE, the set of all problems that can be solved by analternating Turing machine in polynomial space. This is one way to see that PSPACE ⊆ EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine.^[3]

A decision problem is EXPTIME-complete if it is in EXPTIME and every problem in EXPTIME has apolynomial-time many-one reduction to it. In other words, there is a polynomial-timealgorithm that transforms instances of one to instances of the other with the same answer. Problems that are EXPTIME-complete might be thought of as the hardest problems in EXPTIME. Notice that although it is unknown whether NP is equal to P, we do know that EXPTIME-complete problems are not in P; it has been proven that these problems cannot be solved inpolynomial time, by thetime hierarchy theorem.

Incomputability theory, one of the basic undecidable problems is thehalting problem: deciding whether adeterministic Turing machine (DTM) halts. One of the most fundamental EXPTIME-complete problems is a simpler version of this, which asks if a DTM halts on a given input in at mostk steps. It is in EXPTIME because a trivial simulation requires O(k) time, and the inputk is encoded using O(logk) bits which causes exponential number of simulations. It is EXPTIME-complete because, roughly speaking, we can use it to determine if a machine solving an EXPTIME problem accepts in an exponential number of steps; it will not use more.^[4] The same problem with the number of steps written in unary isP-complete.

Other examples of EXPTIME-complete problems include the problem of evaluating a position ingeneralizedchess,^[5]checkers,^[6] orGo (with Japanese ko rules).^[7] These games have a chance of being EXPTIME-complete because games can last for a number of moves that is exponential in the size of the board. In the Go example, the Japanese ko rule is known to imply EXPTIME-completeness, but it is not known if the American or Chinese rules for the game are EXPTIME-complete (they could range from PSPACE to EXPSPACE).

By contrast, generalized games that can last for a number of moves that is polynomial in the size of the board are oftenPSPACE-complete. The same is true of exponentially long games in which non-repetition is automatic.

Another set of important EXPTIME-complete problems relates to succinct circuits. The idea is that if we can exponentially compress the description of a problem that requires polynomial time, then that compressed problem would require exponential time.

As one example, some graphs can be succinctly described by a small Boolean circuit. The circuit has${\displaystyle 2n}$ inputs, 1 output and${\displaystyle \mathsf{p}\mathsf{o}\mathsf{l}\mathsf{y}(n)}$ gates, thus requiring${\displaystyle \mathsf{p}\mathsf{o}\mathsf{l}\mathsf{y}(n)}$ bits to describe. The circuit represents a graph with${\displaystyle 2^n}$ vertices. For each pair of vertices, if the binary code for the two vertices are put into the circuit, then the output of the circuit states whether the two vertices are connected by an edge.

For many naturally occurringP-complete decision problems about graphs, where the graph is expressed in a natural representation such as anadjacency matrix, solving the same problem on a succinct circuit representation is EXPTIME-complete, because the input is exponentially smaller; but this requires nontrivial proof, since succinct circuits can only describe a subclass of graphs.^[8]

Generically, a Boolean circuit with${\displaystyle n}$ inputs and a single output is a succinct representation of a string of${\displaystyle 2^n}$ bits, which can be used to describe some other object, such as a graph, a 3-CNF formula, etc. For essentially all known NP-complete problems, the succinct version of it is NEXP-complete. In particular, SUCCINCT 3-SAT is NEXP-complete under polynomial-time reductions.^[9][10]

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