computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...

, the

complexity class In computational complexity theory, a complexity class is a set (mathematics), set of computational problems of related resource-based computational complexity, complexity. The two most commonly analyzed resources are time complexity, time and spa ...

EXPTIME (sometimes called EXP or DEXPTIME) is the set of all

decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm wheth ...

s that are solvable by a deterministic Turing machine in

exponential time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...

, i.e., in O(2^''p''(''n'')) time, where ''p''(''n'') is a polynomial function of ''n''. EXPTIME is one intuitive class in an

exponential hierarchy In computational complexity theory, the exponential hierarchy is a hierarchy of complexity classes, which is an exponential time analogue of the polynomial hierarchy. As elsewhere in complexity theory, “exponential” is used in two different me ...

of complexity classes with increasingly more complex oracles or quantifier alternations. For example, the class

2-EXPTIME In computational complexity theory, the complexity class 2-EXPTIME (sometimes called 2-EXP) is the set of all decision problems solvable by a deterministic Turing machine in O(22''p''(''n'')) time, where ''p''(''n'') is a polynomial function of ''n ...

is defined similarly to EXPTIME but with a doubly 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 an alternating Turing machine in polynomial space. EXPTIME relates to the other basic time and space complexity classes in the following way: P ⊆ NP ⊆

PSPACE In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. Formal definition If we denote by SPACE(''t''(''n'')), the set of all problems that can b ...

⊆ EXPTIME ⊆

NEXPTIME In computational complexity theory, the complexity class NEXPTIME (sometimes called NEXP) is the set of decision problems that can be solved by a non-deterministic Turing machine using time 2^. In terms of NTIME, :\mathsf = \bigcup_ \mathsf(2^) A ...

⊆ EXPSPACE. Furthemore, by the time hierarchy theorem and the

space hierarchy theorem In computational complexity theory, the space hierarchy theorems are separation results that show that both deterministic and nondeterministic machines can solve more problems in (asymptotically) more space, subject to certain conditions. For exampl ...

, it is known that P ⊊ EXPTIME, NP ⊊ NEXPTIME and PSPACE ⊊ EXPSPACE.

Formal definition

In terms of

DTIME In computational complexity theory, DTIME (or TIME) is the computational resource of computation time for a deterministic Turing machine. It represents the amount of time (or number of computation steps) that a "normal" physical computer would ta ...

Relationships to other classes

It is known that and also, by the time hierarchy theorem and the

, 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. Most experts believe all the inclusions are proper. It is also known that if

P = NP The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved. The informal term ''quickly'', used above ...

, then EXPTIME

, the class of problems solvable in exponential time by a

nondeterministic Turing machine In theoretical computer science, a nondeterministic Turing machine (NTM) is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations. That is, an NTM's next state is ''not'' comp ...

. More precisely, EXPTIME ≠ NEXPTIME if and only if there exist sparse languages in NP that are not in P. EXPTIME can be reformulated as the space class APSPACE, the set of all problems that can be solved by an alternating 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.

EXPTIME-complete

A decision problem is EXPTIME-complete if it is in EXPTIME and every problem in EXPTIME has a polynomial-time many-one reduction to it. In other words, there is a polynomial-time

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

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 in polynomial time, by the time hierarchy theorem. In computability theory, one of the basic undecidable problems is the

halting problem In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a ...

: deciding whether a deterministic Turing machine (DTM) halts. One of the most fundamental EXPTIME-complete problems is a simpler version of this, which asks if a DTM halts in at most ''k'' steps. It is in EXPTIME because a trivial simulation requires O(''k'') time, and the input ''k'' is encoded using O(log ''k'') 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. The same problem with the number of steps written in unary is

P-complete In computational complexity theory, a decision problem is P-complete (complete for the complexity class P) if it is in P and every problem in P can be reduced to it by an appropriate reduction. The notion of P-complete decision problems is usef ...

. Other examples of EXPTIME-complete problems include the problem of evaluating a position in generalized

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...

checkers Checkers (American English), also known as draughts (; British English), is a group of strategy board games for two players which involve diagonal moves of uniform game pieces and mandatory captures by jumping over opponent pieces. Checkers ...

, or Go (with Japanese ko rules). 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 sufficiently intractable to imply EXPTIME-completeness, but it is not known if the more tractable American or Chinese rules for the game are EXPTIME-complete. By contrast, generalized games that can last for a number of moves that is polynomial in the size of the board are often

PSPACE-complete In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (polynomial space) and if every other problem that can be solved in polynomial space can b ...

. 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. Succinct circuits are simple machines used to describe some graphs in exponentially less space. They accept two vertex numbers as input and output whether there is an edge between them. For many natural

graph problems, where the graph is expressed in a natural representation such as an

adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...

, 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.

References

{{ComplexityClasses Complexity classes