# EXPTIME

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

In terms of DTIME,

$\mbox{EXPTIME} = \bigcup_{k\in\mathbb{N}} \mbox{DTIME}\left(2^{n^k}\right).$

We know

P $\subseteq$ NP $\subseteq$ PSPACE $\subseteq$ EXPTIME $\subseteq$ NEXPTIME $\subseteq$ EXPSPACE

and also, by the time hierarchy theorem,

P $\subsetneq$ EXPTIME   (equivalently, PEXPTIME)

so at least one of the first three inclusions on the first line must be proper (most experts believe all the inclusions are proper). It's also known that if P = NP, then EXPTIME = NEXPTIME, the class of problems solvable in exponential time by a nondeterministic Turing machine. 1

EXPTIME can also be reformulated as the space class APSPACE, the problems that can be solved by an alternating Turing machine in polynomial space. This is one way to see that PSPACE $\subseteq$ EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine. 2

## EXPTIME-complete

The complexity class EXPTIME-complete is also a set of decision problems. A decision problem is in 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 that transforms instances of one to instances of the other with the same answer. EXPTIME-complete might be thought of as the hardest problems in EXPTIME. Notice that although we don't know if NP-complete is a subset of P or not, we do know that EXPTIME-complete lies outside P; none of these problems can possibly be solved in polynomial time.

In computability theory, one of the basic undecidable problems is that of deciding whether a deterministic Turing machine (DTM) accepts a particular input. One of the most fundamental EXPTIME-complete problems is a simpler version of this which asks if a DTM accepts an input 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.4 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.

Other examples of EXPTIME-complete problems include the problem of looking at a generalized Chess, Checkers, or Go position, and determining whether the first player can force a win. These games are EXPTIME-complete because games can last for a number of moves that is exponential in the size of the board. 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.

Another set of important EXPTIME-complete problems relates to succinct circuits. Succinct circuits are simple machines used to describe graphs in exponentially less space. They accept two vertex numbers as input and output whether there is an edge between them. If solving a problem on a graph in a natural representation, such as an adjacency matrix, is P-complete, then solving the same problem on a succinct circuit representation is EXPTIME-complete, because the input is exponentially smaller.3

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