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An oracle machine is a Turing machine connected to an oracle. The Turing machine can write on its own tape an input for the oracle, then tell the oracle to execute. In a single step, the oracle computes its function, erases its input, and writes its output to the tape. Sometimes the Turing machine is described as having two tapes, one of which is reserved for oracle inputs and outputs.
The complexity class of decision problems solvable by an algorithm in class A with an oracle for a problem in class B is written AB. For example, the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for a problem in NP is PNP. (This is also the class of problems reducible by polynomial-time Turing reduction to a problem in NP.)
It is obvious that NP ⊆ PNP, but the question of whether NP ⊂ PNP remains open. See polynomial hierarchy for further extensions.
The notation AB also means the class of problems solvable by an algorithm in class A with an oracle for the language B. For example, PSAT is the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for the Boolean satisfiability problem. When language B is complete for some class C, then AB=AC. In particular, since SAT is NP-complete, PSAT=PNP.
Oracle machines are useful for investigating the relationship between complexity classes P and NPComputational complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps does it take to solve a problem) and space (how much m, by considering the relationship between PA and NPA for an oracle A. In particular, it has been shown that there exist languages A and B such that PA=NPA and PB≠NPB (Baker, Gill, Solovay, 1975). When a question such as this has different answers for different oracles, it is said to relativize both ways. The fact that the P=NP question relativizes both ways is taken as evidence that answering this question will be difficult, because any proof technique that relativizes (i.e., is unaffected by the addition of an oracle) will not answer the P=NP question.
It is interesting to consider the case where an oracle is chosen randomly from among all possible oracles. It has been shown that if oracle A is chosen randomly, then with probability 1, PA≠NPA (Bennett, Gill, 1981). When a question is true for almost all oracles, it is said to be true for a random oracle. This is sometimes taken as evidence that P≠NP; unfortunately, it is possible for a statement to be true for a random oracle, but not be true for ordinary Turing machines.
It is possible to posit the existence of an oracle which computes a non-computable function, such as the answer to the halting problem or some equivalent. A machine with an oracle of this sort is a hypercomputer.
Interestingly, the halting paradox still applies to such machines; that is, although they can determine whether particular Turing machines will halt on particular inputs, they cannot determine whether machines with equivalent halting oracles will themselves halt. This fact creates a hierarchyComputability Set theory In mathematical logic, the arithmetical hierarchy (also known as the arithmetic hierarchy classifies the set of all formulas (or functions) according to their degree of solvability. Each formula or function is equivalent to a Turi of machines, each with a more powerful halting oracle and an even harder halting problem.