In computational complexity theory, P/poly is a complexity class representing problems that can be solved by small circuits. More precisely, it is the set of formal languages that have polynomial-size circuit families. It can also be defined equivalently in terms of Turing machines with advice, extra information supplied to the Turing machine along with its input, that may depend on the input length but not on the input itself. In this formulation, P/poly is the class of decision problems that can be solved by a polynomial-time Turing machine with advice strings of length polynomial in the input size. These two different definitions make P/poly central to circuit complexity and non-uniform complexity.
For example, the popular Miller–Rabin primality test can be formulated as a P/poly algorithm: the "advice" is a list of candidate values to test. It is possible to precompute a list of values such that every composite -bit number will be certain to have a witness in the list. For example, to correctly determine the primality of 32-bit numbers, it is enough to test . The existence of short lists of candidate witnesses follows from the fact that for each composite , three out of four candidate values successfully detect that is composite. From this, a simple counting argument similar to the one in the proof that BPP P/poly below shows that there exists a suitable list of candidate values for every input size, and more strongly that most long-enough lists of candidate values will work correctly, although finding a list that is guaranteed to work may be expensive.
P/poly, unlike other polynomial-time classes such as P or BPP, is not generally considered a practical class for computing. Indeed, it contains every undecidable unary language, none of which can be solved in general by real computers. On the other hand, if the input length is bounded by a relatively small number and the advice strings are short, it can be used to model practical algorithms with a separate expensive preprocessing phase and a fast processing phase, as in the Miller–Rabin example.
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In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH.
This is a list of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics. Many of these classes have a 'co' partner which consists of the complements of all languages in the original class. For example, if a language L is in NP then the complement of L is in co-NP. (This does not mean that the complement of NP is co-NP—there are languages which are known to be in both, and other languages which are known to be in neither.
In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible input length. Boolean circuits are defined in terms of the logic gates they contain. For example, a circuit might contain binary AND and OR gates and unary NOT gates, or be entirely described by binary NAND gates. Each gate corresponds to some Boolean function that takes a fixed number of bits as input and outputs a single bit.
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