Concepts inAlgebrization: a new barrier in complexity theory
P/poly
In computational complexity theory, P/poly is the complexity class of languages recognized by a polynomial-time Turing machine with a polynomial-bounded advice function. It is also equivalently defined as the class PSIZE of languages that have a polynomial-size circuits.
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Natural proof
In computational complexity theory, a natural proof is a certain kind of proof establishing that one complexity class differs from another one. While these proofs are in some sense "natural", it can be shown (assuming a widely believed conjecture on the existence of one-way functions) that no such proof can possibly be used to solve the P vs. NP problem.
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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 O(2) for some polynomial p(n), and unlimited space. In terms of NTIME, An important set of NEXPTIME-complete problems relates to succinct circuits. Succinct circuits are simple machines used to describe graphs in exponentially less space.
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IP (complexity)
In computational complexity theory, the class IP (which stands for Interactive Polynomial time) is the class of problems solvable by an interactive proof system. The concept of an interactive proof system was first introduced by Shafi Goldwasser, Silvio Micali, and Charles Rackoff in 1985.
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RP (complexity)
In complexity theory, RP ("randomized polynomial time") is the complexity class of problems for which a probabilistic Turing machine exists with these properties: It always runs in polynomial time in the input size If the correct answer is NO, it always returns NO If the correct answer is YES, then it returns YES with probability at least 1/2 (otherwise, it returns NO). In other words, the algorithm is allowed to flip a truly random coin while it is running.
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Oracle machine
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to decide certain decision problems in a single operation. The problem can be of any complexity class. Even undecidable problems, like the halting problem, can be used.
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Interactive proof system
In computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two parties. The parties, the verifier and the prover, interact by exchanging messages in order to ascertain whether a given string belongs to a language or not. The prover is all-powerful and possesses unlimited computational resources, but cannot be trusted, while the verifier has bounded computation power.
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P versus NP problem
The P versus NP problem is a major unsolved problem in computer science. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. It was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" and is considered by many to be the most important open problem in the field.
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