# The Role of Relativization in Complexity Theory

@article{Fortnow1994TheRO, title={The Role of Relativization in Complexity Theory}, author={Lance Fortnow}, journal={Bull. EATCS}, year={1994}, volume={52}, pages={229-243} }

Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and program checking results and try to understand why they do not relativize. We give some new results that may help us to understand these questions better.

## 87 Citations

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It is concluded that relativization techniques cannot prove any meaningful restrictions on the power of the principle of local checkability, which is a known proof technique in complexity theory.

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The examination of relativized complexity theoretic statements which hold for a measure one set of oracles in the measure defined by putting each string into the oracle with probability 1/2 independent of all other strings (a formal definition is given below).

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An axiomatic approach to "algebrization", which complements and clarifies the approaches of [For94] and [AW08], and presents logical theories formalizing the notion of algebrizing techniques in the following sense.

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This paper systematically goes through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier, and shows that all known non-relativizing results based on arithmetization -- both inclusions such as IP=PSPACE and MIP=NEXP, and separations such as MAEXP not in P/poly -- do indeed algeBrize.

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The question of whether there is a logic capturing P is subject to a relativization barrier is examined and how the question for P differs from those for the other classes by taking a short tour through relativizations, complete problems and recursive enumerations of complexity classes.

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- Mathematics, Computer ScienceTOCT
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This article systematically goes through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier, and shows that almost all of the major open problems---including P versus NP, P versus RP, and NEXP versus P/poly---will require non-algebrizing techniques.

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The main result of this lecture is to show the existence of oracles A,B such that P = NP while P 6= NP . A fancy way of expressing this is to say that the P vs. NP question has contradictory…

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The possibility of using the relatively old technique of diagonalization to separate complexity classes, in particular NL from NP, is discussed and several results in this direction are shown.

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