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The Role of Relativization in Complexity Theory

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The Role of Relativization in Complexity Theory See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/2583799 The Role of Relativization in Complexity Theory ARTICLE · APRIL 1999 Source: CiteSeer CITATIONS READS 46 28 1 AUTHOR: Lance Fortnow Georgia Institute of Technology 252 PUBLICATIONS 5,786 CITATIONS SEE PROFILE Available from: Lance Fortnow Retrieved on: 13 January 2016 The Role of Relativization in Complexity Theory Lance Fortnow University of Chicago Department of Computer Science East th Street Chicago Illinois Abstract Several recent nonrelativizing results in the area of interactive pro ofs have caused many p eople to review the imp ortance of relativization In this pap er we take a lo ok at how complexity theorists use and misuse oracle results We pay sp ecial attention to the new interactive pro of 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 b etter Intro duction The recent result IP PSPACE LFKN Sha surprised the theoretical computer science community in more ways than one A few years earlier Fortnow and Sipser FS created an oracle relative to which coNP did not have interactive pro ofs The IP PSPACE result was honestly a nonrelativizing theorem Several questions immediately p opp ed up Why didnt this result relativize What sp ecic techniques were used that avoided relativization How can we use these techniques to prove other nonrelativizing facts Also much older questions resurfaced What exactly to oracle results mean What should we infer if anything from a relativization Such questions gained even more imp ortance when we discovered the amazing p ower of mul tiple prover interactive pro of systems BFL transparent pro ofs BFLS and probabilistically + checkable pro ofs AS ALM We may not nd satisfactory answers to these questions in the near future However in this pap er we will give some intuition and some theorems ab out oracles that may help shed light on some of these issues In Section we will see that relativization is an extremely p owerful to ol in helping complexity theorists direct their research In Section we will lo ok at early results of provable statements with negative relativization We will lo ok at these various examples and argue that they all lack a prop er oracle access mechanism In section we take a close lo ok at the new unrelativizing results for interactive pro of systems We prove some new results that may help us understand these issues b etter We also argue that the Email fortnowcsuchicagoedu Partially supp orted by NSF grant CCR new interactive pro of system techniques do not relativize b ecause they take advantage of certain algebraic prop erties of complexity classes In section we lo ok at what happ ens when we add an oracle to probabilistically checkable + pro ofs PCP Arora Lund Motwani Sudan and Szegedy ALM show that every language in NP has a PCP using only a logarithmic numb er of random coin tosses and a constant numb er of queries to the pro of We show that in a relativized world for all k such a result do es not k hold even if we allow the verier to use a p olynomial numb er of random bits and n pro of queries Theorem A A Although Heller Hel has created an oracle A relative to which NP EXP we show that A A PCP log n EXP would imply that P NP Theorem Thus nding such an oracle would b e as hard as settling the P NP question Arora Impagliazzo and Vazirani AIV argue that the lo calcheckability prop erty of com plexity classes is a ma jor reason that the results on interactive pro ofs do not relativize In Sec tion we give negative evidence for this thesis by showing that under a reasonable access mech anism lo cal checkability do es in fact relativize We give supp ort instead to the thesis that it is the algebraic prop erties of complexity classes that do not relativize In Section we give evidence for this prop osition by showing that IP PSPACE holds relative to algebraic extensions of arbitrarily complicated languages Finally in Section we give a brief arguement against infering any information from random oracle results We argue that we should only use random oracles as a to ol to combine oracle constructions However we b elieve that generic oracles are a much stronger and sharp er to ol for this purp ose Caveats This pap er contains several opinions on the use and misuse of relativization results We must caution the reader that other complexity theorists may have diering opinions on these matters In this pap er we have tried to give several examples to illustrate various p oints However this pap er is not meant to b e a survey pap er Many imp ortant works in the area have not b een mentioned due to lack of space Notation and Denitions Most of the notation and denitions follow from the standard textb o oks on the eld HU GJ We use to represent the join of two sets A and B ie A B fg A fg B We use FP to represent the p olynomialtime computable functions It is a misnomer to relativize a complexity class C Instead supp ose we take an enumeration of machines for C and give them some access mechanism to an oracle set A We then say the A relativized class C consists of the languages recognized by the acceptance criteria for C applied to the machines using the oracle A Of course this denition may dep end greatly on the sp ecic enumeration of the machines of C as well as the oracle access mechanism The usual access mechanism consists of a separate oracle tap e that the Turing machine can write down queries and learn the answers However as we shall see in Sections and such mo dels may unduly handicap the machines Relativizing Formulae It will b e useful to have relativized NPcomplete and PSPACEcomplete sets Let a relativized CNF formula b e a CNF formula with clauses of the form x x x Ax x i1 i2 i3 j 1 j k where Ax x is true if x x is in A Any of the variables or the Ax x term j 1 j k j 1 j k j 1 j k may b e negated Lemma Let be a relativized CNF formula over an oracle A Let be a closed relativized A A CNF formula with arbitrary rstorder existential and universal quantiers over the variables A Determining whether is satisable is NP complete A A Determining the truth value of is PSPACE complete A Furthermore the completeness reductions do not need access to the oracle Goldsmith and Joseph GJ prove the rst part of the lemma An easy mo dication of their pro of gives us the second part as well Uses of Oracle Results In this section we will discuss some legitimate uses of relativization results As we will see complexity theorists have used relativization as a p owerful to ol in studying complexity theory In Section we will see how theorists use oracles to discover what techniques will not likely work to solve certain problems In Section we will see how relativization allows us to push at a problem in two dierent directions In Section we will argue that that lack of nonrelativizable techniques have caused theorists to lo ok at new directions of research Finally in Section we will see how old relativization results help us to recognize new techniques that can b e applied to other problems Of course theorists must execute extreme care in how one should interpret oracle results In Sections and we lo ok at some computational mo dels where oracle results may not have the exp ected interpretation Limiting techniques Supp ose we can show for some statement S that there exists an oracle A such that S fails relative to A in some oracle mo del Then any pro of that S hold must not relativize in that mo del or otherwise that statement would also hold relative to A If we can also nd an oracle relative to which S holds then no relativizable technique can decide the truth of S Baker Gill and Solovay BGS noted this in their original oracle pap er where they give a relativized world where P NP and another where P NP They also noted that essentially all the known complexity techniques at that time relativize They concluded that current techniques would not solve the P NP question In the nearly two decades since the BakerGillSolovay pap er there have b een literally hundreds of results in complexity theory With the exception of some results in interactive pro of systems see Sections and all of the results in complexity theory have relativized These include several P imp ortant results such as PH P To d and PP is closed under intersection BRS The techniques for interactive pro ofs have not yet proven fruitful towards proving any other theorems ab out complexity theory Thus it really do es app ear that we still need to develop new techniques to settle the hundreds of complexity statements that relativize b oth ways Early on some p eople sp eculated that p erhaps the BakerGillSolovay result indicated that these questions ab out complexity theory may fall outside the axioms of set theory see Har chapter However most researchers no longer subscrib e to this viewp oint anymore b ecause of lack of evidence and some of the examples in Sections and Two directions Often in complexity theory one has a complexity statement S where one can easily show a rela tivized world where S holds but it is op en whether there exists a relativizable pro of of S In order to tackle this problem many complexity theorists lo ok at trying to prove two opp osite directions Trying to prove S or Creating a relativized world where S fails Often by working on a problem in two directions one can often push a failure of a pro of in
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