## The Role of Relativization in Complexity Theory (1994)

Venue: | Bulletin of the European Association for Theoretical Computer Science |

Citations: | 41 - 9 self |

### BibTeX

@ARTICLE{Fortnow94therole,

author = {Lance Fortnow},

title = {The Role of Relativization in Complexity Theory},

journal = {Bulletin of the European Association for Theoretical Computer Science},

year = {1994},

volume = {52},

pages = {52--229}

}

### Years of Citing Articles

### OpenURL

### Abstract

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.

### Citations

11403 | Computers and Intractability: A Guide to the Theory of NP-Completeness - Garey, Johnson - 1979 |

4091 | Introduction to Automata Theory, Languages, and Computation - HOPCROFT, ULLMAN - 1979 |

1079 | The Knowledge Complexity of Interactive Proof Systems
- Goldwasser, Micali, et al.
- 1989
(Show Context)
Citation Context ...e can why these techniques do not relativize. 5 Relativizations of Interactive Proofs Interactive proofs were invented in 1985 simultaneously by Babai [Bab85, BM88] and Goldwasser, Micali and Rackoff =-=[GMR89]-=-. We refer the reader to these papers for descriptions and formal definitions of interactive proofs. In 1986, Fortnow and Sipser [FS88] created a relativized world where some language in co-NP does no... |

823 | The complexity of theorem-proving procedures - Cook - 1971 |

405 | Non-deterministic exponential time has two-prover interactive protocols
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...le results mean? What should we infer, if anything, from a relativization? Such questions gained even more importance when we discovered the amazing power of multiple prover interactive proof systems =-=[BFL91]-=-, transparent proofs [BFLS91] and probabilistically checkable proofs [AS92, ALM + 92]. We may not find satisfactory answers to these questions in the near future. However, in this paper we will give s... |

366 | Probabilistic checking of proofs: A new characterization of NP
- Arora, Safra
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(Show Context)
Citation Context ... the result of Arora, Lund, Motwani, Sudan and Szegedy [ALM + 92] does not relativize in a strong way. We then look at what happens if we look at oracles trying to relate PCP and EXP. Arora and Safra =-=[AS92]-=- define a hierarchy of complexity classes PCP, corresponding to the number of random and query bits required to verify a proof of membership in the language, as follows: A verifier M is a probabilisti... |

315 | Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes - Babai, Moran - 1988 |

309 | Algebraic methods for interactive proof systems
- Lund, Fortnow, et al.
- 1992
(Show Context)
Citation Context ...a known negative relativization [FRS88]. Lund, Fortnow and Karloff analyzed this proof and using Nisan's techniques combined with some new ones showed a single-prover interactive proof system for SAT =-=[LFKN92]-=-. Several other important papers on interactive proof theory followed from extensions of these techniques (e.g. [Sha92, BFL91, BFLS91, ALM + 92]). Babai [Bab90] goes into more detail about the history... |

304 | Trading group theory for randomness - Babai - 1985 |

255 | Checking computations in polylogarithmic time
- Babai, Fortnow, et al.
- 1990
(Show Context)
Citation Context ... we infer, if anything, from a relativization? Such questions gained even more importance when we discovered the amazing power of multiple prover interactive proof systems [BFL91], transparent proofs =-=[BFLS91]-=- and probabilistically checkable proofs [AS92, ALM + 92]. We may not find satisfactory answers to these questions in the near future. However, in this paper we will give some intuition and some theore... |

237 | Almost Optimal Lower Bounds for Small Depth Circuits, volume 5
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- 1989
(Show Context)
Citation Context ...omplexity theory. In fact, circuit complexity has given us the tools to 4 prove some important relativization results, such as a relativized world where the polynomial-time hierarchy is infinite (see =-=[Has89]-=-). 3.4 Recognizing new techniques Suppose we have a proof of a statement S but we also know that there exists a relativized world where S does not hold. We can then analyze the proof of S to find the ... |

199 |
On the computational complexity of algorithms
- Hartmanis, Stearns
- 1965
(Show Context)
Citation Context ...e techniques to separate PCP(log n; 1) from PSPACE. The class EXP does not fall into the same trap. For every oracle A, we have P A 6= EXP A since the deterministic time hierarchy theorem relativizes =-=[HS65]-=-. Heller [Hel81] showed that there exists an oracle A where NP A = EXP A . Since Theorem 5.1 does not relativize, Heller's theorem does not necessarily imply that PCP A (log n; 1) = EXP A . In fact an... |

190 |
PP is as hard as the polynomial-time hierarchy
- Toda
- 1991
(Show Context)
Citation Context ...e exception of some results in interactive proof systems (see 3 Sections 3.4 and 5) all of the results in complexity theory have relativized. These include several important results such as PH ` P #P =-=[Tod91]-=- and PP is closed under intersection [BRS91]. The techniques for interactive proofs have not yet proven fruitful towards proving any other theorems about complexity theory. Thus it really does appear ... |

184 |
Private coins versus public coins in interactive proof systems
- Goldwasser, Sipser
- 1986
(Show Context)
Citation Context ...a relativized world where S fails. Often by working on a problem in two directions, one can often push a failure of a proof in one direction into a proof of the other direction. Goldwasser and Sipser =-=[GS89]-=- used this method in their proof of the equivalence of public and private coins in interactive proof systems. Initially, Goldwasser and Sipser tried to prove that the private coin interactive proof hi... |

135 | Multi-prover interactive proofs: how to remove intractability assumptions
- Ben-Or, Goldwasser, et al.
- 1988
(Show Context)
Citation Context ...very random string, i.e. with probability 1. 2. If x 62 L, then for all proofs \Pi, the probability that M using proof \Pi accepts is bounded by 1=2. In 1988, Ben-Or, Goldwasser, Kilian and Wigderson =-=[BGKW88]-=- defined multiple prover interactive proof systems where the verifier communicates with several provers that cannot communicate among themselves. Fortnow, Rompel and Sipser [FRS88] show that the langu... |

130 | On the power of multi-prover interactive protocols
- Fortnow, Rompel, et al.
- 1994
(Show Context)
Citation Context ...n then, hopefully, apply this technique to other negatively relativized problems. In 1989, Noam Nisan found a multiple-prover interactive proof for SAT, a problem with a known negative relativization =-=[FRS88]-=-. Lund, Fortnow and Karloff analyzed this proof and using Nisan's techniques combined with some new ones showed a single-prover interactive proof system for SAT [LFKN92]. Several other important paper... |

88 | PP is closed under intersection
- Beigel, Reingold, et al.
- 1995
(Show Context)
Citation Context ...roof systems (see 3 Sections 3.4 and 5) all of the results in complexity theory have relativized. These include several important results such as PH ` P #P [Tod91] and PP is closed under intersection =-=[BRS91]-=-. The techniques for interactive proofs have not yet proven fruitful towards proving any other theorems about complexity theory. Thus it really does appear that we still need to develop new techniques... |

75 | Generic oracles and oracle classes - Blum, Impagliazzo - 1987 |

61 |
Relativized circuit complexity
- Wilson
- 1985
(Show Context)
Citation Context ...ed in finding a relativizable proof. The whole area of circuit complexity was developed as a potential method for attacking the hard problems like P 6= NP. Although one can easily relativize circuits =-=[Wil85]-=-, many researchers believed the structure of circuits would allow us to find nonrelativizable techniques to solve some basic complexity questions. Circuit complexity still has a long way to go before ... |

45 | Arithmetization: A new method in structural complexity theory
- BABAI, FORTNOW
- 1991
(Show Context)
Citation Context ...ized reduction f mapping an input x to some relativized 3CNF formula OE A . Part of \Pi will contain the formula OE A as well as as proof that OE A = f(x). 2 9 5.3 Algebraic Oracles Babai and Fortnow =-=[BF91]-=- give an algebraic characterization of various complexity classes and argue that the interactive proof take advantage of this characterization. This algebraic characterization also does not seem to re... |

45 | An oracle builder’s toolkit - Fenner, Fortnow, et al. |

38 | Are there interactive protocols for coNP languages - Fortnow, Sipser - 1988 |

38 | A survey of Russian approaches to Perebor (brute-force search) algorithms - Trakhtenbrot - 1984 |

37 | Universal’nyĭe perebornyĭe zadachi (Universal search problems - Levin - 1973 |

30 |
Oracles for structural properties: The isomorphism problem and public-key cryptography
- Homer, Selman
- 1992
(Show Context)
Citation Context ...know how to prove S or show that S does not hold in some relativized world. We thus would like to end this paper with two interesting exceptions: 1. Does P = UP and NP = co-NP imply that P = NP? (See =-=[HS92]-=-) 2. Does the isomorphism conjecture imply that there are no one-way functions? (See [FFK92]) Acknowledgments This paper grew out of an informal debate with Russell Impagliazzo on relativization resul... |

26 | The isomorphism conjecture holds relative to an oracle
- Fenner, Fortnow, et al.
- 1996
(Show Context)
Citation Context ...s the theory of generic oracles. Generic oracles allow us to combine different oracle requirements in a clean manner. They give us very powerful tools in developing oracles. Fenner, Fortnow and Kurtz =-=[FFK92]-=- use generic oracles to develop a relativized world where the isomorphism conjecture holds, answering a long-standing open question. Space limitations prevent us from giving more details about generic... |

26 | Separability and one-way functions - Fortnow, Rogers |

25 | The random oracle hypothesis is false - Chor, Goldreich, et al. - 1990 |

22 |
On the random oracle hypothesis
- Kurtz
- 1982
(Show Context)
Citation Context ...e class or to use the Landner and Lynch model with a careful eye. 5 Partial Relativizations In these results some, but not all, of the objects involved are allowed to have access to the oracle. Kurtz =-=[Kur83]-=- showed that NP ` P SAT but for a random R, NP R 6` P SAT\PhiR giving a counterexample to the Random Oracle Hypothesis (See Section 6). However, it uses heavily the fact that queries to the SAT oracle... |

20 |
Relative to a random oracle A, P A 6= NP A 6= co \Gamma NP A with probability 1
- Bennett, Gill
- 1981
(Show Context)
Citation Context ... some different assumption, like a suitably strong one-way function, would allow us to find an oracle relative to which PCP(log n; 1) A = EXP A . 6 Random and Generic Oracles In 1981, Bennet and Gill =-=[BG81]-=- looked at what happens when we choose the oracles randomly: decide for each string whether or not it should be in the oracle independently with probability one-half. We say a statement S holds with p... |

19 | E-mail and the unexpected power of interaction
- BABAI
- 1990
(Show Context)
Citation Context ...rover interactive proof system for SAT [LFKN92]. Several other important papers on interactive proof theory followed from extensions of these techniques (e.g. [Sha92, BFL91, BFLS91, ALM + 92]). Babai =-=[Bab90]-=- goes into more detail about the history of these developments. 4 When Oracles Fail Almost since the first relativization results of Baker, Gill and Solovay [BGS75], complexity theorists have look for... |

18 |
Alternations and space–bounded computations
- Buss
- 1988
(Show Context)
Citation Context ...ake the oracle tape a write-only tape that is automatically erased after each query. While this suggestion works well below P, it does not solve the quandary described in the previous paragraph. Buss =-=[Bus88]-=- creates an oracle model that seems to get around this problem but is too cumbersome for use in practice. If one must relativize a space class we suggest to either use the corresponding relativized al... |

12 | Relativizing versus nonrelativizing techniques: the role of local checkability
- Arora, Impagliazzo, et al.
- 1992
(Show Context)
Citation Context ...A = EXP A , we show that PCP A (log n; 1) = EXP A would imply that P 6= NP (Theorem 5.4). Thus finding such an oracle would be as hard as settling the P = NP question. Arora, Impagliazzo and Vazirani =-=[AIV92] argue tha-=-t the "local-checkability" property of complexity classes is a major reason that the results on interactive proofs do not relativize. In Section 5.2, we give negative evidence for this thesi... |

12 | Structural complexity theory: Recent surprises - Hartmanis, Chang, et al. - 1990 |

11 |
Three results on the polynomial isomorphism of complete sets
- Goldsmith, Joseph
- 1986
(Show Context)
Citation Context ...ether OE A is satisfiable is NP A -complete. 2. Determining the truth value of /A is PSPACE A -complete. Furthermore the completeness reductions do not need access to the oracle. Goldsmith and Joseph =-=[GJ86]-=- prove the first part of the lemma. An easy modification of their proof gives us the second part as well. 3 Uses of Oracle Results In this section, we will discuss some legitimate uses of relativizati... |

11 |
Feasible Computations and Provable Complexity Properties
- Hartmanis
- 1978
(Show Context)
Citation Context ... relativize both ways. Early on some people speculated that perhaps the Baker-Gill-Solovay result indicated that these questions about complexity theory may fall outside the axioms of set theory (see =-=[Har78]-=-, chapter 7). However most researchers no longer subscribe to this viewpoint anymore because of lack of evidence and some of the examples in Sections 4 and 5. 3.2 Two directions Often in complexity th... |

11 |
Solvable problems with conflicting relativizations
- Hartmanis
- 1985
(Show Context)
Citation Context ...ounterexample to the Random Oracle Hypothesis (See Section 6). However, it uses heavily the fact that queries to the SAT oracle do not have access to R. In fact, for all A, NP A ` P SAT A . Hartmanis =-=[Har85]-=- has a similar example. Chang [Cha90] has a different example where a function s(n) is not space-constructible in the unrelativized world but s(n) is space-constructible in a relativized world. Howeve... |

9 |
Relativized polynomial hierarchies extending two levels
- Heller
- 1981
(Show Context)
Citation Context ...e show that in a relativized world, for all k, such a result does not hold even if we allow the verifier to use a polynomial number of random bits and n k proof queries (Theorem 5.2). Although Heller =-=[Hel81]-=- has created an oracle A relative to which NP A = EXP A , we show that PCP A (log n; 1) = EXP A would imply that P 6= NP (Theorem 5.4). Thus finding such an oracle would be as hard as settling the P =... |

6 |
Relativization of questions about log-space reducibility
- Ladner, Lynch
- 1976
(Show Context)
Citation Context ...l not relativize because the space-bounded machine could ask exponentially long queries. Hartmanis, Chang, Kadin and Mitchell [HCKM88] have further discussion on these oracle models. Ladner and Lynch =-=[LL76]-=- suggest a reasonable alternative: Do not count the space on the oracle tape but make the oracle tape a write-only tape that is automatically erased after each query. While this suggestion works well ... |

4 | An example of a theorem that has contradictory relativizations and a diagonalization proof
- Chang
- 1990
(Show Context)
Citation Context ...pothesis (See Section 6). However, it uses heavily the fact that queries to the SAT oracle do not have access to R. In fact, for all A, NP A ` P SAT A . Hartmanis [Har85] has a similar example. Chang =-=[Cha90]-=- has a different example where a function s(n) is not space-constructible in the unrelativized world but s(n) is space-constructible in a relativized world. However, if the function s(n) was computed ... |

4 | Some observations about relativizations of space bounded computations
- Hartmanis, Chang, et al.
- 1993
(Show Context)
Citation Context ...s. If we do not count the tape then a result like ATIME(poly) = PSPACE will not relativize because the space-bounded machine could ask exponentially long queries. Hartmanis, Chang, Kadin and Mitchell =-=[HCKM88]-=- have further discussion on these oracle models. Ladner and Lynch [LL76] suggest a reasonable alternative: Do not count the space on the oracle tape but make the oracle tape a write-only tape that is ... |

3 |
Some results on relativized deterministic and nondeterministic time hierarchies
- Moran
- 1981
(Show Context)
Citation Context ... complexity theory is linear speedup: if a program takes t(n) time with t(n) superlinear then for every c, for all but a finite n, there is a another Turing machine that takes only t(n)=c time. Moran =-=[Mor81]-=- noted that this result does not relativize: we need only make the language dependent on strings in the oracle of length say t(n) + 1. This oracle failure occurs because of insufficient oracle access.... |