S. Fenner, L. Fortnow, and S.A. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proc. 33rd IEEE Symposium Foundations of Computer Science, pages 30-39, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... with the IC were those relative to which it is false, including ones constructed by Kurtz [Kur83] Kurtz, Mahaney, and Royer [KMR89] and Hartmanis and Hemachandra [HH91] Finding an oracle relative to which the IC is true was a difficult task finally accomplished by Fenner, Fortnow, and Kurtz [FFK92] in 1992. Their oracle was sp generic, a complicated sort of generic oracle defined using infinite conditions that allowed infinite encoding. The proof that the IC holds relative to this oracle was intricate and technical, relying on the notions of genericity and relativized Kolmogorov complexity. ....

....is exactly what Beigel, Buhrman, and Fortnow did: They created an oracle A that makes P = UP (in fact, P = PhiP) and NP = EXP. Please see their paper for details about the construction. 2 Relative to A, P = UP, so there are no one way functions. A question raised by Fenner, Fortnow, and Kurtz [FFK92] was whether there was an oracle relative to which the IC holds and one way functions exist. This was settled in 1995 in [Rog97] The construction there begins with the oracle in [FFK92] relative to which P = UP, and re relativize to a UP generic oracle. The result is based on the following ....

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S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, pages 30--39. IEEE, New York, 1992.

.... isomorphism type, thus yielding a relativized failure of the Joseph Young Conjecture (JYC) JY85] We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but one way functions exist, which answers an open question of Fenner, Fortnow, and Kurtz [FFK92]. Thus, there are now relativizations realizing every one of the four possible states of affairs between the IC and the existence of one way functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomial time ....

....languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NP complete language is isomorphic to SAT. This conjecture has been neither proven nor refuted and, because there are oracles relative to which it fails [Kur83, KMR89, HH91] and holds [FFK92], doing so will probably require new proof techniques. In the meantime, we can try to demonstrate relationships between the IC and other complexity theoretic propositions. For example, do the IC and the proposition P 6= UP have anything to do with another Grollmann and Selman [GS88] show that P ....

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S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, pages 30--39. IEEE, New York, 1992.

....complete sets, that differentiates between the classes NP and P. Over the last two decades, the Berman Hartmanis Conjecture led to a series of controversial discussions and a sequence of structural results both in support of and against its validity (see, e.g. JY85, GJ86, KMR89, KMR90, You90, FFK92] Moreover, the fascinating idea behind the conjecture inspired many researchers to extensively investigate the structural properties of complete sets for NP and other complexity classes under various types of reducibilities (see, e.g. the surveys of Homer [Hom90] and of Buhrman and Torenvliet ....

S. Fenner, L. Fortnow, and S. Kurtz, The isomorphism conjecture holds relative to an oracle, Proc. 33rd FOCS (IEEE, 1992) 30--39.

....been done by [JY85, KLD86, KMR95] Q: Let me interrupt. You ll never get this article done on time if you are going to survey the entire field. Are there any surveys you can suggest A: Yes. Two very nice ones that spring to mind are [KMR90] and [Yo90] Some exciting developments (such as [FFK96, JPY94, Ro95]) are too recent to be mentioned in the surveys. Q: What does this have to do with factorization A: I m getting to that. In spite of the general feeling that the original BermanHartmanis conjecture is probably false, it was shown in [AAR96] that the BermanHartmanis conjecture for AC 0 ....

S. Fenner, L. Fortnow, and S. Kurtz, The isomorphism conjecture holds relative to an oracle, SIAM Journal on Computing 25 (1996) 193-206.

....of all p m complete sets in NP, and oracle proof that the conjecture might not hold (most prominent, the conjecture fails relative to a random oracle [KMR89] opinion shifted against the idea that the isomorphism conjecture might be true. It was not until 1992 that Fenner, Fortnow and Kurtz [FFK92] proved the existence of an oracle relative to which the isomorphism conjecture holds. In 1985, Joseph and Young [JY85] constructed unnatural sets, the so called k creative sets for which every 1 1 polynomial time computable honest function is a productive function. Hence these sets are very ....

S. Fenner, L. Fortnow, and S.A. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proc. 33rd IEEE Symposium Foundations of Computer Science, pages 30--39, 1992.

....means of efficient renaming procedures of strings, and thus, there is basically only one, unique NP complete set. This simple, plausible statement has attracted many brilliant scientists over the past two decades and has become a central research area in complexity theory. The reader may refer to [KMR90, You90, FFK92] for the current status of the isomorphism conjecture. What does this conjecture suggest First of all, P 6= NP, because finite sets are in P NP but not isomorphic to SAT. Second, more amusingly, sparse sets are not NPcomplete. For a language A, define the census function of A, cens A (n) to ....

S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of the 33rd Symposium on Foundations of Computer Science, pages 30--39. IEEE Computer Society Press, October 1992.

....isomorphism conjecture is true [JY85] There are NP complete sets that cannot be isomorphic under commonly used distributions on instances [WB95] The isomorphism conjecture is an active research topic with considerable work on the subject. For the current status of the subject, see [Sel92, FFK92, KMR90, You90] We will not go into this any further in this article. 3.2 Mahaney s theorem As typical NP complete languages such as SAT are all exponentially dense, and polynomial time isomorphisms cannot change the density from exponential to polynomial, it follows from the isomorphism ....

S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proc. 33rd Symp. on Found. of Comp. Sci., pages 30--39. IEEE, October 1992. 1. Sparse Sets versus Complexity Classes 28

....are essentially one and the same problem. However, there is now considerable doubt whether the BermanHartmanis isomorphism conjecture is true [JY85] The isomorphism conjecture is an active research topic with considerable work on the subject. For the current status of the subject, see [FFK92, KMR90, You90] We will not go into this any further in this article. 3.2 Mahaney s theorem As typical NP complete languages such as SAT are all exponentially dense, and polynomial time isomorphisms cannot change the density from exponential to polynomial, it follows from the isomorphism ....

S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proc. 33rd Symp. on Found. of Comp. Sci., pages 30--39. IEEE, October 1992.

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S. Fenner, L. Fortnow, and S.A. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proc. 33rd IEEE Symposium Foundations of Computer Science, pages 30-39, 1992.

....= EXP then the isomorphism conjecture holds. They created a relativized world where P = UP and Sigma 2 = EXP. However, even a relativized world where P = UP 6= NP = coNP seemed much more difficult to prove. Theorem 1.4 is the first to break this barrier. Fenner, Fortnow and Kurtz [FFK96] used a very different and complicated approach to resolve the relativized isomorphism conjecture. Their oracle is nonconstructive and makes the polynomialtime hierarchy infinite. Theorem 1.4 is the first to fulfill the Homer and Selman approach. The proof is considerably simpler than Fenner, ....

....by getting P = UP and NP = EXP. They showed the following result. Theorem 4.4 (Homer Selman) There exists a relativized world where P = UP and Sigma 2 = EXP. Theorem 4.4 gives the first relativized world where all Sigma 2 complete sets are isomorphic. Later, Fenner, Fortnow and Kurtz [FFK96] used a very different approach to settle the relativized isomorphism conjecture. Theorem 4.5 (Fenner Fortnow Kurtz) There exists a relativized world where all NP complete sets are polynomial time isomorphic. 5 The proof of Fenner, Fortnow and Kurtz requires a complicated, nonconstructive ....

S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. SIAM Journal on Computing, 25(1):193--206, 1996.

....sets p isomorphic in a relativized world by setting P = UP and C = EXP. Homer and Selman [HS92] were able to do this for C = p 2 , and our present oracle works for C = P NP . An oracle for C = NP con rms the isomorphism conjecture [BH77] Oracles making the conjecture true are known since [FFK96]. However, BBF98] gives an oracle relative to which P = UP and NP = EXP. 3 2 Preliminaries We assume the reader familiar with standard notions in structural complexity theory, as are de ned e.g. in [BDG88] Nonetheless, we will in this section, recall some notions that we feel are not common ....

S. Fenner, L. Fortnow, and S.A. Kurtz. The isomorphism conjecture holds relative to an oracle. SIAM Journal on Computing, 25(1):193-206, 1996.

....for example the isomorphism conjecture. Relative to a random oracle the isomorphism conjecture fails (Kurtz, Mahaney and Royer [KMR95] and NP simple sets exist (Vereshchagin [Ver94] On the other hand the isomorphism conjecture holds relative to an sp generic oracle (Fenner, Fortnow and Kurtz [FFK96] and NP simple sets exist (as we will show presently) Hence the assumption that NP simple sets exist will not be enough to show the isomorphism conjecture true or false with relativizing techniques. In the other direction we can use an oracle constructed by Beigel, Buhrman, and Fortnow [BBF98] ....

....definition of all the terms needed in the proof that NP simple sets exist relative to an sp generic oracle. However, the reader not familiar with generic oracles is advised to take a closer look at the paper by Fenner, Fortnow and Kurtz for background. Definition 9. 1 (Fenner, Fortnow, Kurtz [FFK96] An iterated polynomial sequence (a i ) i## fulfills a 0 # 2 and a i 1 = p(a i ) for some polynomial p(n) # n 2 . We call a partial function 21 # : # # # 0, 1 a symmetric perfect forcing condition (sp condition) if # is undefined on strings of length a i (for all i) A partial ....

Stephen Fenner, Lance Fortnow, and Stuart A. Kurtz. The isomorphism conjecture holds relative to an oracle. SIAM Journal on Computing, 25(1):193--206, 1996.

....generic) 1 2. FR94] There exists an oracle A such that NP A 6= coNP A and Q A holds. 3. There exists an oracle B such that NP B = UP B , Q B holds and NP B 6= coNP B . 4. FR94, IN88, CS93] There exists an oracle C such that P C = NP C coNP C and Q 0 C fails. 5. FFK92] There exists an oracle D such that Q D fails and the isomorphism conjecture holds relative to D. 6. KMR89] There exists an oracle E such that Q E fails and the isomorphism conjecture fails relative to E. Proof To prove (3) it is not hard to see that the oracle in (2) can be constructed ....

S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 3039, 1992.

....genericity. 2. FR94] There exists an oracle A such that NP A 6= coNP A and Q A holds. 3. There exists an oracle B such that NP B = UP B , Q B holds and NP B 6= coNP B . 4. FR94, IN88, CS93] There exists an oracle C such that P C = NP C coNP C and Q 0 C fails. 5. FFK92] There exists an oracle D such that Q D fails and the isomorphism conjecture holds relative to D. 6. KMR89] There exists an oracle E such that Q E fails and the isomorphism conjecture fails relative to E. Proof To prove (3) it is not hard to see that the oracle in (2) can be constructed ....

S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 30--39, 1992. To appear in SIAM Journal of Computing.

....that NP A 6= coNP A and Q A holds. 1 See [FR94] for a discussion on sparse genericity. 3. There exists an oracle B such that NP B = UP B , Q B holds and NP B 6= coNP B . 4. FR94, IN88, CS93] There exists an oracle C such that P C = NP C coNP C and Q 0 C fails. 5. FFK92] There exists an oracle D such that Q D fails and the isomorphism conjecture holds relative to D. 6. KMR89] There exists an oracle E such that Q E fails and the isomorphism conjecture fails relative to E. Proof To prove (3) it is not hard to see that the oracle in (2) can be constructed so ....

S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 3039, 1992.

....UP and Sigma p 2 = EXP. However, even a relativized world where P = UP 6= NP = coNP seemed much more difficult to prove. Theorem 1.4 is the first to break this barrier. Corollary 1. 5 There exists an oracle A such that P A = UP A 6= NP A = coNP A = EXP A Fenner, Fortnow and Kurtz [FFK96] used a very different and complicated approach to resolve the relativized isomorphism conjecture. Their oracle is nonconstructive and makes the polynomialtime hierarchy infinite. Theorem 1.4 is the first to fulfill the Homer and Selman approach. The proof is considerably simpler than Fenner, ....

....P = UP and NP = EXP. They showed the following result. Theorem 4.4 (Homer Selman) There exists an oracle relativize to which P = UP and Sigma p 2 = EXP. Theorem 4.4 gives the first relativized world where all Sigma p 2 complete sets are isomorphic. Later, Fenner, Fortnow and Kurtz [FFK96] used a very different approach to settle the relativized isomorphism conjecture. Theorem 4.5 (Fenner Fortnow Kurtz) There exists a relativized world where all NP complete sets are polynomial time isomorphic. The proof of Fenner, Fortnow and Kurtz requires a complicated, nonconstructive ....

S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. SIAM Journal on Computing, 25(1):193--206, 1996.

....in a relativized world by setting P = UP and C = EXP. Homer and Selman [HS92] were able to do this for C = Sigma p 2 , and our present oracle works for C = P NP . An oracle for C = NP would affirm the Isomorphism Conjecture [BH77] Even though oracles making the Conjecture true are known [FFK92], it is still an interesting open question whether there is an oracle relative to which P = UP and NP = EXP. 2 Notation and Definitions We let Sigma = f0; 1g and identify Sigma with the natural numbers 0; 1; 2; in the usual way. We will sometimes use the divider symbol, # , and in ....

S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, pages 30--39, 1992. To appear in SIAM J. Comp.

....a large collection of different generic notions. We then show several specific results: 1. We show how to use standard (Cohen) genericity to create an oracle where the isomorphism conjecture fails in a strong way. 2. We look at sp genericity which was first used by Fenner, Fortnow and Kurtz [FFK92] to create an oracle where the isomorphism conjecture holds. By looking at what happens to a new complexity class AWPP, we can infer several strong collapses for sp generics. 3. We show that given some assumptions or additional information, these same collapses occur for Cohen generics as well. ....

....trees, as used in the construction of a minimal degree, as giving a very different notion of genericity. A third notion, closely related to the second, is the symmetric perfect genericity used to prove the existence of an oracle relative to which the Berman Hartmanis Isomorphism Conjecture holds [FFK92]. Definition 2.2 Let LPA [X ] be the language of Peano Arithmetic, augmented by a unary predicate symbol X. We assume that the only logical operators in LPA [X ] are : and 9; the other standard operators are defined in terms of these three. The G forcing relation fl G on G Theta sent(L PA ....

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S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, pages 30--39. IEEE, New York, 1992.

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S. Fenner, L. Fortnow, and S.A. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proc. 33rd IEEE Symposium Foundations of Computer Science, pages 30--39, 1992.

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