## An oracle builder’s toolkit (2002)

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Citations: | 47 - 10 self |

### BibTeX

@MISC{Fenner02anoracle,

author = {Stephen Fenner and Lance Fortnow and Stuart A. Kurtz and Lide Li},

title = {An oracle builder’s toolkit},

year = {2002}

}

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### Abstract

We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SP-generics, ULIN ∩ co-ULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ co-NP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩co-NP/1 ̸ ⊆ (NP∩co-NP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.

### Citations

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Citation Context ... that may not be widely known. These are mostly counting classes described in, for example, [FFK94, For97, Sch90]. For completeness, we dene them here and give their basic properties. Denition 2.1 ([V=-=a-=-l79]) A function f : ! ! is in #P if there is a nondeterministic polynomialtime Turing machine M such that f(x) is the number of accepting paths of M on input x, for all x 2 . The rest of the info... |

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Citation Context ...x) = x2 c and g 2;c (x) = x 2 c , and forsxed c, g 3;c (x) = 2 2 c log x grows superpolynomially in x. The g i;c form a natural hierarchy of subexponentially growing functions (see, for example, Lutz =-=[Lut92]). -=-The proof of Theorem 7.6 \scales up" in a straightforward way to prove Theorem 7.9 For i; j; k 2 ! with 2 isj, relative to any SP j -generic set, ULIN \ co-ULIN \ ZPLIN 6 DTIME(g i;k ): In parti... |

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Citation Context ...zing by G yields P A = UP A . By Grollmann and Selman [GS88], we see that there can be no one-way functions relative to A. Oursnal comment in the theorem is justied by Berman and Hartmanis's theorem [=-=BH77-=-] that if two sets are equivalent by one-one, length-increasing, invertible functions, then they must be isomorphic. This oracle A collapses several other classes to P simultaneously. See Section 6.5... |

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Citation Context ...icate complexity, and Theorem 6.10 follows immediately from Lemma 6.8. For the remaining collapses we will use the class AWPP described in Section 6.1 and a powerful theorem from Nisan and Szegedy [N=-=S94-=-]. For a Boolean function f(x 1 ; : : : ; xN ) and an input y 2 f0; 1g N let S y f1; : : : ; Ng be the lexicographically least set of minimum size such that f(z) = f(y) for all z agreeing with y on t... |

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Citation Context ...FP RG , as desired. Remark Part of Theorem 4.10 can also be proved with language classes: the existence of one-way functions relative to R is equivalent to P R 6= UP R , by relativizing results in [G=-=S-=-88]. Relativizing Corollary 4.6 to R gives us P RG 6= UP RG , hence there exist one-way functions relative to RG [GS88]. Much of structural complexity, including this work, grew out of the Berman-Hart... |

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Citation Context ...P if there is a nondeterministic polynomialtime Turing machine M such that f(x) is the number of accepting paths of M on input x, for all x 2 . The rest of the information in this section is from [F=-=FK-=-94], which can be consulted for more details. Denition 2.2 A function f : ! Z is in GapP if there are two #P functions f + and f such that f(x) = f + (x) f (x) for all x. Equivalently, f 2 GapP if t... |

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Citation Context ...ative to SP-generic oracles. We also use the notion of certicate complexity and a new complexity class AWPP to present a general criterion for when such collapses occur. Recently, Fortnow and Rogers [=-=FR99]-=- have shown that AWPP contains BQP, the class of all languages decided by a quantum computer in bounded error probabilistic polynomial time [BV97]. Thus relative to SP-generics, P = BQP but PH separat... |

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Citation Context ...cteristic functions with computable domains are pointed perfect sets. 12 notion of genericity R. The fact that randomness may be viewed as a particular form of genericity has been known for some time =-=[Sol-=-70]. The set R consists of those conditions, all of whose branches have positive Lebesgue measure, i.e., 2 R i , its 0- and 1-branches, the 0- and 1-branches of its 0- and 1-branches, etc. all have p... |

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Citation Context ...ng the ideas of Baire category. We mention these as well. In Section 4, we systematically assemble a number of basic facts about Cohen generic sets, many of which are generalizations of known results =-=[BI8-=-7]. We then use these results together with the technique of `rerelativization' to produce a number of oracles that cause the Isomorphism Conjecture to fail in a variety of dierent ways. We dene SP-ge... |

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54 |
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Citation Context ...KP54], see Section 5.1), whose denition is calibrated for complexity theory. Finally, random sets, studied extensively by many people andsrst used as oracles in complexity theory by Bennett and Gill [=-=BG81]-=-, also fall under our scheme as R-generic sets for a particular 3 A pointed perfect set is a subtree of 2 ! that can be computed given any of its branches. For example, all conditions corresponding to... |

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Citation Context ...l q such that x 2 L ) (1 2 r(n) )2 q(n) g(x) 2 q(n) x 62 L ) 0 g(x) 2 r(n) 2 q(n) where n = jxj. Li essentially showed that AWPP is low for the class PP, that is, PP L = PP for every L 2 AWPP [Li9=-=3] (for-=- a published proof, see Fenner [Fen02a]). Fenner also showed that one can \amplify" the ratio g(x)=2 q(n) towards zero or one, so that 2 r(n) may be replaced with a constant, say 1=3, wherever it... |

44 | Nonrelativizing separations
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Citation Context ...h respect to SP-generics (or Cohen generics with help), is it the case that p 2 \ p 2 collapses to P NP ? What happens at higher levels? Both these questions were resolved by Fortnow and Yamakami [F=-=Y9-=-6], who showed that p 2 \ p 2 does not collapse to P NP , and that separations also occur at higher levels as well. Whether their techniques apply to hierarchies involving counting classes is a natu... |

41 | The isomorphism conjecture fails relative to a random oracle
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(Show Context)
Citation Context ...ique: rerelativizing by a generic oracle. The rest of this section illustrates the technique. For background on the notions of \honest," \paddable," \1-li degrees," etc., see for exampl=-=e Kurtz et al. [KMR95-=-] or [KMR90]. Denition 4.9 ([KMR95]) A one-to-one and honest function f 2 FP is annihilating if every P subset of range(f) is sparse; f is scrambling if range(f) does not contain a paddable set. If sc... |

37 |
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Citation Context ...FK96] to establish the existence of an oracle relative to which the Isomorphism Conjecture holds. SP-genericity is really just a variant of what one might call \exact pair forcing" (a la Kleene-P=-=ost [KP54-=-], see Section 5.1), whose denition is calibrated for complexity theory. Finally, random sets, studied extensively by many people andsrst used as oracles in complexity theory by Bennett and Gill [BG81... |

34 |
On degrees of recursive unsolvability
- Spector
- 1956
(Show Context)
Citation Context ...ty over !) has been extensively studied in computability theory (see Jockusch [Joc80] for a slightly dated but very useful survey) and computer science (e.g., [Meh73, BI87, AS96] and others). Spector =-=[Spe56]-=- implicitly used forcing with computable trees to construct a minimal Turing degree. Sacks [Sac71] considered forcing with arithmetical and pointed perfect sets. 3 See Odifreddi [Odi83] for a detailed... |

33 |
Decision trees downward closure
- Impagliazzo, Naor
- 1988
(Show Context)
Citation Context ...e seen that generic oracles do not only separate classes, but in some cases they in fact collapse them. In this section we will extend the results of Blum and Impagliazzo [BI87], Impagliazzo and Naor =-=[IN88]-=-, and Fenner, Fortnow and Kurtz [FFK96] to show general collapses for classes into P. We will show that the following complexity classes equal P relative to SP-generic oracles: UP, FewP, SPP, BPP, BQP... |

32 | Two queries
- Buhrman, Fortnow
- 1996
(Show Context)
Citation Context ...ogers [FR94] to get simultaneous collapses and separations of various subclasses of NP. Buhrman and Fortnow also used a UP-generic set tosnd an oracle where NP 6= co-NP but P NP[1] = P NP[2] = PSPACE =-=[BF99-=-]. In a clever use of the rerelativization technique, Rogers [Rog97] dened a generic oracle where both P 6= UP and the Isomorphism Conjecture holds. This oracle is a UP-generic oracle built \on top of... |

30 | The structure of complete degrees
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(Show Context)
Citation Context ...ativizing by a generic oracle. The rest of this section illustrates the technique. For background on the notions of \honest," \paddable," \1-li degrees," etc., see for example Kurtz et =-=al. [KMR95] or [KMR90-=-]. Denition 4.9 ([KMR95]) A one-to-one and honest function f 2 FP is annihilating if every P subset of range(f) is sparse; f is scrambling if range(f) does not contain a paddable set. If scrambling fu... |

29 |
Oracles for structural properties: The isomorphism problem and public-key cryptography
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Citation Context ... [BBF98]. Relative to their oracle, NP = EXP and P = UP = P (see Denition 2.3 for a denition of P). It is known that thesrst two identities together relativizably imply the Isomorphism Conjecture [HS9=-=2]-=-. We will show collapses to P of a variety of other complexity classes in Section 6. In Section 7.3 we show that these collapses are optimal in some sense: we cannot stratify these collapses in time c... |

27 | Resource-bounded genericity - Ambos-Spies - 1996 |

27 |
Degrees of generic sets
- Jockusch
- 1981
(Show Context)
Citation Context ... the guise ofsnite function forcing by Feferman [Fef65]. Finite function forcing (and the derived notion of Cohen genericity over !) has been extensively studied in computability theory (see Jockusch =-=[Joc80]-=- for a slightly dated but very useful survey) and computer science (e.g., [Meh73, BI87, AS96] and others). Spector [Spe56] implicitly used forcing with computable trees to construct a minimal Turing d... |

26 | The isomorphism conjecture holds relative to an oracle
- Fenner, Fortnow, et al.
- 1996
(Show Context)
Citation Context ...er turns out to be armative, an explicit oracle construction is unnecessary. This was exactly the line of attack used to discover an oracle for which the Berman-Hartmanis Isomorphism Conjecture holds =-=[FFK9-=-6]. The notion of SP-genericity (see Section 5) was dened and many of its properties studied before its eect on the Conjecture was known. It was natural to try to determine the status of the Conjectur... |

26 | Separability and one-way functions
- Fortnow, Rogers
(Show Context)
Citation Context ...d a specialized form of generic, an MA-generic, to solve a specic problem. Other specialized generic sets, including UP-generic and (NP \ co-NP)-generic sets, have been applied by Fortnow and Rogers [=-=FR94]-=- to get simultaneous collapses and separations of various subclasses of NP. Buhrman and Fortnow also used a UP-generic set tosnd an oracle where NP 6= co-NP but P NP[1] = P NP[2] = PSPACE [BF99]. In a... |

24 |
Graph Isomorphism in SPP
- Arvind, Kurur
- 2002
(Show Context)
Citation Context ... C=P \ co-C =P C=P [ co-C =P PP: It is known that the Graph Isomorphism problem is in LWPP, and Graph Automorphism is in SPP [KST92]. More recently Graph Isomorphism has been shown to be in SPP [AK=-=02]. T-=-he class AWPP (\approximate WPP"), dened in Section 6, contains WPP, but probably does not contain C=P or co-C =P. All the denitions above relativize in the usual way. 3 Abstract Genericity This ... |

24 | NP might not be as easy as detecting unique solutions
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- 1998
(Show Context)
Citation Context ...l Isomorphism oracle result [FFK96], Beigel, Buhrman, and Fortnow constructed an oracle making the Isomorphism Conjecture hold using an entirely dierent construction employing coding via polynomials [=-=BB-=-F98]. Relative to their oracle, NP = EXP and P = UP = P (see Denition 2.3 for a denition of P). It is known that thesrst two identities together relativizably imply the Isomorphism Conjecture [HS92]. ... |

24 | Graph isomorphism is low for PP
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- 1992
(Show Context)
Citation Context ... inclusions are known [FFK94]: P UP FewP Few SPP LWPP WPP C=P \ co-C =P C=P [ co-C =P PP: It is known that the Graph Isomorphism problem is in LWPP, and Graph Automorphism is in SPP [KST92]. =-=M-=-ore recently Graph Isomorphism has been shown to be in SPP [AK02]. The class AWPP (\approximate WPP"), dened in Section 6, contains WPP, but probably does not contain C=P or co-C =P. All the deni... |

22 |
Some applications of the notion of forcing and generic sets, Fund
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- 1965
(Show Context)
Citation Context ...uced the notion of forcing in set theory to establish the independence of the continuum hypothesis from ZFC. His ideas were transferred to arithmetic in the guise ofsnite function forcing by Feferman =-=[Fef65]-=-. Finite function forcing (and the derived notion of Cohen genericity over !) has been extensively studied in computability theory (see Jockusch [Joc80] for a slightly dated but very useful survey) an... |

17 | On the size of sets of computable functions - Mehlhorn - 1973 |

16 |
One-way functions and the nonisomorphism of NP-complete sets
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- 1991
(Show Context)
Citation Context ...he Berman-Hartmanis Isomorphism Conjecture. The rerelativization technique can be used to give several novel failures of this conjecture. The following theorem was proven by Hartmanis and Hemachandra =-=[HH91]-=- to refute a conjecture by Kurtz, Mahaney, and Royer that the Isomorphism Conjecture might be equivalent to the nonexistence of one-way functions. Their original proof combined two dicult construction... |

12 | The power of counting - Schoning - 1990 |

10 |
Forcing with perfect closed sets
- Sacks
- 1971
(Show Context)
Citation Context ...ly dated but very useful survey) and computer science (e.g., [Meh73, BI87, AS96] and others). Spector [Spe56] implicitly used forcing with computable trees to construct a minimal Turing degree. Sacks =-=[Sac71-=-] considered forcing with arithmetical and pointed perfect sets. 3 See Odifreddi [Odi83] for a detailed technical development of Feferman's and Sacks's ideas in a context dierent from that presented i... |

9 | The isomorphism conjecture holds and one-way functions exist relative to an oracle
- Rogers
- 1995
(Show Context)
Citation Context ...ous subclasses of NP. Buhrman and Fortnow also used a UP-generic set tosnd an oracle where NP 6= co-NP but P NP[1] = P NP[2] = PSPACE [BF99]. In a clever use of the rerelativization technique, Rogers =-=[Rog97] den-=-ed a generic oracle where both P 6= UP and the Isomorphism Conjecture holds. This oracle is a UP-generic oracle built \on top of" an SP-generic. Perhaps other specialized forms of generics may ha... |

8 |
Forcing and reducibilities, The
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- 1983
(Show Context)
Citation Context ...hers). Spector [Spe56] implicitly used forcing with computable trees to construct a minimal Turing degree. Sacks [Sac71] considered forcing with arithmetical and pointed perfect sets. 3 See Odifreddi =-=[Odi83-=-] for a detailed technical development of Feferman's and Sacks's ideas in a context dierent from that presented in the present paper. Two notions of genericity|similar to each other|were used by Slama... |

6 | The independence of the continuum hypothesis II - Cohen - 1964 |

4 |
The isomorphism conjecture fails relative to a generic oracle
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(Show Context)
Citation Context ...nnihilating functions do not. To prove Theorem 4.10, we use the next lemma, whose proof is a modication of an earlier proof showing that the Isomorphism Conjecture fails relative to a generic oracle [=-=Kur88]-=-. Lemma 4.11 If G is generic, then there are no annihilating functions relative to G. Proof: Fix a deterministic ptime oracle transducer T , and for any oracle A, let f A 2 FP A be the function comput... |

4 | personal communication - Böttcher - 2007 |

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