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On the Cutting Edge of Relativization: The Resource - Bounded Injury Method (Correct)
....of oracles has dramatically lost popularity in the complexity theory community. Relations between complexity classes relativizing both ways were formerly considered a demonstration of the difficulty of proving a relation between these complexity classes in the real world. Random relativizations [BG81] relativize only one way, but unfortunately the random oracle hypothesis was also provided with counterexamples soon after it s statement. A relatively new insight [AIV93] connecting local computation to oracle results, shows that there may be hope yet for oracles satisfying certain conditions. ....
C. Bennett and J. Gill. Relative to a random oracle A, P
Almost-Everywhere Superiority - For Quantum Polynomial-Time (Correct)
....So, as all extensions have the same measure, G # i (G # i ) Then (G # ) This ends the proof of Claim 2.4. q 2. 3 Lower Bound For Error bounded Probabilistic Machines To move to bounded error probability machines, we invoke the techniques that Bennett and Gill [BG81] used to prove P = BPP relative to a random oracle. An adaptation of their method demonstrates Claim 2.5. For completeness and since there are some di#erences between our context and the one in the paper of Bennett and Gill [BG81] we will prove Claim 2.5 in detail. In the proof, we will ....
....machines, we invoke the techniques that Bennett and Gill [BG81] used to prove P = BPP relative to a random oracle. An adaptation of their method demonstrates Claim 2.5. For completeness and since there are some di#erences between our context and the one in the paper of Bennett and Gill [BG81], we will prove Claim 2.5 in detail. In the proof, we will assume that all the oracles A are in A. Proof of Claim 2.5. In the construction, we will first decrease the error probability of a probabilistic machine. The standard way to decrease the error probability of a probabilistic machine is ....
C. Bennett and J. Gill. Relative to a random oracle A, P
Almost-Everywhere Superiority for Quantum.. - Hemaspaandra.. (2001) (2 citations) (Correct)
....same measure, G B 4 ) Then (G B 4 ) B 4 ) 4 (G x ) 4 (G ) This ends the proof of Claim 2.4. 2. 3 Lower Bound For Error bounded Probabilistic Machines To move to bounded error probability machines, we invoke the techniques that Bennett and Gill [BG81] used to prove P = BPP relative to a random oracle. An adaptation of their method demonstrates Claim 2.5. For completeness and since there are some di erences between our context and the one in the paper of Bennett and Gill [BG81] we will prove Claim 2.5 in detail. In the proof, we will ....
....machines, we invoke the techniques that Bennett and Gill [BG81] used to prove P = BPP relative to a random oracle. An adaptation of their method demonstrates Claim 2.5. For completeness and since there are some di erences between our context and the one in the paper of Bennett and Gill [BG81], we will prove Claim 2.5 in detail. In the proof, we will assume that all the oracles A are in A. Proof of Claim 2.5. In the construction, we will rst decrease the error probability of a probabilistic machine. The standard way to decrease the error probability of a probabilistic machine is the ....
C. Bennett and J. Gill. Relative to a random oracle A, P
Small Spans in Scaled Dimension - Hitchcock (2004) (Correct)
....is a suitable complexity class. If (C = 0, then is a negligible subset of D. Theorem 1.1. Juedes and Lutz [10] Let EXP . For every A = 0. In particular, deg = 0. That is, at least one of the upper or lower spans of A is small within D. Using a result of Bennett and Gill [4], Juedes and Lutz [10] noted that strengthening Theorem 1.1 from m reductions to reductions would achieve the separation BPP EXP. However, small span theorems for reductions of progressively increasing strength between have been obtained by Linder [11] Ambos Spies, Neis, and Terwijn ....
C. H. Bennett and J. Gill. Relative to a random oracle A, P
NP-hard Sets are P-Superterse Unless R = NP - Beigel (1988) (25 citations) (Correct)
....R = NP [56] iv. If R = NP then NP P=poly [1] so the polynomial time hierarchy collapses [61] Corollary 10. All NP hard sets (under tt reductions) are p superterse relative to almost all oracles B. Proof: The preceding results relativize. R relative to almost all oracles [20]. It is also known that P 6= UP for almost all oracles B [12, 21] 3.1. Extensions to Incomplete Sets In this section we consider two kinds of sets believed not to be NP complete. The techniques in the preceding section generalize to show that these sets are p superterse unless certain ....
....is tt reducible to a non p superterse set, then in fact the problem belongs to P. Definition 13. Bounded probability Reduction] A tt B if there is a probabilistic polynomial time oracle Turing machine M computing a 1=4 otherwise: By amplification techniques (using polling) [20], we may assume for any polynomial p that M accepts with probability 1 Gamma 2 if x 2 A, and with probability 2 if x = 2 A. Theorem 14. All NP hard sets under ( tt reductions) are p superterse unless R = NP. Proof: Assume that A is not p superterse; then every function in PF n tt is ....
C. H. Bennett and J. Gill. Relative to a random oracle A, P
The Relativized Relationship between Probabilistically.. - Russell, Sundaram (1993) (Correct)
....holds. Thus, the presentation of contradictory relativizations of a relationship between two complexity classes has been a standard tool for arguing the difficulty of precisely determining that relationship. The notion of relativization was strengthened by the consideration of random oracles [5]. In the words of Bennett and Gill: E mail address: acr theory.lcs.mit.edu. Supported by an NSF Graduate Fellowship and grants NSF 92 12184, AFOSR F49620 92 J 0125, and DARPA N00014 92 1799 E mail address: koods theory.lcs.mit.edu. Supported by grants NSF 92 12184, AFOSR F49620 92 J 0125, ....
....benign and less likely to distort the relations among complexity classes than the other oracles used in complexity theory and recursive function theory, which are usually designed expressly to help or frustrate some class of computations. This led them to formulate the random oracle hypothesis [5]: the relationship between two natural complexity classes is preserved with probability 1 under relativization by a random oracle. In this new framework, a conjectured relationship may be supported by showing that it holds with probability 1 relative to a random oracle. Clearly, this framework ....
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C. Bennett and J. Gill. Relative to a random oracle A, P
The Polynomial Method in Circuit Complexity - Beigel (1995) (26 citations) (Correct)
....d = o( n) and ffl 0, there is no degree d polynomial p such that sign(p(x 1 ; x n ) parity(x 1 ; x n ) on a ffl fraction of all inputs. This implies [4] that relative to a random oracle PARITYP 6 PP and PSPACE 6 PP, theorems that were conjectured by Bennett and Gill [22]. Barrington and Straubing [10] proved a nontrivial extension of Theorem 6. Let MOD i;k (x 1 ; x n ) 0 if the number of x i s that are true is congruent to i modulo q; MOD i;k (x 1 ; x n ) 1, otherwise. Define the MOD k function from Z k into ftrue; falseg by MOD k (x 1 ; ....
C. H. Bennett and J. Gill. Relative to a random oracle A, P
When are k+1 Queries Better Than k? - Beigel (1995) (Correct)
....In the next section, we will show that most (in the sense of Lebesgue measure) sets are supportive and parallel supportive. 29 4.8. Almost All Sets A measure can be defined on the set of all sets of strings by treating a set of strings as a real number and using Lebesgue measure on reals [BG81]. In this section we show that almost all (i.e. a measure 1 set of) sets of strings are supportive and parallel supportive. Lemma 61 For almost all A, PARITY n 1 = 2 Q(n; A) Proof: Let M be an oracle Turing machine that computes with at most n queries to its oracle A. We show that M correctly ....
Charles H. Bennett and John Gill. Relative to a random oracle A,
Relativized Counting Classes: Relations among Thresholds, Parity, .. - Beigel (1991) (11 citations) (Correct)
....We describe an initial segment construction that will form the skeleton of each construction in this paper. Let C and D be complexity classes defined in terms of C machines and D machines. Suppose we want to construct an oracle A such that C say that L 0 is an oracle property (c.f. [5]) if there is a predicate Q such that ( Q(A Our first step is to define an oracle property such that for all A, L construct A = lim i A i in stages so that L = 2 D . Since L Gamma D , we obtain C . Stage 0: Let A 0 = and n = 0. sufficiently large for the ....
....exists an oracle A such that PhiP 6 has a long history. Although they never discussed relativizations, an oracle construction can be obtained as a corollary to Minsky and Papert s Theorem 3.1. 1 in [17] The separation was claimed to hold relative to almost all oracles by Bennett and Gill [5]; however their proof is incorrect. A correct oracle construction is given by Tor an in [25] We present a construction that exploits a simple combinatorial symmetry: Every nonempty set S has as many subsets with odd cardinality as with even cardinality. Why Fix any string x 2 S. A subset of T ....
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C. H. Bennett and J. Gill. Relative to a random oracle A, P
Logical Depth and Physical Complexity - Charles Bennett Ibm (1988) (2 citations) Self-citation (Bennett) (Correct)
....their chance of being produced slowly. The question of whether such strings exist is probably hard to answer because it does not relativize uniformly. Deterministic and probabilistic depths are not very di#erent relative to a random coin toss oracle A (this can be shown by a proof similar to that [4] of the equality of random oracle relativized deterministic and probabilistic polynomial time complexity classes) but they can be very di#erent relative to an oracle B deliberately designed to hide information from deterministic computations (this parallels Hunt s proof [23] that deterministic ....
....but unproven assumptions at the low end of complexity theory. Motivated by the finding that many open questions of complexity theory can be easily shown to have the answers one would like them to have in the relativized context of a random oracle (e.g. P PSPACE ) Bennett and Gill [4] informally conjectured that pseudorandom functions that behave like random oracles exist absolutely, and therefore that all natural mathematical statements (such as P NP ) true relative to a random oracle should be true absolutely. Their attempt to formalize the latter idea (by defining a ....
C.H. Bennett and J. Gill, "Relative to a Random Oracle A, P
On the Structure of Valiant's Complexity Classes - Bürgisser (1999) Self-citation (Gill) (Correct)
....from Cor. 7.6 on the preceding page(a) that (D n (h n ) is VNP complete. As (D n (h n ) is also contained in VP , it follows that VP Up to now we have not succeeded in establishing a p family h such that VP . A promising approach for this is as follows (compare Bennett and Gill [4]) For each n choose independently h n 2 k[X 1 ; X n ] of degree most n at random according to some probability distribution. Since the classes are invariant under finite variation of h, the event E = fh j VP g is a so called tail event. Kolmogorov s zero one law (cf. Feller [14, ....
C.H. Bennett and J. Gill, Relative to a random oracle A, P
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Bennett C., Gill J. Relative to a Random Oracle A, P
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