34 research outputs found
Circuit Depth Relative to a Random Oracle
The study of separation of complexity classes with respect to random oracles was initiated by Bennett and Gill and continued by many other authors. Wilson defined relativized circuit depth and constructed various oracles A for which P^A ¬ NC^A NC^A_k ¬ NC^A_k+varepsilon, AC^A_k ¬ AC^A_k+varepsilon, AC^A_k ¬ subset= AC^A_k+1-varepsilon, and NC^A_k not subset= AC^A_ k-varepsilon,for all positive rational k and varepsilon, thus separating those classes for which no trivial argument shows inclusion. In this note we show that as a consequence of a single lemma, these separations (or improvements of them) hold with respect to a random oracle A
Quantum Computation Relative to Oracles
The study of the power and limitations of quantum computation remains a major challenge in complexity theory. Key questions revolve around the quantum complexity classes EQP, BQP, NQP, and their derivatives. This paper presents new relativized worlds in which (i) co-RP is not a subset of NQE, (ii) P=BQP and UP=EXP, (iii) P=EQP and RP=EXP, and (iv) EQP is not a subset of the union of Sigma{p}{2} and Pi{p}{2}. We also show a partial answer to the question of whether Almost-BQP=BQP
Strengths and Weaknesses of Quantum Computing
Recently a great deal of attention has focused on quantum computation
following a sequence of results suggesting that quantum computers are more
powerful than classical probabilistic computers. Following Shor's result that
factoring and the extraction of discrete logarithms are both solvable in
quantum polynomial time, it is natural to ask whether all of NP can be
efficiently solved in quantum polynomial time. In this paper, we address this
question by proving that relative to an oracle chosen uniformly at random, with
probability 1, the class NP cannot be solved on a quantum Turing machine in
time . We also show that relative to a permutation oracle chosen
uniformly at random, with probability 1, the class cannot be
solved on a quantum Turing machine in time . The former bound is
tight since recent work of Grover shows how to accept the class NP relative to
any oracle on a quantum computer in time .Comment: 18 pages, latex, no figures, to appear in SIAM Journal on Computing
(special issue on quantum computing
Thoughts on the Riemann hypothesis
The simultaneous appearance in May 2003 of four books on the Riemann
hypothesis (RH) provoked these reflections. We briefly discuss whether the RH
should be added as a new axiom, or whether a proof of the RH might involve the
notion of randomness
Resource Bounded Immunity and Simplicity
Revisiting the thirty years-old notions of resource-bounded immunity and
simplicity, we investigate the structural characteristics of various immunity
notions: strong immunity, almost immunity, and hyperimmunity as well as their
corresponding simplicity notions. We also study limited immunity and
simplicity, called k-immunity and feasible k-immunity, and their simplicity
notions. Finally, we propose the k-immune hypothesis as a working hypothesis
that guarantees the existence of simple sets in NP.Comment: This is a complete version of the conference paper that appeared in
the Proceedings of the 3rd IFIP International Conference on Theoretical
Computer Science, Kluwer Academic Publishers, pp.81-95, Toulouse, France,
August 23-26, 200
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The universality of polynomial time Turing equivalence
We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees
Generic oracles, uniform machines, and codes
AbstractThe basic properties of generic oracles are reviewed, and proofs given that they separate P and NP and are weakly incompressible. A new notion of generic oracle, called t-generic, is defined. It is shown that t-generic oracles do not exist, and consequently a nondeterministic oracle machine which for any oracle X accepts the tautologies relativized to X when running with oracle X does not run in polynomial time at any oracle. A weak form of t-generic oracle, called r-generic, is shown to exist, and it is shown that if there exists an r-generic oracle X at which the r-query relativized tautologies are not in co NPX then NP ≠ co NP. The notion of a code for the Boolean functions is defined, and it is shown that generic oracles do not have short codes in any code. Universal circuits of size O(n log4 n) are shown to exist, and it is shown that increasing the number of ⋏, ⋎ gates from g to 2g + 1 allows the computation of new Boolean functions