12,601 research outputs found
Rational weak mixing in infinite measure spaces
Rational weak mixing is a measure theoretic version of Krickeberg's strong
ratio mixing property for infinite measure preserving transformations. It
requires "{\tt density}" ratio convergence for every pair of measurable sets in
a dense hereditary ring. Rational weak mixing implies weak rational ergodicity
and (spectral) weak mixing. It is enjoyed for example by Markov shifts with
Orey's strong ratio limit property. The power, subsequence version of the
property is generic.Comment: Typos in the definitions of "rational weak mixing" and "weak rational
ergodicity" (p.5) are correcte
NP-complete Problems and Physical Reality
Can NP-complete problems be solved efficiently in the physical universe? I
survey proposals including soap bubbles, protein folding, quantum computing,
quantum advice, quantum adiabatic algorithms, quantum-mechanical
nonlinearities, hidden variables, relativistic time dilation, analog computing,
Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and
"anthropic computing." The section on soap bubbles even includes some
"experimental" results. While I do not believe that any of the proposals will
let us solve NP-complete problems efficiently, I argue that by studying them,
we can learn something not only about computation but also about physics.Comment: 23 pages, minor correction
Quantum Certificate Complexity
Given a Boolean function f, we study two natural generalizations of the
certificate complexity C(f): the randomized certificate complexity RC(f) and
the quantum certificate complexity QC(f). Using Ambainis' adversary method, we
exactly characterize QC(f) as the square root of RC(f). We then use this result
to prove the new relation R0(f) = O(Q2(f)^2 Q0(f) log n) for total f, where R0,
Q2, and Q0 are zero-error randomized, bounded-error quantum, and zero-error
quantum query complexities respectively. Finally we give asymptotic gaps
between the measures, including a total f for which C(f) is superquadratic in
QC(f), and a symmetric partial f for which QC(f) = O(1) yet Q2(f) = Omega(n/log
n).Comment: 9 page
Is Quantum Mechanics An Island In Theoryspace?
This recreational paper investigates what happens if we change quantum
mechanics in several ways. The main results are as follows. First, if we
replace the 2-norm by some other p-norm, then there are no nontrivial
norm-preserving linear maps. Second, if we relax the demand that norm be
preserved, we end up with a theory that allows rapid solution of PP-complete
problems (as well as superluminal signalling). And third, if we restrict
amplitudes to be real, we run into a difficulty much simpler than the usual one
based on parameter-counting of mixed states.Comment: 9 pages, minor correction
Relative complexity of random walks in random sceneries
Relative complexity measures the complexity of a probability preserving
transformation relative to a factor being a sequence of random variables whose
exponential growth rate is the relative entropy of the extension. We prove
distributional limit theorems for the relative complexity of certain zero
entropy extensions: RWRSs whose associated random walks satisfy the
\alpha-stable CLT (). The results give invariants for relative
isomorphism of these.Comment: Published in at http://dx.doi.org/10.1214/11-AOP688 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Oracles Are Subtle But Not Malicious
Theoretical computer scientists have been debating the role of oracles since
the 1970's. This paper illustrates both that oracles can give us nontrivial
insights about the barrier problems in circuit complexity, and that they need
not prevent us from trying to solve those problems.
First, we give an oracle relative to which PP has linear-sized circuits, by
proving a new lower bound for perceptrons and low- degree threshold
polynomials. This oracle settles a longstanding open question, and generalizes
earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More
importantly, it implies the first nonrelativizing separation of "traditional"
complexity classes, as opposed to interactive proof classes such as MIP and
MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does
not have circuits of size n^k for any fixed k. We present an alternative proof
of this fact, which shows that PP does not even have quantum circuits of size
n^k with quantum advice. To our knowledge, this is the first nontrivial lower
bound on quantum circuit size.
Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean
circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be
parallelized by any relativizing technique, by giving an oracle relative to
which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand,
we also show that the NP queries could be parallelized if P=NP. Thus, classes
such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish
between relativizing and black-box techniques. Our results on this subject have
implications for computational learning theory as well as for the circuit
minimization problem.Comment: 20 pages, 1 figur
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