1,040 research outputs found
Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read
A Boolean function of n bits is balanced if it takes the value 1 with
probability 1/2. We exhibit a balanced Boolean function with a randomized
evaluation procedure (with probability 0 of making a mistake) so that on
uniformly random inputs, no input bit is read with probability more than
Theta(n^{-1/2} sqrt{log n}). We give a balanced monotone Boolean function for
which the corresponding probability is Theta(n^{-1/3} log n). We then show that
for any randomized algorithm for evaluating a balanced Boolean function, when
the input bits are uniformly random, there is some input bit that is read with
probability at least Theta(n^{-1/2}). For balanced monotone Boolean functions,
there is some input bit that is read with probability at least Theta(n^{-1/3}).Comment: 11 page
Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes
Some No-go Theorems for String Duals of Non-relativistic Lifshitz-like Theories
We study possibilities of string theory embeddings of the gravity duals for
non-relativistic Lifshitz-like theories with anisotropic scale invariance. We
search classical solutions in type IIA and eleven-dimensional supergravities
which are expected to be dual to (2+1)-dimensional Lifshitz-like theories.
Under reasonable ansaetze, we prove that such gravity duals in the
supergravities are not possible. We also discuss a possible physical reason
behind this.Comment: 18 pages, Latex, flux conditions clarified (v2), brief summary of
results added (v3
Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation
We present an efficient proof system for Multipoint Arithmetic Circuit
Evaluation: for every arithmetic circuit of size and
degree over a field , and any inputs ,
the Prover sends the Verifier the values and a proof of length, and
the Verifier tosses coins and can check the proof in about time, with probability of error less than .
For small degree , this "Merlin-Arthur" proof system (a.k.a. MA-proof
system) runs in nearly-linear time, and has many applications. For example, we
obtain MA-proof systems that run in time (for various ) for the
Permanent, Circuit-SAT for all sublinear-depth circuits, counting
Hamiltonian cycles, and infeasibility of - linear programs. In general,
the value of any polynomial in Valiant's class can be certified
faster than "exhaustive summation" over all possible assignments. These results
strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed
by Russell Impagliazzo and others.
We also give a three-round (AMA) proof system for quantified Boolean formulas
running in time, nearly-linear time MA-proof systems for
counting orthogonal vectors in a collection and finding Closest Pairs in the
Hamming metric, and a MA-proof system running in -time for
counting -cliques in graphs.
We point to some potential future directions for refuting the
Nondeterministic Strong ETH.Comment: 17 page
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
Entanglement Wedge Cross Sections Require Tripartite Entanglement
We argue that holographic CFT states require a large amount of tripartite
entanglement, in contrast to the conjecture that their entanglement is mostly
bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp
conflict with two well-supported conjectures about the entanglement wedge cross
section surface . If is related to either the CFT's reflected
entropy or its entanglement of purification, then those quantities can differ
from the mutual information at . We prove that this
implies holographic CFT states must have amounts
of tripartite entanglement. This proof involves a new Fannes-type inequality
for the reflected entropy, which itself has many interesting applications.Comment: 20 pages, 5 figures, comments added in v
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