128 research outputs found
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
Violating Constant Degree Hypothesis Requires Breaking Symmetry
The Constant Degree Hypothesis was introduced by Barrington et. al. (1990) to
study some extensions of -groups by nilpotent groups and the power of these
groups in a certain computational model. In its simplest formulation, it
establishes exponential lower bounds for circuits computing AND of unbounded arity (for
constant integers and a prime ). While it has been proved in some
special cases (including ), it remains wide open in its general form for
over 30 years.
In this paper we prove that the hypothesis holds when we restrict our
attention to symmetric circuits with being a prime. While we build upon
techniques by Grolmusz and Tardos (2000), we have to prove a new symmetric
version of their Degree Decreasing Lemma and apply it in a highly non-trivial
way. Moreover, to establish the result we perform a careful analysis of
automorphism groups of subcircuits and
study the periodic behaviour of the computed functions.
Finally, our methods also yield lower bounds when is treated as a
function of
効率的な秘匿情報検索法の提案
学位の種別: 課程博士審査委員会委員 : (主査)東京大学准教授 國廣 昇, 東京大学教授 山本 博資, 東京大学教授 杉山 将, 東京大学客員教授 Phong Nguyen, 筑波大学教授 佐久間 淳University of Tokyo(東京大学
Ball arithmetic
33 pagesThe Mathemagix project aims at the development of a ''computer analysis'' system, in which numerical computations can be done in a mathematically sound manner. A major challenge for such systems is to conceive algorithms which are both efficient, reliable and available at any working precision. In this paper, we survey several older and newer such algorithms. We mainly concentrate on the automatic and efficient computation of high quality error bounds, based on a variant of interval arithmetic which we like to call ''ball arithmetic''
- …