212 research outputs found
Hardness Amplification for Non-Commutative Arithmetic Circuits
We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire.
This is part of a recent line of inquiry into why arithmetic circuit complexity, despite being a heavily restricted version of Boolean complexity, still cannot prove super-linear lower bounds on general devices. One can view our work as positive news (it suffices to prove weak lower bounds to get strong ones) or negative news (it is as hard to prove weak lower bounds as it is to prove strong ones). We leave it to the reader to determine their own level of optimism
Arithmetic Circuits and the Hadamard Product of Polynomials
Motivated by the Hadamard product of matrices we define the Hadamard product
of multivariate polynomials and study its arithmetic circuit and branching
program complexity. We also give applications and connections to polynomial
identity testing. Our main results are the following. 1. We show that
noncommutative polynomial identity testing for algebraic branching programs
over rationals is complete for the logspace counting class \ceql, and over
fields of characteristic the problem is in \ModpL/\Poly. 2.We show an
exponential lower bound for expressing the Raz-Yehudayoff polynomial as the
Hadamard product of two monotone multilinear polynomials. In contrast the
Permanent can be expressed as the Hadamard product of two monotone multilinear
formulas of quadratic size.Comment: 20 page
Near-optimal Bootstrapping of Hitting Sets for Algebraic Models
The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel
[Ore22,DL78,Zip79,Sch80] states that any nonzero polynomial of degree at most will evaluate to a nonzero value at some point on a
grid with . Thus, there is an explicit
hitting set for all -variate degree , size algebraic circuits of size
.
In this paper, we prove the following results:
- Let be a constant. For a sufficiently large constant and
all , if we have an explicit hitting set of size
for the class of -variate degree polynomials that are computable by
algebraic circuits of size , then for all , we have an explicit hitting
set of size for -variate circuits of
degree and size . That is, if we can obtain a barely non-trivial
exponent compared to the trivial sized hitting set even for
constant variate circuits, we can get an almost complete derandomization of
PIT.
- The above result holds when "circuits" are replaced by "formulas" or
"algebraic branching programs".
This extends a recent surprising result of Agrawal, Ghosh and Saxena [AGS18]
who proved the same conclusion for the class of algebraic circuits, if the
hypothesis provided a hitting set of size at most
(where is any constant). Hence, our work significantly weakens the
hypothesis of Agrawal, Ghosh and Saxena to only require a slightly non-trivial
saving over the trivial hitting set, and also presents the first such result
for algebraic branching programs and formulas.Comment: The main result has been strengthened significantly, compared to the
older version of the paper. Additionally, the stronger theorem now holds even
for subclasses of algebraic circuits, such as algebraic formulas and
algebraic branching program
New Lower Bounds Against Homogeneous Non-Commutative Circuits
We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree d which requires homogeneous non-commutative circuit of size ?(d/log d). For an n-variate polynomial with n > 1, the result can be improved to ?(nd), if d ? n, or ?(nd (log n)/(log d)), if d ? n. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial
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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 developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes
Algebraic and Combinatorial Methods in Computational Complexity
At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The PCP characterization of NP and the Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples. Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques. This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings
Recommended from our members
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
Hardness magnification for natural problems
We show that for several natural problems of interest, complexity lower bounds that are barely non-trivial imply super-polynomial or even exponential lower bounds in strong computational models. We term this phenomenon "hardness magnification". Our examples of hardness magnification include: 1. Let MCSP be the decision problem whose YES instances are truth tables of functions with circuit complexity at most s(n). We show that if MCSP[2^√n] cannot be solved on average with zero error by formulas of linear (or even sub-linear) size, then NP does not have polynomial-size formulas. In contrast, Hirahara and Santhanam (2017) recently showed that MCSP[2^√n] cannot be solved in the worst case by formulas of nearly quadratic size. 2. If there is a c > 0 such that for each positive integer d there is an ε > 0 such that the problem of checking if an n-vertex graph in the adjacency matrix representation has a vertex cover of size (log n)^c cannot be solved by depth-d AC^0 circuits of size m^1+ε, where m = Θ(n^2), then NP does not have polynomial-size formulas. 3. Let (α, β)-MCSP[s] be the promise problem whose YES instances are truth tables of functions that are α-approximable by a circuit of size s(n), and whose NO instances are truth tables of functions that are not β-approximable by a circuit of size s(n). We show that for arbitrary 1/2 ≺ β ≺ α ≤ 1, if (α, β)-MCSP[2^√n] cannot be solved by randomized algorithms with random access to the input running in sublinear time, then NP is not contained in BPP. 4. If for each probabilistic quasi-linear time machine M using poly-logarithmic many random bits that is claimed to solve Satisfiability, there is a deterministic polynomial-time machine that on infinitely many input lengths n either identifies a satisfiable instance of bit-length n on which M does not accept with high probability or an unsatisfiable instance of bit-length n on which M does not reject with high probability, then NEXP is not contained in BPP. 5. Given functions s, c N → N where s ≻ c, let MKtP[c, s] be the promise problem whose YES instances are strings of Kt complexity at most c(N) and NO instances are strings of Kt complexity greater than s(N). We show that if there is a δ ≻ 0 such that for each ε ≻ 0, MKtP[N^ε, N^ε + 5 log(N)] requires Boolean circuits of size N^1+δ, then EXP is not contained in SIZE (poly). For each of the cases of magnification above, we observe that standard hardness assumptions imply much stronger lower bounds for these problems than we require for magnification. We further explore magnification as an avenue to proving strong lower bounds, and argue that magnification circumvents the "natural proofs" barrier of Razborov and Rudich (1997). Examining some standard proof techniques, we find that they fall just short of proving lower bounds via magnification. As one of our main open problems, we ask whether there are other meta-mathematical barriers to proving lower bounds that rule out approache
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