370 research outputs found
Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes
We associate to each Boolean language complexity class C the algebraic class a·C consisting of families of polynomials {fn} for which the evaluation problem over Z is in C. We prove the following lower bound and randomness-to-hardness results: 1. If polynomial identity testing (PIT) is in NSUBEXP then a·NEXP does not have poly size constant-free arithmetic circuits. 2. a·NEXP RP does not have poly size constant-free arithmetic circuits. 3. For every fixed k, a·MA does not have arithmetic circuits of size nk. Items 1 and 2 strengthen two results due to Kabanets and Impagliazzo [7]. The third item improves a lower bound due to Santhanam [11]. We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case. Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT â i.o-NTIME[2no(1)]/no(1) if and only if there exists a family of multilinear polynomials in a·NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree
Tighter Connections between Derandomization and Circuit Lower Bounds
We tighten the connections between circuit lower bounds and derandomization for each of the following three types of derandomization:
- general derandomization of promiseBPP (connected to Boolean circuits),
- derandomization of Polynomial Identity Testing (PIT) over fixed finite fields (connected to arithmetic circuit lower bounds over the same field), and
- derandomization of PIT over the integers (connected to arithmetic circuit lower bounds over the integers).
We show how to make these connections uniform equivalences, although at the expense of using somewhat less common versions of complexity classes and for a less studied notion of inclusion.
Our main results are as follows:
1. We give the first proof that a non-trivial (nondeterministic subexponential-time) algorithm for PIT over a fixed finite field yields arithmetic circuit lower bounds.
2. We get a similar result for the case of PIT over the integers, strengthening a result of Jansen and Santhanam [JS12] (by removing the need for advice).
3. We derive a Boolean circuit lower bound for NEXP intersect coNEXP from the assumption of sufficiently strong non-deterministic derandomization of promiseBPP (without advice), as well as from the assumed existence of an NP-computable non-empty property of Boolean functions useful for proving superpolynomial circuit lower bounds (in the sense of natural proofs of [RR97]); this strengthens the related results of [IKW02].
4. Finally, we turn all of these implications into equivalences for appropriately defined promise classes and for a notion of robust inclusion/separation (inspired by [FS11]) that lies between the classical "almost everywhere" and "infinitely often" notions
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Proof Complexity and Beyond
Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form âhow difficult is it to prove certain mathematical facts?â The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples
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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 randomness extraction. 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
Applications of Derandomization Theory in Coding
Randomized techniques play a fundamental role in theoretical computer science
and discrete mathematics, in particular for the design of efficient algorithms
and construction of combinatorial objects. The basic goal in derandomization
theory is to eliminate or reduce the need for randomness in such randomized
constructions. In this thesis, we explore some applications of the fundamental
notions in derandomization theory to problems outside the core of theoretical
computer science, and in particular, certain problems related to coding theory.
First, we consider the wiretap channel problem which involves a communication
system in which an intruder can eavesdrop a limited portion of the
transmissions, and construct efficient and information-theoretically optimal
communication protocols for this model. Then we consider the combinatorial
group testing problem. In this classical problem, one aims to determine a set
of defective items within a large population by asking a number of queries,
where each query reveals whether a defective item is present within a specified
group of items. We use randomness condensers to explicitly construct optimal,
or nearly optimal, group testing schemes for a setting where the query outcomes
can be highly unreliable, as well as the threshold model where a query returns
positive if the number of defectives pass a certain threshold. Finally, we
design ensembles of error-correcting codes that achieve the
information-theoretic capacity of a large class of communication channels, and
then use the obtained ensembles for construction of explicit capacity achieving
codes.
[This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
Distributional PAC-Learning from Nisan's Natural Proofs
Carmosino et al. (2016) demonstrated that natural proofs of circuit lower
bounds for imply efficient algorithms for learning
-circuits, but only over \textit{the uniform distribution}, with
\textit{membership queries}, and provided \AC^0[p] \subseteq \Lambda. We
consider whether this implication can be generalized to \Lambda \not\supseteq
\AC^0[p], and to learning algorithms which use only random examples and learn
over arbitrary example distributions (Valiant's PAC-learning model).
We first observe that, if, for any circuit class , there is an
implication from natural proofs for to PAC-learning for ,
then standard assumptions from lattice-based cryptography do not hold. In
particular, we observe that depth-2 majority circuits are a (conditional)
counter example to the implication, since Nisan (1993) gave a natural proof,
but Klivans and Sherstov (2009) showed hardness of PAC-learning under
lattice-based assumptions. We thus ask: what learning algorithms can we
reasonably expect to follow from Nisan's natural proofs?
Our main result is that all natural proofs arising from a type of
communication complexity argument, including Nisan's, imply PAC-learning
algorithms in a new \textit{distributional} variant (i.e., an ``average-case''
relaxation) of Valiant's PAC model. Our distributional PAC model is stronger
than the average-case prediction model of Blum et al. (1993) and the heuristic
PAC model of Nanashima (2021), and has several important properties which make
it of independent interest, such as being \textit{boosting-friendly}. The main
applications of our result are new distributional PAC-learning algorithms for
depth-2 majority circuits, polytopes and DNFs over natural target
distributions, as well as the nonexistence of encoded-input weak PRFs that can
be evaluated by depth-2 majority circuits.Comment: Added discussio
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