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REPORTS > KEYWORD > CIRCUIT COMPLEXITY:
Reports tagged with circuit complexity:
TR94-012 | 12th December 1994

Bounds for the Computational Power and Learning Complexity of Analog Neural Nets

It is shown that high order feedforward neural nets of constant depth with piecewise
polynomial activation functions and arbitrary real weights can be simulated for boolean
inputs and outputs by neural nets of a somewhat larger size and depth with heaviside
gates and weights ... more >>>


TR95-027 | 30th May 1995
Frederic Green

Lower Bounds for Circuits with Mod Gates and One Exact Threshold Gate

We say an integer polynomial $p$, on Boolean inputs, weakly
$m$-represents a Boolean function $f$ if $p$ is non-constant and is zero (mod
$m$), whenever $f$ is zero. In this paper we prove that if a polynomial
weakly $m$-represents the Mod$_q$ function on $n$ inputs, where $q$ and $m$
more >>>


TR95-051 | 16th October 1995
Pascal Koiran

VC Dimension in Circuit Complexity

The main result of this paper is a Omega(n^{1/4}) lower bound
on the size of a sigmoidal circuit computing a specific AC^0_2 function.
This is the first lower bound for the computation model of sigmoidal
circuits with unbounded weights. We also give upper and lower bounds for
the ... more >>>


TR96-002 | 10th January 1996
Manindra Agrawal, Eric Allender

An Isomorphism Theorem for Circuit Complexity

We show that all sets complete for NC$^1$ under AC$^0$
reductions are isomorphic under AC$^0$-computable isomorphisms.

Although our proof does not generalize directly to other
complexity classes, we do show that, for all complexity classes C
closed under NC$^1$-computable many-one reductions, the sets ... more >>>


TR96-004 | 18th January 1996
Shiva Chaudhuri, Jaikumar Radhakrishnan

Deterministic Restrictions in Circuit Complexity

We study the complexity of computing Boolean functions using AND, OR
and NOT gates. We show that a circuit of depth $d$ with $S$ gates can
be made to output a constant by setting $O(S^{1-\epsilon(d)})$ (where
$\epsilon(d) = 4^{-d}$) of its input values. This implies a
superlinear size lower bound ... more >>>


TR96-023 | 21st March 1996
Eric Allender

A Note on Uniform Circuit Lower Bounds for the Counting Hierarchy

Comments: 1

A very recent paper by Caussinus, McKenzie, Therien, and Vollmer
[CMTV] shows that ACC^0 is properly contained in ModPH, and TC^0
is properly contained in the counting hierarchy. Thus, [CMTV] shows
that there are problems in ModPH that require superpolynomial-size
uniform ACC^0 ... more >>>


TR96-024 | 21st March 1996
Eric Allender, Robert Beals, Mitsunori Ogihara

The complexity of matrix rank and feasible systems of linear equations

We characterize the complexity of some natural and important
problems in linear algebra. In particular, we identify natural
complexity classes for which the problems of (a) determining if a
system of linear equations is feasible and (b) computing the rank of
an integer matrix, ... more >>>


TR96-030 | 31st March 1996
Meera Sitharam

Approximation from linear spaces and applications to complexity


We develop an analytic framework based on
linear approximation and point out how a number of complexity
related questions --
on circuit and communication
complexity lower bounds, as well as
pseudorandomness, learnability, and general combinatorics
of Boolean functions --
fit neatly into this framework. ... more >>>


TR96-041 | 24th July 1996
Oded Goldreich, Avi Wigderson

On the Circuit Complexity of Perfect Hashing

Revisions: 1 , Comments: 2

We consider the size of circuits which perfectly hash
an arbitrary subset $S\!\subset\!\bitset^n$ of cardinality $2^k$
into $\bitset^m$.
We observe that, in general, the size of such circuits is
exponential in $2k-m$,
and provide a matching upper bound.

more >>>

TR96-049 | 9th September 1996
Per Enflo, Meera Sitharam

Stable basis families and complexity lower bounds

--
Scalar product estimates have so far been used in
proving several unweighted threshold lower bounds.
We show that if a basis set of Boolean functions satisfies
certain weak stability conditions, then
scalar product estimates
yield lower bounds for the size of weighted thresholds
of these basis functions.
Stable ... more >>>


TR96-055 | 22nd October 1996
Alexander E. Andreev, Andrea E. F. Clementi, Jose' D. P. Rolim

Hitting Properties of Hard Boolean Operators and their Consequences on BPP

Revisions: 1 , Comments: 1

We present the first worst-case hardness conditions
on the circuit complexity of EXP functions which are
sufficient to obtain P=BPP. In particular, we show that
from such hardness conditions it is possible to construct
quick Hitting Sets Generators with logarithmic prize.
... more >>>


TR98-056 | 31st August 1998
Anna Bernasconi, Igor E. Shparlinski

Circuit Complexity of Testing Square-Free Numbers

In this paper we extend the area of applications of
the Abstract Harmonic Analysis to the field of Boolean function complexity.
In particular, we extend the class of functions to which
a spectral technique developed in a series of works of the first author
can be applied.
This extension ... more >>>


TR98-067 | 12th November 1998
Paul Beame

Propositional Proof Complexity: Past, Present and Future

Proof complexity, the study of the lengths of proofs in
propositional logic, is an area of study that is fundamentally connected
both to major open questions of computational complexity theory and
to practical properties of automated theorem provers. In the last
decade, there have been a number of significant advances ... more >>>


TR98-070 | 7th December 1998
Rüdiger Reischuk

Can Large Fanin Circuits Perform Reliable Computations in the Presence of Noise?


For ordinary circuits with a fixed upper bound on the maximal fanin
of gates it has been shown that logarithmic redundancy is necessary and
sufficient to overcome random hardware faults.
Here, we consider the same question for unbounded fanin circuits that
in the noiseless case can compute ... more >>>


TR01-009 | 5th January 2001
Ronen Shaltiel

Towards proving strong direct product theorems

A fundamental question of complexity theory is the direct product
question. Namely weather the assumption that a function $f$ is hard on
average for some computational class (meaning that every algorithm from
the class has small advantage over random guessing when computing $f$)
entails that computing $f$ on ... more >>>


TR01-067 | 18th September 2001
Hubie Chen

Polynomial Programs and the Razborov-Smolensky Method

Representations of boolean functions as polynomials (over rings) have
been used to establish lower bounds in complexity theory. Such
representations were used to great effect by Smolensky, who
established that MOD q \notin AC^0[MOD p] (for distinct primes p, q)
by representing AC^0[MOD p] functions as low-degree multilinear
polynomials over ... more >>>


TR01-070 | 24th October 2001
Robert Albin Legenstein

Total Wire Length as a Salient Circuit Complexity Measure for Sensory Processing

We introduce em total wire length as salient complexity measure
for analyzing the circuit complexity of sensory processing in
biological neural systems and neuromorphic engineering. The new
complexity measure is applied in this paper to two basic
computational problems that arise in translation- and
scale-invariant pattern recognition, and hence appear ... more >>>


TR01-071 | 24th October 2001
Robert Albin Legenstein

Neural Circuits for Pattern Recognition with Small Total Wire Length

One of the most basic pattern recognition problems is whether a
certain local feature occurs in some linear array to the left of
some other local feature. We construct in this article circuits that
solve this problem with an asymptotically optimal number of
threshold gates. Furthermore it is shown that ... more >>>


TR01-095 | 12th November 2001
Hubie Chen

Arithmetic Versions of Constant Depth Circuit Complexity Classes

The boolean circuit complexity classes
AC^0 \subseteq AC^0[m] \subseteq TC^0 \subseteq NC^1 have been studied
intensely. Other than NC^1, they are defined by constant-depth
circuits of polynomial size and unbounded fan-in over some set of
allowed gates. One reason for interest in these classes is that they
contain the ... more >>>


TR03-067 | 14th August 2003
Ran Raz

Multi-Linear Formulas for Permanent and Determinant are of Super-Polynomial Size

An arithmetic formula is multi-linear if the polynomial computed
by each of its sub-formulas is multi-linear. We prove that any
multi-linear arithmetic formula for the permanent or the
determinant of an $n \times n$ matrix is of size super-polynomial
in $n$.

more >>>

TR04-042 | 21st May 2004
Ran Raz

Multilinear-$NC_1$ $\ne$ Multilinear-$NC_2$

An arithmetic circuit or formula is multilinear if the polynomial
computed at each of its wires is multilinear.
We give an explicit example for a polynomial $f(x_1,...,x_n)$,
with coefficients in $\{0,1\}$, such that over any field:
1) $f$ can be computed by a polynomial-size multilinear circuit
of depth $O(\log^2 ... more >>>


TR04-044 | 1st June 2004
Eric Allender, Harry Buhrman, Michal Koucky

What Can be Efficiently Reduced to the Kolmogorov-Random Strings?

We investigate the question of whether one can characterize complexity
classes (such as PSPACE or NEXP) in terms of efficient
reducibility to the set of Kolmogorov-random strings R_C.
We show that this question cannot be posed without explicitly dealing
with issues raised by the choice of universal
machine in the ... more >>>


TR04-047 | 22nd April 2004
Xiaoyang Gu

A note on dimensions of polynomial size circuits

In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every $i\geq 0$, $\Ppoly$ has $i$th order scaled $\pthree$-strong dimension $0$. We also show that $\Ppoly^\io$ has $\pthree$-dimension $1/2$, $\pthree$-strong dimension $1$. Our results improve previous measure results of Lutz (1992) and dimension ... more >>>


TR04-056 | 1st July 2004
N. V. Vinodchandran

A note on the circuit complexity of PP

In this short note we show that for any integer k, there are
languages in the complexity class PP that do not have Boolean
circuits of size $n^k$.

more >>>

TR04-108 | 24th November 2004
Eric Allender, Samir Datta, Sambuddha Roy

Topology inside NC^1

We show that ACC^0 is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC^0. Thus polylogarithmic genus provides no additional computational power in this model.
We consider other generalizations of ... more >>>


TR05-018 | 6th February 2005
Oded Goldreich

On Promise Problems (a survey in memory of Shimon Even [1935-2004])


The notion of promise problems was introduced and initially studied
by Even, Selman and Yacobi
(Information and Control, Vol.~61, pages 159-173, 1984).
In this article we survey some of the applications that this
notion has found in the twenty years that elapsed.
These include the notion ... more >>>


TR05-049 | 1st April 2005
Joan Boyar, rene peralta

The Exact Multiplicative Complexity of the Hamming Weight Function

We consider the problem of computing the Hamming weight of an n-bit vector using a circuit with gates for GF2 addition and multiplication only. We show the number of multiplications necessary and sufficient to build such a circuit is n - |n| where |n| is the Hamming weight of the ... more >>>


TR05-070 | 6th July 2005
Mahdi Cheraghchi

On Matrix Rigidity and the Complexity of Linear Forms

The rigidity function of a matrix is defined as the minimum number of its entries that need to be changed in order to reduce the rank of the matrix to below a given parameter. Proving a strong enough lower bound on the rigidity of a matrix implies a nontrivial lower ... more >>>


TR06-049 | 9th April 2006
Guy Wolfovitz

The Complexity of Depth-3 Circuits Computing Symmetric Boolean Functions

Comments: 1

We give tight lower bounds for the size of depth-3 circuits with limited bottom fanin computing symmetric Boolean functions. We show that any depth-3 circuit with bottom fanin $k$ which computes the Boolean function $EXACT_{n/(k+1)}^{n}$, has at least $(1+1/k)^{n+\O(\log n)}$ gates. We show that this lower bound is tight, by ... more >>>


TR06-107 | 26th August 2006
Arkadev Chattopadhyay

An improved bound on correlation between polynomials over Z_m and MOD_q

Revisions: 1

Let m,q > 1 be two integers that are co-prime and A be any subset of Z_m. Let P be any multi-linear polynomial of degree d in n variables over Z_m. We show that the MOD_q boolean function on n variables has correlation at most exp(-\Omega(n/(m2^{m-1})^d)) with the boolean function ... more >>>


TR06-138 | 13th November 2006
Kei Uchizawa, Rodney Douglas

Energy Complexity and Entropy of Threshold Circuits

Circuits composed of threshold gates (McCulloch-Pitts neurons, or
perceptrons) are simplified models of neural circuits with the
advantage that they are theoretically more tractable than their
biological counterparts. However, when such threshold circuits are
designed to perform a specific computational task they usually
differ ... more >>>


TR08-038 | 4th April 2008
Eric Allender, Michal Koucky

Amplifying Lower Bounds by Means of Self-Reducibility

Revisions: 2

We observe that many important computational problems in NC^1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC^0 circuits if and only if it has TC^0 circuits of size n^{1+\epsilon} for every \epsilon > 0 (counting the ... more >>>


TR09-011 | 31st January 2009
Mark Braverman

Poly-logarithmic independence fools AC0 circuits

We prove that poly-sized AC0 circuits cannot distinguish a poly-logarithmically independent distribution from the uniform one. This settles the 1990 conjecture by Linial and Nisan [LN90]. The only prior progress on the problem was by Bazzi [Baz07], who showed that O(log^2 n)-independent distributions fool poly-size DNF formulas. Razborov [Raz08] has ... more >>>


TR09-051 | 2nd July 2009
Eric Allender, Michal Koucky, Detlef Ronneburger, Sambuddha Roy

The Pervasive Reach of Resource-Bounded Kolmogorov Complexity in Computational Complexity Theory

We continue an investigation into resource-bounded Kolmogorov complexity \cite{abkmr}, which highlights the close connections between circuit complexity and Levin's time-bounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity.
The Kolmogorov measures that have been ... more >>>


TR09-085 | 14th September 2009
Christoph Behle, Andreas Krebs, Stephanie Reifferscheid

An Approach to characterize the Regular Languages in TC0 with Linear Wires

Revisions: 1

We consider the regular languages recognized by weighted threshold circuits with a linear number of wires.
We present a simple proof to show that parity cannot be computed by such circuits.
Our proofs are based on an explicit construction to restrict the input of the circuit such that the value ... more >>>


TR09-146 | 29th December 2009
Dan Gutfreund, Akinori Kawachi

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

We show that if Arthur-Merlin protocols can be derandomized, then there is a Boolean function computable in deterministic exponential-time with access to an NP oracle, that cannot be computed by Boolean circuits of exponential size. More formally, if $\mathrm{prAM}\subseteq \mathrm{P}^{\mathrm{NP}}$ then there is a Boolean function in $\mathrm{E}^{\mathrm{NP}}$ that requires ... more >>>


TR11-095 | 22nd June 2011
Christoph Behle, Andreas Krebs, Klaus-Joern Lange, Pierre McKenzie

Low uniform versions of NC1

Revisions: 1

In the setting known as DLOGTIME-uniformity,
the fundamental complexity classes
$AC^0\subset ACC^0\subseteq TC^0\subseteq NC^1$ have several
robust characterizations.
In this paper we refine uniformity further and examine the impact
of these refinements on $NC^1$ and its subclasses.
When applied to the logarithmic circuit depth characterization of $NC^1$,
some refinements leave ... more >>>


TR11-128 | 21st September 2011
Michael Elberfeld, Andreas Jakoby, Till Tantau

Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

An algorithmic meta theorem for a logic and a class $C$ of structures states that all problems expressible in this logic can be solved efficiently for inputs from $C$. The prime example is Courcelle's Theorem, which states that monadic second-order (MSO) definable problems are linear-time solvable on graphs of bounded ... more >>>


TR11-145 | 2nd November 2011
Alexander A. Sherstov

The Multiparty Communication Complexity of Set Disjointness

We study the set disjointness problem in the number-on-the-forehead model.

(i) We prove that $k$-party set disjointness has randomized and nondeterministic
communication complexity $\Omega(n/4^k)^{1/4}$ and Merlin-Arthur complexity $\Omega(n/4^k)^{1/8}.$
These bounds are close to tight. Previous lower bounds (2007-2008) for $k\geq3$ parties
were weaker than $n^{1/(k+1)}/2^{k^2}$ in all ... more >>>


TR11-158 | 25th November 2011
Matthew Anderson, Dieter van Melkebeek, Nicole Schweikardt, Luc Segoufin

Locality from Circuit Lower Bounds

We study the locality of an extension of first-order logic that captures graph queries computable in AC$^0$, i.e., by families of polynomial-size constant-depth circuits. The extension considers first-order formulas over relational structures which may use arbitrary numerical predicates in such a way that their truth value is independent of the ... more >>>


TR12-059 | 14th May 2012
Rahul Santhanam, Ryan Williams

Uniform Circuits, Lower Bounds, and QBF Algorithms

We explore the relationships between circuit complexity, the complexity of generating circuits, and circuit-analysis algorithms. Our results can be roughly divided into three parts:

1. Lower Bounds Against Medium-Uniform Circuits. Informally, a circuit class is ``medium uniform'' if it can be generated by an algorithmic process that is somewhat complex ... more >>>


TR13-021 | 5th February 2013
Kristoffer Arnsfelt Hansen, Vladimir Podolskii

Polynomial threshold functions and Boolean threshold circuits

We study the complexity of computing Boolean functions on general
Boolean domains by polynomial threshold functions (PTFs). A typical
example of a general Boolean domain is $\{1,2\}^n$. We are mainly
interested in the length (the number of monomials) of PTFs, with
their degree and weight being of secondary interest. We ... more >>>


TR13-102 | 17th July 2013
Andreas Krebs, Nutan Limaye, Meena Mahajan, Karteek Sreenivasaiah

Small Depth Proof Systems

A proof system for a language $L$ is a function $f$ such that Range$(f)$ is exactly $L$. In this paper, we look at proofsystems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by ... more >>>


TR14-120 | 16th September 2014
Olaf Beyersdorff, Leroy Chew, Mikolas Janota

Proof Complexity of Resolution-based QBF Calculi

Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of important
QBF solvers. However, the proof complexity of these proof systems is currently not well understood and in particular
lower bound techniques are missing. In this paper we exhibit a new and elegant proof technique ... more >>>


TR15-018 | 31st January 2015
Eric Allender, Ian Mertz

Complexity of Regular Functions

Revisions: 1

We give complexity bounds for various classes of functions computed by cost register automata.

more >>>

TR15-030 | 6th March 2015
Mahdi Cheraghchi, Elena Grigorescu, Brendan Juba, Karl Wimmer, Ning Xie

${\mathrm{AC}^{0} \circ \mathrm{MOD}_2}$ lower bounds for the Boolean Inner Product

Revisions: 1

$\mathrm{AC}^{0} \circ \mathrm{MOD}_2$ circuits are $\mathrm{AC}^{0}$ circuits augmented with a layer of parity gates just above the input layer. We study the $\mathrm{AC}^{0} \circ \mathrm{MOD}_2$ circuit lower bound for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have ... more >>>


TR15-188 | 24th November 2015
Daniel Kane, Ryan Williams

Super-Linear Gate and Super-Quadratic Wire Lower Bounds for Depth-Two and Depth-Three Threshold Circuits

In order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower bounds and the first super-quadratic wire lower bounds for ... more >>>


TR16-022 | 22nd February 2016
Alexander Golovnev, Alexander Kulikov, Alexander Smal, Suguru Tamaki

Circuit size lower bounds and #SAT upper bounds through a general framework

Revisions: 2

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the ... more >>>


TR16-071 | 1st May 2016
Jan Krajicek, Igor Carboni Oliveira

Unprovability of circuit upper bounds in Cook's theory PV

We establish unconditionally that for every integer $k \geq 1$ there is a language $L \in P$ such that it is consistent with Cook's theory PV that $L \notin SIZE(n^k)$. Our argument is non-constructive and does not provide an explicit description of this language.

more >>>

TR16-095 | 7th June 2016
Arkadev Chattopadhyay, Nikhil Mande

Small Error Versus Unbounded Error Protocols in the NOF Model

Revisions: 1 , Comments: 1

We show that a simple function has small unbounded error communication complexity in the $k$-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with sub-exponential advantage over random guessing has cost essentially $\Omega\left(\frac{\sqrt{n}}{4^k}\right)$ bits. Such a separation was first shown for $k=2$ independently by Buhrman et al. ['07] ... more >>>


TR16-110 | 19th July 2016
Alexander Golovnev, Oded Regev, Omri Weinstein

The Minrank of Random Graphs

Revisions: 1

The minrank of a graph $G$ is the minimum rank of a matrix $M$ that can be obtained from the adjacency matrix of $G$ by switching ones to zeros (i.e., deleting edges) and setting all diagonal entries to one. This quantity is closely related to the fundamental information-theoretic problems of ... more >>>


TR16-158 | 9th October 2016
Alexander Kulikov, Vladimir Podolskii

Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates

We study the following computational problem: for which values of $k$, the majority of $n$ bits $\text{MAJ}_n$ can be computed with a depth two formula whose each gate computes a majority function of at most $k$ bits? The corresponding computational model is denoted by $\text{MAJ}_k \circ \text{MAJ}_k$. We observe that ... more >>>


TR17-019 | 8th February 2017
Andreas Krebs, Nutan Limaye, Michael Ludwig

A Unified Method for Placing Problems in Polylogarithmic Depth

Revisions: 2

In this work we consider the term evaluation problem which involves, given a term over some algebra and a valid input to the term, computing the value of the term on that input. This is a classical problem studied under many names such as formula evaluation problem, formula value problem ... more >>>


TR17-022 | 13th February 2017
Benjamin Rossman, Srikanth Srinivasan

Separation of AC$^0[\oplus]$ Formulas and Circuits

This paper gives the first separation between the power of {\em formulas} and {\em circuits} of equal depth in the $\mathrm{AC}^0[\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) \le O(\frac{\log n}{\log\log n})$, that there exist {\em polynomial-size depth-$d$ circuits} that are not equivalent ... more >>>


TR17-092 | 10th May 2017
Shuichi Hirahara

A Duality Between Depth-Three Formulas and Approximation by Depth-Two

We establish an explicit link between depth-3 formulas and one-sided approximation by depth-2 formulas, which were previously studied independently. Specifically, we show that the minimum size of depth-3 formulas is (up to a factor of n) equal to the inverse of the maximum, over all depth-2 formulas, of one-sided-error correlation ... more >>>


TR17-109 | 22nd June 2017
Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Pierre McKenzie, Shadab Romani

Does Looking Inside a Circuit Help?

The Black-Box Hypothesis, introduced by Barak et al. (JACM, 2012), states that any property of boolean functions decided efficiently (e.g., in BPP) with inputs represented by circuits can also be decided efficiently in the black-box setting, where an algorithm is given an oracle access to the input function and an ... more >>>


TR17-149 | 7th October 2017
Or Meir, Avi Wigderson

Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds

Revisions: 5

Consider a random sequence of $n$ bits that has entropy at least $n-k$, where $k\ll n$. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random”. In this work, we prove a stronger result that ... more >>>


TR17-188 | 22nd December 2017
Cody Murray, Ryan Williams

Circuit Lower Bounds for Nondeterministic Quasi-Polytime: An Easy Witness Lemma for NP and NQP

We prove that if every problem in $NP$ has $n^k$-size circuits for a fixed constant $k$, then for every $NP$-verifier and every yes-instance $x$ of length $n$ for that verifier, the verifier's search space has an $n^{O(k^3)}$-size witness circuit: a witness for $x$ that can be encoded with a circuit ... more >>>


TR17-191 | 15th December 2017
Alexander Smal, Navid Talebanfard

Prediction from Partial Information and Hindsight, an Alternative Proof

Revisions: 2

Let $X$ be a random variable distributed over $n$-bit strings with $H(X) \ge n - k$, where $k \ll n$. Using subadditivity we know that a random coordinate looks random. Meir and Wigderson [TR17-149] showed a random coordinate looks random to an adversary who is allowed to query around $n/k$ ... more >>>


TR18-004 | 3rd January 2018
Aayush Ojha, Raghunath Tewari

Circuit Complexity of Bounded Planar Cutwidth Graph Matching

Recently, perfect matching in bounded planar cutwidth bipartite graphs
$BGGM$ was shown to be in ACC$^0$ by Hansen et al.. They also conjectured that
the problem is in AC$^0$.
In this paper, we disprove their conjecture by showing that the problem is
not in AC$^0[p^{\alpha}]$ for every prime $p$. ... more >>>


TR18-139 | 10th August 2018
Igor Carboni Oliveira, Rahul Santhanam

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$[s]$ be the decision problem whose YES instances are ... more >>>


TR19-064 | 23rd April 2019
Igor Carboni Oliveira

Randomness and Intractability in Kolmogorov Complexity

We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin's notion of Kolmogorov complexity from 1984. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability.

This complexity measure gives rise to a ... more >>>


TR19-073 | 17th May 2019
Igor Carboni Oliveira, Rahul Santhanam, Srikanth Srinivasan

Parity helps to compute Majority

We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC$^0[\oplus]$ circuits. Razborov (1987) and Smolensky (1987, 1993) showed that Majority requires depth-$d$ AC$^0[\oplus]$ circuits of size $2^{\Omega(n^{1/2(d-1)})}$. By using a divide-and-conquer approach, it is easy to show that Majority can ... more >>>


TR19-077 | 30th May 2019
Jan Bydzovsky, Igor Carboni Oliveira, Jan Krajicek

Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $P^{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere ... more >>>


TR19-128 | 24th September 2019
Anna Gal, Robert Robere

Lower Bounds for (Non-monotone) Comparator Circuits

Comparator circuits are a natural circuit model for studying the concept of bounded fan-out computations, which intuitively corresponds to whether or not a computational model can make "copies" of intermediate computational steps. Comparator circuits are believed to be weaker than general Boolean circuits, but they can simulate Branching Programs and ... more >>>


TR19-142 | 23rd October 2019
Yaroslav Alekseev, Dima Grigoriev, Edward Hirsch, Iddo Tzameret

Semi-Algebraic Proofs, IPS Lower Bounds and the $\tau$-Conjecture: Can a Natural Number be Negative?

Revisions: 1

We introduce the `binary value principle' which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional superpolynomial lower bounds on the Ideal Proof System (IPS) refutation size of this instance, ... more >>>


TR19-168 | 20th November 2019
Igor Carboni Oliveira, Lijie Chen, Shuichi Hirahara, Ján Pich, Ninad Rajgopal, Rahul Santhanam

Beyond Natural Proofs: Hardness Magnification and Locality

Hardness magnification reduces major complexity separations (such as $EXP \not\subseteq NC^1$) to proving lower bounds for some natural problem $Q$ against weak circuit models. Several recent works [OS18, MMW19, CT19, OPS19, CMMW19, Oli19, CJW19a] have established results of this form. In the most intriguing cases, the required lower bound is ... more >>>


TR20-152 | 7th October 2020
Prasad Chaugule, Nutan Limaye, Shourya Pandey

Variants of the Determinant polynomial and VP-completeness

The determinant is a canonical VBP-complete polynomial in the algebraic complexity setting. In this work, we introduce two variants of the determinant polynomial which we call $StackDet_n(X)$ and $CountDet_n(X)$ and show that they are VP and VNP complete respectively under $p$-projections. The definitions of the polynomials are inspired by a ... more >>>


TR21-006 | 18th January 2021
Susanna de Rezende, Jakob Nordström, Marc Vinyals

How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity)

We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constant-size coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large ... more >>>


TR21-040 | 15th March 2021
Lijie Chen, Zhenjian Lu, Xin Lyu, Igor Carboni Oliveira

Majority vs. Approximate Linear Sum and Average-Case Complexity Below NC1

We develop a general framework that characterizes strong average-case lower bounds against circuit classes $\mathcal{C}$ contained in $\mathrm{NC}^1$, such as $\mathrm{AC}^0[\oplus]$ and $\mathrm{ACC}^0$. We apply this framework to show:

- Generic seed reduction: Pseudorandom generators (PRGs) against $\mathcal{C}$ of seed length $\leq n -1$ and error $\varepsilon(n) = n^{-\omega(1)}$ can ... more >>>


TR21-095 | 8th July 2021
Marco Carmosino, Valentine Kabanets, Antonina Kolokolova, Igor Oliveira

LEARN-Uniform Circuit Lower Bounds and Provability in Bounded Arithmetic

We investigate randomized LEARN-uniformity, which captures the power of randomness and equivalence queries (EQ) in the construction of Boolean circuits for an explicit problem. This is an intermediate notion between P-uniformity and non-uniformity motivated by connections to learning, complexity, and logic. Building on a number of techniques, we establish the ... more >>>


TR21-125 | 23rd August 2021
Zhiyuan Fan, Jiatu Li, Tianqi Yang

The Exact Complexity of Pseudorandom Functions and Tight Barriers to Lower Bound Proofs

Revisions: 1

How much computational resource do we need for cryptography? This is an important question of both theoretical and practical interests. In this paper, we study the problem on pseudorandom functions (PRFs) in the context of circuit complexity. Perhaps surprisingly, we prove extremely tight upper and lower bounds in various circuit ... more >>>


TR21-127 | 30th August 2021
Ron D. Rothblum, Michael Ezra

Small Circuits Imply Efficient Arthur-Merlin Protocols

Revisions: 1

The inner product function $\langle x,y \rangle = \sum_i x_i y_i \bmod 2$ can be easily computed by a (linear-size) ${AC}^0(\oplus)$ circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates. But what if we impose the restriction that the parity gates can only be on ... more >>>


TR21-158 | 12th November 2021
Noah Fleming, Toniann Pitassi, Robert Robere

Extremely Deep Proofs

Revisions: 1

We further the study of supercritical tradeoffs in proof and circuit complexity, which is a type of tradeoff between complexity parameters where restricting one complexity parameter forces another to exceed its worst-case upper bound. In particular, we prove a new family of supercritical tradeoffs between depth and size for Resolution, ... more >>>


TR21-159 | 15th November 2021
Lijie Chen, Ce Jin, Rahul Santhanam, Ryan Williams

Constructive Separations and Their Consequences

For a complexity class $C$ and language $L$, a constructive separation of $L \notin C$ gives an efficient algorithm (also called a refuter) to find counterexamples (bad inputs) for every $C$-algorithm attempting to decide $L$. We study the questions: Which lower bounds can be made constructive? What are the consequences ... more >>>


TR21-171 | 2nd December 2021
Bruno Pasqualotto Cavalar, Zhenjian Lu

Algorithms and Lower Bounds for Comparator Circuits from Shrinkage

Comparator circuits are a natural circuit model for studying bounded fan-out computation whose power sits between nondeterministic branching programs and general circuits. Despite having been studied for nearly three decades, the first superlinear lower bound against comparator circuits was proved only recently by Gál and Robere (ITCS 2020), who established ... more >>>


TR22-025 | 15th February 2022
Oliver Korten

Efficient Low-Space Simulations From the Failure of the Weak Pigeonhole Principle

Revisions: 3

A recurring challenge in the theory of pseudorandomness and circuit complexity is the explicit construction of ``incompressible strings,'' i.e. finite objects which lack a specific type of structure or simplicity. In most cases, there is an associated NP search problem which we call the ``compression problem,'' where we are given ... more >>>


TR22-050 | 12th April 2022
Klim Efremenko, Bernhard Haeupler, Yael Kalai, Pritish Kamath, Gillat Kol, Nicolas Resch, Raghuvansh Saxena

Circuits Resilient to Short-Circuit Errors

Given a Boolean circuit $C$, we wish to convert it to a circuit $C'$ that computes the same function as $C$ even if some of its gates suffer from adversarial short circuit errors, i.e., their output is replaced by the value of one of their inputs [KLM97]. Can we ... more >>>


TR22-141 | 20th October 2022
Sam Buss, Noah Fleming, Russell Impagliazzo

TFNP Characterizations of Proof Systems and Monotone Circuits

Revisions: 2

Connections between proof complexity and circuit complexity have become major tools for obtaining lower bounds in both areas. These connections -- which take the form of interpolation theorems and query-to-communication lifting theorems -- translate efficient proofs into small circuits, and vice versa, allowing tools from one area to be applied ... more >>>


TR23-022 | 11th March 2023
Jiatu Li, Igor Carboni Oliveira

Unprovability of strong complexity lower bounds in bounded arithmetic

While there has been progress in establishing the unprovability of complexity statements in lower fragments of bounded arithmetic, understanding the limits of Jerabek's theory $\textbf{APC}_1$ (2007) and of higher levels of Buss's hierarchy $\textbf{S}^i_2$ (1986) has been a more elusive task. Even in the more restricted setting of Cook's theory ... more >>>


TR23-045 | 13th April 2023
Vinayak Kumar

Tight Correlation Bounds for Circuits Between AC0 and TC0

Revisions: 1

We initiate the study of generalized $AC^0$ circuits comprised of arbitrary unbounded fan-in gates which only need to be constant over inputs of Hamming weight $\ge k$ (up to negations of the input bits), which we denote $GC^0(k)$. The gate set of this class includes biased LTFs like the $k$-$OR$ ... more >>>


TR23-069 | 11th May 2023
Bruno Pasqualotto Cavalar, Igor Carboni Oliveira

Constant-depth circuits vs. monotone circuits

We establish new separations between the power of monotone and general (non-monotone) Boolean circuits:

- For every $k \geq 1$, there is a monotone function in ${\rm AC^0}$ (constant-depth poly-size circuits) that requires monotone circuits of depth $\Omega(\log^k n)$. This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai ... more >>>


TR23-171 | 15th November 2023
Shuichi Hirahara, Rahul Ilango, Ryan Williams

Beating Brute Force for Compression Problems

A compression problem is defined with respect to an efficient encoding function $f$; given a string $x$, our task is to find the shortest $y$ such that $f(y) = x$. The obvious brute-force algorithm for solving this compression task on $n$-bit strings runs in time $O(2^{\ell} \cdot t(n))$, where $\ell$ ... more >>>


TR23-184 | 22nd November 2023
Gabriel Bathie, Ryan Williams

Towards Stronger Depth Lower Bounds

A fundamental problem in circuit complexity is to find explicit functions that require large depth to compute. When considering the natural DeMorgan basis of $\{\text{OR},\text{AND}\}$, where negations incur no cost, the best known depth lower bounds for an explicit function in NP have the form $(3-o(1))\log_2 n$, established by H{\aa}stad ... more >>>


TR23-186 | 28th November 2023
Ce Jin, Ryan Williams, Nathaniel Young

A VLSI Circuit Model Accounting For Wire Delay

Given the need for ever higher performance, and the failure of CPUs to keep providing single-threaded performance gains, engineers are increasingly turning to highly-parallel custom VLSI chips to implement expensive computations. In VLSI design, the gates and wires of a logical circuit are placed on a 2-dimensional chip with a ... more >>>




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