51,704 research outputs found
Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity
We say that a circuit over a field functionally computes an
-variate polynomial if for every we have that . This is in contrast to syntactically computing , when as
formal polynomials. In this paper, we study the question of proving lower
bounds for homogeneous depth- and depth- arithmetic circuits for
functional computation. We prove the following results :
1. Exponential lower bounds homogeneous depth- arithmetic circuits for a
polynomial in .
2. Exponential lower bounds for homogeneous depth- arithmetic circuits
with bounded individual degree for a polynomial in .
Our main motivation for this line of research comes from our observation that
strong enough functional lower bounds for even very special depth-
arithmetic circuits for the Permanent imply a separation between and
. Thus, improving the second result to get rid of the bounded individual
degree condition could lead to substantial progress in boolean circuit
complexity. Besides, it is known from a recent result of Kumar and Saptharishi
[KS15] that over constant sized finite fields, strong enough average case
functional lower bounds for homogeneous depth- circuits imply
superpolynomial lower bounds for homogeneous depth- circuits.
Our proofs are based on a family of new complexity measures called shifted
evaluation dimension, and might be of independent interest
Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits
In this paper, we prove superpolynomial lower bounds for the class of
homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP
of degree in variables such that any homogeneous depth 4 arithmetic
circuit computing it must have size .
Our results extend the works of Nisan-Wigderson [NW95] (which showed
superpolynomial lower bounds for homogeneous depth 3 circuits),
Gupta-Kamath-Kayal-Saptharishi and Kayal-Saha-Saptharishi [GKKS13, KSS13]
(which showed superpolynomial lower bounds for homogeneous depth 4 circuits
with bounded bottom fan-in), Kumar-Saraf [KS13a] (which showed superpolynomial
lower bounds for homogeneous depth 4 circuits with bounded top fan-in) and
Raz-Yehudayoff and Fournier-Limaye-Malod-Srinivasan [RY08, FLMS13] (which
showed superpolynomial lower bounds for multilinear depth 4 circuits). Several
of these results in fact showed exponential lower bounds.
The main ingredient in our proof is a new complexity measure of {\it bounded
support} shifted partial derivatives. This measure allows us to prove
exponential lower bounds for homogeneous depth 4 circuits where all the
monomials computed at the bottom layer have {\it bounded support} (but possibly
unbounded degree/fan-in), strengthening the results of Gupta et al and Kayal et
al [GKKS13, KSS13]. This new lower bound combined with a careful "random
restriction" procedure (that transforms general depth 4 homogeneous circuits to
depth 4 circuits with bounded support) gives us our final result
Static Data Structure Lower Bounds Imply Rigidity
We show that static data structure lower bounds in the group (linear) model
imply semi-explicit lower bounds on matrix rigidity. In particular, we prove
that an explicit lower bound of on the cell-probe
complexity of linear data structures in the group model, even against
arbitrarily small linear space , would already imply a
semi-explicit () construction of rigid matrices with
significantly better parameters than the current state of art (Alon, Panigrahy
and Yekhanin, 2009). Our results further assert that polynomial () data structure lower bounds against near-optimal space, would
imply super-linear circuit lower bounds for log-depth linear circuits (a
four-decade open question). In the succinct space regime , we show
that any improvement on current cell-probe lower bounds in the linear model
would also imply new rigidity bounds. Our results rely on a new connection
between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak,
2006), and on a new reduction from worst-case to average-case rigidity, which
is of independent interest
Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework
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 known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function.
In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To~obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration.
We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework.
Using this framework, we prove the following new bounds: average case lower bounds of 3.24n and 2.59n for circuits over U_2 and B_2, respectively (though the lower bound for the basis B_2 is given for a quadratic disperser whose explicit construction is not currently known), and faster than 2^n #SAT-algorithms for circuits over U_2 and B_2 of size at most 3.24n and 2.99n, respectively. Here by B_2 we mean the set of all bivariate Boolean functions, and by U_2 the set of all bivariate Boolean functions except for parity and its complement
Minimal Universal Two-qubit Quantum Circuits
We give quantum circuits that simulate an arbitrary two-qubit unitary
operator up to global phase. For several quantum gate libraries we prove that
gate counts are optimal in worst and average cases. Our lower and upper bounds
compare favorably to previously published results. Temporary storage is not
used because it tends to be expensive in physical implementations.
For each gate library, best gate counts can be achieved by a single universal
circuit. To compute gate parameters in universal circuits, we only use
closed-form algebraic expressions, and in particular do not rely on matrix
exponentials. Our algorithm has been coded in C++.Comment: 8 pages, 2 tables and 4 figures. v3 adds a discussion of asymetry
between Rx, Ry and Rz gates and describes a subtle circuit design problem
arising when Ry gates are not available. v2 sharpens one of the loose bounds
in v1. Proof techniques in v2 are noticeably revamped: they now rely less on
circuit identities and more on directly-computed invariants of two-qubit
operators. This makes proofs more constructive and easier to interpret as
algorithm
Lower Bounds for Symmetric Circuits for the Determinant
Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is that the circuits which take a matrix input are unchanged by a permutation applied simultaneously to the rows and columns of the matrix. Under such restrictions we have polynomial-size circuits for computing the determinant but no subexponential size circuits for the permanent. Here, we consider a more stringent symmetry requirement, namely that the circuits are unchanged by arbitrary even permutations applied separately to rows and columns, and prove an exponential lower bound even for circuits computing the determinant. The result requires substantial new machinery. We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games. These are applied to the determinant problem with a novel construction of matrices that are bi-adjacency matrices of graphs based on the CFI construction. Our general framework opens the way to exploring a variety of symmetry restrictions and studying trade-offs between symmetry and other resources used by arithmetic circuits
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 depth-two linear threshold circuits with arbitrary weights, and depth-three
majority circuits computing an explicit function.
We prove that for all , the
linear-time computable Andreev's function cannot be computed on a
-fraction of -bit inputs by depth-two linear threshold
circuits of gates, nor can it be computed with
wires. This establishes an average-case
``size hierarchy'' for threshold circuits, as Andreev's function is computable
by uniform depth-two circuits of linear threshold gates, and by
uniform depth-three circuits of majority gates.
We present a new function in based on small-biased sets, which
we prove cannot be computed by a majority vote of depth-two linear threshold
circuits with gates, nor with
wires.
We give tight average-case (gate and wire) complexity results for
computing PARITY with depth-two threshold circuits; the answer turns out to be
the same as for depth-two majority circuits.
The key is a new random restriction lemma for linear threshold functions. Our
main analytical tool is the Littlewood-Offord Lemma from additive
combinatorics
A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits
We give a nontrivial algorithm for the satisfiability problem for cn-wire
threshold circuits of depth two which is better than exhaustive search by a
factor 2^{sn} where s= 1/c^{O(c^2)}. We believe that this is the first
nontrivial satisfiability algorithm for cn-wire threshold circuits of depth
two. The independently interesting problem of the feasibility of sparse 0-1
integer linear programs is a special case. To our knowledge, our algorithm is
the first to achieve constant savings even for the special case of Integer
Linear Programming. The key idea is to reduce the satisfiability problem to the
Vector Domination Problem, the problem of checking whether there are two
vectors in a given collection of vectors such that one dominates the other
component-wise.
We also provide a satisfiability algorithm with constant savings for depth
two circuits with symmetric gates where the total weighted fan-in is at most
cn.
One of our motivations is proving strong lower bounds for TC^0 circuits,
exploiting the connection (established by Williams) between satisfiability
algorithms and lower bounds. Our second motivation is to explore the connection
between the expressive power of the circuits and the complexity of the
corresponding circuit satisfiability problem
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