19,701 research outputs found
On Uniformity and Circuit Lower Bounds
AbstractâWe explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: 1. Lower Bounds Against Medium-Uniform Circuits. Informally, a circuit class is âmedium uniform â if it can be generated by an algorithmic process that is somewhat complex (stronger than LOGTIME) but not infeasible. Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium uniform circuit classes, including: âą For all k, P is not contained in P-uniform SIZE(n k). That is, for all k there is a language Lk â P that does not have O(n k)-size circuits constructible in polynomial time. This improves Kannanâs lower bound from 1982 that NP is not in P-uniform SIZE(n k) for any fixed k
AND and/or OR: Uniform Polynomial-Size Circuits
We investigate the complexity of uniform OR circuits and AND circuits of
polynomial-size and depth. As their name suggests, OR circuits have OR gates as
their computation gates, as well as the usual input, output and constant (0/1)
gates. As is the norm for Boolean circuits, our circuits have multiple sink
gates, which implies that an OR circuit computes an OR function on some subset
of its input variables. Determining that subset amounts to solving a number of
reachability questions on a polynomial-size directed graph (which input gates
are connected to the output gate?), taken from a very sparse set of graphs.
However, it is not obvious whether or not this (restricted) reachability
problem can be solved, by say, uniform AC^0 circuits (constant depth,
polynomial-size, AND, OR, NOT gates). This is one reason why characterizing the
power of these simple-looking circuits in terms of uniform classes turns out to
be intriguing. Another is that the model itself seems particularly natural and
worthy of study.
Our goal is the systematic characterization of uniform polynomial-size OR
circuits, and AND circuits, in terms of known uniform machine-based complexity
classes. In particular, we consider the languages reducible to such uniform
families of OR circuits, and AND circuits, under a variety of reduction types.
We give upper and lower bounds on the computational power of these language
classes. We find that these complexity classes are closely related to tallyNL,
the set of unary languages within NL, and to sets reducible to tallyNL.
Specifically, for a variety of types of reductions (many-one, conjunctive truth
table, disjunctive truth table, truth table, Turing) we give characterizations
of languages reducible to OR circuit classes in terms of languages reducible to
tallyNL classes. Then, some of these OR classes are shown to coincide, and some
are proven to be distinct. We give analogous results for AND circuits. Finally,
for many of our OR circuit classes, and analogous AND circuit classes, we prove
whether or not the two classes coincide, although we leave one such inclusion
open.Comment: In Proceedings MCU 2013, arXiv:1309.104
Consistency of circuit lower bounds with bounded theories
Proving that there are problems in 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 circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
Quantum algorithms for testing properties of distributions
Suppose one has access to oracles generating samples from two unknown
probability distributions P and Q on some N-element set. How many samples does
one need to test whether the two distributions are close or far from each other
in the L_1-norm ? This and related questions have been extensively studied
during the last years in the field of property testing. In the present paper we
study quantum algorithms for testing properties of distributions. It is shown
that the L_1-distance between P and Q can be estimated with a constant
precision using approximately N^{1/2} queries in the quantum settings, whereas
classical computers need \Omega(N) queries. We also describe quantum algorithms
for testing Uniformity and Orthogonality with query complexity O(N^{1/3}). The
classical query complexity of these problems is known to be \Omega(N^{1/2}).Comment: 20 page
The Complexity of Bisimulation and Simulation on Finite Systems
In this paper the computational complexity of the (bi)simulation problem over
restricted graph classes is studied. For trees given as pointer structures or
terms the (bi)simulation problem is complete for logarithmic space or NC,
respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and
S\'antha. Furthermore, if only one of the input graphs is required to be a
tree, the bisimulation (simulation) problem is contained in AC (LogCFL). In
contrast, it is also shown that the simulation problem is P-complete already
for graphs of bounded path-width
An Introduction to Quantum Complexity Theory
We give a basic overview of computational complexity, query complexity, and
communication complexity, with quantum information incorporated into each of
these scenarios. The aim is to provide simple but clear definitions, and to
highlight the interplay between the three scenarios and currently-known quantum
algorithms.Comment: 28 pages, LaTeX, 11 figures within the text, to appear in "Collected
Papers on Quantum Computation and Quantum Information Theory", edited by C.
Macchiavello, G.M. Palma, and A. Zeilinger (World Scientific
On Symmetric Circuits and Fixed-Point Logics
We study properties of relational structures such as graphs that are decided
by families of Boolean circuits. Circuits that decide such properties are
necessarily invariant to permutations of the elements of the input structures.
We focus on families of circuits that are symmetric, i.e., circuits whose
invariance is witnessed by automorphisms of the circuit induced by the
permutation of the input structure. We show that the expressive power of such
families is closely tied to definability in logic. In particular, we show that
the queries defined on structures by uniform families of symmetric Boolean
circuits with majority gates are exactly those definable in fixed-point logic
with counting. This shows that inexpressibility results in the latter logic
lead to lower bounds against polynomial-size families of symmetric circuits.Comment: 22 pages. Full version of a paper to appear in STACS 201
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