1,754 research outputs found
Logics for complexity classes
A new syntactic characterization of problems complete via Turing reductions
is presented. General canonical forms are developed in order to define such
problems. One of these forms allows us to define complete problems on ordered
structures, and another form to define them on unordered non-Aristotelian
structures. Using the canonical forms, logics are developed for complete
problems in various complexity classes. Evidence is shown that there cannot be
any complete problem on Aristotelian structures for several complexity classes.
Our approach is extended beyond complete problems. Using a similar form, a
logic is developed to capture the complexity class which very
likely contains no complete problem.Comment: This article has been accepted for publication in Logic Journal of
IGPL Published by Oxford University Press; 23 pages, 2 figure
Processing Succinct Matrices and Vectors
We study the complexity of algorithmic problems for matrices that are
represented by multi-terminal decision diagrams (MTDD). These are a variant of
ordered decision diagrams, where the terminal nodes are labeled with arbitrary
elements of a semiring (instead of 0 and 1). A simple example shows that the
product of two MTDD-represented matrices cannot be represented by an MTDD of
polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by
allowing componentwise symbolic addition of variables (of the same dimension)
in rules. It is shown that accessing an entry, equality checking, matrix
multiplication, and other basic matrix operations can be solved in polynomial
time for MTDD_+-represented matrices. On the other hand, testing whether the
determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the
same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing
a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of
CSR 201
Reconfiguration in bounded bandwidth and treedepth
We show that several reconfiguration problems known to be PSPACE-complete
remain so even when limited to graphs of bounded bandwidth. The essential step
is noticing the similarity to very limited string rewriting systems, whose
ability to directly simulate Turing Machines is classically known. This
resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show
that a large class of reconfiguration problems becomes tractable on graphs of
bounded treedepth, and that this result is in some sense tight.Comment: 14 page
The Computational Complexity of Estimating Convergence Time
An important problem in the implementation of Markov Chain Monte Carlo
algorithms is to determine the convergence time, or the number of iterations
before the chain is close to stationarity. For many Markov chains used in
practice this time is not known. Even in cases where the convergence time is
known to be polynomial, the theoretical bounds are often too crude to be
practical. Thus, practitioners like to carry out some form of statistical
analysis in order to assess convergence. This has led to the development of a
number of methods known as convergence diagnostics which attempt to diagnose
whether the Markov chain is far from stationarity. We study the problem of
testing convergence in the following settings and prove that the problem is
hard in a computational sense: Given a Markov chain that mixes rapidly, it is
hard for Statistical Zero Knowledge (SZK-hard) to distinguish whether starting
from a given state, the chain is close to stationarity by time t or far from
stationarity at time ct for a constant c. We show the problem is in AM
intersect coAM. Second, given a Markov chain that mixes rapidly it is coNP-hard
to distinguish whether it is close to stationarity by time t or far from
stationarity at time ct for a constant c. The problem is in coAM. Finally, it
is PSPACE-complete to distinguish whether the Markov chain is close to
stationarity by time t or far from being mixed at time ct for c at least 1
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