45,357 research outputs found
Reachability problems for products of matrices in semirings
We consider the following matrix reachability problem: given square
matrices with entries in a semiring, is there a product of these matrices which
attains a prescribed matrix? We define similarly the vector (resp. scalar)
reachability problem, by requiring that the matrix product, acting by right
multiplication on a prescribed row vector, gives another prescribed row vector
(resp. when multiplied at left and right by prescribed row and column vectors,
gives a prescribed scalar). We show that over any semiring, scalar reachability
reduces to vector reachability which is equivalent to matrix reachability, and
that for any of these problems, the specialization to any is
equivalent to the specialization to . As an application of this result and
of a theorem of Krob, we show that when , the vector and matrix
reachability problems are undecidable over the max-plus semiring
. We also show that the matrix, vector, and scalar
reachability problems are decidable over semirings whose elements are
``positive'', like the tropical semiring .Comment: 21 page
Maximal Cost-Bounded Reachability Probability on Continuous-Time Markov Decision Processes
In this paper, we consider multi-dimensional maximal cost-bounded
reachability probability over continuous-time Markov decision processes
(CTMDPs). Our major contributions are as follows. Firstly, we derive an
integral characterization which states that the maximal cost-bounded
reachability probability function is the least fixed point of a system of
integral equations. Secondly, we prove that the maximal cost-bounded
reachability probability can be attained by a measurable deterministic
cost-positional scheduler. Thirdly, we provide a numerical approximation
algorithm for maximal cost-bounded reachability probability. We present these
results under the setting of both early and late schedulers
On the complexity of the chip-firing reachability problem
In this paper, we study the complexity of the chip-firing reachability
problem. We show that for Eulerian digraphs, the reachability problem can be
decided in strongly polynomial time, even if the digraph has multiple edges. We
also show a special case when the reachability problem can be decided in
polynomial time for general digraphs: if the target distribution is recurrent
restricted to each strongly connected component. As a further positive result,
we show that the chip-firing reachability problem is in co-NP for general
digraphs. We also show that the chip-firing halting problem is in co-NP for
Eulerian digraphs
Constraint-based reachability
Iterative imperative programs can be considered as infinite-state systems
computing over possibly unbounded domains. Studying reachability in these
systems is challenging as it requires to deal with an infinite number of states
with standard backward or forward exploration strategies. An approach that we
call Constraint-based reachability, is proposed to address reachability
problems by exploring program states using a constraint model of the whole
program. The keypoint of the approach is to interpret imperative constructions
such as conditionals, loops, array and memory manipulations with the
fundamental notion of constraint over a computational domain. By combining
constraint filtering and abstraction techniques, Constraint-based reachability
is able to solve reachability problems which are usually outside the scope of
backward or forward exploration strategies. This paper proposes an
interpretation of classical filtering consistencies used in Constraint
Programming as abstract domain computations, and shows how this approach can be
used to produce a constraint solver that efficiently generates solutions for
reachability problems that are unsolvable by other approaches.Comment: In Proceedings Infinity 2012, arXiv:1302.310
Vector Reachability Problem in
The decision problems on matrices were intensively studied for many decades
as matrix products play an essential role in the representation of various
computational processes. However, many computational problems for matrix
semigroups are inherently difficult to solve even for problems in low
dimensions and most matrix semigroup problems become undecidable in general
starting from dimension three or four.
This paper solves two open problems about the decidability of the vector
reachability problem over a finitely generated semigroup of matrices from
and the point to point reachability (over rational
numbers) for fractional linear transformations, where associated matrices are
from . The approach to solving reachability problems
is based on the characterization of reachability paths between points which is
followed by the translation of numerical problems on matrices into
computational and combinatorial problems on words and formal languages. We also
give a geometric interpretation of reachability paths and extend the
decidability results to matrix products represented by arbitrary labelled
directed graphs. Finally, we will use this technique to prove that a special
case of the scalar reachability problem is decidable
Temporal Reachability Graphs
While a natural fit for modeling and understanding mobile networks,
time-varying graphs remain poorly understood. Indeed, many of the usual
concepts of static graphs have no obvious counterpart in time-varying ones. In
this paper, we introduce the notion of temporal reachability graphs. A
(tau,delta)-reachability graph} is a time-varying directed graph derived from
an existing connectivity graph. An edge exists from one node to another in the
reachability graph at time t if there exists a journey (i.e., a spatiotemporal
path) in the connectivity graph from the first node to the second, leaving
after t, with a positive edge traversal time tau, and arriving within a maximum
delay delta. We make three contributions. First, we develop the theoretical
framework around temporal reachability graphs. Second, we harness our
theoretical findings to propose an algorithm for their efficient computation.
Finally, we demonstrate the analytic power of the temporal reachability graph
concept by applying it to synthetic and real-life datasets. On top of defining
clear upper bounds on communication capabilities, reachability graphs highlight
asymmetric communication opportunities and offloading potential.Comment: In proceedings ACM Mobicom 201
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