15,584 research outputs found
On the logical definability of certain graph and poset languages
We show that it is equivalent, for certain sets of finite graphs, to be
definable in CMS (counting monadic second-order logic, a natural extension of
monadic second-order logic), and to be recognizable in an algebraic framework
induced by the notion of modular decomposition of a finite graph. More
precisely, we consider the set of composition operations on graphs
which occur in the modular decomposition of finite graphs. If is a subset
of , we say that a graph is an \calF-graph if it can be
decomposed using only operations in . A set of -graphs is recognizable if
it is a union of classes in a finite-index equivalence relation which is
preserved by the operations in . We show that if is finite and its
elements enjoy only a limited amount of commutativity -- a property which we
call weak rigidity, then recognizability is equivalent to CMS-definability.
This requirement is weak enough to be satisfied whenever all -graphs are
posets, that is, transitive dags. In particular, our result generalizes Kuske's
recent result on series-parallel poset languages
A simple linear-time algorithm for finding path-decompositions of small width
We described a simple algorithm running in linear time for each fixed
constant , that either establishes that the pathwidth of a graph is
greater than , or finds a path-decomposition of of width at most
. This provides a simple proof of the result by Bodlaender that many
families of graphs of bounded pathwidth can be recognized in linear time.Comment: 9 page
Concurrent Kleene Algebra: Free Model and Completeness
Concurrent Kleene Algebra (CKA) was introduced by Hoare, Moeller, Struth and
Wehrman in 2009 as a framework to reason about concurrent programs. We prove
that the axioms for CKA with bounded parallelism are complete for the semantics
proposed in the original paper; consequently, these semantics are the free
model for this fragment. This result settles a conjecture of Hoare and
collaborators. Moreover, the techniques developed along the way are reusable;
in particular, they allow us to establish pomset automata as an operational
model for CKA.Comment: Version 2 includes an overview section that outlines the completeness
proof, as well as some extra discussion of the interpolation lemma. It also
includes better typography and a number of minor fixes. Version 3
incorporates the changes by comments from the anonymous referees at ESOP.
Among other things, these include a worked example of computing the syntactic
closure by han
An Efficient Algorithm for Computing Network Reliability in Small Treewidth
We consider the classic problem of Network Reliability. A network is given
together with a source vertex, one or more target vertices, and probabilities
assigned to each of the edges. Each edge appears in the network with its
associated probability and the problem is to determine the probability of
having at least one source-to-target path. This problem is known to be NP-hard.
We present a linear-time fixed-parameter algorithm based on a parameter
called treewidth, which is a measure of tree-likeness of graphs. Network
Reliability was already known to be solvable in polynomial time for bounded
treewidth, but there were no concrete algorithms and the known methods used
complicated structures and were not easy to implement. We provide a
significantly simpler and more intuitive algorithm that is much easier to
implement.
We also report on an implementation of our algorithm and establish the
applicability of our approach by providing experimental results on the graphs
of subway and transit systems of several major cities, such as London and
Tokyo. To the best of our knowledge, this is the first exact algorithm for
Network Reliability that can scale to handle real-world instances of the
problem.Comment: 14 page
Equivalence checking for weak bi-Kleene algebra
Pomset automata are an operational model of weak bi-Kleene algebra, which
describes programs that can fork an execution into parallel threads, upon
completion of which execution can join to resume as a single thread. We
characterize a fragment of pomset automata that admits a decision procedure for
language equivalence. Furthermore, we prove that this fragment corresponds
precisely to series-rational expressions, i.e., rational expressions with an
additional operator for bounded parallelism. As a consequence, we obtain a new
proof that equivalence of series-rational expressions is decidable
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