5,762 research outputs found
Observable Graphs
An edge-colored directed graph is \emph{observable} if an agent that moves
along its edges is able to determine his position in the graph after a
sufficiently long observation of the edge colors. When the agent is able to
determine his position only from time to time, the graph is said to be
\emph{partly observable}. Observability in graphs is desirable in situations
where autonomous agents are moving on a network and one wants to localize them
(or the agent wants to localize himself) with limited information. In this
paper, we completely characterize observable and partly observable graphs and
show how these concepts relate to observable discrete event systems and to
local automata. Based on these characterizations, we provide polynomial time
algorithms to decide observability, to decide partial observability, and to
compute the minimal number of observations necessary for finding the position
of an agent. In particular we prove that in the worst case this minimal number
of observations increases quadratically with the number of nodes in the graph.
From this it follows that it may be necessary for an agent to pass through
the same node several times before he is finally able to determine his position
in the graph. We then consider the more difficult question of assigning colors
to a graph so as to make it observable and we prove that two different versions
of this problem are NP-complete.Comment: 15 pages, 8 figure
On the Minimal Revision Problem of Specification Automata
As robots are being integrated into our daily lives, it becomes necessary to
provide guarantees on the safe and provably correct operation. Such guarantees
can be provided using automata theoretic task and mission planning where the
requirements are expressed as temporal logic specifications. However, in
real-life scenarios, it is to be expected that not all user task requirements
can be realized by the robot. In such cases, the robot must provide feedback to
the user on why it cannot accomplish a given task. Moreover, the robot should
indicate what tasks it can accomplish which are as "close" as possible to the
initial user intent. This paper establishes that the latter problem, which is
referred to as the minimal specification revision problem, is NP complete. A
heuristic algorithm is presented that can compute good approximations to the
Minimal Revision Problem (MRP) in polynomial time. The experimental study of
the algorithm demonstrates that in most problem instances the heuristic
algorithm actually returns the optimal solution. Finally, some cases where the
algorithm does not return the optimal solution are presented.Comment: 23 pages, 16 figures, 2 tables, International Joural of Robotics
Research 2014 Major Revision (submitted
A weighted pair graph representation for reconstructibility of Boolean control networks
A new concept of weighted pair graphs (WPGs) is proposed to represent a new
reconstructibility definition for Boolean control networks (BCNs), which is a
generalization of the reconstructibility definition given in [Fornasini &
Valcher, TAC2013, Def. 4]. Based on the WPG representation, an effective
algorithm for determining the new reconstructibility notion for BCNs is
designed with the help of the theories of finite automata and formal languages.
We prove that a BCN is not reconstructible iff its WPG has a complete subgraph.
Besides, we prove that a BCN is reconstructible in the sense of [Fornasini &
Valcher, TAC2013, Def. 4] iff its WPG has no cycles, which is simpler to be
checked than the condition in [Fornasini & Valcher, TAC2013, Thm. 4].Comment: 20 pages, 10 figures, accepted by SIAM Journal on Control and
Optimizatio
CAIR: Using Formal Languages to Study Routing, Leaking, and Interception in BGP
The Internet routing protocol BGP expresses topological reachability and
policy-based decisions simultaneously in path vectors. A complete view on the
Internet backbone routing is given by the collection of all valid routes, which
is infeasible to obtain due to information hiding of BGP, the lack of
omnipresent collection points, and data complexity. Commonly, graph-based data
models are used to represent the Internet topology from a given set of BGP
routing tables but fall short of explaining policy contexts. As a consequence,
routing anomalies such as route leaks and interception attacks cannot be
explained with graphs.
In this paper, we use formal languages to represent the global routing system
in a rigorous model. Our CAIR framework translates BGP announcements into a
finite route language that allows for the incremental construction of minimal
route automata. CAIR preserves route diversity, is highly efficient, and
well-suited to monitor BGP path changes in real-time. We formally derive
implementable search patterns for route leaks and interception attacks. In
contrast to the state-of-the-art, we can detect these incidents. In practical
experiments, we analyze public BGP data over the last seven years
A Trichotomy for Regular Trail Queries
Regular path queries (RPQs) are an essential component of graph query languages. Such queries consider a regular expression r and a directed edge-labeled graph G and search for paths in G for which the sequence of labels is in the language of r. In order to avoid having to consider infinitely many paths, some database engines restrict such paths to be trails, that is, they only consider paths without repeated edges. In this paper we consider the evaluation problem for RPQs under trail semantics, in the case where the expression is fixed. We show that, in this setting, there exists a trichotomy. More precisely, the complexity of RPQ evaluation divides the regular languages into the finite languages, the class T_tract (for which the problem is tractable), and the rest. Interestingly, the tractable class in the trichotomy is larger than for the trichotomy for simple paths, discovered by Bagan et al. [Bagan et al., 2013]. In addition to this trichotomy result, we also study characterizations of the tractable class, its expressivity, the recognition problem, closure properties, and show how the decision problem can be extended to the enumeration problem, which is relevant to practice
Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems
A complete classification of the computational complexity of the fixed-point
existence problem for boolean dynamical systems, i.e., finite discrete
dynamical systems over the domain {0, 1}, is presented. For function classes F
and graph classes G, an (F, G)-system is a boolean dynamical system such that
all local transition functions lie in F and the underlying graph lies in G. Let
F be a class of boolean functions which is closed under composition and let G
be a class of graphs which is closed under taking minors. The following
dichotomy theorems are shown: (1) If F contains the self-dual functions and G
contains the planar graphs then the fixed-point existence problem for (F,
G)-systems with local transition function given by truth-tables is NP-complete;
otherwise, it is decidable in polynomial time. (2) If F contains the self-dual
functions and G contains the graphs having vertex covers of size one then the
fixed-point existence problem for (F, G)-systems with local transition function
given by formulas or circuits is NP-complete; otherwise, it is decidable in
polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24
versio
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A Combinatorial Parametric Engineering Model for Solid Freeform Fabrication
Fabricated parts are often represented as compact connected smooth 3-manifolds with
boundary, where the boundaries consist of compact smooth 2-manifolds. This class of mathematical
structures includes topological spaces with enclosed voids and tunnels. Useful information about these
structures are coded into level functions (Morse functions) which map points in the 3-manifold onto their
height above a fixed plane. By definition, Morse functions are smooth functions, all of whose critical
points are nondegenerate. This information is presented by the Reeb graph construction that develops a
topologically informative skeleton of the manifold whose nodes are the critical points of the Morse function
and whose edges are associated with the connected components between critical slices. This approach
accurately captures the SFF process: using a solid geometric model of the part, defining surface
boundaries; selecting a part orientation; forming planar slices, decomposing the solid into a sequence of
thin cross-sectional polyhedral layers; and then fabricating the part by producing the polyhedra by additive
manufacturing. This note will define a qualitative and combinatorial parametric engineering model of the
SFF part design process. The objects under study will be abstract simplicial complexes K with boundary
∂K. Systems of labeled 2-surfaces in K, called slices, will be associated with the cross-sectional polyhedral
layers. The labeled slices are mapped into a family of digraph automata, which, unlike cellular automata,
are defined not on regular lattices with simple connectivities (cells usually have either 4 or 8 cell
neighborhoods) but on unrestricted digraphs whose connectivities are irregular and more complicated.Mechanical Engineerin
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