847 research outputs found
1-Safe Petri nets and special cube complexes: equivalence and applications
Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net
unfolds into an event structure . By a result of Thiagarajan
(1996 and 2002), these unfoldings are exactly the trace regular event
structures. Thiagarajan (1996 and 2002) conjectured that regular event
structures correspond exactly to trace regular event structures. In a recent
paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on
the striking bijection between domains of event structures, median graphs, and
CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we
proved that Thiagarajan's conjecture is true for regular event structures whose
domains are principal filters of universal covers of (virtually) finite special
cube complexes.
In the current paper, we prove the converse: to any finite 1-safe Petri net
one can associate a finite special cube complex such that the
domain of the event structure (obtained as the unfolding of
) is a principal filter of the universal cover of .
This establishes a bijection between 1-safe Petri nets and finite special cube
complexes and provides a combinatorial characterization of trace regular event
structures.
Using this bijection and techniques from graph theory and geometry (MSO
theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet
another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that
the monadic second order logic of a 1-safe Petri net is decidable if and only
if its unfolding is grid-free.
Our counterexample is the trace regular event structure
which arises from a virtually special square complex . The domain of
is grid-free (because it is hyperbolic), but the MSO
theory of the event structure is undecidable
A counterexample to Thiagarajan's conjecture on regular event structures
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002)
that regular event structures correspond exactly to event structures obtained
as unfoldings of finite 1-safe Petri nets. The same counterexample is used to
disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999)
that domains of regular event structures with bounded -cliques are
recognizable by finite trace automata. Event structures, trace automata, and
Petri nets are fundamental models in concurrency theory. There exist nice
interpretations of these structures as combinatorial and geometric objects.
Namely, from a graph theoretical point of view, the domains of prime event
structures correspond exactly to median graphs; from a geometric point of view,
these domains are in bijection with CAT(0) cube complexes.
A necessary condition for both conjectures to be true is that domains of
regular event structures (with bounded -cliques) admit a regular nice
labeling. To disprove these conjectures, we describe a regular event domain
(with bounded -cliques) that does not admit a regular nice labeling.
Our counterexample is derived from an example by Wise (1996 and 2007) of a
nonpositively curved square complex whose universal cover is a CAT(0) square
complex containing a particular plane with an aperiodic tiling. We prove that
other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way
deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009).
On the positive side, using breakthrough results by Agol (2013) and Haglund
and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's
conjecture is true for regular event structures whose domains occur as
principal filters of hyperbolic CAT(0) cube complexes which are universal
covers of finite nonpositively curved cube complexes
On two conjectures of Maurer concerning basis graphs of matroids
We characterize 2-dimensional complexes associated canonically with basis
graphs of matroids as simply connected triangle-square complexes satisfying
some local conditions. This proves a version of a (disproved) conjecture by
Stephen Maurer (Conjecture 3 of S. Maurer, Matroid basis graphs I, JCTB 14
(1973), 216-240). We also establish Conjecture 1 from the same paper about the
redundancy of the conditions in the characterization of basis graphs. We
indicate positive-curvature-like aspects of the local properties of the studied
complexes. We characterize similarly the corresponding 2-dimensional complexes
of even -matroids.Comment: 28 page
Direct generation of a multi-transverse mode non-classical state of light
Quantum computation and communication protocols require quantum resources
which are in the continuous variable regime squeezed and/or quadrature
entangled optical modes. To perform more and more complex and robust protocols,
one needs sources that can produce in a controlled way highly multimode quantum
states of light. One possibility is to mix different single mode quantum
resources. Another is to directly use a multimode device, either in the spatial
or in the frequency domain. We present here the first experimental
demonstration of a device capable of producing simultanuously several squeezed
transverse modes of the same frequency and which is potentially scalable. We
show that this device, which is an Optical Parametric Oscillator using a
self-imaging cavity, produces a multimode quantum resource made of three
squeezed transverse modes
Microscopic Description of Coherent Transport by Thermal Phonons
We demonstrate the coherent transport of thermal energy in superlattices by
introducing a microscopic definition of the phonon coherence length. We
demonstrate how to distinguish a coherent transport regime from diffuse
interface scattering and discuss how these can be specifically controlled by
several physical parameters. Our approach provides a convenient framework for
the interpretation of previous experiments and thermal conductivity
calculations and paves the way for the design of a new class of thermal
interface materials.Comment: 5 pages, 6 figures, 1 tables The method which is described is too
sensitive to numerical noise. A new method has been developed and published
in http://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.01430
Cop and robber game and hyperbolicity
In this note, we prove that all cop-win graphs G in the game in which the
robber and the cop move at different speeds s and s' with s'<s, are
\delta-hyperbolic with \delta=O(s^2). We also show that the dependency between
\delta and s is linear if s-s'=\Omega(s) and G obeys a slightly stronger
condition. This solves an open question from the paper (J. Chalopin et al., Cop
and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25
(2011) 333-359). Since any \delta-hyperbolic graph is cop-win for s=2r and
s'=r+2\delta for any r>0, this establishes a new - game-theoretical -
characterization of Gromov hyperbolicity. We also show that for weakly modular
graphs the dependency between \delta and s is linear for any s'<s. Using these
results, we describe a simple constant-factor approximation of the
hyperbolicity \delta of a graph on n vertices in O(n^2) time when the graph is
given by its distance-matrix
Election and rendezvous with incomparable labels
International audienceIn “Can we elect if we cannot compare” (SPAA'03), Barrière, Flocchini, Fraigniaud and Santoro consider a qualitative model of distributed computing, where the labels of the entities are distinct but mutually incomparable. They study the leader election problem in a distributed mobile environment and they wonder whether there exists an algorithm such that for each distributed mobile environment, it either states that the problem cannot be solved in this environment, or it successfully elects a leader. In this paper, we give a positive answer to this question. We also give a characterization of the distributed mobile environments where election and rendezvous can be solved
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