65 research outputs found
Nice labeling problem for event structures: a counterexample
In this note, we present a counterexample to a conjecture of Rozoy and
Thiagarajan from 1991 (called also the nice labeling problem) asserting that
any (coherent) event structure with finite degree admits a labeling with a
finite number of labels, or equivalently, that there exists a function such that an event structure with degree
admits a labeling with at most labels. Our counterexample is based on
the Burling's construction from 1965 of 3-dimensional box hypergraphs with
clique number 2 and arbitrarily large chromatic numbers and the bijection
between domains of event structures and median graphs established by
Barth\'elemy and Constantin in 1993
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
A Counterexample to Thiagarajan\u27s Conjecture on Regular Event Structures
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular prime event structures correspond exactly to those 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 natural-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 and both conjectures can be reformulated in this framework. Namely, the domains of prime event structures correspond exactly to pointed median graphs; from a geometric point of view, these domains are in bijection with pointed CAT(0) cube complexes.
A necessary condition for both conjectures to be true is that domains of respective regular event structures admit a regular nice labeling. To disprove these conjectures, we describe a regular event domain (with bounded natural-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
A Polynomial Time Algorithm to Compute Geodesics in CAT(0) Cubical Complexes
This paper presents the first polynomial time algorithm to compute geodesics in a CAT(0) cubical complex in general dimension. The algorithm is a simple iterative method to update breakpoints of a path joining two points using Miller, Owen and Provan\u27s algorithm (Adv. in Appl. Math, 2015) as a subroutine. Our algorithm is applicable to any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes
On embeddings of CAT(0) cube complexes into products of trees
We prove that the contact graph of a 2-dimensional CAT(0) cube complex of maximum degree can be coloured with at most
colours, for a fixed constant . This implies
that (and the associated median graph) isometrically embeds in the
Cartesian product of at most trees, and that the event
structure whose domain is admits a nice labeling with
labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire
- …