1-Safe Petri nets and special cube complexes: equivalence and applications

Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net $N$ unfolds into an event structure $\mathcal{E}_N$. 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 $N$ one can associate a finite special cube complex ${X}_N$ such that the domain of the event structure $\mathcal{E}_N$ (obtained as the unfolding of $N$) is a principal filter of the universal cover $\widetilde{X}_N$ of $X_N$. 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 $\mathcal{\dot E}_Z$ which arises from a virtually special square complex $\dot Z$. The domain of $\mathcal{\dot E}_Z$ is grid-free (because it is hyperbolic), but the MSO theory of the event structure $\mathcal{\dot E}_Z$ 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 $\natural$-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. 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 $\natural$-cliques) 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. 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

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

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 $\Delta$-matroids.Comment: 28 page

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

A Bridge Between the Asynchronous Message Passing Model and Local Computations in Graphs

International audienceA distributed system is a collection of processes that can interact. Three major process interaction models in distributed systems have principally been considered: - the message passing model, - the shared memory model, - the local computation model. In each model the processes are represented by vertices of a graph and the interactions are represented by edges. In the message passing model and the shared memory model, processes interact by communication primitives: messages can be sent along edges or atomic read/write operations can be performed on registers associated with edges. In the local computation model interactions are defined by labelled graph rewriting rules; supports of rules are edges or stars. These models (and their sub-models) reflect different system architectures, different levels of synchronization and different levels of abstraction. Understanding the power of various models, the role of structural network properties and the role of the initial knowledge enhances our understanding of basic distributed algorithms. This is done with some typical problems in distributed computing: election, naming, spanning tree construction, termination detection, network topology recognition, consensus, mutual exclusion. Furthermore, solutions to these problems constitute primitive building blocks for many other distributed algorithms. A survey may be found in [FR03], this survey presents some links with several parameters of the models including synchrony, communication media and randomization. An important goal in the study of these models is to understand some relationships between them. This paper is a contribution to this goal; more precisely we establish a bridge between tools and results presented in [YK96] for the message passing model and tools and results presented in [Ang80, BCG+96, Maz97, CM04, CMZ04, Cha05] for the local computation mode

Election and Local Computations on Edges

International audienceThe point of departure and the motivation for this paper are the results of Angluin [1] which has introduced a tool to analyze the election algorithm: the coverings, Yamashita and Kameda [21] and Mazurkiewicz [15] which have obtained characterizations of graphs in which election is possible under two different models of distributed computations. Our aim is twofold. First it is to obtain characterizations of graphs in which election is possible under intermediate models between the models of Yamashita-Kameda and of Mazurkiewicz. Our second aim is to understand the implications of the models for the borderline between positive and negative results for distributed computations. In this work, characterizations are obtained under three different models