311 research outputs found
Strongly primitive species with potentials I: Mutations
Motivated by the mutation theory of quivers with potentials developed by
Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster
algebras it provides, we propose a mutation theory of species with potentials
for species that arise from skew-symmetrizable matrices that admit a
skew-symmetrizer with pairwise coprime diagonal entries. The class of
skew-symmetrizable matrices covered by the mutation theory proposed here
contains a class of matrices that do not admit global unfoldings, that is,
unfoldings compatible with all possible sequences of mutations.Comment: 51 page
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
Petri nets, probability and event structures
Models of true concurrency have gained a lot of interest over the last decades as models
of concurrent or distributed systems which avoid the well-known problem of state
space explosion of the interleaving models. In this thesis, we study such models from
two perspectives.
Firstly, we study the relation between Petri nets and stable event structures. Petri nets
can be considered as one of the most general and perhaps wide-spread models of true
concurrency. Event structures on the other hand, are simpler models of true concurrency
with explicit causality and conflict relations. Stable event structures expand the
class of event structures by allowing events to be enabled in more than one way. While
the relation between Petri nets and event structures is well understood, the relation between
Petri nets and stable event structures has not been studied explicitly. We define
a new and more compact unfoldings of safe Petri nets which is directly translatable
to stable event structures. In addition, the notion of complete finite prefix is defined
for compact unfoldings, making the existing model checking algorithms applicable to
them. We present algorithms for constructing the compact unfoldings and their complete
finite prefix.
Secondly, we study probabilistic models of true concurrency. We extend the definition
of probabilistic event structures as defined by Abbes and Benveniste to a newly defined
class of stable event structures, namely, jump-free stable event structures arising
from Petri nets (characterised and referred to as net-driven). This requires defining
the fundamental concept of branching cells in probabilistic event structures, for jump-free
net-driven stable event structures, and by proving the existence of an isomorphism
among the branching cells of these systems, we show that the latter benefit from the
related results of the former models. We then move on to defining a probabilistic
logic over probabilistic event structures (PESL). To our best knowledge, this is the first
probabilistic logic of true concurrency. We show examples of expressivity achieved by
PESL, which in particular include properties related to synchronisation in the system.
This is followed by the model checking algorithm for PESL for finite event structures.
Finally, we present a logic over stable event structures (SEL) along with an account of
its expressivity and its model checking algorithm for finite stable event structures
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
Stability of pullbacks of foliations on weighted projective spaces
We show a stability-type theorem for foliations on projective spaces which
arise as pullbacks of foliations with a split tangent sheaf on weighted
projective spaces. As a consequence, we will be able to construct many
irreducible components of the corresponding spaces of foliations, most of them
being previously unknown. This result also provides an alternative and unified
proof for the stability of other families of foliations
On the Transformation of Petri Nets into BPMN Models
Antud magistritöö käsitleb Petri võrkude teisendamist samaväärseteksBPMN mudeliteks. Täpsemalt öeldes keskendub see Petri võrkude alamklassilenimega töövoo võrgud. Antud lõputöös implementeeriti teisendaja, kasutadesselleks mitmeid tehnikaid nagu näiteks Petri võrkude lahti pakkimineja modulaarsed dekompositsiooni puud. Sellest tulenevalt pakub antud magistritöö välja täieliku teisendusalgoritmi, mis suudab käsitleda sümmeetrilisi segadusi Petri võrkudes. See on antud valdkonnas üks esimesi teisendamise meetodeid, mis katab ka seda klassi. Hetkel oleme teadlikud ainult ühest teosest, mis illustreerib mõlemasuunalist teisendamist töövoo võrkude ja graafide vahel. Lisaks, esitleme me käitumissõltuvuste maatriksi arvutamise meetodi.Käsitleme ka erijuhtumeid, kus peame BPMN-is lisama tau sündmuse, ettegemist oleks samakujulise mudeliga.This thesis addresses the problem of translating a Petri net into an equivalentBPMN process model. This is fundamental problem with implications on theunderstanding of the semantics of the notation and that has potential applications in areas such process model discovery from event logs and structuring of process models. In previous work, it has been shown that the well-known family of free-choice Petri nets can be bidirectionally mapped into the subset of BPMN process models constructed solely with tasks and exclusive/parallel getaways. In contrast, this work searches at lifting the restriction to a larger family of Petri nets by proposing a translation that covers also the case of nets with symmetric confusion. The approach has been implemented in a prototype which has allowed us to conduct a preliminary performance study
Thermodynamics and Self-Gravitating Systems
This work assembles some basic theoretical elements on thermal equilibrium,
stability conditions, and fluctuation theory in self-gravitating systems
illustrated with a few examples. Thermodynamics deals with states that have
settled down after sufficient time has gone by. Time dependent phenomena are
beyond the scope of this paper. While thermodynamics is firmly rooted in
statistical physics, equilibrium configurations, stability criteria and the
destabilizing effect of fluctuations are all expressed in terms of
thermodynamic functions. The work is not a review paper but a pedagogical
introduction which may interest theoreticians in astronomy and astrophysicists.
It contains sufficient mathematical details for the reader to redo all
calculations. References are only to seminal works or readable reviews. (Except
in the ultimate sub-section.) Delicate mathematical problems are mentioned but
are not discussed in detail.Comment: 35 pages,6 figure
Cluster Algebras: Network Science and Machine Learning
Cluster algebras have recently become an important player in mathematics and
physics. In this work, we investigate them through the lens of modern data
science, specifically with techniques from network science and
machine-learning. Network analysis methods are applied to the exchange graphs
for cluster algebras of varying mutation types. The analysis indicates that
when the graphs are represented without identifying by permutation equivalence
between clusters an elegant symmetry emerges in the quiver exchange graph
embedding. The ratio between number of seeds and number of quivers associated
to this symmetry is computed for finite Dynkin type algebras up to rank 5, and
conjectured for higher ranks. Simple machine learning techniques successfully
learn to differentiate cluster algebras from their seeds. The learning
performance exceeds 0.9 accuracies between algebras of the same mutation type
and between types, as well as relative to artificially generated data.Comment: 38 pages, 27 figure
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