95 research outputs found
Formal Relationships Between Geometrical and Classical Models for Concurrency
A wide variety of models for concurrent programs has been proposed during the
past decades, each one focusing on various aspects of computations: trace
equivalence, causality between events, conflicts and schedules due to resource
accesses, etc. More recently, models with a geometrical flavor have been
introduced, based on the notion of cubical set. These models are very rich and
expressive since they can represent commutation between any bunch of events,
thus generalizing the principle of true concurrency. While they seem to be very
promising - because they make possible the use of techniques from algebraic
topology in order to study concurrent computations - they have not yet been
precisely related to the previous models, and the purpose of this paper is to
fill this gap. In particular, we describe an adjunction between Petri nets and
cubical sets which extends the previously known adjunction between Petri nets
and asynchronous transition systems by Nielsen and Winskel
Marked petri nets within a categorial framework
Well know categories of Petri nets lack coproducts and some re strictioné on nets, morphisms or initial markings are required in or der to guarantee the existence of colimits. Categories of Petri nets equipped with a set of initial markings (instead of a single initial marking) are introduced. It is shown that the proposed categories of nets are complete and cocomplete. Moreover,interpretations of limits and colimits are adequate for expressing semantics of concurrent sys tems. Examples ofstructuring and modeling of behavior of nets using categoria! constructions based on limits and colimits are provided
Models for Concurrency
Revised version of DAIMI PB-429Â This is, we believe, the final version of a chapter for the Handbook of Logic and the Foundations of Computer Science, vol. IV, Oxford University Press.It surveys a range of models for parallel computation to include interleaving models like transition systems, synchronisation trees and languages (often called Hoare traces in this context), and models like Petri nets, asynchronous transition systems, event structures, pomsets and Mazurkiewicz traces where concurrency is represented more explicitly by a form of causal independence.The presentation is unified by casting the models in a category-theoretic framework. One aim is to use category theory to provide abstract characterisations of constructions like parallel composition valid throughout a range of different models and to provide formal means for translating between different models. A knowledge of basic category theory is assumed, up to an acquaintance with the notion of adjunction
A categorical framework for concurrent, anticipatory systems
A categorical semantic domain is constructed for Petri nets which satisfies the diagonal compositionality requirement with respect to anticipations, i.e., Petri nets are equipped with a compositional anticipation mechanism (vertical compositionality) that distributes through net combinators (horizontal compositionality). The anticipation mechanism is based on graph transformations (single pushout approach). A finitely bicomplete category of partial Petri nets and partial morphisms is introduced. Classes of transformations stand for anticipations. The composition of anticipations (i.e., composition of pushouts) is defined, leading to a category of nets and anticipations which is also complete and cocomplete. Since the anticipation operation composes, the vertical compositionality requirement of Petri nets is achieved. Then, it is proven that the anticipation also satisfies the horizontal compositionality requirement. A specification grammar stands for a system specification and the corresponding induced subcategory of nets and anticipation's stands for ali possible dynamic anticipation's ofthe system (objects) and their relationship (morphims)
Homology groups of cubical sets
The paper is devoted to homology groups of cubical sets with coefficients in
contravariant systems of Abelian groups. The study is based on the proof of the
assertion that the homology groups of the category of cubes with coefficients
in the diagram of Abelian groups are isomorphic to the homology groups of
normalized complex of the cubical Abelian group corresponding to this diagram.
The main result shows that the homology groups of a cubical set with
coefficients in a contravariant system of Abelian groups are isomorphic to the
values of left derived functors of the colimit functor on this contravariant
system. This is used to obtain the isomorphism criterion for homology groups of
cubical sets with coefficients in contravariant systems, and also to construct
spectral sequences for the covering of a cubical set and for a morphism between
cubical sets.Comment: 24 page
Higher Dimensional Transition Systems
We introduce the notion of higher dimensional transition systems as a model of concurrency providing an elementary, set-theoretic formalisation of the idea of higher dimensional transition. We show an embedding of the category of higher dimensional transition systems into that of higher dimensional automata which cuts down to an equivalence when we restrict to non-degenerate automata. Moreover, we prove that the natural notion of bisimulation for such structures is a generalisation of the strong history preserving bisimulation, and provide an abstract categorical account of it via open maps. Finally, we define a notion of unfolding for higher dimensional transition systems and characterise the structures so obtained as a generalisation of event structures
Homotopy Bisimilarity for Higher-Dimensional Automata
We introduce a new category of higher-dimensional automata in which the
morphisms are functional homotopy simulations, i.e. functional simulations up
to concurrency of independent events. For this, we use unfoldings of
higher-dimensional automata into higher-dimensional trees. Using a notion of
open maps in this category, we define homotopy bisimilarity. We show that
homotopy bisimilarity is equivalent to a straight-forward generalization of
standard bisimilarity to higher dimensions, and that it is finer than split
bisimilarity and incomparable with history-preserving bisimilarity.Comment: Heavily revised version of arXiv:1209.492
Bisimulations and Unfolding in P-Accessible Categorical Models
In this paper, we propose a categorical framework for bisimulations and unfoldings that unifies the classical approach from Joyal and al. via open maps and unfoldings. This is based on a notion of categories accessible with respect to a subcategory of path shapes, i.e., for which one can define a nice notion of trees as glueing of paths. We prove that transitions systems and pre sheaf models are a particular case of our framework. We also prove that in our framework, several characterizations of bisimulation coincide, in particular an "operational one" akin to the standard definition in transition systems. Also, accessibility is preserved by coreflexions. We then design a notion of unfolding, which has good properties in the accessible case: its is a right adjoint and is a universal covering, i.e., initial among the morphisms that have the unique lifting property with respect to path shapes. As an application, we prove that the universal covering of a groupoid, a standard construction in algebraic topology, coincides with an unfolding, when the category of path shapes is well chosen
- …