114 research outputs found
On the Category of Petri Net Computations
We introduce the notion of strongly concatenable process as a refinement of concatenable processes [DMM89] which can be expressed axiomatically via a functor from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net , the strongly concatenable processes of are isomorphic to the arrows of . In addition, we identify a coreflection right adjoint to and characterize its replete image, thus yielding an axiomatization of the category of net computations
Modelling Concurrency with Comtraces and Generalized Comtraces
Comtraces (combined traces) are extensions of Mazurkiewicz traces that can
model the "not later than" relationship. In this paper, we first introduce the
novel notion of generalized comtraces, extensions of comtraces that can
additionally model the "non-simultaneously" relationship. Then we study some
basic algebraic properties and canonical reprentations of comtraces and
generalized comtraces. Finally we analyze the relationship between generalized
comtraces and generalized stratified order structures. The major technical
contribution of this paper is a proof showing that generalized comtraces can be
represented by generalized stratified order structures.Comment: 49 page
Functorial Semantics for Petri Nets under the Individual Token Philosophy
Although the algebraic semantics of place/transition Petri nets under the collective token philosophy has been fully explained in terms of (strictly) symmetric (strict) monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory because it lacks universality and also functoriality. We introduce the notion of pre-net to recover these aspects, obtaining a fully satisfactory categorical treatment centered on the notion of adjunction. This allows us to present a purely logical description of net behaviours under the individual token philosophy in terms of theories and theory morphisms in partial membership equational logic, yielding a complete match with the theory developed by the authors for the collective token view of net
An Approach to the Category of Net Computations
We introduce the notion of strongly concatenable process as a refinement of concatenable processes [DMM89] which can be expressed axiomatically via a functor from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net , the strongly concatenable processes of are isomorphic to the arrows of . In addition, we identify a coreflection right adjoint to and characterize its replete image, thus yielding an axiomatization of the category of net computations
Petri Nets and Other Models of Concurrency
This paper retraces, collects, and summarises contributions of the authors --- in collaboration with others --- on the theme of Petri nets and their categorical relationships to other models of concurrency
Axiomatizing Petri Net Concatenable Processes
The concatenable processes of a Petri net can be characterized abstractly as the arrows of a symmetric monoidal category . Yet, this is only a partial axiomatization, since is built on a concrete, ad hoc chosen, category of symmetries. In this paper we give a fully equational description of the category of concatenable processes of , thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets
Event structures for Petri nets with persistence
Event structures are a well-accepted model of concurrency. In a seminal paper by Nielsen, Plotkin and Winskel, they are used to establish a bridge between the theory of domains and the approach to concurrency proposed by Petri. A basic role is played by an unfolding construction that maps (safe) Petri nets into a subclass of event structures, called prime event structures, where each event has a uniquely determined set of causes. Prime event structures, in turn, can be identified with their domain of configurations. At a categorical level, this is nicely formalised by Winskel as a chain of coreflections. Contrary to prime event structures, general event structures allow for the presence of disjunctive causes, i.e., events can be enabled by distinct minimal sets of events. In this paper, we extend the connection between Petri nets and event structures in order to include disjunctive causes. In particular, we show that, at the level of nets, disjunctive causes are well accounted for by persistent places. These are places where tokens, once generated, can be used several times without being consumed and where multiple tokens are interpreted collectively, i.e., their histories are inessential. Generalising the work on ordinary nets, Petri nets with persistence are related to a new subclass of general event structures, called locally connected, by means of a chain of coreflections relying on an unfolding construction
Strong Concatenable Processes: An Approach to the Category of Petri Net Computations
We introduce the notion of strong concatenable process for Petri nets as the least refinement of non-sequential (concatenable) processes which can be expressed abstractly by means of a functor Q[_] from the category of Petri nets to an appropriate category of symmetric strict monoidal categories with free non-commutative monoids of objects, in the precise sense that, for each net N, the strong concatenable processes of N are isomorphic to the arrows of Q[N]. This yields an axiomatization of the causal behaviour of Petri nets in terms of symmetric strict monoidal categories. In addition, we identify a coreflection right adjoint to Q[_] and we characterize its replete image in the category of symmetric monoidal categories, thus yielding an abstract description of the category of net computations
Asynchronous wreath product and cascade decompositions for concurrent behaviours
We develop new algebraic tools to reason about concurrent behaviours modelled
as languages of Mazurkiewicz traces and asynchronous automata. These tools
reflect the distributed nature of traces and the underlying causality and
concurrency between events, and can be said to support true concurrency. They
generalize the tools that have been so efficient in understanding, classifying
and reasoning about word languages. In particular, we introduce an asynchronous
version of the wreath product operation and we describe the trace languages
recognized by such products (the so-called asynchronous wreath product
principle). We then propose a decomposition result for recognizable trace
languages, analogous to the Krohn-Rhodes theorem, and we prove this
decomposition result in the special case of acyclic architectures. Finally, we
introduce and analyze two distributed automata-theoretic operations. One, the
local cascade product, is a direct implementation of the asynchronous wreath
product operation. The other, global cascade sequences, although conceptually
and operationally similar to the local cascade product, translates to a more
complex asynchronous implementation which uses the gossip automaton of Mukund
and Sohoni. This leads to interesting applications to the characterization of
trace languages definable in first-order logic: they are accepted by a
restricted local cascade product of the gossip automaton and 2-state
asynchronous reset automata, and also by a global cascade sequence of 2-state
asynchronous reset automata. Over distributed alphabets for which the
asynchronous Krohn-Rhodes theorem holds, a local cascade product of such
automata is sufficient and this, in turn, leads to the identification of a
simple temporal logic which is expressively complete for such alphabets
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