Compositional Reachability in Petri Nets

Rathke, Julian; Sobociński, Paweł; Stephens, Owen

doi:10.1007/978-3-319-11439-2_18

Cited by 13 publications

(15 citation statements)

References 26 publications

(38 reference statements)

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“…Note also that the definition of monoidal width is not tied to ABUV and makes sense in any monoidal theory, which solves the problem of generality outlined in the Introduction. Our claims are that (i) ABUV is a canonical, compositional algebra of simple graphs, and (ii) monoidal width is a robust concept that we believe will be useful in a number of different applications areas [1,4,7,10,19] where monoidal theories are used.…”

Section: Example 58 [Clique]mentioning

confidence: 99%

“…Walters and collaborators [11,12]. Although we do not concentrate on applications in this paper, we believe that our theory will be relevant in several settings as there has been a recent surge in the applications of symmetric monoidal theories: amongst other works we mention signal flow graphs [1,4], Petri nets [19][20][21], asynchronous circuits [10] and quantum circuits [5,7].…”

Section: Introductionmentioning

confidence: 95%

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Towards Compositional Graph Theory

Chantawibul

Sobociński

2015

Electronic Notes in Theoretical Computer Science

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Decomposing graphs into simpler graphs is one of the central concerns of graph theory. Investigations have revealed deep concepts such as modular decomposition, tree width or rank width, which measure-in different ways-the structural complexity of a graph's topology. Courcelle and others have shown that such concepts can be used to obtain efficient algorithms for families of graphs that are amenable to decomposition (e.g. those that have bounded tree-width). These algorithms, in turn, are of course of use in computer science, where graphs are ubiquitous. In this paper we take the first steps towards understanding notions of decomposition in graph theory compositionally, and more generally, in a categorical setting: category theory, after all, is the mathematics of compositionality. We introduce the concept of ∪−matrices (cup-matrices). Like ordinary matrices, ∪-matrices are the arrows of a PROP: we give a presentation, extending the work of Lafont, and Bonchi, Zanasi and the second author. A variant of ∪-matrices is then used in the development of a novel algebra of simple graphs, the lingua franca of graph theory. The algebra is that of a certain symmetric monoidal theory: ∪-matrices-akin to adjacency matrices-encode the graphs' topology.

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Section: Example 58 [Clique]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Towards Compositional Graph Theory

Chantawibul

Sobociński

2015

Electronic Notes in Theoretical Computer Science

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“…Using compositionality, the automaton for (10) can be obtained by composing (11) with the automaton ? 0/ε 1/ε (13) which is the semantics of the rightmost component in the composition (10).…”

Section: Compositional Reachability Checkingmentioning

confidence: 99%

“…In several of these Penrose, unsurprisingly, beats its competitors which do not have access and do not take advantage of the high-level component-wise specification. See [11,13] for a detailed account of the experimental results. In fact, many (most?)…”

Section: Compositional Reachability Checkingmentioning

confidence: 99%

Compositional model checking of concurrent systems, with Petri nets

Sobociński

2016

Electron. Proc. Theor. Comput. Sci.

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Compositionality and process equivalence are both standard concepts of process algebra. Compositionality means that the behaviour of a compound system relies only on the behaviour of its components, i.e. there is no emergent behaviour. Process equivalence means that the explicit statespace of a system takes a back seat to its interaction patterns: the information that an environment can obtain though interaction.Petri nets are a classical, widely used and understood, model of concurrency. Nevertheless, they have often been described as a non-compositional model, and tools tend to deal with monolithic, globally-specified models.This tutorial paper concentrates on Petri Nets with Boundaries (PNB): a compositional, graphical algebra of 1-safe nets, and its applications to reachability checking within the tool Penrose. The algorithms feature the use of compositionality and process equivalence, a powerful combination that can be harnessed to improve the performance of checking reachability and coverability in several common examples where Petri nets model realistic concurrent systems.

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“…Recently, more attention has been paid to a compositional treatment in which Petri nets can be assembled from smaller "open" Petri nets (Baez and Pollard 2017;Baldan et al 2005Baldan et al , 2015Bruni et al 2011Bruni et al , 2013Bruni et al , 2001). In particular, the reachability problem for Petri nets, which asks whether one marking of a Petri net can be obtained from another via a sequence of transitions, can be studied compositionally (Rathke et al 2014;Sassone and Sobociński 2005;Sobociński and Stephens 2013). Here, we seek to give this line of work a firmer footing in category theory.…”

Section: Introductionmentioning

confidence: 99%

Open Petri nets

Baez

Master

2020

Math. Struct. Comp. Sci.

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The reachability semantics for Petri nets can be studied using open Petri nets. For us an 'open' Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category Open(Petri), which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category Open(Petri). Various choices of semantics for open Petri nets can be described using symmetric monoidal double functors out of Open(Petri). Here we describe the reachability semantics, which assigns to each open Petri net the relation saying which markings of the outputs can be obtained from a given marking of the inputs via a sequence of transitions. We show this semantics gives a symmetric monoidal lax double functor from Open(Petri) to the double category of relations. A key step in the proof is to treat Petri nets as presentations of symmetric monoidal categories; for this we use the work of Meseguer, Montanari, Sassone and others.

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Compositional Reachability in Petri Nets

Cited by 13 publications

References 26 publications

Towards Compositional Graph Theory

Towards Compositional Graph Theory

Compositional model checking of concurrent systems, with Petri nets

Open Petri nets

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