Relational Semantics of Non-Deterministic Dataflow

We recast dataflow in a modern categorical light using profunctors as a generalization of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preservemuch of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially theusual axioms of monotone IO automata are read off from the definition of profunctors, (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps and (3) a treatment of higher-order dataflow as a biproduct,essentially by following the geometry of interaction programme

Unfolding-Based Process Discovery

This paper presents a novel technique for process discovery. In contrast to the current trend, which only considers an event log for discovering a process model, we assume two additional inputs: an independence relation on the set of logged activities, and a collection of negative traces. After deriving an intermediate net unfolding from them, we perform a controlled folding giving rise to a Petri net which contains both the input log and all independence-equivalent traces arising from it. Remarkably, the derived Petri net cannot execute any trace from the negative collection. The entire chain of transformations is fully automated. A tool has been developed and experimental results are provided that witness the significance of the contribution of this paper.Comment: This is the unabridged version of a paper with the same title appearead at the proceedings of ATVA 201

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

On the construction of pullbacks for safe Petri nets

The product of safe Petri nets is a well known operation: it generalizes to concurrent systems the usual synchronous product of automata. In this short note, we consider the definition of pullbacks of safe PNs, another categorical construction. Pullbacks generalize the product to nets which interact both by synchronized transitions and by a shared sub-net. \\ Le produit de réseaux de Petri saufs (éventuellement à labels) est une opération bien connue~: on peut la voir comme une généralisation du produit synchrone d'automates à des systèmes concurrents. Dans cette note, on s'intéresse à la construction de pullbacks de réseaux saufs, une autre construction catégorique. Les pullbacks généralisent le produit de réseaux en permettant une interaction non seulement par la synchronisation de transitions, mais aussi par partage de places et de transitions

A Linear Specification Language for Petri Nets

This paper defines a category GNet with object set all Petri nets. A morphism in GNet from a net N to a net N' gives a precise way of simulating every evolution of N by an evolution of N'. We exhibit a morphism from a simple message handler to one with error-correction, showing that the more refined message handler can simulate any behaviour of its simple counterpart. The existence of such a morphism proves the correctness of the refinement

Profunctors, Open Maps and Bisimulation

This paper studies fundamental connections between profunctors (i.e., distributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells. A guiding idea is the view that profunctors, and colimit preserving functors, are linear maps in a model of classical linear logic. But profunctors, and colimit preserving functors, as linear maps, are too restrictive for many applications. This leads to a study of a range of pseudo-comonads and how non-linear maps in their co-Kleisli bicategories preserve open maps and bisimulation. The pseudo-comonads considered are based on finite colimit completion, ``lifting'', and indexed families. The paper includes an appendix summarising the key results on coends, left Kan extensions and the preservation of colimits. One motivation for this work is that it provides a mathematical framework for extending domain theory and denotational semantics of programming languages to the more intricate models, languages and equivalences found in concurrent computation. But the results are likely to have more general applicability because of the ubiquitous nature of profunctors

Models for Concurrency

Revised version of DAIMI PB-429 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

An algebra of high level petri nets

PhD ThesisPetri nets were introduced by C.A. Petri as a theoretical model of concurrency in which the causal relationship between actions, rather than just their temporal ordering, can be represented. As a theoretical model of concurrency, Petri nets have been widely successful. Moreover, Petri nets are popular with practitioners, providing practical tools for the designer and developer of real concurrent and distributed systems. However, it is from this second context that perhaps the most widely voiced criticism of Petri nets comes. It is that Petri nets lack any algebraic structure or modularity, and this results in large, unstructured models of real systems, which are consequently often intractable. Although this is not a criticism of Petri nets per se, but rather of the uses to which Petri nets are put, the criticism is well taken. We attempt to answer this criticism in this work. To do this we return to the view of Petri nets as a model of concurrency and consider how other models of concurrency counter this objection. The foremost examples are then the synchronisation trees of Milner, and the traces of Hoare, (against which such criticism is rarely, if ever, levelled). The difference between the models is clear, and is to be found in the richness of the algebraic characterisations which have been made for synchronisation trees in Milner's Calculus of Communicating Systems (CCS), and for traces in Hoare's Communicating Sequential Processes (CSP). With this in mind we define, in this thesis, a class of high level Petri nets, High Level Petri Boxes, and provide for them a very general algebraic description language, the High Level Petri Box Algebra, with novel ideas for synchronisation, and including both refinement and recursion among its operators. We also begin on the (probably open-ended task of the) algebraic characterisation of High Level Petri Boxes. The major contribution of this thesis is a full behavioural characterisation of the High Level Petri Boxes which form the semantic domain of the algebra. Other contributions are: a very general method of describing communication protocols which extend the synchronisation algebras of Winskel; a recursive operator that preserves finiteness of state (the best possible, given the generality of the algebra); a refinement operator that is syntactic in nature, and for which the recursive construct is a behavioural fix-point; and a notion of behavioural equivalence which is a congruence with respect to a major part of the High Level Petri Box Algebra