Lucia Pomello

A survey of equivalence notions for net based systems

This paper surveys various notions of equivalence for concurrent systems in the framework of Elementary Net Systems, a fundamental class in the family of Petri Net models. Two types of equivalences are considered: equivalences based on observations of actions defined in the framework of interleaving, step and partial order semantics; and equivalences based on state spaces and state observability.

Some equivalence notions for concurrent systems. An overview

String-equivalence, Behaviour-equivalence, Observation-equivalence, Exhibited-Behaviour-equivalence, Failure-equivalence and Testing equivalence are re-defined and discussed on Petri Net models of concurrent systems. Since the previous notions, being based on sequences of events, do not distinguish real concurrency from its sequential non-deterministic simulation, they are refined, in order to capture the concurrency degree of the systems, in terms of the subset languages approach. The equivalence notions introduced are then compared w.r.t. their strength.

Superposed Automata Nets

The capability of representing in an highly communicable way the concurrency among autonomous subsystems is frequently crucial in real systems modelling, whichever the nature of the system can be.

Modelling mobility with petri hypernets

Petri hypernets, a novel framework for modeling mobile agents based on nets-within-nets paradigm is presented. Hypernets employ a local and finitary character of interactions between agents, and provide means for a modular and hierarchical description. They are capable of modelling mobile agents tfrahat can dynamically change their hierarchy, and can communicate with each other and with the outside world by exchanging messages, i.e., other mobile agents.

A state transformation preorder over a class of EN systems

The paper introduces a notion of preorder between Elementary Net Systems which is based on the observability of places. This latter allows one to define the notion of Observable Local States whose transformations are the main concern of the proposed preorder, called State Transformation (ST) preorder. ST preorder compares systems with a different level of granularity in local states transformation by requiring an injective morphism between their algebraic structures of observable local states. In addition, it allows one to define in a standard way a notion of ST equivalence. This latter is related to a previous notion, EF equivalence, for which the existence of canonical representatives has been previously proved. Finally, ST preorder and equivalence over a subclass of EN systems (the S-observable systems) are discussed in the framework of system development.

Modular system development with pullbacks

Two, seemingly different modular techniques for concurrent system development are investigated from a categorical perspective. A novel approach is presented in which they turn out to be merely special instances of pullback, a general categorical limit construction. Interestingly, the approach is based on truly concurrent semantics of systems.

MILNER'S COMMUNICATING SYSTEMS AND PETRI NETS

The comparison between Milner’s CCS and Petri nets leads to prove that CCS defines a subclass of the concurrent systems definable by means of net theory, i.e. the class of systems composed of interacting sequential automata. The proof is based on (the construction of) an isomorphism between Milner’s communicating systems class and a subclass of Petri nets.The paper fully presents the proof for a subclass of communicating systems (where neither value passing nor finite unlimited instances of agents are allowed) and a specific subclass of nets, the SA2 PT class (a restriction of the Superposed Automata nets class). Some hints how to extend the proof to the whole CCS conclude the paper. This result allows to transfer analytic concepts and techniques from a theory to the other, as for example the observation equivalence notion defined inside the CCS. The paper is fully self-consistent.

Formal Tools for Modular System Development

The dualities event–condition and local state–global state in basic net theory are exploited in order to equip system designers with formal tools supporting modular system development. In the framework of categories with suitable morphisms of Elementary Net systems, Elementary Transition systems and Orthomodular posets of system local states (regions), some operations of composition and refinement are discussed as tools for modularity and abstraction.

Observation equivalences for the semantics of inheritance

In the object-oriented languages, inheritance is a fundamental relation among classes, that originally indicated the is-a relation; however it has been often used for code-reuse. The integration between object-orientation and concurrency gives the possibility of using theoretical tools, such as the notions of action and state observation equivalences and preorders, developed within concurrency theories (Petri nets, CCS). In this paper we propose a semantic characterisation of two forms of inheritance in concurrent context: in particular, we propose a preorder based on action observation to give the semantics of inheritance used as is-a relation (type inheritance), and a preorder based on state observation for the semantics of inheritance as code-reuse (implementation inheritance).

An algebraic model of observable properties in distributed systems

We propose orthomodular posets, algebraic models of quantum logic, as a formal tool in concurrency theory. We discuss their characteristics and study mutual relations with two other models of distributed systems: condition event net systems, a basic class of Petri nets, and the transition systems modelling CE net system behaviour. Central results are an adjointness situation among the three models and a strict relationship between fundamental notions in the different considered frameworks such as the relations of incompatibility and concurrency. Furthermore, substructures of orthomodular posets, like Boolean subalgebras or centres are interpreted, respectively, as state machine components of CE net systems or synchronization structures.