No abstract

Three basic operations on labelled net structures are proposed: synchronised union, synchronised intersection and synchronised difference. The first of them is a version of known parallel composition with synchronised actions identically labelled. The operations work analogously to the ordinary union, intersection and difference on sets. It is shown that the universe of net structures with these operations is a distributive lattice and -if infinite pre/post sets of transitions are allowed -even a Boolean algebra. As a consequence, some representation theorems of this algebra are stated. The primitive objects are atomic net structures containing one transition with at most one pre-place or post-place (but not both). A simple example of a production system constructed by making use of the operations (and its transformations) is given. Some remarks on behavioural properties of compound nets are stated, in particular, how some constructing strategies may help to infer liveness. The latter issue is limited to semantics of place/transition nets without weights on arrows and with unbounded capacity of places and is not extensively investigated, since the main objective is focused on a calculus of net structures.

In this paper we define a certain class of process languages viewing processes as bipartite graphs with an associative operation (sequential composition) on them. They describe finite evolutions of Petri nets. When extended to sets, we get an ω-complete semiring such that rational, linear, and algebraic sets of such processes can be defined as least fixed points of systems of equations. With a norm of processes also iteration lemmata can be obtained. Finally, we also present a related structure of directed acyclic graphs.

Synthesis of elementary Petri net from a given process language (given by an expression in an algebra resembling the algebra of regular languages) is posed and solved. A necessary and sufficient condition for existence of such net is proved.

IntroductionIn [Cza 2006] fix-point equations specifying synchronous ("hand-shaking") communication in distributed systems have been proposed. Their solution yielded a communication network of agents, directly presented as a Petri net-like structure, and determined the global state of the specified system. The netplaces represented agents, while transitions -transfer of messages. A special algebra being a semi-ring with "addition" (nondeterministic choice) and "multiplication" (simultaneity) was a formal basis for the equations and their solving procedure. Here, the equations are modified to specify asynchronous communication, that is, such that the senders, after sending message, continue their performance without waiting for reception. This required introducing a new type of objects called buffers or mailboxesapart from the agents (senders/receivers). In the asynchronous communication, solution to the fix-point equations should determine that: (1) the sender can send message as soon as mailboxes of its simultaneous receivers can store the message, no matter whether the receivers are ready to get it or not. (2) in the resulting net, the mailboxes are included as net-places too, each one collecting messages from its senders (possibly a number of senders) and transferring them to its (exactly one) receiver for which it is the unique mailbox. The proposed modelling of communication takes some (but only some!) ideas from CSP [6], [7], CCS [8], [9], [10] (e.g. a concept of agents, ports, synchronization between senders and mailboxes or its absence: no synchronization between senders and receivers, communication media or channels), Petri nets [11] (e.g. graphical presentation of solution to the communication equations) or practice of computer networks and distributed systems [3] (e.g. multicasting and broadcasting, blocking/non-blocking, connection-oriented/connectionless communication mode). The communication equations may be treated as specifying interconnection system among agents, while net-structures resulted from their solution -as "implementation" of this system. The correctness of such implementation is ensured by identity of each group of receivers from a given sender (and, symmetrically, of each group of senders to a given receiver), both in specification and implementation.