Antoni W. Mazurkiewicz

Semantics of concurrent systems: a modular fixed-point trace approach

A method for finding the set of processes generated by a concurrent system (the behaviour of a system) in modular way is presented. A system is decomposed into modules with behaviours assumed to be known and then the behaviours are successively put together giving finally the initial system behaviour. It is shown that there is much of freedom in choice of modules; in extreme case atoms of a system, i.e. subsystems containing only one resource, can be taken as modules; each atom has its behaviour defined a proiri. The basic operation used for composing behaviours is the synchronization operation defined in the paper. The fixed point method of describing sets of processes is extensively applied, with processes regarded as traces rather than strings of actions.

Basic notions of trace theory

The concept of traces has been introduced for describing non-sequential behaviour of concurrent systems via its sequential observations. Traces represent concurrent processes in the same way as strings represent sequential ones. The theory of traces can be used as a tool for reasoning about nets and it is hoped that applying this theory one can get a calculus of the concurrent processes analogous to that available for sequential systems. The following topics will be discussed: algebraic properties of traces, trace models of some concurrency phenomena, fixed-point calculus for finding the behaviour of nets, modularity, and some applications of the presented theory.

Concurrent systems and inevitability

Abstract Concurrent systems viewed as partially ordered sets of states are considered. A property of system states is called inevitable, if the system will eventually reach a state with this property. This notion is discussed within the partial order framework.

Concurrency, Modularity, and Synchronization

There is a number of methods used for concurrent systems behaviour description. In the paper an algebra of place-transition nets is defined and chosen as a concurrent system description language; on the other hand, algebras of strings, of pomsets and of so-called multitrees, are presented and serve as examples of semantics algebras. The behaviour is then defined as a mapping from the description language into a semantic algebra; the behaviour is compositional (modular), if it is a homomorphism from the description language into the semantic algebra in question. In all above algebras the operation to be preserved by homomorphism is synchronization. It is claimed that the synchronization is a basic operation of composing event oriented concurrent systems. It is shown that all of these algebras have the same expressive power in describing the behaviour of nets.

Parallel recursive program schemes

In the paper a notion of parallel program scheme is introduced. Interpretations of such schemes are defined in two steps: first, independently of instructions meaning, to each scheme a language over the set of all instructions is assigned; second, for a given interpretation of instructions, to each such a language a function is assigned, being the intended interpretation of the scheme. This construction allows to obtain a satisfactory interpretation of parallel components of recursive schemes. The parallel composition of languages is introduced and some properties of this notion are given. It is shown how the theory of recursive procedures can be extended to procedures containing parallel operations. An example shows an application of the above approach.

Connectedness and synchronization

Given a description language for concurrent systems, the specification of its semantics amounts usually to assigning to each description an object from a hopefully well understood domain D whose elements are called semantical processes. Here is a general accepted “taxonomy” of event oriented processes: (1) Linear-process: a set of runs, where a run is a sequence of events (actions) [3]. (2) Pomset process: a set of node-labelled partial orders, henceforth referred to as pomsets [lo, 12, 51. (3) Automaton process: a rooted transition diagram whose edges are labelled by actions [7]. (4) Event structure [9]; also behavior structure [13]. This list reflects the alternatives: l “Linear” time (1,2) vs branching time (3,4). l Interleaving (1,3) vs causality (“true concurrency”) (2,4). The four domains support different levels of abstraction in semantical specifications: linear processes are the coarsest, whereas event structures (behavior structures), which integrate branching and causality are the most discriminating. On each of these levels synchronization is a basic operation of composing event oriented concurrent systems. In general, by synchronization we mean a composition

Petri nets without tokens

For more than 40 years Petri nets [1] serve as an efficient formal model of concurrent behavior of complex discrete systems. There exists a rich bibliography of books and works devoted to these methods and many applications of nets have been already created. The model is extremely simple: it uses three basic concepts, of places, of transitions, and of a flow relation. The behavior of a net is represented by changing distribution of tokens situated in the net places, according to some simple rules (so-called firing rules). Non-negative integers play essential role in the description of tokens distribution, indicating the number of tokens contained in nets places and . Transitions determine way of changing the distribution, taking off a number of tokens from entry places and putting a number of tokens in exit places of an active transition. Formally, to any transition some operations on numbers stored in places are assigned and therefore the behavior of nets is described by means of a simple arithmetic with adding or subtracting operations on non-negative integers.

Distributed disassembly of mosaics

Abstract Mosaics are planar graphs with vertices labelled with nonnegative integers. In a mosaic, vertices labelled with 0 represent empty places (holes); vertices labelled with positive integers are interpreted as places occupied by pieces with colour indicated by their labels. A distributed algorithm for choosing a single piece of mosaic is given, provided the mosaic is finite, triangular, and has nonempty and connected sets of holes and pieces. Correctness of the algorithm of removing pieces from mosaics is proved.