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1

Behaviour algebras

100%

Winkowski J.

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tom Nr 992

1-36

EN

The paper is concerned with algebras whose elements can be used to represent runs of a system, called processes. These algebras, called behaviour algebras, are categories with respect to a partial binary operation called seąuential composition, and they arę partial monoids with respect to a partial binary operation called parallel composition. They are characterized by axioms such that their elements and operations can be represented as specific pomsets and specific operations on such pomsets. The respective representation theorem is universal in the sense that it is obtained without assuming a discrete nature of represented elements. In particular, it remains true for behaviour algebras with infinitely divisible elements, and thus also with elements which can represent continuous processes. An important consequence of the representation theorem is that elements of some subalgebras of behaviour algebras can be endowed in a consistent way with structures such as a graph structure etc.

PL

Praca dotyczy algebr częściowych, których elementy mogą służyć do reprezentowania procesów, tzn. przebiegów zachowania się systemów. Algebry te, zwane algebrami zachowań, są kategoriami ze względu na częściową operację binarną składania szeregowego, oraz są monoidami częściowymi ze względu na częściową operację binarną składania równoległego. Zostały one scharakteryzowane aksjomatycznie tak, że ich elementy i operacje można reprezentować klasami izomorficznych poetykietowanych zbiorów częściowo uporządkowanych i operacjami na takich klasach. Orzekające o tym twierdzenie o reprezentacji jest uniwersalne w tym sensie, że zostało uzyskane bez zakładania dyskretnej natury reprezentowanych elementów. W szczególności pozostaje prawdziwe dla algebr zachowań z nieskończenie podzielnymi elementami, które mogą reprezentować procesy ciągłe. Ważną konsekwencją tego twierdzenia jest też to, że elementy pewnych podalgebr algebr zachowań można wyposażać sensownie w struktury takie jak struktury grafowe itp.

2

Behaviour Algebras

100%

Winkowski J.

|

tom Vol. 75, nr 1-4

537-560

EN

The paper is concerned with algebras whose elements can be used to represent runs of a system, called processes. These algebras, called behaviour algebras, are categories with respect to a partial binary operation called sequential composition, and they are partial monoids with respect to a partial binary operation called parallel composition. They are characterized by axioms such that their elements and operations can be represented by labelled posets and operations on such posets. The respective representation is obtained without assuming a discrete nature of represented elements. In particular, it remains true for behaviour algebras with infinitely divisible elements, and thus also with elements which can represent continuous and partially continuous processes. An important consequence of the representation of elements of behaviour algebras by labelled posets is that elements of some subalgebras of behaviour algebras can be endowed in a consistent way with structures such as a graph structure etc.

3

An Algebraic Framework for Defining Behaviours of Concurrent Systems. Part 2: The Axiomatic Presentation

67%

Winkowski J.

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tom Vol. 97, nr 4

439-470

EN

The paper is the second part of two-part paper that contributeswith a concept of a process, with operations allowing to define complex processes in terms of their components, with the respective algebras, and with the idea of using the formal tools thus obtained to describe the behaviours of concurrent systems. In the first part a universal model of a process have been introduced and operations on processes have been defined. A process is viewed as a model of a run of a system (discrete, continuous, or of a mixed type). A process may have an initial state (a source), a final state (a target), or both. A process can be represented by a partially ordered multiset. Processes of which one is a continuation of the other can be composed sequentially. Independent processes, can be composed in parallel. Processes may be prefixes, i.e. independent components of initial segments of other processes. It has been shown that processes in a universe of objects and operations on such processes form a partial algebra, called algebra of processes, that is a specific partial category with respect to the sequential composition, and a specific partial monoid with respect to the parallel composition. In the second part the properties of algebras of processes described in the first part are regarded as axioms defining a class of abstract partial algebras, called behaviour-oriented algebras, and properties of such algebras are investigated. In particular, it is shown how some of the behaviour-oriented algebras can be represented as algebras of processes, and how to use them to describe phenomena with states and processes provided with specific structures.

4

An Algebraic Framework for Defining Behaviours of Concurrent Systems. Part 1: The Constructive Presentation

67%

Winkowski J.

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tom Vol. 97, nr 1/2

235-273

EN

The paper is the first part of a two-part paper that contributes with a concept of a process viewed as a model of a run of a phenomenon (discrete, continuous, or of a mixed type), with operations allowing to define complex processes in terms of their components, with the respective algebras, and with the idea of using the formal tools thus obtained to describe the behaviours of concurrent systems. A process may have an initial state (a source), a final state (a target), or both. A process can be represented by a partially ordered multiset. Processes of which one can be a continuation of the other can be composed sequentially. Independent processes, i.e. processes which do not disturb each other, can be composed in parallel. Processes may be prefixes, i.e. independent components of initial segments of other processes. Processes in a universe of objects and operations on such processes form a partial algebra, called algebra of processes. Algebras of processes are partial categories with respect to the sequential composition, and partial monoids with respect to the parallel composition. Algebras of processes can be used to define behaviours of concurrent systems. The behaviour of a system can be defined as the set of possible processes of this system with a structure on this set. Such a set is prefix-closed. The structure on this set reflects the prefix order and, possibly, specific features of the behaviour like observability, the relation to flow of real time, etc. Algebras of processes can also be used to define behaviours, to define operations on behaviours similar to those in the existing calculi of behaviours, and to define random behaviours. The first part of the whole paper investigates algebras of processes and their applications to describing behaviours of systems. In the second part the properties of algebras of processes described in the first part are regarded as axioms defining a class of abstract partial algebras, called behaviouroriented algebras, and they initiate a theory of such algebras.