Quotient Monoids and Concurrent Behaviours

Janicki, R; Kleijn, J; Koutny, M

Quotient Monoids and Concurrent Behaviours

Lookup NU author(s): Professor Maciej Koutny ORCiD

Downloads

Full text for this publication is not currently held within this repository. Alternative links are provided below where available.

Abstract

In this paper, we discuss fundamental mathematical abstractions which can be used to capture and analyse operational semantics of concurrent systems. We focus our attention on the issues involved in the description of runs or behaviours of such systems. Assuming the discrete nature of system executions, in the most basic case, they can be represented as sequences of symbols, each symbol corresponding to the execution of a basic atomic action. By taking into account only the essentialcausal relationships between the executed actions one can then group together different runs which only differ by the ordering of causally unrelated actions. In the resulting model of Mazurkiewicz traces, each abstract execution (trace) is an equivalence class of the quotient monoid of sequential executions which can be represented by a (causal) partial order on the actions involved in these executions.Starting from this initial setting, the paper then considers behaviours which are (step) sequences of sets of simultaneously executed actions, and takes into account other relationships between pairs of executed actions, such as weak causality. The resulting abstract behaviours can again be expressed in terms of suitable quotient monoids step sequences or, equivalently, relational structures generalising causal partial orders. We then show how concrete system models coming from the Perti net domain can be treated using the above abstract notions of concurrent behaviour.