Adding Action Refinement to a Finite Process Algebra

AbstractIn this paper we present a Process Algebra for the specification of concurrent, communicating processes which incorporates operators for the refinement of actions by processes, in addition to the usual operators for communication, nondeterminism, internal actions, and restrictions, and study a suitable notion of semantic equivalence for it. We argue that action refinements should not, in some formal sense, interfere with the internal evolution of processes and their application to processes should consider the restriction operator as a "binder." We show that, under the above assumptions, the weak version of the refine equivalence introduced by Aceto and Hennessy ((1993) Inform. and Comput.103, 204-269) is preserved by action refinements and, moreover, is the largest such equivalence relation contained in weak bismulation equivalence. We also discuss an example showing that, contrary to what happens in Aceto and Hennessy ((1993) Inform. and Comput.103, 204-269), refine equivalence and timed equivalence are different notions of equivalence over the language considered in this paper

Contexts, refinement and determinism

In this paper we have been influenced by those who take an “engineering view” of the problem of designing systems, i.e. a view that is motivated by what someone designing a real system will be concerned with, and what questions will arise as they work on their design. Specifically, we have borrowed from the testing work of Hennessy, de Nicola and van Glabbeek, e.g. [13, 5, 21, 40, 39]. Here we concentrate on one fundamental part of the engineering view and where consideration of it leads. The aspects we are concerned with are computational entities in contexts, observed by users. This leads to formalising design steps that are often left informal, and that in turn gives insights into non-determinism and ultimately leads to being able to use refinement in situations where existing techniques fail

Atomic components

There has been much interest in components that combine the best of state-based and event-based approaches. The interface of a component can be thought of as its specification and substituting components with the same interface cannot be observed by any user of the components. Here we will define the semantics of atomic components where both states and event can be part of the interface. The resulting semantics is very similar to that of (event only) processes. But it has two main novelties: one, it does not need recursion or unique fixed points to model nontermination; and two, the behaviour of divergence is modelled by abstraction, i.e. the construction of the observational semantics

Constructing programs or processes

We define interacting sequential programs, motivated originally by constructivist considerations. We use them to investigate notions of implementation and determinism. Process algebras do not define what can be implemented and what cannot. As we demonstrate it is problematic to do so on the set of all processes. Guided by constructivist notions we have constructed interacting sequential programs which we claim can be readily implemented and are a subset of processes

Comparability in the graph monoid

Let $\Gamma$ be the infinite cyclic group on a generator $x.$ To avoid confusion when working with $\mathbb Z$-modules which also have an additional $\mathbb Z$-action, we consider the $\mathbb Z$-action to be a $\Gamma$-action instead. Starting from a directed graph $E$, one can define a cancellative commutative monoid $M_E^\Gamma$ with a $\Gamma$-action which agrees with the monoid structure and a natural order. The order and the action enable one to label each nonzero element as being exactly one of the following: comparable (periodic or aperiodic) or incomparable. We comprehensively pair up these element features with the graph-theoretic properties of the generators of the element. We also characterize graphs such that every element of $M_E^\Gamma$ is comparable, periodic, graphs such that every nonzero element of $M_E^\Gamma$ is aperiodic, incomparable, graphs such that no nonzero element of $M_E^\Gamma$ is periodic, and graphs such that no element of $M_E^\Gamma$ is aperiodic. The Graded Classification Conjecture can be formulated to state that $M_E^\Gamma$ is a complete invariant of the Leavitt path algebra $L_K(E)$ of $E$ over a field $K.$ Our characterizations indicate that the Graded Classification Conjecture may have a positive answer since the properties of $E$ are well reflected by the structure of $M_E^\Gamma.$ Our work also implies that some results of [R. Hazrat, H. Li, The talented monoid of a Leavitt path algebra, J. Algebra, 547 (2020) 430-455] hold without requiring the graph to be row-finite.Comment: This version contains some modifications based on the input of a referee for the New York Journal of Mathematic