Concurrent Program Schemes and their Interpretations

Abstract

Schemes of concurrent programs are considered. The result of a scheme is defined as a set of traces, where each trace is a partially ordered set of symbol occurrences. It is shown that to each scheme corresponds a set of equations determining the result of the scheme; it is shown how these equations can be solved and that the solutions of these equations are regular trace languages. Next, a notion of action systems is introduced; an action consists of its resources and its transformation. Some properties of action systems are shown. Interpretations of schemes are defined as mappings which assign actions to scheme symbols. Interpreted schemes can be regarded as concurrent programs. It is shown how the results of schemes can be lifted (via interpretations) to the results of programs. Some examples of applications of the described methods to prove concurrent programs correct are given.

Petri nets and other models of concurrency

“…The theory of nets was originally a strong source of inspiration behind the introduction of traces by Mazurkiewicz in [43]. Also, the relationship between traces and nets have been extensively studied, see in particular the survey papers by Rozenberg and Thiagarajan in [73,87].…”

“…Other models include the event structures of Winskel [94], the trace structures of Mazurkiewicz [43], the asynchronous and the concurrent transition systems of Bednarczyk [4], Shields [83] and Stark [84], just to name a few. Similarly, concurrency deals with an abundance of notions for behavioural equivalence, with the bisimulation of Milner [50], trace equivalence of Hoare [30], and pomset equivalence of Pratt [69] as prime examples.…”

Optimal Zielonka-Type Construction of Deterministic Asynchronous Automata

“…Concurrent executions of systems with shared actions given by a distributed alphabet (Σ, dom), are readily modeled by Mazurkiewicz traces [9]. The idea is that the distribution of the alphabet defines an independence relation among actions I ⊆ Σ × Σ, by setting (a, b) ∈ I if and only if dom(a) ∩ dom(b) = ∅.…”

“…This model has a solid theoretical foundation based on the theory of Mazurkiewicz traces [9,4]. The key property of asynchronous automata, known as Zielonka's theorem, is that every regular trace language can be represented by a deterministic asynchronous automaton [15].…”

On Aperiodic and Star-free Formal Power Series in Partially Commuting Variables

“…A trace monoid M is therefore defined as the quotient of the free monoid Σ * modulo the congruence generated by the relations ab ∼ ba if a I b. These monoids were introduced by Mazurkiewicz [18,19] as an important mathematical model for the behavior of concurrent systems, see also [3,5,4] for their well-developed theory. Now let K be an arbitrary semiring, and let K M be the collection of all formal power series S = m∈M (S, m) · m. These can also be viewed as series with entries (S, m) from K in which certain of the variables (= elements of Σ) are allowed to commute, as indicated by the relation I.…”