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1

Invariant Structures and Dependence Relations

Janicki R., Kleijn J., Koutny M., Mikulski Ł.

Fundamenta Informaticae

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2017

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Vol. 155, nr 1/2

1--29

EN

A step trace is an equivalence class of step sequences which can be thought of as different observations of the same underlying concurrent history. Equivalence is determined on basis of a step alphabet that describes the relations between events in terms of potential simultaneity and sequentialisability. Step traces cannot be represented by standard partial orders, but require so-called invariant structures, extended order structures that capture the phenomena of mutual exclusion and weak causality. In this paper, we present an effective way of deciding whether an invariant structure represents a step trace over a given step alphabet. We also describe a method by which one can check whether a given invariant structure can represent a step trace over any step alphabet. Moreover, if the answer is positive, the method provides a suitable step alphabet.

2

Alphabets of Acyclic Invariant Structures

Janicki R., Kleijn J., Koutny M., Mikulski Ł.

Fundamenta Informaticae

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2017

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Vol. 154, nr 1/4

207--224

EN

A step trace is an equivalence class of step sequences, where the equivalence is determined by dependencies between pairs of actions expressed as potential simultaneity and sequentialisability. Step traces can be represented by invariant structures with two relations: mutual exclusion and (possibly cyclic) weak causality. An important issue concerning invariant structures is to decide whether an invariant structure represents a step trace over a given step alphabet. For the general case this problem has been solved and an effective decision procedure has been proposed. In this paper, we restrict the class of order structures being considered with the aim of achieving a better characterisation. Requiring that the weak causality relation is acyclic, makes it possible to solve the problem in a purely local way, by considering pairs of events, rather than whole structures.

3

Characterising Concurrent Histories

Janicki R., Kleijn J., Koutny M., Mikulski Ł.

Fundamenta Informaticae

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2015

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Vol. 139, nr 1

21--42

EN

Non-interleaving semantics of concurrent systems is often expressed using posets, where causally related events are ordered and concurrent events are unordered. Each causal poset describes a unique concurrent history, i.e., a set of executions, expressed as sequences or step sequences, that are consistent with it. Moreover, a poset captures all precedence-based invariant relationships between the events in the executions belonging to its concurrent history. However, concurrent histories in general may be too intricate to be described solely in terms of causal posets. In this paper, we introduce and investigate generalised mutex order structures which can capture the invariant causal relationships in any concurrent history consisting of step sequence executions. Each such structure comprises two relations, viz. interleaving/mutex and weak causality. As our main result we prove that each generalised mutex order structure is the intersection of the step sequence executions which are consistent with it.

4

Mutex Causality in Processes and Traces of General Elementary Nets

Kleijn J., Koutny M.

Fundamenta Informaticae

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2013

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Vol. 122, nr 1-2

119--146

EN

A concurrent history represented by a causality structure that captures the intrinsic, invariant dependencies between its actions, can be interpreted as defining a set of closely related observations (e.g., step sequences). Depending on the relationships observed in the histories of a system, the concurrency paradigm to which it adheres may be identified, with different concurrency paradigms underpinned by different kinds of causality structures. Elementary net systems with inhibitor arcs and mutex arcs (ENIM-systems) are a system model that through its process semantics and associated causality structures fits the least restrictive concurrency paradigm. One can also investigate the abstract behaviour of an ENIM-system by grouping together step sequences in equivalence classes (generalised comtraces) using the structural relations between its transitions. The resulting concurrent histories of the ENIM-system are consistent with the generalised stratified order structures underlying its processes. The paper establishes a link between ENIM-systems and trace theory allowing one to discuss different observations of concurrent behaviour in a way that is consistent with the causality semantics defined by the operationally defined processes.

5

Membrane Systems with Qualitative Evolution Rules

Kleijn J., Koutny M.

Fundamenta Informaticae

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2011

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Vol. 110, nr 1-4

217-230

EN

In membrane systems, biochemical reactions taking place in the compartments of a cell are abstracted to evolution rules that specify which and how many objects are consumed and produced. The recently proposed reaction systems also investigate processes carried by biochemical reactions, but the resulting computational model is remarkably different. A key difference is that in reaction systems, biochemical reactions are modeled using a qualitative rather than a quantitative approach. In this paper, we introduce so-called set membrane systems, a variant of membrane systems with qualitative evolution rules inspired by reaction systems. We then relate set membrane systems to Petri nets which leads to a new class of Petri nets: set-nets with localities. This Petri net model provides a faithful match with the operational semantics of set membrane systems.

6

Associativity of Infinite Synchronized Shuffles and Team Automata

ter Beek M.H., Kleijn J.

Fundamenta Informaticae

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2009

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Vol. 91, nr 3-4

4437-461

EN

Motivated by different ways to obtain team automata from synchronizing component automata, we consider various definitions of synchronized shuffles of words. A shuffle of two words is an interleaving of their symbol occurrenceswhich preserves the original order of these occurrences within each of the two words. In a synchronized shuffle, however, also two occurrences of one symbol, each from a different word, may be identified as a single occurrence. In case at least one of the words involved is infinite, a (synchronized) shuffle can also be unfair in the sense that an infinite word may prevail fromsome point onwards even when the other word still has occurrences to contribute to the shuffle. We prove that for the synchronized shuffle operations under consideration, every (fair or unfair) synchronized shuffle can be obtained as a limit of synchronized shuffles of the finite prefixes of the words involved. In addition, it is shown that with the exception of one, all synchronized shuffle operations that we consider satisfy a natural notion of associativity, also in case of unfairness. Finally, using these results, some compositionality results for team automata are established.

7

Processes of Petri Nets with Range Testing

Kleijn J., Koutny M.

Fundamenta Informaticae

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2007

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Vol. 80, nr 1-3

199-219

EN

We are concerned with causality semantics in the executions of Petri nets with range arcs. Range arcs combine (and subsume) the distinctive features of inhibitor and activator arcs, and each such arc provides a means of specifying a range (a finite or infinite interval of non-negative integers) for the number of tokens in a place which makes enabling of a given transition possible. We demonstrate that the existing treatment of causality developed for Petri nets with inhibitor arcs based on structures generalising partial orders can also be applied to nets with range arcs.