Lukasz Mikulski

Hasse Diagrams of Combined Traces

One of the standard ways to represent concurrent behaviours is to use concepts originating from language theory, such as traces and comtraces. Traces can express notions such as concurrency and causality, whereas comtraces can also capture weak causality and simultaneity. This paper is concerned with the development of efficient data structures and algorithms for manipulating comtraces. We introduce Hasse diagrams for comtraces which are a generalisation of Hasse diagrams defined for partial orders and traces, and develop an efficient algorithm for deriving them from language theoretic representations of comtraces. We also explain how the new representation of comtraces can be used to implement efficiently some basic operations on comtraces.

Reduction of Order Structures

Relational order structures are used to describe and investigate properties of concurrent systems. To reduce the complexity of order structures, one typically considers only their essential components, which, in the case of partial orders, leads to the notion of Hasse diagrams. We lift this notion to the level of generalised mutex order structures, which are used to model not only causal dependencies but also weak causality and mutual exclusion. We provide a new and more concise axiomatic definition of these structures, investigate their important properties, and present efficient algorithms for computing their reduction and closure. The algorithms are implemented in a publicly available software tool with graphical user interface.

A Precise Characterisation of Step Traces and Their Concurrent Histories

Step traces are an extension of Mazurkiewicz traces where each equivalence class (trace) consists of sequences of steps instead of sequences of atomic actions. Relations between the actions of the system are defined statically, as parameters of a concurrent step alphabet. By allowing only some of the possible relationships between actions, subclasses of step alphabets can be derived in a natural way. Properties of these classes can then be investigated in terms of invariant structures, i.e., the relational structures that represent the causal invariants that underlie the corresponding step traces. In this paper, we refine an earlier classification of subclasses of step alphabets and add eight new subclasses to this hierarchy. We divide these eight classes into three families on basis of the absence of a specific behavioural relation and then characterise the corresponding invariant structures.

Reaction Systems, Transition Systems, and Equivalences

Reaction systems originated as a formal model for processes inspired by the functioning of the living cell. The underlying idea of this model is that the functioning of the living cell is determined by the interactions of biochemical reactions and these interactions are based on the mechanisms of facilitation and inhibition. Since their inception, reaction systems became a well-investigated novel model of computation. Following this line of research, in this paper we discuss a systematic framework for investigating a whole range of equivalence notions for reaction systems. Some of the equivalences are defined directly on reaction systems while some are defined through transition systems associated with reaction systems. In this way we establish a new bridge between reaction systems and transition systems. In order to define equivalences which capture various ways of interacting with an environment, we also introduce models of the environment which evolve in a finite-state fashion.

Lexicographical Generations of Combined Traces

Combined traces are intrinsic mathematical model for studying concurrent systems behaviors. They can be used to describe and investigate processes of elementary net systems with inhibitor arcs and allow to describe weak causality and simultaneity of actions. We provide several algorithms for manipulating combined traces using their language theoretic representations. In particular, we propose two methods of enumeration related to combined traces, supported by a collection of auxiliary procedures. First, for a specified combined trace we iterate the set of all its representatives (namely step sequences). Next, we use the lexicographical order on step sequences to list all combined traces of a fixed size. We also discuss the time complexity of all presented algorithms.

Algorithmics of Posets Generated by Words Over Partially Commutative Alphabets (Extended Version)

It is natural to try to relate partially ordered sets (posets in short) and classes of equivalent words over partially commutative alphabets. Their common graphical representation are Hasse diagrams. We will investigate this relation in detail and propose an efficient on-line algorithm that decompresses a string to Hasse diagram. Further we propose a definition of the canonical representatives of classes of equivalent words. The advantage of this representation lies in the fact that we can enumerate the classes of equivalent words in a lexicographical order. We will give an algorithm which enumerates all distinct classes of words over partially commutative alphabets by their lexicographically minimal representatives.