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The Topology of Mazurkiewicz Traces
"... The present paper characterizes the topological structure of real traces. This is done in terms of graphtheoretic properties of the underlying dependence alphabet, which may be innite. The topological space of real traces is shown to be homeomorphic to the direct product of (at most) the full b ..."
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The present paper characterizes the topological structure of real traces. This is done in terms of graphtheoretic properties of the underlying dependence alphabet, which may be innite. The topological space of real traces is shown to be homeomorphic to the direct product of (at most) the full binary tree and the full countably branching tree and one higherdimensional grid. The occurrence of each of these factors depends on the existence of nite nontrivial and of innite connected components and on the number of isolated letters of the dependence alphabet. 1 Introduction Trace monoids were introduced by Cartier and Foata [3], who investigated combinatorial problems concerning the rearrangement of words, and by Mazurkiewicz [14], who was motivated to provide a mathematical model for concurrent systems. Since then trace theory has become a very popular topic, see the recent surveys [5, 6]. Corresponding author. This work was written while the second author worked at th...
Representation of Computations in Concurrent Automata by Dependence Orders
 THEORETICAL COMP. SCIENCE
, 1997
"... An automaton with concurrency relations A is a labeled transition system with a collection of binary relations indicating when two actions in a given state of the automaton can occur independently of each other. The concurrency relations induce a natural equivalence relation for finite computatio ..."
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An automaton with concurrency relations A is a labeled transition system with a collection of binary relations indicating when two actions in a given state of the automaton can occur independently of each other. The concurrency relations induce a natural equivalence relation for finite computation sequences. We investigate two graphtheoretic representations of the equivalence classes of computation sequences and obtain that under suitable assumptions on A they are isomorphic. Furthermore, the graphs are shown to carry a monoid operation reflecting precisely the composition of computations. This generalizes fundamental graphtheoretical representation results due to Mazurkiewicz in trace theory.
Type Theory via Exact Categories (Extended Abstract)
 In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science LICS '98
, 1998
"... Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why ..."
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Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...
Concurrency in Computational Networks
 in MEMOCODE 2003, 1th International Conference on Formal Methods and Models for CoDesign, Proc
, 2002
"... Future generations of embedded multimedia systems will have an increasing need for compute platforms that combine high compute power with low energy consumption. To meet with these requirements, multiprocessor systems must be used. These systems are in nature concurrent, and this concurrency in th ..."
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Future generations of embedded multimedia systems will have an increasing need for compute platforms that combine high compute power with low energy consumption. To meet with these requirements, multiprocessor systems must be used. These systems are in nature concurrent, and this concurrency in the architecture should be exploited. This requires that the concurrency is used in the mapping trajectory from the system specification to the hardware architecture. The concurrency in an application should therefore be extracted, and made explicit, in the models that are used to specify a system. To support the extraction of concurrency from an application, the model must contain a concurrency model. This concurrency model should support formal reasoning about concurrency.
J.C.: Modelchecking games for fixpoint logics with partial order models
 In: Proceedings of CONCUR’09. Volume 5710 of LNCS
, 2009
"... Abstract. We introduce modelchecking games that allow local secondorder power on sets of independent transitions in the underlying partial order models where the games are played. Since the onestep interleaving semantics of such models is not considered, some problems that may arise when using int ..."
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Abstract. We introduce modelchecking games that allow local secondorder power on sets of independent transitions in the underlying partial order models where the games are played. Since the onestep interleaving semantics of such models is not considered, some problems that may arise when using interleaving semantics are avoided and new decidability results for partial orders are achieved. The games are shown to be sound and complete, and therefore determined. While in the interleaving case they coincide with the local modelchecking games for the µcalculus, Lµ, in a noninterleaving setting they verify properties of Separation Fixpoint Logic (SFL), a logic that can specify in partial orders properties not expressible with Lµ. The games underpin a novel decision procedure for modelchecking all temporal properties of a class of infinite and regular event structures, thus improving previous results in the literature. Keywords: Modal and temporal logics; Modelchecking games; Hintikka game semantics; Partial order models of concurrency; Process algebras. 1
A Nice Labelling for TreeLike Event Structures of Degree 3
, 2009
"... We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the index of an event structure of degree 3 is bounded by a linear function of the height. The main theorem of the paper states that event structures of degree 3 who ..."
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We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the index of an event structure of degree 3 is bounded by a linear function of the height. The main theorem of the paper states that event structures of degree 3 whose causality order is a tree have a nice labelling with 3 colors. We exemplify how to use this theorem to construct upper bounds for the index of other event structures of degree 3.
Pomsets for Local Trace Languages  Recognizability, Logic & Petri Nets
, 2000
"... Mazurkiewicz traces can be seen as equivalence classes of words or as pomsets. Their generalisation by local traces was formalized by Hoogers, Kleijn and Thiagarajan as equivalence classes of step ring sequences. First we introduce a pomset representation for local traces. Extending Büchi's The ..."
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Mazurkiewicz traces can be seen as equivalence classes of words or as pomsets. Their generalisation by local traces was formalized by Hoogers, Kleijn and Thiagarajan as equivalence classes of step ring sequences. First we introduce a pomset representation for local traces. Extending Büchi's Theorem and a previous generalisation to Mazurkiewicz traces, we show then that a local trace language is recognized by a finite step transition system if and only if its class of pomsets is bounded and definable in the Monadic Second Order logic. Finally, using Zielonka's Theorem, we show that each recognizable local trace language is described by a finite safe labelled Petri net.
On an Open Problem of Amadio and Curien: the Finite Antichain Condition 1 Abstract
"... More than a dozen years ago, Amadio [1] (see Amadio and Curien [2] as well) raised the question of whether the category of stable bifinite domains of AmadioDroste [1,6,7] is the largest cartesian closed full subcategory of the category of ωalgebraic meetcpos with stable functions. A solution to ..."
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More than a dozen years ago, Amadio [1] (see Amadio and Curien [2] as well) raised the question of whether the category of stable bifinite domains of AmadioDroste [1,6,7] is the largest cartesian closed full subcategory of the category of ωalgebraic meetcpos with stable functions. A solution to this problem has two major steps: (1) Show that for any ωalgebraic meetcpo D, if all higherorder stable function spaces built from D are ωalgebraic, then D is finitary (i.e., it satisfies the socalled axiom I); (2) Show that for any ωalgebraic meetcpo D, if D violates MI ∞ , then [D → D] violates either M or I. We solve the first part of the problem in this paper, i.e., for any ωalgebraic meetcpo D, if the stable function space [D → D] satisfies M, then D is finitary. Our notion of (mub, meet)closed set, which is introduced for step 1, will also be used for treating some example cases in step 2. 1