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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 category-theoretic 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...

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 graph-theoretic 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 graph-theoretical representation results due to Mazurkiewicz in trace theory.

Future generations of embedded multi-media systems will have an increasing need for compute platforms that combine high compute power with low energy consumption. To meet with these requirements, multi-processor 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.

The present paper characterizes the topological structure of real traces. This is done in terms of graph-theoretic 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 higher-dimensional grid. The occurrence of each of these factors depends on the existence of nite non-trivial 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...

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.

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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. 1