# On the concatenation of infinite traces

- First Online:

DOI: 10.1007/BFb0020791

- Cite this paper as:
- Diekert V. (1991) On the concatenation of infinite traces. In: Choffrut C., Jantzen M. (eds) STACS 91. STACS 1991. Lecture Notes in Computer Science, vol 480. Springer, Berlin, Heidelberg

## Abstract

There is a straightforward generalization of traces to infinite traces as dependence graphs where every vertex has finitely many predecessors, or what is the same, as a backward closed and directed set of traces with respect to prefix ordering. However, this direct approach has a drawback since it allows no good notion of a concatenation. We solve this problem by adding to an infinite trace a second component. This second component is a finite alphabetic information which is called the *alphabet at infinity*. We obtain a compact and complete ultra-metric space where the concatenation is uniformly continuous and where the set of finite traces is an open, discrete, and dense subset. Our objects arise in a natural way from the consideration of dependence graphs where the induced partial order is well-founded. Such a graph splits into a so-called *real part* and a *transfinite* (or *imaginary*) *part*. From the transfinite part only the alphabet is of importance.

Our approach is a non-trivial generalization of the well-known construction for direct products of free monoids and yields a convenient semantics for infinite concurrent processes.

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