# On the logical definability of some rational trace languages

## Abstract

Trace monoids are obtained from free monoids by defining a subset *I* of pairs of letters that are allowed to commute. Most of the work of this theory is an attempt to relate the properties of these monoids to the properties of *I*. Following the work initiated by Büchi we show that when *I* is an equivalence relation (the trace monoid is then a free product of free commutative monoids) it is possible to define a second order logic whose models are the traces viewed as dependence graphs and which characterizes exactly the sets of traces that are rational. This logic essentially utilizes a predicate based on the ordering defined by the dependence graph and a predicate related to a restricted use of the comparison of cardinality.

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