Trace
Let denote the free monoid, that is, the set of all strings written in the alphabet . Here, the asterisk denotes, as usual, the Kleene star. An independency relation then induces a binary relation on, where if and only if there exist, and a pair such that and . Here, and are understood to be strings (elements of ), while and are letters (elements of ).
The trace is defined as the symmetric, reflexive and transitive closure of . The trace is thus an equivalence relation on, and is denoted by . The subscript D on the equivalence simply denotes that the equivalence is obtained from the independency I induced by the dependency D. Clearly, different dependencies will give different equivalence relations.
The transitive closure simply implies that if and only if there exists a sequence of strings such that and and for all .
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