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We provide syntactic derivative-like operations, defined by recursion onregular expressions, in the styles of both Brzozowski and Antimirov, for traceclosures of regular languages. Just as the Brzozowski and Antimirov derivativeoperations for regular languages, these syntactic reordering derivativeoperations yield deterministic and nondeterministic automata respectively. Buttrace closures of regular languages are in general not regular, hence theseautomata cannot generally be finite. Still, as we show, for star-connectedexpressions, the Antimirov and Brzozowski automata, suitably quotiented, arefinite. We also define a refined version of the Antimirov reordering derivativeoperation where parts-of-derivatives (states of the automaton) are nonemptylists of regular expressions rather than single regular expressions. We definethe uniform scattering rank of a language and show that, for a regexp whoselanguage has finite uniform scattering rank, the truncation of the (generallyinfinite) refined Antimirov automaton, obtained by removing long states, isfinite without any quotienting, but still accepts the trace closure. We alsoshow that star-connected languages have finite uniform scattering rank.

Reordering Derivatives of Trace Closures of Regular Languages (Full Version)