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Reordering Derivatives of Trace Closures of Regular Languages

Authors Hendrik Maarand , Tarmo Uustalu

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Hendrik Maarand
  • Department of Software Science, Tallinn University of Technology, Estonia
Tarmo Uustalu
  • School of Computer Science, Reykjavik University, Iceland
  • Department of Software Science, Tallinn University of Technology, Estonia


We thank Pierre-Louis Curien, Jacques Sakarovitch, Simon Doherty, Georg Struth and Ralf Hinze for inspiring discussions, and our anonymous reviewers for the exceptionally thorough and constructive feedback they gave us.

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Hendrik Maarand and Tarmo Uustalu. Reordering Derivatives of Trace Closures of Regular Languages. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We provide syntactic derivative-like operations, defined by recursion on regular expressions, in the styles of both Brzozowski and Antimirov, for trace closures of regular languages. Just as the Brzozowski and Antimirov derivative operations for regular languages, these syntactic reordering derivative operations yield deterministic and nondeterministic automata respectively. But trace closures of regular languages are in general not regular, hence these automata cannot generally be finite. Still, as we show, for star-connected expressions, the Antimirov and Brzozowski automata, suitably quotiented, are finite. We also define a refined version of the Antimirov reordering derivative operation where parts-of-derivatives (states of the automaton) are nonempty lists of regular expressions rather than single regular expressions. We define the uniform scattering rank of a language and show that, for a regexp whose language has finite uniform scattering rank, the truncation of the (generally infinite) refined Antimirov automaton, obtained by removing long states, is finite without any quotienting, but still accepts the trace closure. We also show that star-connected languages have finite uniform scattering rank.

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Theory of computation → Concurrency
  • Mazurkiewicz traces
  • trace closure
  • regular languages
  • finite automata
  • language derivatives
  • scattering rank
  • star-connected expressions


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