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Axiomatization using locality and free choice

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Salomaa gave an axiomatization for equivalence of regular expressions, and we know that checking equivalence is complete for polynomial space. Regular languages can be described more succinctly by using additional operations like synchronized shuffle (also called merge), renaming and hiding. Equivalence of these expressions for traces is axiomatized using Milner’s expansion law (or Bergstra and Klop’s left and right merge operations), and checking equivalence is complete for exponential space. If we disallow nesting of shuffle, renaming and hiding, checking equivalence is still in polynomial space. We give a proof system for a fragment. We do not use the expansion law. The syntax matches languages corresponding to 1-bounded free choice Petri nets.

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