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This survey reviews some of the most recent achievements in the saga of theaxiomatisation of parallel composition, along with some classic results. Wefocus on the recursion, relabelling and restriction free fragment of CCS and wediscuss the solutions to three problems that were open for many years. Thefirst problem concerns the status of Bergstra and Klop's auxiliary operatorsleft merge and communication merge in the finite axiomatisation of parallelcomposition modulo bisimiliarity: We argue that, under some naturalassumptions, the addition of a single auxiliary binary operator to CCS does notyield a finite axiomatisation of bisimilarity. Then we delineate the boundarybetween finite and non-finite axiomatisability of the congruences in vanGlabbeek's linear time-branching time spectrum over CCS. Finally, we present anovel result to the effect that rooted weak bisimilarity has no finite completeaxiomatisation over CCS.

In search of lost time: Axiomatising parallel composition in process algebras