Mixed choice in session types

Session types provide a flexible programming style for structuring interaction, and are used to guarantee a safe and consistent composition of distributed processes. Traditional session types include only one-directional input (external) and output (internal) guarded choices. This prevents the session-processes to explore the full expressive power of the π-calculus where mixed choice was proved more expressive. Recently Casal, Mordido, and Vasconcelos proposed binary session types with mixed choices (${\mathrm{CMV}}^{+}$). Surprisingly, in spite of an inclusion of unrestricted channels with mixed choice, ${\mathrm{CMV}}^{+}$'s mixed choice is rather separate and not mixed. We prove this negative result using two methodologies (using either the leader election problem or a synchronisation pattern as distinguishing feature), showing that there exists no good encoding from the π-calculus into ${\mathrm{CMV}}^{+}$, preserving distribution. We then close their open problem on the encoding from ${\mathrm{CMV}}^{+}$ into $\mathrm{CMV}$ (without mixed choice), proving its soundness.