Games, mobile processes, and functionss -- alternating, concurrent, and well-bracketed semantics

We establish a tight connection between two models of the $\lambda$-calculus, namely Milner's encoding into the $\pi$-calculus (precisely, the Internal $\pi$-calculus), and operational game semantics (OGS). We first investigate the operational correspondence between the behaviours of the encoding provided by $\pi$ and OGS. We do so for various LTSs: the standard LTS for $\pi$ and a new `concurrent' LTS for OGS; an `output-prioritised' LTS for $\pi$ and the standard alternating LTS for OGS. We then show that the equivalences induced on $\lambda$-terms by all these LTSs (for $\pi$ and OGS) coincide. We also prove that when equivalence is based on complete traces, the `concurrent' and `alternating' variants of OGS also coincide with the `well-bracketed' variant. These connections allow us to transfer results and techniques between $\pi$ and OGS. In particular: we import up-to techniques from $\pi$ onto OGS; we derive congruence and compositionality results for OGS from those of $\pi$; we transport the notion of complete traces from OGS onto $\pi$, obtaining a new behavioural equivalence that yields a full abstraction result for the encoding of $\lambda$-terms with respect to contexts written in a $\lambda$-calculus extended with store. The study is illustrated for both call-by-value and call-by-name.