Towards Nominal Abramsky

Since the discovery of fully abstract models of PCF in the early 1990s, game semantics has expanded to a wide range of programming paradigms, covering effects like state, control, general references, non-determinism, probability and concurrency. Those models revealed an interesting phenomenon referred to as Abramsky’s cube: starting from the PCF model and relaxing each of its combinatorial conditions, one was led to capture a corresponding impure effect. In this paper we initiate the construction of an analogous cube for nominal games, a strand of game semantics developed in the last ten years that incorporates names as semantic atoms and captures generative effects without using “bad-object” constructors. In particular, we examine the stateful axis of the cube: starting from games for higher-order references we move to full ground references, where strategies respect visibility, and from there to purely functional behaviour and innocent strategies.