Complete trace models of state and control

We consider a hierarchy of four typed call-by-value languages with either higher-order or ground-type references and with either [Formula: see text] or no control operator. Our first result is a fully abstract trace model for the most expressive setting, featuring both higher-order references and [Formula: see text] , constructed in the spirit of operational game semantics. Next we examine the impact of suppressing higher-order references and callcc in contexts and provide an operational explanation for the game-semantic conditions known as visibility and bracketing respectively. This allows us to refine the original model to provide fully abstract trace models of interaction with contexts that need not use higher-order references or [Formula: see text] . Along the way, we discuss the relationship between error- and termination-based contextual testing in each case, and relate the two to trace and complete trace equivalence respectively. Overall, the paper provides a systematic development of operational game semantics for all four cases, which represent the state-based face of the so-called semantic cube.