Relational parametricity and control

We study the equational theory of Parigot's second-order /spl lambda//spl mu/-calculus in connection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order /spl lambda/-calculus. It is observed that the relational parametricity on the target calculus induces a natural notion of equivalence on the /spl lambda//spl mu/-terms. On the other hand, the unconstrained relational parametricity on the /spl lambda//spl mu/-calculus turns out to be inconsistent with this CPS semantics. Following these facts, we propose to formulate the relational parametricity on the /spl lambda//spl mu/-calculus in a constrained way, which might be called "focal parametricity".