Axioms for control operators in the CPS hierarchy

A CPS translation is a syntactic translation of programs, which is useful for describing their operational behavior. By iterating the standard callby-value CPS translation, Danvy and Filinski discovered the CPS hierarchy and proposed a family of control operators, shift and reset, that make it possible to capture successive delimited continuations in a CPS hierarchy. Although shift and reset have found their applications in several areas such as partial evaluation, most studies in the literature have been devoted to the base level of the hierarchy, namely, to level-1 shift and reset. In this article, we investigate the whole family of shift and reset. We give a simple calculus with level-n shift and level-n reset for an arbitrary n > 0. We then give a set of equational axioms for them, and prove that these axioms are sound and complete with respect to the CPS translation. The resulting set of axioms is concise and a natural extension of those for level-1 shift and reset.