Reasoning with Continuations III : A Complete Calculus of ControlAmr

The-calculus is the simple and elegant, yet powerful basis for the semantic manipulation of functional programs by programmers and programming tools (compilers, optimizers, semantics-based editors) alike. The theory provides a simple system for evaluating programs in a symbolic manner; the typed version is a complete system of program equality for the pure core language. Felleisen et al. 6, 9, 8] and Talcott 24, 25] have recently developed extensions of the-calculus for reasoning about languages with powerful control operators, the v-C-calculus and the rst-order logic IOCC, respectively, but they did not address the issue of completeness. The central result of this paper is an extension of the v-C-calculus for a complete set of control operators and a correspondence theorem between the new theory and the-calculus. Technically, the theorem shows that the two calculi prove the same equations with respect to the well-known continuation-passing style translation (and its inverse), which is the standard tool for deening control operators via translation into a functional language. As a corollary, the calculus proves all program equivalences between terms over the pure language extended with control operators. We believe that this work has important consequences for the direct-style compilation of programming languages with control operators.