Plotkin's v -calculus for call-by-value programs is weaker than the fij- calculus for the same programs in continuation-passing style (CPS). To identify the callby -value axioms that correspond to fij on CPS terms, we define a new CPS transformation and an inverse mapping, both of which are interesting in their own right. Using the new CPS transformation, we determine the precise language of CPS terms closed under fij-transformations, as well as the call-by-value axioms that correspond to the so-called administrative fij-reductions on CPS terms. Using the inverse mapping, we map the remaining fi and j equalities on CPS terms to axioms on call-by-value terms. On the pure (constant free) set of-terms, the resulting set of axioms is equivalent to Moggi's computational -calculus. If the call-by-value language includes the control operators abort and call-with-current-continuation, the axioms are equivalent to an extension of Felleisen et al.'s v-C-calculus and to the equational subtheory of Talcott's logic IOCC. Contents 1 Compiling with and without Continuations 4 2 : Calculi and Semantics 7 3 The Origins and Practice of CPS 10 3.1 The Original Encoding : : : : : : : : : : : : : : : : : : : : : 10 3.2 The Universe of CPS Terms : : : : : : : : : : : : : : : : : : 11 4 A Compacting CPS Transformation 13

Compilers for higher-order programming languages like Scheme, ML, and Lisp can be broadly characterized as either "direct compilers" or "continuation-passing style (CPS) compilers", depending on their main intermediate representation. Our central result is a precise correspondence between the two compilation strategies. Starting from

Type systems for object-oriented programming languages have been studied a great deal over the past few years. Since Smalltalk was one of the earliest object-oriented languages, it is not surprising that there have been several attempts to provide a type system for it. Unfortunately, none of the attempts have been completely successful. In particular, none of the proposed type systems are both type-safe and capable of type-checking most common Smalltalk programs. Smalltalk violates many of the assumptions on which most object-oriented type systems are based, and a successful type system for Smalltalk is necessarily different from those for other languages. We have designed a new type system for Smalltalk. The biggest difference between our type system and others is that most type systems for object-oriented programming languages equate classes with types and subclassing with subtyping. In our type system, types are based on classes (i.e. each class defines a type or a family of types)...