Reasoning with Continuations II : Full Abstraction for Models

A fully abstract model of a programming language assigns the same meaning to two terms if and only if they have the same operational behavior. Such models are well-known for functional languages but little is known about extended functional languages with sophisticated control structures. We show that a direct model with error values and the conventional continuation model are adequate for functional languages augmented with rst-and higher-order control facilities, respectively. Furthermore , both models become fully abstract on adding a control delimiter and a parallel conditional to the programming languages. 1 The Control Structure of Lisp Many programming languages, in particular Lisp 17, 24] and Scheme 22, 26], contain sophisticated control structures with constructs for labeling the control state at an arbitrary point for later reuse. Since Lisp is an expression-oriented language , a jump to a labeled control state|just like a procedure|may also pass a parameter so that it is possible to communicate results between control points. In most Lisp systems, control points have a special status, distinct from all other values. The 0 Scheme dialect treats its control points|referred to as continuations|as ordinary, rst-class procedural objects and even uses the same syntax for continuation invocations and procedure calls. For many programming tasks, control facilities are an important means of abstraction. They avoid the cumbersome encoding of control mechanisms through boolean predicates and tests, and expli-cate the purpose of otherwise unreadable programming patterns. Moreover, by incorporating explicit statements about the control behavior, a program provides clues to the compiler for possible optimizations. However, control constructs also invalidate some common laws of reasoning that hold for functional programming languages like pure Lisp 6, 18, 27]. More technically, the operational equivalence relation for the functional subset of Lisp is not a subset of the operational equivalence relation for Lisp with control facilities. Given the importance of operational equivalence laws for reasoning about programs, this raises two crucial questions: 1. How can we reason with control facilities? 2. How can we relate the reasoning system for functional languages to the system for extended languages with control facilities? Talcott's 27, 28] and our own 6, 7, 11] results on functional languages with rst-class control abstractions provide a partial answer to the rst question. 1 There is a small number of fundamental laws for the operational equivalence relation of such languages that play a crucial role in equivalence 1 The recent work by Aiken …