Observable Sequentiality and Full Abstraction 1 Full Abstraction and Sequentiality

One of the major challenges in denotational semantics is the construction of fully abstract models for sequential programming languages. For the past fteen years, research on this problem has focused on developing models for PCF, an idealized functional programming language based on the typed lambda calculus. Unlike most practical languages, PCF has no facilities for observing and exploiting the evaluation order of arguments in procedures. Since we believe that such facilities are crucial for understanding the nature of sequential computation , this paper focuses on a sequential extension of PCF (called SPCF) that includes two classes of control operators: error generators and escape handlers. These new control operators enable us to construct a fully abstract model for SPCF that interprets higher types as sets of error-sensitive functions instead of continuous functions. The error-sensitive functions form a Scott domain that is isomorphic to a domain of decision trees. We believe that the same construction will yield fully abstract models for functional languages with diierent control operators for observing the order of evaluation. A denotational semantics for a programming language determines two natural equivalence relations on program phrases. The rst relation, denotational equivalence , equates two phrases if and only if they denote the same element in the model. The second relation, observational equivalence, equates two phrases if and only if they have the same \observable behavior". In the denotational framework, observable behavior refers to the concrete output produced by entire programs. Thus, two phrases are observationally equivalent if and only if they can be interchanged in an arbitrary program without aaecting the output. These two equivalence relations are diierent in general. Denotational equivalence implies observational equivalence because denotational models are composi-tional. However, the converse rarely holds for conventional models of practical programming languages; some observationally equivalent phrases inevitably have different meanings. A semantics in which denotational equivalence and observational equivalence coincide is said to be fully abstract. Observational equivalence is the fundamental constraint governing program optimization. Since the users of a program are primarily interested in its input-output behavior, a compiler can optimize a program by interchanging observationally equivalent phrases. If a deno-tational semantics is fully abstract, then observational equivalence reduces to denotational equivalence, which can be analyzed with traditional mathematical tools. Consequently, one of the primary goals of research in de-notational semantics has been the construction of fully abstract models for practical languages. The following example shows why models based …