We propose a novel approach to the view update problem for tree-structured data: a domainspecific programming language in which all expressions denote bi-directional transformations on trees. In one direction, these transformations—dubbed lenses—map a “concrete ” tree into a simplified “abstract view”; in the other, they map a modified abstract view, together with the original concrete tree, to a correspondingly modified concrete tree. Our design emphasizes both robustness and ease of use, guaranteeing strong well-behavedness and totality properties for welltyped lenses. We identify a natural mathematical space of well-behaved bi-directional transformations over arbitrary structures, study definedness and continuity in this setting, and state a precise connection with the classical theory of “update translation under a constant complement ” from databases. We then instantiate this semantic framework in the form of a collection of lens combinators that can be assembled to describe transformations on trees. These combinators include familiar constructs from functional programming (composition, mapping, projection, conditionals, recursion) together with some novel primitives for manipulating trees (splitting, pruning, copying, merging, etc.). We illustrate the expressiveness of these combinators by developing a number of bi-directional listprocessing transformations as derived forms. An extended example shows how our combinators can be used to define a lens that translates between a native HTML representation of browser bookmarks and a generic abstract bookmark format.

We present an internal language with equivalent expressive power to Standard ML, and discuss its formalization in LF and the machine-checked verification of its type safety in Twelf. The internal language is intended to serve as the target of elaboration in an elaborative semantics for Standard ML in the style of Harper and Stone. Therefore, it includes all the programming mechanisms necessary to implement Standard ML, including translucent modules, abstraction, polymorphism, higher kinds, references, exceptions, recursive types, and recursive functions. Our successful formalization of the proof involved a careful interplay between the precise formulations of the various mechanisms, and required the invention of new representation and proof techniques of general interest.

The need for subtyping in type-systems with dependent types has been realized for some years. But it is hard to prove that systems combining the two features have fundamental properties such as subject reduction. Here we investigate a subtyping extension of the system *P, which is an abstract version of the type system of the Edinburgh Logical Framework LF. By using an equivalent formulation, we establish some important properties of the new system *P^, including subject reduction. Our analysis culminates in a complete and terminating algorithm which establishes the decidability of type-checking.

We develop an imperative calculus that provides a formal model for both single and mixin inheritance. By introducing classes and mixins as the basic object-oriented constructs in a -calculus with records and references, we obtain a system with an intuitive operational semantics. New

This paper presents an imperative object calculus designed to support class-based programming via a combination of extensible objects and encapsulation. This calculus simplifies the language presented in [17] in that, like C++ and Java, it chooses to support an imperative semantics instead of method specialization. We show how Java-style classes and "mixins" may be coded in this calculus, prove a type soundness theorem (via a subject reduction property), and give a sound and complete typing algorithm.

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In this paper we present an extension to basic type theory to allow a uniform construction of abstract data types (ADTs) having many of the properties of objects, including abstraction, subtyping, and inheritance. The extension relies on allowing type dependencies for function types to range over a well-founded domain. Using the propositions--as--types correspondence, abstract data types can be identified with logical theories, and proofs of the theories are the objects that inhabit the corresponding ADT. 1 Introduction In the past decade, there has been considerable progress in developing formal account of a theory of objects. One property of object oriented languages that make them popular is that they attack the problem of scale: all object oriented languages provide mechanisms for providing software modularity and reuse. In addition, the mechanisms are intuitive enough to be followed easily by novice programmers. During the same decade, the body of formal mathematics has be...

Many object-oriented languages used in practice descend from Algol. With this motivation, we study the theoretical issues underlying such languages via the theory of Algollike languages. It is shown that the basic framework of this theory extends cleanly and elegantly to the concepts of objects and classes. An important idea that comes to light is that classes are abstract data types, whose theory corresponds to that of existential types. Equational and Hoare-like reasoning methods, and relational parametricity provide powerful formal tools for reasoning about Algol-like object-oriented programs. 1

We develop a semantic foundation and a core programming language for bi-directional transformations on treestuctured data. In one direction, these transformations, called lenses, map a complex "concrete" tree into a simplified "abstract" one; in the other, they map a modified abstract tree, together with the original concrete tree, to a correspondingly modified concrete tree.

Coercive subtyping offers a general approach to subtyping and inheritance by introducing a simple abbreviational mechanism to constructive type theories. In this paper, we study coercion completion in coercive subtyping and prove that the formal extension with coercive subtyping of a type theory such as Martin-Lof's type theory and UTT is a conservative extension. The importance of coherence conditions for the conservativity result is also discussed.