Hazelnut: a bidirectionally typed structure editor calculus

Structure editors allow programmers to edit the tree structure of a program directly. This can have cognitive benefits, particularly for novice and end-user programmers. It also simplifies matters for tool designers, because they do not need to contend with malformed program text. This paper introduces Hazelnut, a structure editor based on a small bidirectionally typed lambda calculus extended with holes and a cursor. Hazelnut goes one step beyond syntactic well-formedness: its edit actions operate over statically meaningful incomplete terms. Naïvely, this would force the programmer to construct terms in a rigid "outside-in" manner. To avoid this problem, the action semantics automatically places terms assigned a type that is inconsistent with the expected type inside a hole. This meaningfully defers the type consistency check until the term inside the hole is finished. Hazelnut is not intended as an end-user tool itself. Instead, it serves as a foundational account of typed structure editing. To that end, we describe how Hazelnut's rich metatheory, which we have mechanized using the Agda proof assistant, serves as a guide when we extend the calculus to include binary sum types. We also discuss various interpretations of holes, and in so doing reveal connections with gradual typing and contextual modal type theory, the Curry-Howard interpretation of contextual modal logic. Finally, we discuss how Hazelnut's semantics lends itself to implementation as an event-based functional reactive program. Our simple reference implementation is written using js_of_ocaml.