Modularity in the LF Logical Framework

Formal deductive systems play an important role in computer science, particularly in the areas of logic and semantics of programming languages. They are employed in three different, but obviously related roles. Firstly, they are used to specify logics, type systems, operational semantics and other aspects of languages. Secondly, they form the basis for the implementation of such deductive systems. Thirdly, they provide an appropriate language for the formulation and proof of metatheorems of programming languages. The LF Logical Framework [8] has been designed to provide an appropriate language for the high-level specification of deductive systems as they occur in logic and computer science. Its basic principle is often summarized by saying that judgments (the basic unit of deductive systems) are represented as types and deductions as objects. The framework was intentionally kept weak (by excluding, for example, polymorphism and impredicative constructs) in order to better support mechanization and to allow a simple meta-theory. This has proved auspicious: algorithms for unification have been developed [5, 21] and the type theory underlying LF has been amenable to an operational interpretation which is realized in the Elf programming language [18, 19]. Furthermore, it also seems possible to express a wide range of meta-theoretic properties of deductive systems within LF, though this line of research is only in its initial stages [15]. We believe that for all three tasks, specification, implementation, and meta-theory of deductive systems, substantial benefits can be derived from explicit structuring mechanisms for the presentation of such systems. In this paper we make a concrete proposal for a module system for the Elf language which attempts to address those three central issues. Various approaches to the static and dynamic semantics of such a module calculus are possible, but beyond the scope of this paper. Here we provide only informal discussions of the meanings of various language constructs and properties. As an extended example throughout the paper we will use two formulations of minimal propositional calculus with implication and conjunction: an axiom system in the style of Hilbert and Gentzen’s calculus of natural deduction. The problem of modularity in the presentation of theories and logical system has been addressed from the semantical [10, 9] and the type-theoretic [3, 4, 25] point of view. Our design has been guided by these ideas and the pragmatic principles of the ML module system [14, 17]. For further discussion of related work, the reader is referred to Section 7.