A Logical Foundation for Environment Classifiers

Taha and Nielsen have developed a multi-stage calculus {\lambda}{\alpha} with a sound type system using the notion of environment classifiers. They are special identifiers, with which code fragments and variable declarations are annotated, and their scoping mechanism is used to ensure statically that certain code fragments are closed and safely runnable. In this paper, we investigate the Curry-Howard isomorphism for environment classifiers by developing a typed {\lambda}-calculus {\lambda}|>. It corresponds to multi-modal logic that allows quantification by transition variables---a counterpart of classifiers---which range over (possibly empty) sequences of labeled transitions between possible worlds. This interpretation will reduce the "run" construct---which has a special typing rule in {\lambda}{\alpha}---and embedding of closed code into other code fragments of different stages---which would be only realized by the cross-stage persistence operator in {\lambda}{\alpha}---to merely a special case of classifier application. {\lambda}|> enjoys not only basic properties including subject reduction, confluence, and strong normalization but also an important property as a multi-stage calculus: time-ordered normalization of full reduction. Then, we develop a big-step evaluation semantics for an ML-like language based on {\lambda}|> with its type system and prove that the evaluation of a well-typed {\lambda}|> program is properly staged. We also identify a fragment of the language, where erasure evaluation is possible. Finally, we show that the proof system augmented with a classical axiom is sound and complete with respect to a Kripke semantics of the logic

Tracking Data-Flow with Open Closure Types

Type systems hide data that is captured by function closures in function types. In most cases this is a beneficial design that favors simplicity and compositionality. However, some applications require explicit information about the data that is captured in closures. This paper introduces open closure types, that is, function types that are decorated with type contexts. They are used to track data-flow from the environment into the function closure. A simply-typed lambda calculus is used to study the properties of the type theory of open closure types. A distinctive feature of this type theory is that an open closure type of a function can vary in different type contexts. To present an application of the type theory, it is shown that a type derivation establishes a simple non-interference property in the sense of information-flow theory. A publicly available prototype implementation of the system can be used to experiment with type derivations for example programs.Comment: Logic for Programming Artificial Intelligence and Reasoning (2013

Polymorphic Context for Contextual Modality

Through the Curry-Howard isomorphism between logics and calculi, necessity modality in logic is interpreted as types representing program code. Particularly, \lamcirc, which was proposed in influential work by Davies, and its successors have been widely used as a logical foundation for syntactic meta-programming. However, it is less known how to extend calculi based on modal type theory to handle more practical operations including manipulation of variable binding structures. This paper constructs such a modal type theory in two steps. First, we reconstruct contextual modal type theory by Nanevski, et al.\ as a Fitch-style system, which introduces hypothetical judgment with hierarchical context. The resulting type theory, \multilayer contextual modal type theory \fcmtt, is generalized to accommodate not only S4 but also K, T, and K4 modalities, and proven to enjoy many desired properties. Second, we extend \fcmtt with polymorphic context, which is an internalization of contextual weakening, to obtain a novel modal type theory \envpoly. Despite the fact that it came from observation in logic, polymorphic context allows both binding manipulation and hygienic code generation. We claim this by showing a sound translation from \lamcirc to \envpoly

Program logics for homogeneous meta-programming.

A meta-program is a program that generates or manipulates another program; in homogeneous meta-programming, a program may generate new parts of, or manipulate, itself. Meta-programming has been used extensively since macros were introduced to Lisp, yet we have little idea how formally to reason about metaprograms. This paper provides the first program logics for homogeneous metaprogramming – using a variant of MiniMLe by Davies and Pfenning as underlying meta-programming language.We show the applicability of our approach by reasoning about example meta-programs from the literature. We also demonstrate that our logics are relatively complete in the sense of Cook, enable the inductive derivation of characteristic formulae, and exactly capture the observational properties induced by the operational semantics

A dependently typed multi-stage calculus

Programming Languages and Systems: 17th Asian Symposium, APLAS 2019, Nusa Dua, Bali, Indonesia, December 1–4, 2019. Part of the Lecture Notes in Computer Science book series (LNCS, volume 11893). Also part of the Programming and Software Engineering book sub series (LNPSE, volume 11893).We study a dependently typed extension of a multi-stage programming language à la MetaOCaml, which supports quasi-quotation and cross-stage persistence for manipulation of code fragments as first-class values and an evaluation construct for execution of programs dynamically generated by this code manipulation. Dependent types are expected to bring to multi-stage programming enforcement of strong invariant—beyond simple type safety—on the behavior of dynamically generated code. An extension is, however, not trivial because such a type system would have to take stages of types—roughly speaking, the number of surrounding quotations—into account. To rigorously study properties of such an extension, we develop λMD, which is an extension of Hanada and Igarashi’s typed calculus λ▹% with dependent types, and prove its properties including preservation, confluence, strong normalization for full reduction, and progress for staged reduction. Motivated by code generators that generate code whose type depends on a value from outside of the quotations, we argue the significance of cross-stage persistence in dependently typed multi-stage programming and certain type equivalences that are not directly derived from reduction rules