A Logical Foundation for Environment Classifiers

Abstract

Abstract. Taha and Nielsen have developed a multi-stage calculus λ α 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 λ-calculus λ ⊲ . 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 λ α -and embedding of closed code into other code fragments of different stages-which would be only realized by the cross-stage persistence operator in λ α -to merely a special case of classifier application. λ ⊲ enjoys not only basic properties including subject reduction, confluence, and strong normalization but also an important property as a multi-stage calculus: timeordered normalization of full reduction.Then, we develop a big-step evaluation semantics for an ML-like language based on λ ⊲ with its type system and prove that the evaluation of a well-typed λ ⊲ 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.

Dynamic type inference for gradual Hindley–Milner typing

“…Shields et al [1998] propose a combination of dynamic typing and staged computation [Davies and Pfenning 2001;Taha and Sheard 2000] in a statically typed language. In their language, all code values are given a single code type ⟨⟩ (without information on the static type of the code as is typical in type systems for staged computation [Davies 1996;Davies and Pfenning 2001;Kim et al 2006;Taha and Nielsen 2003;Taha and Sheard 2000;Tsukada and Igarashi 2010]) and they are typechecked only when it is executed by eval. Such a "typecheck before eval" strategy is called staged type inference.…”

On the Semantics of Intensionality

“…This language has explicit annotations of the stage at which each computation is taking place, and is also able to handle open code. This led to a flurry of developments, and in particular the very influential work of Sheard [1997, 2000] on MetaML, and then the environment classifiers of Taha and Nielsen [2003], which have recently been improved by Tsukada and Igarashi [2010]. See [Kavvos, 2016, §6] for further references, and for a discussion of the modal aspects used.…”

Intensionality, Intensional Recursion, and the Gödel-Löb axiom

“…It refers to the ability given to a programmer to quote code, and carry it around as a datum; see [4] for an instance of that in LISP. This ability can be used for metaprogramming, which is the activity of writing programs that write other programs; indeed, this is what LISP macros excel at [10], and this is what the metaprogramming community has been studying for a long time: see [24,26]. But we want go even further.…”