Software engineering demands generality and abstraction, performance demands specialization and concretization. Generative programming can provide both, but the effort required to develop high-quality program generators likely offsets their benefits, even if a multi-stage programming language is used.We present lightweight modular staging, a library-based multi-stage programming approach that breaks with the tradition of syntactic quasi-quotation and instead uses only types to distinguish between binding times. Through extensive use of component technology, lightweight modular staging makes an optimizing compiler framework available at the library level, allowing programmers to tightly integrate domain-specific abstractions and optimizations into the generation process.We argue that lightweight modular staging enables a form of language virtualization, i.e. allows to go from a pure-library embedded language to one that is practically equivalent to a stand-alone implementation with only modest effort.

Lightweight modular staging: a pragmatic approach to runtime code generation and compiled DSLs

The intuitionistic modal logic of necessity is based on the judgmental notion of categorical truth. In this article we investigate the consequences of relativizing these concepts to explicitly specified contexts. We obtain contextual modal logic and its type-theoretic analogue. Contextual modal type theory provides an elegant, uniform foundation for understanding metavariables and explicit substitutions. We sketch some applications in functional programming and logical frameworks.

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Contextual modal type theory

Nominal unification

The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Lofs universe a la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0, which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated.In this paper we argue that there is an underlying general notion of simultaneous inductive-recursive definition which is implicit in Martin-Lof's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model in the style of Allen.

A general formulation of simultaneous inductive-recursive definitions in type theory

Higher-order abstract syntax (HOAS) is a simple, powerful technique for implementing object languages, since it directly supports common and tricky routines dealing with variables, such as capture-avoiding substitution and renaming. This is achieved by representing binders in the object-language via binders in the meta-language. However, enriching functional programming languages with direct support for HOAS has been a major challenge, because recursion over HOAS encodings requires one to traverse lambda-abstractions and necessitates programming with open objects.We present a novel type-theoretic foundation based on contextual modal types which allows us to recursively analyze open terms via higher-order pattern matching. By design, variables occurring in open terms can never escape their scope. Using several examples, we demonstrate that our framework provides a name-safe foundation to operations typically found in nominal systems. In contrast to nominal systems however, we also support capture-avoiding substitution operations and even provide first-class substitutions to the programmer. The main contribution of this paper is a syntax-directed bi-directional type system where we distinguish between the data language and the computation language together with the progress and preservation proof for our language.

A type-theoretic foundation for programming with higher-order abstract syntax and first-class substitutions

This paper presents a polymorphic type system for a language with delimited control operators, shift and reset. Based on the monomorphic type system by Danvy and Filinski, the proposed type system allows pure expressions to be polymorphic. Thanks to the explicit presence of answer types, our type system satisfies various important properties, including strong type soundness, existence of principal types and an inference algorithm, and strong normalization. Relationship to CPS translation as well as extensions to impredicative polymorphism are also discussed. These technical results establish the foundation of polymorphic delimited continuations.

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Polymorphic delimited continuations

The shift and reset operators, proposed by Danvy and Filinski, are powerful control primitives for capturing delimited continuations. Delimited continuation is a similar concept as the standard (unlimited) continuation, but it represents part of the rest of the computation, rather than the whole rest of computation. In the literature, the semantics of shift and reset has been given by a CPS-translation only. This paper gives a direct axiomatization of calculus with shift and reset, namely, we introduce a set of equations, and prove that it is sound and complete with respect to the CPS-translation. We also introduce a calculus with control operators which is as expressive as the calculus with shift and reset, has a sound and complete axiomatization, and is conservative over Sabry and Felleisen's theory for first-class continuations.

/pdf/a-sound-and-complete-axiomatization-of-delimited-34g5vi4h05.pdf

A sound and complete axiomatization of delimited continuations

It is often hard to write programs that are efficient yet reusable. For example, an efficient implementation of Gaussian elimination should be specialized to the structure and known static properties of the input matrix. The most profitable optimizations, such as choosing the best pivoting or memoization, cannot be expected of even an advanced compiler because they are specific to the domain, but expressing these optimizations directly makes for ungainly source code. Instead, a promising and popular way to reconcile efficiency with reusability is for a domain expert to write code generators.Two pillars of this approach are types and effects. Typed multilevel languages such as MetaOCaml ensure _safety_: a well-typed code generator neither goes wrong nor generates code that goes wrong. Side effects such as state and control ease correctness: an effectful generator can resemble the textbook presentation of an algorithm, as is familiar to domain experts, yet insert let for memoization and if for bounds-checking, as is necessary for efficiency. However, adding effects blindly renders multilevel types unsound.We introduce the first two-level calculus with control effects and a sound type system. We give small-step operational semantics as well as a continuation-passing style (CPS) translation. For soundness, our calculus restricts the code generator's effects to the scope of generated binders. Even with this restriction, we can finally write efficient code generators for dynamic programming and numerical methods in direct style, like in algorithm textbooks, rather than in CPS or monadic style.

Shifting the stage: staging with delimited control

Code generation lets us write well-abstracted programs without performance penalty. Writing a correct code generator is easier than building a full-scale compiler but still hard. Typed multistage languages such as MetaOCaml help in two ways: they provide simple annotations to express code generation, and they assure that the generated code is well-typed and well-scoped. Unfortunately, the assurance only holds without side effects such as state and control. Without effects, generators often have to be written in a continuation-passing or monadic style that has proved inconvenient. It is thus a pressing open problem to combine effects with staging in a sound type system.This paper takes a first step towards solving the problem, by translating the staging away. Our source language models MetaOCaml restricted to one future stage. It is a call-by-value language, with a sound type system and a small-step operational semantics, that supports building open code, running closed code, cross-stage persistence, and non-termination effects. We translate each typing derivation from this source language to the unstaged System F with constants. Our translation represents future-stage code using closures, yet preserves the typing, α-equivalence (hygiene), and (we conjecture) termination and evaluation order of the staged program.To decouple evaluation from scope (a defining characteristic of staging), our translation weakens the typing environment of open code using a term coercion reminiscent of Goedel's translation from intuitionistic to modal logic. By converting open code to closures with typed environments, our translation establishes a framework in which to study staging with effects and to prototype staged languages. It already makes scope extrusion a type error.

Closing the stage: from staged code to typed closures

It is often hard to write programs that are efficient yet reusable. For example, an efficient implementation of Gaussian elimination should be specialized to the structure and known static properties of the input matrix. The most profitable optimizations, such as choosing the best pivoting or memoization, cannot be expected of even an advanced compiler because they are specific to the domain, but expressing these optimizations directly makes for ungainly source code. Instead, a promising and popular way to reconcile efficiency with reusability is for a domain expert to write code generators.

Two pillars of this approach are types and effects. Typed multilevel languages such as MetaOCaml ensure safety and early error reporting: a well-typed code generator neither goes wrong nor generates code that goes wrong. Side effects such as state and control ease correctness and expressivity: An effectful generator can resemble the textbook presentation of an algorithm, as is familiar to domain experts, yet insert let for memoization and if for bounds checking, as is necessary for efficiency. Together, types and effects enable structuring code generators as compositions of modules with well-defined interfaces, and hence scaling to large programs. However, blindly adding effects renders multilevel types unsound.

We introduce the first multilevel calculus with control effects and a sound type system. We give small-step operational semantics as well as a one-pass continuation-passing-style translation. For soundness, our calculus restricts the code generator's effects to the scope of generated binders. Even with this restriction, we can finally write efficient code generators for dynamic programming and numerical methods in direct style, like in algorithm textbooks, rather than in continuation-passing or monadic style.

/pdf/shifting-the-stage-staging-with-delimited-control-2doia4upat.pdf

Yukiyoshi Kameyama

Papers

Polymorphic delimited continuations

A sound and complete axiomatization of delimited continuations

Shifting the stage: staging with delimited control

Closing the stage: from staged code to typed closures

Shifting the stage: Staging with delimited control