Continuation-passing Style Translation Research Articles

On continuation-passing transformations and expected cost analysis

We define a continuation-passing style (CPS) translation for a typed λ-calculus with probabilistic choice, unbounded recursion, and a tick operator — for modeling cost. The target language is a (non-probabilistic) λ-calculus, enriched with a type of extended positive reals and a fixpoint operator. We then show that applying the CPS transform of an expression M to the continuation λ v . 0 yields the expected cost of M . We also introduce a formal system for higher-order logic, called EHOL, prove it sound, and show it can derive tight upper bounds on the expected cost of classic examples, including Coupon Collector and Random Walk. Moreover, we relate our translation to Kaminski et al.’s ert-calculus, showing that the latter can be recovered by applying our CPS translation to (a generalization of) the classic embedding of imperative programs into λ-calculus. Finally, we prove that the CPS transform of an expression can also be used to compute pre-expectations and to reason about almost sure termination.

Effect handlers via generalised continuations

AbstractPlotkin and Pretnar’s effect handlers offer a versatile abstraction for modular programming with user-defined effects. This paper focuses on foundations for implementing effect handlers, for the three different kinds of effect handlers that have been proposed in the literature: deep, shallow, and parameterised. Traditional deep handlers are defined by folds over computation trees and are the original construct proposed by Plotkin and Pretnar. Shallow handlers are defined by case splits (rather than folds) over computation trees. Parameterised handlers are deep handlers extended with a state value that is threaded through the folds over computation trees. We formulate the extensions both directly and via encodings in terms of deep handlers and illustrate how the direct implementations avoid the generation of unnecessary closures. We give two distinct foundational implementations of all the kinds of handlers we consider: a continuation-passing style (CPS) transformation and a CEK-style abstract machine. In both cases, the key ingredient is a generalisation of the notion of continuation to accommodate stacks of effect handlers. We obtain our CPS translation through a series of refinements as follows. We begin with a first-order CPS translation into untyped lambda calculus which manages a stack of continuations and handlers as a curried sequence of arguments. We then refine the initial CPS translation by uncurrying it to yield a properly tail-recursive translation and then moving towards more and more intensional representations of continuations in order to support different kinds of effect handlers. Finally, we make the translation higher order in order to contract administrative redexes at translation time. Our abstract machine design then uses the same generalised continuation representation as the CPS translation. We have implemented both the abstract machine and the CPS transformation (plus extensions) as backends for the Links web programming language.

Compiling with continuations, or without? whatever.

What makes a good compiler IR? In the context of functional languages, there has been an extensive debate on the advantages and disadvantages of continuation-passing-style (CPS). The consensus seems to be that some form of explicit continuations is necessary to model jumps in a functional style, but that they should have a 2nd-class status, separate from regular functions, to ensure efficient code generation. Building on this observation, a recent study from PLDI 2017 proposed a direct-style IR with explicit join points, which essentially represent local continuations, i.e., functions that do not return or escape. While this IR can work well in practice, as evidenced by the implementation of join points in the Glasgow Haskell Compiler (GHC), there still seems to be room for improvement, especially with regard to the way continuations are handled in the course of optimization. In this paper, we contribute to the CPS debate by developing a novel IR with the following features. First, we integrate a control operator that resembles Felleisen’s C , eliminating certain redundant rewrites observed in the previous study. Second, we treat the non-returning and non-escaping aspects of continuations separately, allowing efficient compilation of well-behaved functions defined by the user. Third, we define a selective CPS translation of our IR, which erases control operators while preserving the meaning and typing of programs. These features enable optimizations in both direct style and full CPS, as well as in any intermediate style with selectively exposed continuations. Thus, we change the spectrum of available options from “CPS yes or no” to “as much or as little CPS as you want, when you want it”.

Type Directed Partial Evaluation for Level-1 Shift and Reset

We present an implementation in the Coq proof assistant of type directed partial evaluation (TDPE) algorithms for call-by-name and call-by-value versions of shift and reset delimited control operators, and in presence of strong sum types. We prove that the algorithm transforms well-typed programs to ones in normal form. These normal forms can not always be arrived at using the so far known equational theories. The typing system does not allow answer-type modification for function types and allows delimiters to be set on at most one atomic type. The semantic domain for evaluation is expressed in Constructive Type Theory as a dependently typed monadic structure combining Kripke models and continuation passing style translations.

Proofs, tests and continuation passing style

The concept of syntactical duality is central in logic. In particular, the duality defined by classical negation, or more syntactically by left and right in sequents, has been widely used to relate logic and computations. We study the proof/test duality proposed by Girard in his 1999 paper on the meaning of logical rules. In detail, starting from the notion of “test” proposed by Girard, we develop a notion of test for intuitionistic logic and we give a complete deductive system whose computational interpretation is the target language of the call-by-value and call-by-name continuation passing style translations.

The essence of compiling with continuations

In order to simplify the compilation process, many compilers for higher-order languages use the continuation-passing style (CPS) transformation in a first phase to generate an intermediate representation of the source program. The salient aspect of this intermediate form is that all procedures take an argument that represents the rest of the computation (the "continuation"). Since the naïve CPS transformation considerably increases the size of programs, CPS compilers perform reductions to produce a more compact intermediate representation. Although often implemented as a part of the CPS transformation, this step is conceptually a second phase. Finally, code generators for typical CPS compilers treat continuations specially in order to optimize the interpretation of continuation parameters.A thorough analysis of the abstract machine for CPS terms shows that the actions of the code generator invert the naïve CPS translation step. Put differently, the combined effect of the three phases is equivalent to a source-to-source transformation that simulates the compaction phase. Thus, fully developed CPS compilers do not need to employ the CPS transformation but can achieve the same results with a simple source-level transformation.

CPS Translations and Applications: The Cube and Beyond

Continuation passing style (CPS) translations of typed λ-calculi have numerous applications. However, the range of these applications has been confined by the fact that CPS translations are known for non-dependent type systems only, thus excluding well-known systems like the calculus of constructions (CC) and the logical frameworks (LF). This paper presents techniques for CPS translating systems with dependent types, with an emphasis on pure type-theoretical applications. In the first part of the paper we review several lines of work in which the need for CPS translations of dependent type systems has arisen, and discuss the difficulties involved with CPS translating such systems. One way of overcoming these difficulties is to work with so-called domain-free type systems. Thus, instead of Barendregt‘s λ-cube we shall consider the domain-free λ-cube, and instead of traditional pure type systems, we shall consider domain-free pure type systems. We therefore begin the second part by reviewing the domain-free λ-cube, which includes domain-free versions of CC and LF, and then present CPS translations for all the systems of the domain-free λ-cube. We also introduce Direct Style (DS) (i.e., inverse CPS) translations for all the systems of the domain-free λ-cube; such DS translations, which have been used in a number of applications, were previously formulated for untyped and simply-typed languages only. In the third part we review domain-free pure type systems and generalize the CPS translations of the domain-free λ-cube to a large class of domain-free pure type systems which includes most of the systems that appear in the literature, including those of the domain-free λ-cube. Many translations that appear in the literature arise as special cases of ours. In the fourth part of the paper we present two approaches to CPS translations of traditional pure type systems. The first, indirect, technique lifts the CPS translation of domain-free pure type systems to the analogous class of traditional pure type systems by using results that relate derivations in domain-free and traditional pure type systems. The second, direct, approach translates derivations, requiring a certain order on derivations to be well-founded. Both techniques yield translations for most of the systems that appear in the literature, including those of Barendregt‘s λ-cube.

A reflection on call-by-value

A number of compilers exploit the following strategy: translate a term to continuation-passing style (CPS) and optimize the resulting term using a sequence of reductions. Recent work suggests that an alternative strategy is superior: optimize directly in an extended source calculus. We suggest that the appropriate relation between the source and target calculi may be captured by a special case of a Galois connection known as a reflection . Previous work has focused on the weaker notion of an equational correspondence , which is based on equality rather than reduction. We show that Moggi's monad translation and Plotkin's CPS translation can both be regarded as reflections, and thereby strengthen a number of results in the literature.

The essence of compiling with continuations

In order to simplify the compilation process, many compilers for higher-order languages use the continuation-passing style (CPS) transformation in a first phase to generate an intermediate representation of the source program. The salient aspect of this intermediate form is that all procedures take an argument that represents the rest of the computation (the “continuation”). Since the nai¨ve CPS transformation considerably increases the size of programs, CPS compilers perform reductions to produce a more compact intermediate representation. Although often implemented as a part of the CPS transformation, this step is conceptually a second phase. Finally, code generators for typical CPS compilers treat continuations specially in order to optimize the interpretation of continuation parameters. A thorough analysis of the abstract machine for CPS terms show that the actions of the code generator invert the nai¨ve CPS translation step. Put differently, the combined effect of the three phases is equivalent to a source-to-source transformation that simulates the compaction phase. Thus, fully developed CPS compilers do not need to employ the CPS transformation but can achieve the same results with a simple source-level transformation.