Dariusz Biernacki

A functional correspondence between evaluators and abstract machines

We bridge the gap between functional evaluators and abstract machines for the λ-calculus, using closure conversion, transformation into continuation-passing style, and defunctionalization.We illustrate this approach by deriving Krivines abstract machine from an ordinary call-by-name evaluator and by deriving an ordinary call-by-value evaluator from Felleisen et al.s CEK machine. The first derivation is strikingly simpler than what can be found in the literature. The second one is new. Together, they show that Krivines abstract machine and the CEK machine correspond to the call-by-name and call-by-value facets of an ordinary evaluator for the λ-calculus.We then reveal the denotational content of Hannan and Millers CLS machine and of Landins SECD machine. We formally compare the corresponding evaluators and we illustrate some degrees of freedom in the design spaces of evaluators and of abstract machines for the λ-calculus with computational effects.Finally, we consider the Categorical Abstract Machine and the extent to which it is more of a virtual machine than an abstract machine.

An Operational Foundation for Delimited Continuations in the CPS Hierarchy

We present an abstract machine and a reduction semantics for the lambda-calculus extended with control operators that give access to delimited continuations in the CPS hierarchy. The abstract machine is derived from an evaluator in continuation-passing style (CPS); the reduction semantics (i.e., a small-step operational semantics with an explicit representation of evaluation contexts) is constructed from the abstract machine; and the control operators are the shift and reset family. We also present new applications of delimited continuations in the CPS hierarchy: finding list prefixes and normalization by evaluation for a hierarchical language of units and products.

Clock-directed modular code generation for synchronous data-flow languages

The compilation of synchronous block diagrams into sequential imperative code has been addressed in the early eighties and can now be considered as folklore. However, separate, or modular, code generation, though largely used in existing compilers and particularly in industrial ones, has never been precisely described or entirely formalized. Such a formalization is now fundamental in the long-term goal to develop a mathematically certified compiler for a synchronous language as well as in simplifying existing implementations. This article presents in full detail the modular compilation of synchronous block diagrams into sequential code. We consider a first-order functional language reminiscent of LUSTRE, which it extends with a general n-ary merge operator, a reset construct, and a richer notion of clocks. The clocks are used to express activation of computations in the program and are specifically taken into account during the compilation process to produce efficient imperative code. We introduce a generic machine-based intermediate language to represent transition functions, and we present a concise clock-directed translation from the source to this intermediate language. We address the target code generation phase by describing a translation from the intermediate language to JAVA and C.

From Interpreter to Logic Engine by Defunctionalization

Starting from a continuation-based interpreter for a simple logic programming language, propositional Prolog with cut, we derive the corresponding logic engine in the form of an abstract machine. The derivation originates in previous work (our article at PPDP 2003) where it was applied to the lambda-calculus. The key transformation here is Reynoldss defunctionalization that transforms a tail-recursive, continuation-passing interpreter into a transition system, i.e., an abstract machine. Similar denotational and operational semantics were studied by de Bruin and de Vink (their article at TAPSOFT 1989), and we compare their study with our derivation. Additionally, we present a direct-style interpreter of propositional Prolog expressed with control operators for delimited continuations.

A systematic derivation of the STG machine verified in Coq

Shared Term Graph (STG) is a lazy functional language used as an intermediate language in the Glasgow Haskell Compiler (GHC). In this article, we present a natural operational semantics for STG and we mechanically derive a lazy abstract machine from this semantics, which turns out to coincide with Peyton-Jones and Salkilds Spineless Tagless G-machine (STG machine) used in GHC. Unlike other constructions of STG-like machines present in the literature, ours is based on a systematic and scalable derivation method (inspired by Danvy et al.s functional correspondence between evaluators and abstract machines) and it leads to an abstract machine that differs from the original STG machine only in inessential details. In particular, it handles non-trivial update scenarios and partial applications identically as the STG machine. The entire derivation has been formalized in the Coq proof assistant. Thus, in effect, we provide a machine checkable proof of the correctness of the STG machine with respect to the natural semantics.

Subtyping delimited continuations

We present a type system with subtyping for first-class delimited continuations that generalizes Danvy and Filinskis type system for shift and reset by maintaining explicit information about the types of contexts in the metacontext. We exploit this generalization by considering the control operators known as shift0 and reset0 that can access arbitrary contexts in the metacontext. We use subtyping to control the level of information about the metacontext the expression actually requires and in particular to coerce pure expressions into effectful ones. For this type system we prove strong type soundness and termination of evaluation and we present a provably correct type reconstruction algorithm. We also introduce two CPS translations for shift0 and reset0: one targeting the untyped lambda calculus, and another - type-directed - targeting the simply-typed lambda calculus. The latter translation preserves typability and is selective in that it keeps pure expressions in direct style.

A Dynamic Interpretation of the CPS Hierarchy

The CPS hierarchy of control operators shift i /reset i of Danvy and Filinski is a natural generalization of the shift and reset static control operators that allow for abstracting delimited control in a structured and CPS-guided manner. In this article we show that a dynamic variant of shift/reset, known as shift 0/reset 0, where the discipline of static access to the stack of delimited continuations is relaxed, can fully express the CPS hierarchy. This result demonstrates the expressive power of shift 0 /reset 0 and it offers a new perspective on practical applications of the CPS hierarchy.

Context-based proofs of termination for typed delimited-control operators

We present direct proofs of termination of evaluation for typed delimited-control operators shift and reset using a variant of Taits method with context-based reducibility predicates. We address both call by value and call by name, and for each reduction strategy we consider a type-and-effect system a la Danvy and Filinski as well as a system with a fixed answer type. The call-by-value type-and-effect system we present is a refinement of Danvy and Filinskis original type system, whereas the call-by-name type-and-effect system is new. From the normalization proofs, we extract call-by-value and call-by-name evaluators in continuation-passing style with two layers of continuations; by construction, these evaluators are instances of normalization by evaluation.

On the dynamic extent of delimited continuations

We show that breadth-first traversal exploits the difference between the static delimited-control operator shift (alias S) and the dynamic delimited-control operator control (alias F). For the last 15 years, this difference has been repeatedly mentioned in the literature but it has only been illustrated with one-line toy examples. Breadth-first traversal fills this vacuum.

THEORETICAL PEARL: A simple proof of a folklore theorem about delimited control

We formalize and prove the folklore theorem that the static delimited-control operators shift and reset can be simulated in terms of the dynamic delimited-control operators control and prompt. The proof is based on small-step operational semantics.