A Rational Deconstruction of Landin's SECD Machine with the J Operator

Landin's SECD machine was the first abstract machine for applicative expressions, i.e., functional programs. Landin's J operator was the first control operator for functional languages, and was specified by an extension of the SECD machine. We present a family of evaluation functions corresponding to this extension of the SECD machine, using a series of elementary transformations (transformation into continu-ation-passing style (CPS) and defunctionalization, chiefly) and their left inverses (transformation into direct style and refunctionalization). To this end, we modernize the SECD machine into a bisimilar one that operates in lockstep with the original one but that (1) does not use a data stack and (2) uses the caller-save rather than the callee-save convention for environments. We also identify that the dump component of the SECD machine is managed in a callee-save way. The caller-save counterpart of the modernized SECD machine precisely corresponds to Thielecke's double-barrelled continuations and to Felleisen's encoding of J in terms of call/cc. We then variously characterize the J operator in terms of CPS and in terms of delimited-control operators in the CPS hierarchy. As a byproduct, we also present several reduction semantics for applicative expressions with the J operator, based on Curien's original calculus of explicit substitutions. These reduction semantics mechanically correspond to the modernized versions of the SECD machine and to the best of our knowledge, they provide the first syntactic theories of applicative expressions with the J operator

A Rational Deconstruction of Landin's SECD Machine

Landin's SECD machine was the first abstract machine for the lambda-calculus viewed as a programming language. Both theoretically as a model of computation and practically as an idealized implementation, it has set the tone for the subsequent development of abstract machines for functional programming languages. However, and even though variants of the SECD machine have been presented, derived, and invented, the precise rationale for its architecture and modus operandi has remained elusive. In this article, we deconstruct the SECD machine into a lambda-interpreter, i.e., an evaluation function, and we reconstruct lambda-interpreters into a variety of SECD-like machines. The deconstruction and reconstructions are transformational: they are based on equational reasoning and on a combination of simple program transformations--mainly closure conversion, transformation into continuation-passing style, and defunctionalization. The evaluation function underlying the SECD machine provides a precise rationale for its architecture: it is an environment-based eval-apply evaluator with a callee-save strategy for the environment, a data stack of intermediate results, and a control delimiter. Each of the components of the SECD machine (stack, environment, control, and dump) is therefore rationalized and so are its transitions. The deconstruction and reconstruction method also applies to other abstract machines and other evaluation functions. It makes it possible to systematically extract the denotational content of an abstract machine in the form of a compositional evaluation function, and the (small-step) operational content of an evaluation function in the form of an abstract machine

An interpretation of the Sigma-2 fragment of classical Analysis in System T

We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0 and 1 by showing how to avoid using bar recursion altogether. Our result is proved via a conservative extension of System T with an operator for composable continuations from the theory of programming languages due to Danvy and Filinski. The fragment of Analysis is therefore essentially constructive, even in presence of the full Axiom of Choice schema: Weak Church's Rule holds of it in spite of the fact that it is strong enough to refute the formal arithmetical version of Church's Thesis

On Computational Small Steps and Big Steps: Refocusing for Outermost Reduction

We study the relationship between small-step semantics, big-step semantics and abstract machines, for programming languages that employ an outermost reduction strategy, i.e., languages where reductions near the root of the abstract syntax tree are performed before reductions near the leaves.In particular, we investigate how Biernacka and Danvy's syntactic correspondence and Reynolds's functional correspondence can be applied to inter-derive semantic specifications for such languages.The main contribution of this dissertation is three-fold:First, we identify that backward overlapping reduction rules in the small-step semantics cause the refocusing step of the syntactic correspondence to be inapplicable.Second, we propose two solutions to overcome this in-applicability: backtracking and rule generalization.Third, we show how these solutions affect the other transformations of the two correspondences.Other contributions include the application of the syntactic and functional correspondences to Boolean normalization.In particular, we show how to systematically derive a spectrum of normalization functions for negational and conjunctive normalization

Calculating correct compilers

In this article we present a new approach to the problem of calculating compilers. In particular, we develop a simple but general technique that allows us to derive correct compilers from high- level semantics by systematic calculation, with all details of the implementation of the compilers falling naturally out of the calculation process. Our approach is based upon the use of standard equational reasoning techniques, and has been applied to calculate compilers for a wide range of language features and their combination, including arithmetic expressions, exceptions, state, various forms of lambda calculi, bounded and unbounded loops, non-determinism, and interrupts. All the calculations in the article have been formalised using the Coq proof assistant, which serves as a convenient interactive tool for developing and verifying the calculations

An Analytical Approach to Programs as Data Objects

This essay accompanies a selection of 32 articles (referred to in bold face in the text and marginally marked in the bibliographic references) submitted to Aarhus University towards a Doctor Scientiarum degree in Computer Science.The author's previous academic degree, beyond a doctoral degree in June 1986, is an "Habilitation à diriger les recherches" from the Université Pierre et Marie Curie (Paris VI) in France; the corresponding material was submitted in September 1992 and the degree was obtained in January 1993.The present 32 articles have all been written since 1993 and while at DAIMI.Except for one other PhD student, all co-authors are or have been the author's students here in Aarhus

Foundations for programming and implementing effect handlers

First-class control operators provide programmers with an expressive and efficient means for manipulating control through reification of the current control state as a first-class object, enabling programmers to implement their own computational effects and control idioms as shareable libraries. Effect handlers provide a particularly structured approach to programming with first-class control by naming control reifying operations and separating from their handling. This thesis is composed of three strands of work in which I develop operational foundations for programming and implementing effect handlers as well as exploring the expressive power of effect handlers. The first strand develops a fine-grain call-by-value core calculus of a statically typed programming language with a structural notion of effect types, as opposed to the nominal notion of effect types that dominates the literature. With the structural approach, effects need not be declared before use. The usual safety properties of statically typed programming are retained by making crucial use of row polymorphism to build and track effect signatures. The calculus features three forms of handlers: deep, shallow, and parameterised. They each offer a different approach to manipulate the control state of programs. Traditional deep handlers are defined by folds over computation trees, and are the original con-struct 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. To demonstrate the usefulness of effects and handlers as a practical programming abstraction I implement the essence of a small UNIX-style operating system complete with multi-user environment, time-sharing, and file I/O. The second strand studies continuation passing style (CPS) and abstract machine semantics, which are foundational techniques that admit a unified basis for implementing deep, shallow, and parameterised effect handlers in the same environment. The CPS translation is obtained through a series of refinements of a basic first-order CPS translation for a fine-grain call-by-value language into an untyped language. Each refinement moves toward a more intensional representation of continuations eventually arriving at the notion of generalised continuation, which admit simultaneous support for deep, shallow, and parameterised handlers. The initial refinement adds support for deep handlers by representing stacks of continuations and handlers as a curried sequence of arguments. The image of the resulting translation is not properly tail-recursive, meaning some function application terms do not appear in tail position. To rectify this the CPS translation is refined once more to obtain an uncurried representation of stacks of continuations and handlers. Finally, the translation is made higher-order in order to contract administrative redexes at translation time. The generalised continuation representation is used to construct an abstract machine that provide simultaneous support for deep, shallow, and parameterised effect handlers. kinds of effect handlers. The third strand explores the expressiveness of effect handlers. First, I show that deep, shallow, and parameterised notions of handlers are interdefinable by way of typed macro-expressiveness, which provides a syntactic notion of expressiveness that affirms the existence of encodings between handlers, but it provides no information about the computational content of the encodings. Second, using the semantic notion of expressiveness I show that for a class of programs a programming language with first-class control (e.g. effect handlers) admits asymptotically faster implementations than possible in a language without first-class control