Realizability Interpretation and Normalization of Typed Call-by-Need λ\lambda-calculus With Control

We define a variant of realizability where realizers are pairs of a term and a substitution. This variant allows us to prove the normalization of a simply-typed call-by-need \lambda$-$calculus with control due to Ariola et al. Indeed, in such call-by-need calculus, substitutions have to be delayed until knowing if an argument is really needed. In a second step, we extend the proof to a call-by-need \lambda$$-calculus equipped with a type system equivalent to classical second-order predicate logic, representing one step towards proving the normalization of the call-by-need classical second-order arithmetic introduced by the second author to provide a proof-as-program interpretation of the axiom of dependent choice

Distilling Abstract Machines (Long Version)

It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between big-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.Comment: 63 page

A calculus of expandable stores: Continuation-and-environment-passing style translations

LICS 2020 will be held onlineInternational audienceThe call-by-need evaluation strategy for the λ-calculus is an evaluation strategy that lazily evaluates arguments only if needed, and if so, shares computations across all places where it is needed. To implement this evaluation strategy, abstract machines require some form of global environment. While abstract machines usually lead to a better understanding of the flow of control during the execution, facilitating in particular the definition of continuation-passing style translations , the case of machines with global environments turns out to be much more subtle. The main purpose of this paper is to understand how to type a continuation-and-environment-passing style translation , that is to say how to soundly translate in continuation-passing style a calculus with global environment. To this end, we introduce Fϒ , a generic calculus to define the target of such translations. In particular, Fϒ features a data type for typed stores and a mechanism of explicit coercions witnessing store extensions along environment-passing style translations. On the logical side, this broadly amounts to a Kripke forcing-like translation mixed with a negative translation (for the continuation-passing part). Since Fϒ allows for the definition of such translations for different source calculi (call-by-need, call-by-name, call-by-value) with different type systems (simple types, system F), we claim that it precisely captures the computational content of continuation-and-environment-passing style translations

On the Relative Usefulness of Fireballs

In CSL-LICS 2014, Accattoli and Dal Lago showed that there is an implementation of the ordinary (i.e. strong, pure, call-by-name) $\lambda$-calculus into models like RAM machines which is polynomial in the number of $\beta$-steps, answering a long-standing question. The key ingredient was the use of a calculus with useful sharing, a new notion whose complexity was shown to be polynomial, but whose implementation was not explored. This paper, meant to be complementary, studies useful sharing in a call-by-value scenario and from a practical point of view. We introduce the Fireball Calculus, a natural extension of call-by-value to open terms for which the problem is as hard as for the ordinary lambda-calculus. We present three results. First, we adapt the solution of Accattoli and Dal Lago, improving the meta-theory of useful sharing. Then, we refine the picture by introducing the GLAMoUr, a simple abstract machine implementing the Fireball Calculus extended with useful sharing. Its key feature is that usefulness of a step is tested---surprisingly---in constant time. Third, we provide a further optimization that leads to an implementation having only a linear overhead with respect to the number of $\beta$-steps.Comment: Technical report for the LICS 2015 submission with the same titl

Continuation-and-environment-passing style translations: a focus on call-by-need

The call-by-need evaluation strategy for the $λ$-calculus is an evaluation strategy that lazily evaluates arguments only if needed, and if so, shares computations across all places where it is needed. To implement this evaluation strategy, abstract machines require some form of global environment. While abstract machines usually lead to a better understanding of the flow of control during the execution, easing in particular the definition of continuation-passing style translations, the case of machines with global environments turns out to be much more subtle. The main purpose of this paper is to understand how to type a continuation-and-environment-passing style translations, that it to say how to soundly translate a classical calculus with environment into a calculus that does not have these features. To this end, we focus on a sequent calculus presentation of a call-by-need $λ$-calculus with classical control for which Ariola et. al already defined an untyped translation [5] and which we equipped with a system of simple types in a previous paper [32]. We present here a type system for the target language of their translation, which highlights a variant of Kripke forcing related to the environment-passing part of the translation. Finally, we show that our construction naturally handles the cases of call-by-name and call-by-value calculi with environment, encompassing in particular the Milner Abstract Machine, a machine with global environments for the call-by-name $λ$-calculus

Beta-Conversion, Efficiently

Type-checking in dependent type theories relies on conversion, i.e. testing given lambda-terms for equality up to beta-evaluation and alpha-renaming. Computer tools based on the lambda-calculus currently implement conversion by means of algorithms whose complexity has not been identified, and in some cases even subject to an exponential time overhead with respect to the natural cost models (number of evaluation steps and size of input lambda-terms). This dissertation shows that in the pure lambda-calculus it is possible to obtain conversion algorithms with bilinear time complexity when evaluation is carried following evaluation strategies that generalize Call-by-Value to the stronger case required by conversion

Practical implementation of a dependently typed functional programming language

Types express a program's meaning, and checking types ensures that a program has the intended meaning. In a dependently typed programming language types are predicated on values, leading to the possibility of expressing invariants of a program's behaviour in its type. Dependent types allow us to give more detailed meanings to programs, and hence be more confident of their correctness. This thesis considers the practical implementation of a dependently typed programming language, using the Epigram notation defined by McBride and McKinna. Epigram is a high level notation for dependently typed functional programming elaborating to a core type theory based on Lu๙s UTT, using Dybjer's inductive families and elimination rules to implement pattern matching. This gives us a rich framework for reasoning about programs. However, a naive implementation introduces several run-time overheads since the type system blurs the distinction between types and values; these overheads include the duplication of values, and the storage of redundant information and explicit proofs. A practical implementation of any programming language should be as efficient as possible; in this thesis we see how the apparent efficiency problems of dependently typed programming can be overcome and that in many cases the richer type information allows us to apply optimisations which are not directly available in traditional languages. I introduce three storage optimisations on inductive families; forcing, detagging and collapsing. I further introduce a compilation scheme from the core type theory to G-machine code, including a pattern matching compiler for elimination rules and a compilation scheme for efficient run-time implementation of Peano's natural numbers. We also see some low level optimisations for removal of identity functions, unused arguments and impossible case branches. As a result, we see that a dependent type theory is an effective base on which to build a feasible programming language

The modular compilation of effects

The introduction of new features to a programming language often requires that its compiler goes to the effort of ensuring they are introduced in a manner that does not interfere with the existing code base. Engineers frequently find themselves changing code that has already been designed, implemented and (ideally) proved correct, which is bad practice from a software engineering point of view. This thesis addresses the issue of constructing a compiler for a source language that is modular in the computational features that it supports. Utilising a minimal language that allows us to demonstrate the underlying techniques, we go on to introduce a significant range of effectful features in a modular manner, showing that their syntax can be compiled independently, and that source languages containing multiple features can be compiled by making use of a fold. In the event that new features necessitate changes in the underlying representation of either the source language or that of the compiler, we show that our framework is capable of incorporating these changes with minimal disruption. Finally, we show how the framework we have developed can be used to define both modular evaluators and modular virtual machines