13 research outputs found
A Strong Distillery
Abstract machines for the strong evaluation of lambda-terms (that is, under
abstractions) are a mostly neglected topic, despite their use in the
implementation of proof assistants and higher-order logic programming
languages. This paper introduces a machine for the simplest form of strong
evaluation, leftmost-outermost (call-by-name) evaluation to normal form,
proving it correct, complete, and bounding its overhead. Such a machine, deemed
Strong Milner Abstract Machine, is a variant of the KAM computing normal forms
and using just one global environment. Its properties are studied via a special
form of decoding, called a distillation, into the Linear Substitution Calculus,
neatly reformulating the machine as a standard micro-step strategy for explicit
substitutions, namely linear leftmost-outermost reduction, i.e., the extension
to normal form of linear head reduction. Additionally, the overhead of the
machine is shown to be linear both in the number of steps and in the size of
the initial term, validating its design. The study highlights two distinguished
features of strong machines, namely backtracking phases and their interactions
with abstractions and environments.Comment: Accepted at APLAS 201
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
Useful Open Call-By-Need
This paper studies useful sharing, which is a sophisticated optimization for ?-calculi, in the context of call-by-need evaluation in presence of open terms. Useful sharing turns out to be harder in call-by-need than in call-by-name or call-by-value, because call-by-need evaluates inside environments, making it harder to specify when a substitution step is useful. We isolate the key involved concepts and prove the correctness and the completeness of useful sharing in this setting
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)
-calculus into models like RAM machines which is polynomial in the
number of -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 -steps.Comment: Technical report for the LICS 2015 submission with the same titl
The Dynamic Geometry of Interaction Machine: A Token-Guided Graph Rewriter
In implementing evaluation strategies of the lambda-calculus, both
correctness and efficiency of implementation are valid concerns. While the
notion of correctness is determined by the evaluation strategy, regarding
efficiency there is a larger design space that can be explored, in particular
the trade-off between space versus time efficiency. Aiming at a unified
framework that would enable the study of this trade-off, we introduce an
abstract machine, inspired by Girard's Geometry of Interaction (GoI), a machine
combining token passing and graph rewriting. We show soundness and completeness
of our abstract machine, called the \emph{Dynamic GoI Machine} (DGoIM), with
respect to three evaluations: call-by-need, left-to-right call-by-value, and
right-to-left call-by-value. Analysing time cost of its execution classifies
the machine as ``efficient'' in Accattoli's taxonomy of abstract machines.Comment: arXiv admin note: text overlap with arXiv:1802.0649
Efficient implementation of evaluation strategies via token-guided graph rewriting
In implementing evaluation strategies of the lambda-calculus, both
correctness and efficiency of implementation are valid concerns. While the
notion of correctness is determined by the evaluation strategy, regarding
efficiency there is a larger design space that can be explored, in particular
the trade-off between space versus time efficiency. We contributed to the study
of this trade-off by the introduction of an abstract machine for call-by-need,
inspired by Girard's Geometry of Interaction, a machine combining token passing
and graph rewriting. This work presents an extension of the machine, to
additionally accommodate left-to-right and right-to-left call-by-value
strategies. We show soundness and completeness of the extended machine with
respect to each of the call-by-need and two call-by-value strategies. Analysing
time cost of its execution classifies the machine as "efficient" in Accattoli's
taxonomy of abstract machines.Comment: In Proceedings WPTE 2017, arXiv:1802.0586
Shared-Environment Call-by-Need
Call-by-need semantics formalize the wisdom that work should be done at most once. It frees programmers to focus more on the correctness of their code, and less on the operational details. Because of this property, programmers of lazy functional languages rely heavily on their compiler to both preserve correctness and generate high-performance code for high level abstractions. In this dissertation I present a novel technique for compiling call-by-need semantics by using shared environments to share results of computation. I show how the approach enables a compiler that generates high-performance code, while staying simple enough to lend itself to formal reasoning. The dissertation is divided into three main contributions. First, I present an abstract machine, the \ce machine, which formalizes the approach. Second, I show that it can be implemented as a native code compiler with encouraging performance results. Finally, I present a verified compiler, implemented in the Coq proof assistant, demonstrating how the simplicity of the approach enables formal verification