5,347 research outputs found
Multi-level Contextual Type Theory
Contextual type theory distinguishes between bound variables and
meta-variables to write potentially incomplete terms in the presence of
binders. It has found good use as a framework for concise explanations of
higher-order unification, characterize holes in proofs, and in developing a
foundation for programming with higher-order abstract syntax, as embodied by
the programming and reasoning environment Beluga. However, to reason about
these applications, we need to introduce meta^2-variables to characterize the
dependency on meta-variables and bound variables. In other words, we must go
beyond a two-level system granting only bound variables and meta-variables.
In this paper we generalize contextual type theory to n levels for arbitrary
n, so as to obtain a formal system offering bound variables, meta-variables and
so on all the way to meta^n-variables. We obtain a uniform account by
collapsing all these different kinds of variables into a single notion of
variabe indexed by some level k. We give a decidable bi-directional type system
which characterizes beta-eta-normal forms together with a generalized
substitution operation.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
(Leftmost-Outermost) Beta Reduction is Invariant, Indeed
Slot and van Emde Boas' weak invariance thesis states that reasonable
machines can simulate each other within a polynomially overhead in time. Is
lambda-calculus a reasonable machine? Is there a way to measure the
computational complexity of a lambda-term? This paper presents the first
complete positive answer to this long-standing problem. Moreover, our answer is
completely machine-independent and based over a standard notion in the theory
of lambda-calculus: the length of a leftmost-outermost derivation to normal
form is an invariant cost model. Such a theorem cannot be proved by directly
relating lambda-calculus with Turing machines or random access machines,
because of the size explosion problem: there are terms that in a linear number
of steps produce an exponentially long output. The first step towards the
solution is to shift to a notion of evaluation for which the length and the
size of the output are linearly related. This is done by adopting the linear
substitution calculus (LSC), a calculus of explicit substitutions modeled after
linear logic proof nets and admitting a decomposition of leftmost-outermost
derivations with the desired property. Thus, the LSC is invariant with respect
to, say, random access machines. The second step is to show that LSC is
invariant with respect to the lambda-calculus. The size explosion problem seems
to imply that this is not possible: having the same notions of normal form,
evaluation in the LSC is exponentially longer than in the lambda-calculus. We
solve such an impasse by introducing a new form of shared normal form and
shared reduction, deemed useful. Useful evaluation avoids those steps that only
unshare the output without contributing to beta-redexes, i.e. the steps that
cause the blow-up in size. The main technical contribution of the paper is
indeed the definition of useful reductions and the thorough analysis of their
properties.Comment: arXiv admin note: substantial text overlap with arXiv:1405.331
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
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