6,058 research outputs found
The weak call-by-value λ-calculus is reasonable for both time and space
We study the weak call-by-value -calculus as a model for computational complexity theory and establish the
natural measures for time and space Ð the number of beta-reduction steps and the size of the largest term
in a computation Ð as reasonable measures with respect to the invariance thesis of Slot and van Emde Boas
from 1984. More precisely, we show that, using those measures, Turing machines and the weak call-by-value
-calculus can simulate each other within a polynomial overhead in time and a constant factor overhead in
space for all computations terminating in (encodings of) łtruež or łfalsež. The simulation yields that standard
complexity classes like , NP, PSPACE, or EXP can be defined solely in terms of the -calculus, but does not
cover sublinear time or space.
Note that our measures still have the well-known size explosion property, where the space measure of
a computation can be exponentially bigger than its time measure. However, our result implies that this
exponential gap disappears once complexity classes are considered instead of concrete computations.
We consider this result a first step towards a solution for the long-standing open problem of whether the
natural measures for time and space of the -calculus are reasonable. Our proof for the weak call-by-value
-calculus is the first proof of reasonability (including both time and space) for a functional language based on
natural measures and enables the formal verification of complexity-theoretic proofs concerning complexity
classes, both on paper and in proof assistants.
The proof idea relies on a hybrid of two simulation strategies of reductions in the weak call-by-value
-calculus by Turing machines, both of which are insufficient if taken alone. The first strategy is the most naive
one in the sense that a reduction sequence is simulated precisely as given by the reduction rules; in particular,
all substitutions are executed immediately. This simulation runs within a constant overhead in space, but the
overhead in time might be exponential. The second strategy is heap-based and relies on structure sharing,
similar to existing compilers of eager functional languages. This strategy only has a polynomial overhead in
time, but the space consumption might require an additional factor of log, which is essentially due to the
size of the pointers required for this strategy. Our main contribution is the construction and verification of a
space-aware interleaving of the two strategies, which is shown to yield both a constant overhead in space and
a polynomial overhead in time
(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
Space-Aware Ambients and Processes
Resource control has attracted increasing interest in foundational research on distributed systems. This paper focuses on space control and develops an analysis of space usage in the context of an ambient-like calculus with bounded capacities and weighed processes, where migration and activation require space. A type system complements the dynamics of the calculus by providing static guarantees that the intended capacity bounds are preserved throughout the computation
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