11,060 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
Beta Reduction is Invariant, Indeed (Long Version)
Slot and van Emde Boas' weak invariance thesis states that reasonable
machines can simulate each other within a polynomially overhead in time. Is
-calculus a reasonable machine? Is there a way to measure the
computational complexity of a -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 -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 -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 modelled
after linear logic and 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 -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 -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
-redexes, i.e., the steps that cause the blow-up in size.Comment: 29 page
An Invariant Cost Model for the Lambda Calculus
We define a new cost model for the call-by-value lambda-calculus satisfying
the invariance thesis. That is, under the proposed cost model, Turing machines
and the call-by-value lambda-calculus can simulate each other within a
polynomial time overhead. The model only relies on combinatorial properties of
usual beta-reduction, without any reference to a specific machine or evaluator.
In particular, the cost of a single beta reduction is proportional to the
difference between the size of the redex and the size of the reduct. In this
way, the total cost of normalizing a lambda term will take into account the
size of all intermediate results (as well as the number of steps to normal
form).Comment: 19 page
Arbitrage Bounds for Prices of Weighted Variance Swaps
We develop robust pricing and hedging of a weighted variance swap when market
prices for a finite number of co--maturing put options are given. We assume the
given prices do not admit arbitrage and deduce no-arbitrage bounds on the
weighted variance swap along with super- and sub- replicating strategies which
enforce them. We find that market quotes for variance swaps are surprisingly
close to the model-free lower bounds we determine. We solve the problem by
transforming it into an analogous question for a European option with a convex
payoff. The lower bound becomes a problem in semi-infinite linear programming
which we solve in detail. The upper bound is explicit.
We work in a model-independent and probability-free setup. In particular we
use and extend F\"ollmer's pathwise stochastic calculus. Appropriate notions of
arbitrage and admissibility are introduced. This allows us to establish the
usual hedging relation between the variance swap and the 'log contract' and
similar connections for weighted variance swaps. Our results take form of a
FTAP: we show that the absence of (weak) arbitrage is equivalent to the
existence of a classical model which reproduces the observed prices via
risk-neutral expectations of discounted payoffs.Comment: 25 pages, 4 figure
The Weak Call-By-Value {\lambda}-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 -- the number of beta-reductions and the size of the largest term in a
computation -- as reasonable measures with respect to the invariance thesis of
Slot and van Emde Boas [STOC~84]. 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 that terminate in (encodings) of 'true'
or 'false'. We consider this result as a solution to the long-standing open
problem, explicitly posed by Accattoli [ENTCS~18], of whether the natural
measures for time and space of the -calculus are reasonable, at least
in case of weak call-by-value evaluation.
Our proof 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 , 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
Fractional constant elasticity of variance model
This paper develops a European option pricing formula for fractional market
models. Although there exist option pricing results for a fractional
Black-Scholes model, they are established without accounting for stochastic
volatility. In this paper, a fractional version of the Constant Elasticity of
Variance (CEV) model is developed. European option pricing formula similar to
that of the classical CEV model is obtained and a volatility skew pattern is
revealed.Comment: Published at http://dx.doi.org/10.1214/074921706000001012 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Theories of truth based on four-valued infectious logics
Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at least some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems
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