4,471 research outputs found
Taylor expansion for Call-By-Push-Value
The connection between the Call-By-Push-Value lambda-calculus introduced by Levy and Linear Logic introduced by Girard has been widely explored through a denotational view reflecting the precise ruling of resources in this language. We take a further step in this direction and apply Taylor expansion introduced by Ehrhard and Regnier. We define a resource lambda-calculus in whose terms can be used to approximate terms of Call-By-Push-Value. We show that this approximation is coherent with reduction and with the translations of Call-By-Name and Call-By-Value strategies into Call-By-Push-Value
Normalizing the Taylor expansion of non-deterministic {\lambda}-terms, via parallel reduction of resource vectors
It has been known since Ehrhard and Regnier's seminal work on the Taylor
expansion of -terms that this operation commutes with normalization:
the expansion of a -term is always normalizable and its normal form is
the expansion of the B\"ohm tree of the term. We generalize this result to the
non-uniform setting of the algebraic -calculus, i.e.
-calculus extended with linear combinations of terms. This requires us
to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's
techniques rely heavily on the uniform, deterministic nature of the ordinary
-calculus, and thus cannot be adapted; second is the absence of any
satisfactory generic extension of the notion of B\"ohm tree in presence of
quantitative non-determinism, which is reflected by the fact that the Taylor
expansion of an algebraic -term is not always normalizable. Our
solution is to provide a fine grained study of the dynamics of
-reduction under Taylor expansion, by introducing a notion of reduction
on resource vectors, i.e. infinite linear combinations of resource
-terms. The latter form the multilinear fragment of the differential
-calculus, and resource vectors are the target of the Taylor expansion
of -terms. We show the reduction of resource vectors contains the
image of any -reduction step, from which we deduce that Taylor expansion
and normalization commute on the nose. We moreover identify a class of
algebraic -terms, encompassing both normalizable algebraic
-terms and arbitrary ordinary -terms: the expansion of these
is always normalizable, which guides the definition of a generalization of
B\"ohm trees to this setting
On the Taylor expansion of probabilistic \u3bb-terms
We generalise Ehrhard and Regnier\u2019s Taylor expansion from pure to probabilistic \u3bb-terms. We prove that the Taylor expansion is adequate when seen as a way to give semantics to probabilistic \u3bb-terms, and that there is a precise correspondence with probabilistic B\uf6hm trees, as introduced by the second author. We prove this adequacy through notions of probabilistic resource terms and explicit Taylor expansion
Taylor subsumes Scott, Berry, Kahn and Plotkin
The speculative ambition of replacing the old theory of program approximation based on syntactic continuity with the theory of resource consumption based on Taylor expansion and originating from the differential γ-calculus is nowadays at hand. Using this resource sensitive theory, we provide simple proofs of important results in γ-calculus that are usually demonstrated by exploiting Scott's continuity, Berry's stability or Kahn and Plotkin's sequentiality theory. A paradigmatic example is given by the Perpendicular Lines Lemma for the Böhm tree semantics, which is proved here simply by induction, but relying on the main properties of resource approximants: strong normalization, confluence and linearity
Counting and Generating Terms in the Binary Lambda Calculus (Extended version)
In a paper entitled Binary lambda calculus and combinatory logic, John Tromp
presents a simple way of encoding lambda calculus terms as binary sequences. In
what follows, we study the numbers of binary strings of a given size that
represent lambda terms and derive results from their generating functions,
especially that the number of terms of size n grows roughly like 1.963447954.
.. n. In a second part we use this approach to generate random lambda terms
using Boltzmann samplers.Comment: extended version of arXiv:1401.037
- …