Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\lambda$-terms that this operation commutes with normalization:
the expansion of a $\lambda$-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 $\lambda$-calculus, i.e.
$\lambda$-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
$\lambda$-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 $\lambda$-term is not always normalizable. Our
solution is to provide a fine grained study of the dynamics of
$\beta$-reduction under Taylor expansion, by introducing a notion of reduction
on resource vectors, i.e. infinite linear combinations of resource
$\lambda$-terms. The latter form the multilinear fragment of the differential
$\lambda$-calculus, and resource vectors are the target of the Taylor expansion
of $\lambda$-terms. We show the reduction of resource vectors contains the
image of any $\beta$-reduction step, from which we deduce that Taylor expansion
and normalization commute on the nose. We moreover identify a class of
algebraic $\lambda$-terms, encompassing both normalizable algebraic
$\lambda$-terms and arbitrary ordinary $\lambda$-terms: the expansion of these
is always normalizable, which guides the definition of a generalization of
B\"ohm trees to this setting.

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The categories of this paper are: econ
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph