6 research outputs found
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
Gluing resource proof-structures: inhabitation and inverting the Taylor expansion
A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be
expanded into a set of resource proof-structures: its Taylor expansion. We
introduce a new criterion characterizing those sets of resource
proof-structures that are part of the Taylor expansion of some MELL
proof-structure, through a rewriting system acting both on resource and MELL
proof-structures. As a consequence, we also prove semi-decidability of the type
inhabitation problem for cut-free MELL proof-structures.Comment: arXiv admin note: substantial text overlap with arXiv:1910.0793
Finitary Simulation of Infinitary -Reduction via Taylor Expansion, and Applications
Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion
of -terms has been broadly used as a tool to approximate the terms of
several variants of the -calculus. Many results arise from a
Commutation theorem relating the normal form of the Taylor expansion of a term
to its B\"ohm tree. This led us to consider extending this formalism to the
infinitary -calculus, since the version of
this calculus has B\"ohm trees as normal forms and seems to be the ideal
framework to reformulate the Commutation theorem.
We give a (co-)inductive presentation of . We define
a Taylor expansion on this calculus, and state that the infinitary
-reduction can be simulated through this Taylor expansion. The target
language is the usual resource calculus, and in particular the resource
reduction remains finite, confluent and terminating. Finally, we state the
generalised Commutation theorem and use our results to provide simple proofs of
some normalisation and confluence properties in the infinitary
-calculus
Genericity Through Stratification
A fundamental issue in the -calculus is to find appropriate notions
for meaningfulness. It is well-known that in the call-by-name
-calculus (CbN) the meaningful terms can be identified with the
solvable ones, and that this notion is not appropriate in the call-by-value
-calculus (CbV). This paper validates the challenging claim that yet
another notion, previously introduced in the literature as potential
valuability, appropriately represents meaningfulness in CbV. Akin to CbN, this
claim is corroborated by proving two essential properties. The first one is
genericity, stating that meaningless subterms have no bearing on evaluating
normalizing terms. To prove this (which was an open problem), we use a novel
approach based on stratified reduction, indifferently applicable to CbN and
CbV, and in a quantitative way. The second property concerns consistency of the
smallest congruence relation resulting from equating all meaningless terms.
While the consistency result is not new, we provide the first direct
operational proof of it. We also show that such a congruence has a unique
consistent and maximal extension, which coincides with a well-known notion of
observational equivalence. Our results thus supply the formal concepts and
tools that validate the informal notion of meaningfulness underlying CbN and
CbV
Gluing resource proof-structures: inhabitation and inverting the Taylor expansion
A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be
expanded into a set of resource proof-structures: its Taylor expansion. We
introduce a new criterion characterizing (and deciding in the finite case)
those sets of resource proof-structures that are part of the Taylor expansion
of some MELL proof-structure, through a rewriting system acting both on
resource and MELL proof-structures. We also prove semi-decidability of the type
inhabitation problem for cut-free MELL proof-structures