8 research outputs found
Non uniform (hyper/multi)coherence spaces
In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs.
Intuitively, vertices represent results of computations and the edge relation
witnesses the ability of being assembled into a same piece of data or a same
(strongly) stable function, at arrow types. In (hyper)coherence semantics, the
argument of a (strongly) stable functional is always a (strongly) stable
function. As a consequence, comparatively to the relational semantics, where
there is no edge relation, some vertices are missing. Recovering these vertices
is essential for the purpose of reconstructing proofs/terms from their
interpretations. It shall also be useful for the comparison with other
semantics, like game semantics. In [BE01], Bucciarelli and Ehrhard introduced a
so called non uniform coherence space semantics where no vertex is missing. By
constructing the co-free exponential we set a new version of this last
semantics, together with non uniform versions of hypercoherences and
multicoherences, a new semantics where an edge is a finite multiset. Thanks to
the co-free construction, these non uniform semantics are deterministic in the
sense that the intersection of a clique and of an anti-clique contains at most
one vertex, a result of interaction, and extensionally collapse onto the
corresponding uniform semantics.Comment: 32 page
Categorical models of Linear Logic with fixed points of formulas
We develop a denotational semantics of muLL, a version of propositional
Linear Logic with least and greatest fixed points extending David Baelde's
propositional muMALL with exponentials. Our general categorical setting is
based on the notion of Seely category and on strong functors acting on them. We
exhibit two simple instances of this setting. In the first one, which is based
on the category of sets and relations, least and greatest fixed points are
interpreted in the same way. In the second one, based on a category of sets
equipped with a notion of totality (non-uniform totality spaces) and relations
preserving them, least and greatest fixed points have distinct interpretations.
This latter model shows that muLL enjoys a denotational form of normalization
of proofs.Comment: arXiv admin note: text overlap with arXiv:1906.0559
Taylor expansion in linear logic is invertible
Each Multiplicative Exponential Linear Logic (MELL) proof-net can be expanded
into a differential net, which is its Taylor expansion. We prove that two
different MELL proof-nets have two different Taylor expansions. As a corollary,
we prove a completeness result for MELL: We show that the relational model is
injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the
relational model is exactly axiomatized by cut-elimination
An Indexed Linear Logic for Idempotent Intersection Types (Long version)
Indexed Linear Logic has been introduced by Ehrhard and Bucciarelli, it can
be seen as a logical presentation of non-idempotent intersection types extended
through the relational semantics to the full linear logic. We introduce an
idempotent variant of Indexed Linear Logic. We give a fine-grained
reformulation of the syntax by exposing implicit parameters and by unifying
several operations on formulae via the notion of base change. Idempotency is
achieved by means of an appropriate subtyping relation. We carry on an in-depth
study of indLL as a logic, showing how it determines a refinement of classical
linear logic and establishing a terminating cut-elimination procedure.
Cut-elimination is proved to be confluent up to an appropriate congruence
induced by the subtyping relation
From Differential Linear Logic to Coherent Differentiation
In this survey, we present in a unified way the categorical and syntactical
settings of coherent differentiation introduced recently, which shows that the
basic ideas of differential linear logic and of the differential
lambda-calculus are compatible with determinism. Indeed, due to the Leibniz
rule of the differential calculus, differential linear logic and the
differential lambda-calculus feature an operation of addition of proofs or
terms operationally interpreted as a strong form of nondeterminism. The main
idea of coherent differentiation is that these sums can be controlled and kept
in the realm of determinism by means of a notion of summability, upon enforcing
summability restrictions on the derivatives which can be written in the models
and in the syntax
Differentials and distances in probabilistic coherence spaces
In probabilistic coherence spaces, a denotational model of probabilistic
functional languages, morphisms are analytic and therefore smooth. We explore
two related applications of the corresponding derivatives. First we show how
derivatives allow to compute the expectation of execution time in the weak head
reduction of probabilistic PCF (pPCF). Next we apply a general notion of
"local" differential of morphisms to the proof of a Lipschitz property of these
morphisms allowing in turn to relate the observational distance on pPCF terms
to a distance the model is naturally equipped with. This suggests that
extending probabilistic programming languages with derivatives, in the spirit
of the differential lambda-calculus, could be quite meaningful