470 research outputs found
Collapsing non-idempotent intersection types
We proved recently that the extensional collapse of the relational model of linear logic coincides with its Scott model, whose objects are preorders and morphisms are downwards closed relations. This result is obtained by the construction of a new model whose objects can be understood as preorders equipped with a realizability predicate. We present this model, which features a new duality, and explain how to use it for reducing normalization results in idempotent intersection types (usually proved by reducibility) to purely combinatorial methods. We illustrate this approach in the case of the call-by-value lambda-calculus, for which we introduce a new resource calculus, but it can be applied in the same way to many different calculi
Indexed linear logic and higher-order model checking
In recent work, Kobayashi observed that the acceptance by an alternating tree
automaton A of an infinite tree T generated by a higher-order recursion scheme
G may be formulated as the typability of the recursion scheme G in an
appropriate intersection type system associated to the automaton A. The purpose
of this article is to establish a clean connection between this line of work
and Bucciarelli and Ehrhard's indexed linear logic. This is achieved in two
steps. First, we recast Kobayashi's result in an equivalent infinitary
intersection type system where intersection is not idempotent anymore. Then, we
show that the resulting type system is a fragment of an infinitary version of
Bucciarelli and Ehrhard's indexed linear logic. While this work is very
preliminary and does not integrate key ingredients of higher-order
model-checking like priorities, it reveals an interesting and promising
connection between higher-order model-checking and linear logic.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
Call-by-value non-determinism in a linear logic type discipline
We consider the call-by-value lambda-calculus extended with a may-convergent
non-deterministic choice and a must-convergent parallel composition. Inspired
by recent works on the relational semantics of linear logic and non-idempotent
intersection types, we endow this calculus with a type system based on the
so-called Girard's second translation of intuitionistic logic into linear
logic. We prove that a term is typable if and only if it is converging, and
that its typing tree carries enough information to give a bound on the length
of its lazy call-by-value reduction. Moreover, when the typing tree is minimal,
such a bound becomes the exact length of the reduction
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow
Let be a Calabi-Yau -fold, and consider compact, graded Lagrangians
in . Thomas and Yau math.DG/0104196, math.DG/0104197 conjectured that
there should be a notion of "stability" for such , and that if is stable
then Lagrangian mean curvature flow with
should exist for all time, and should be the
unique special Lagrangian in the Hamiltonian isotopy class of . This paper
is an attempt to update the Thomas-Yau conjectures, and discuss related issues.
It is a folklore conjecture that there exists a Bridgeland stability
condition on the derived Fukaya category of
, such that an isomorphism class in is -semistable if (and possibly only if) it contains a special Lagrangian,
which must then be unique.
We conjecture that if is an object in an enlarged version of
, where is a compact, graded Lagrangian in (possibly
immersed, or with "stable singularities"), a rank one local system,
and a bounding cochain for in Lagrangian Floer cohomology, then
there is a unique family such that
, and in
for all , and satisfies Lagrangian MCF with
surgeries at singular times and in graded Lagrangian integral
currents we have , where is a
special Lagrangian integral current of phase for
, and correspond to
the decomposition of into -semistable objects.
We also give detailed conjectures on the nature of the singularities of
Lagrangian MCF that occur at the finite singular times Comment: 63 pages. (v2) new section 4 added, discussing exact Lagrangians, and
the Kahler-Einstein case. To appear in EMS Surveys in Mathematical Science
On union ultrafilters
We present some new results on union ultrafilters. We characterize stability
for union ultrafilters and, as the main result, we construct a new kind of
unordered union ultrafilter
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