9,195 research outputs found
On Equivalence and Canonical Forms in the LF Type Theory
Decidability of definitional equality and conversion of terms into canonical
form play a central role in the meta-theory of a type-theoretic logical
framework. Most studies of definitional equality are based on a confluent,
strongly-normalizing notion of reduction. Coquand has considered a different
approach, directly proving the correctness of a practical equivalance algorithm
based on the shape of terms. Neither approach appears to scale well to richer
languages with unit types or subtyping, and neither directly addresses the
problem of conversion to canonical.
In this paper we present a new, type-directed equivalence algorithm for the
LF type theory that overcomes the weaknesses of previous approaches. The
algorithm is practical, scales to richer languages, and yields a new notion of
canonical form sufficient for adequate encodings of logical systems. The
algorithm is proved complete by a Kripke-style logical relations argument
similar to that suggested by Coquand. Crucially, both the algorithm itself and
the logical relations rely only on the shapes of types, ignoring dependencies
on terms.Comment: 41 page
A dependent nominal type theory
Nominal abstract syntax is an approach to representing names and binding
pioneered by Gabbay and Pitts. So far nominal techniques have mostly been
studied using classical logic or model theory, not type theory. Nominal
extensions to simple, dependent and ML-like polymorphic languages have been
studied, but decidability and normalization results have only been established
for simple nominal type theories. We present a LF-style dependent type theory
extended with name-abstraction types, prove soundness and decidability of
beta-eta-equivalence checking, discuss adequacy and canonical forms via an
example, and discuss extensions such as dependently-typed recursion and
induction principles
Canonical form of Euler-Lagrange equations and gauge symmetries
The structure of the Euler-Lagrange equations for a general Lagrangian theory
is studied. For these equations we present a reduction procedure to the
so-called canonical form. In the canonical form the equations are solved with
respect to highest-order derivatives of nongauge coordinates, whereas gauge
coordinates and their derivatives enter in the right hand sides of the
equations as arbitrary functions of time. The reduction procedure reveals
constraints in the Lagrangian formulation of singular systems and, in that
respect, is similar to the Dirac procedure in the Hamiltonian formulation.
Moreover, the reduction procedure allows one to reveal the gauge identities
between the Euler-Lagrange equations. Thus, a constructive way of finding all
the gauge generators within the Lagrangian formulation is presented. At the
same time, it is proven that for local theories all the gauge generators are
local in time operators.Comment: 27 pages, LaTex fil
The theory of graph-like Legendrian unfoldings and its applications
This is mainly a survey article on the recent development of the theory of
graph-like Legendrian unfoldings and its applications. The notion of big
Legendrian submanifolds was introduced by Zakalyukin for describing the wave
front propagations. Graph-like Legendrian unfoldings belong to a special class
of big Legendrian submanifolds. Although this is a survey article, some new
original results and the corrected proofs of some results are given.Comment: 30 pages,9 figure
Grothendieck duality made simple
It has long been accepted that the foundations of Grothendieck duality are
complicated. This has changed recently. By "Grothendieck duality" we mean what,
in the old literature, used to go by the name "coherent duality". This isn't to
be confused with what is nowadays called "Verdier duality", and used to pass as
"-adic duality".Comment: Revised to incorporate improvements suggested by a few people, most
notably an anonymous refere
Riemann-Hilbert correspondence for unit -crystals on embeddable algebraic varieties
For a separated scheme of finite type over a perfect field of
characteristic which admits an immersion into a proper smooth scheme over
the truncated Witt ring , we define the bounded derived category of
locally finitely generated unit -crystals with finite Tor-dimension on
over , independently of the choice of the immersion. Then we prove the
anti-equivalence of this category with the bounded derived category of
constructible \'etale sheaves of -modules with
finite Tor dimension. We also discuss the relationship of -structures on
these derived categories when . Our result is a generalization of the
Riemann-Hilbert correspondence for unit -crystals due to Emerton-Kisin to
the case of (possibly singular) embeddable algebraic varieties in
characteristic .Comment: This is the final version, to appear in Annales de l'Institut Fourie
A local Langlands correspondence for unipotent representations
Let G be a connected reductive group over a non-archimedean local field K,
and assume that G splits over an unramified extension of K. We establish a
local Langlands correspondence for irreducible unipotent representations of G.
It comes as a bijection between the set of such representations and the
collection of enhanced L-parameters for G, which are trivial on the inertia
subgroup of the Weil group of K. We show that this correspondence has many of
the expected properties, for instance with respect to central characters,
tempered representations, the discrete series, cuspidality and parabolic
induction.
The core of our strategy is the investigation of affine Hecke algebras on
both sides of the local Langlands correspondence. When a Bernstein component of
G-representations is matched with a Bernstein component of enhanced
L-parameters, we prove a comparison theorem for the two associated affine Hecke
algebras.
This generalizes work of Lusztig in the case of adjoint K-groups.Comment: Minor changes in version 2, in particular Lemma 1.1.
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