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 "$\ell$-adic duality".Comment: Revised to incorporate improvements suggested by a few people, most notably an anonymous refere

Riemann-Hilbert correspondence for unit FF-crystals on embeddable algebraic varieties

For a separated scheme $X$ of finite type over a perfect field $k$ of characteristic $p>0$ which admits an immersion into a proper smooth scheme over the truncated Witt ring $W_{n}$, we define the bounded derived category of locally finitely generated unit $F$-crystals with finite Tor-dimension on $X$ over $W_{n}$, 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 ${\mathbb Z}/{p^{n}{\mathbb Z}}$-modules with finite Tor dimension. We also discuss the relationship of $t$-structures on these derived categories when $n=1$. Our result is a generalization of the Riemann-Hilbert correspondence for unit $F$-crystals due to Emerton-Kisin to the case of (possibly singular) embeddable algebraic varieties in characteristic $p>0$.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.