Proof-Relevant Resolution for Elaboration of Programming Languages

Proof-relevant resolution is a new variant of resolution in Horn-clause logic and its extensions. We propose proof-relevant resolution as a systematic approach to elaboration in programming languages that is close to formal specification and hence allows for analysis of semantics of the language. We demonstrate the approach on two case studies; we describe a novel, proof-relevant approach to type inference and term synthesis in dependently types languages and we show how proof-relevant resolution allows for analysis of inductive and coinductive soundness of type class resolution. We conclude by a discussion of overall contributions of our current work

Operational Semantics of Resolution and Productivity in Horn Clause Logic

This paper presents a study of operational and type-theoretic properties of different resolution strategies in Horn clause logic. We distinguish four different kinds of resolution: resolution by unification (SLD-resolution), resolution by term-matching, the recently introduced structural resolution, and partial (or lazy) resolution. We express them all uniformly as abstract reduction systems, which allows us to undertake a thorough comparative analysis of their properties. To match this small-step semantics, we propose to take Howard's System H as a type-theoretic semantic counterpart. Using System H, we interpret Horn formulas as types, and a derivation for a given formula as the proof term inhabiting the type given by the formula. We prove soundness of these abstract reduction systems relative to System H, and we show completeness of SLD-resolution and structural resolution relative to System H. We identify conditions under which structural resolution is operationally equivalent to SLD-resolution. We show correspondence between term-matching resolution for Horn clause programs without existential variables and term rewriting.Comment: Journal Formal Aspect of Computing, 201

Richardson Varieties Have Kawamata Log Terminal Singularities

Let $X^v_w$ be a Richardson variety in the full flag variety $X$ associated to a symmetrizable Kac-Moody group $G$. Recall that $X^v_w$ is the intersection of the finite dimensional Schubert variety $X_w$ with the finite codimensional opposite Schubert variety $X^v$. We give an explicit \bQ-divisor $\Delta$ on $X^v_w$ and prove that the pair $(X^v_w, \Delta)$ has Kawamata log terminal singularities. In fact, $-K_{X^v_w} - \Delta$ is ample, which additionally proves that $(X^v_w, \Delta)$ is log Fano. We first give a proof of our result in the finite case (i.e., in the case when $G$ is a finite dimensional semisimple group) by a careful analysis of an explicit resolution of singularities of $X^v_w$ (similar to the BSDH resolution of the Schubert varieties). In the general Kac-Moody case, in the absence of an explicit resolution of $X^v_w$ as above, we give a proof that relies on the Frobenius splitting methods. In particular, we use Mathieu's result asserting that the Richardson varieties are Frobenius split, and combine it with a result of N. Hara and K.-I. Watanabe relating Frobenius splittings with log canonical singularities.Comment: 15 pages, improved exposition and explanation. To appear in the International Mathematics Research Notice

The Gravitational Lensing in the QSO 1208+10 from the Proximity Effect in its Lyman alpha Forest

The quasar Q1208+1011 (z_{em}=3.8) is the second highest redshift double quasar ever detected. Several indications point toward it being a gravitational lensed system, although a definitive proof is still lacking. We present new evidence of its lensed nature based on the weakness of the ``proximity effect'' measured in the high resolution Lyman absorption spectrum of the QSO. A luminosity amplification as large as 22 has been derived from this analysis. Indications on the redshift of the lensing galaxy can be obtained from the analysis of the intervening heavy element absorption systems discovered in the QSO high resolution spectrum. On statistical and dynamical grounds a MgII system present at z=1.13 appears as the most likely candidate for the lensing galaxy. We compare the observed parameters with a simple isothermal model for the lens to derive the properties of the lensing galaxy. The resulting magnification factor is smaller, although marginally consistent with that derived by the analysis of the proximity effect.Comment: 11 pages, 2 Postscript figures, ApJ in pres

Elemental surface analysis at ambient pressure by electron-induced x-ray fluorescence

The development of a portable surface elemental analysis tool, based on the excitation of characteristic x rays from samples at ambient pressure with a focused electron beam is described. This instrument relies on the use of a thin electron transmissive membrane to isolate the vacuum of the electron source from the ambient atmosphere. The major attributes of this instrument include rapid (several minutes) spectrum acquisition, nondestructive evaluation of elemental composition, no sample preparation, and high-to-medium (several hundreds µm) spatial resolution. The instrument proof-of-principle has been demonstrated in a laboratory setup by obtaining energy dispersive x-ray spectra from metal and mineral samples

Well-posedness of the plasma-vacuum interface problem

We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations. Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev spaces. The proof is based on the results proved in the companion paper arXiv:1112.3101, about the well-posedness of the homogeneous linearized problem and the proof of a basic a priori energy estimate. The proof of the resolution of the nonlinear problem given in the present paper follows from the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash-Moser iteration.Comment: 58 page

The Singular Supports of IC sheaves on Quasimaps' Spaces are Irreducible

Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha\in N[I]$ of positive integers one can consider the space $Q_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_\alpha$ (Quasimaps), $Q^L_\alpha$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $\pi: Q^L_\alpha\to Q^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_\alpha$ on the Quasimaps' space $Q^D_\alpha$. Namely, we prove that this singular support $SS(IC_\alpha)$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $\pi: Q^L_\alpha\to Q^D_\alpha$.Comment: 8 pages, AmsLatex 1.