Parametricity and Dependent Types

Reynolds' abstraction theorem shows how a typing judgement in System F can be translated into a relational statement (in second order predicate logic) about inhabitants of the type. We (in second order predicate logic) about inhabitants of the type. We obtain a similar result for a single lambda calculus (a pure type system), in which terms, types and their relations are expressed. Working within a single system dispenses with the need for an interpretation layer, allowing for an unusually simple presentation. While the unification puts some constraints on the type system (which we spell out), the result applies to many interesting cases, including dependently-typed ones

Microlensing optical depth towards the Galactic bulge from MOA observations during 2000 with Difference Image Analysis

We analyze the data of the gravitational microlensing survey carried out by by the MOA group during 2000 towards the Galactic Bulge (GB). Our observations are designed to detect efficiently high magnification events with faint source stars and short timescale events, by increasing the the sampling rate up to 6 times per night and using Difference Image Analysis (DIA). We detect 28 microlensing candidates in 12 GB fields corresponding to 16 deg^2. We use Monte Carlo simulations to estimate our microlensing event detection efficiency, where we construct the I-band extinction map of our GB fields in order to find dereddened magnitudes. We find a systematic bias and large uncertainty in the measured value of the timescale $t_{\rm Eout}$ in our simulations. They are associated with blending and unresolved sources, and are allowed for in our measurements. We compute an optical depth tau = 2.59_{-0.64}^{+0.84} \times 10^{-6} towards the GB for events with timescales 0.3<t_E<200 days. We consider disk-disk lensing, and obtain an optical depth tau_{bulge} = 3.36_{-0.81}^{+1.11} \times 10^{-6}[0.77/(1-f_{disk})] for the bulge component assuming a 23% stellar contribution from disk stars. These observed optical depths are consistent with previous measurements by the MACHO and OGLE groups, and still higher than those predicted by existing Galactic models. We present the timescale distribution of the observed events, and find there are no significant short events of a few days, in spite of our high detection efficiency for short timescale events down to t_E = 0.3 days. We find that half of all our detected events have high magnification (>10). These events are useful for studies of extra-solar planets.Comment: 65 pages and 30 figures, accepted for publication in ApJ. A systematic bias and uncertainty in the optical depth measurement has been quantified by simulation

`What is a Thing?': Topos Theory in the Foundations of Physics

The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger's timeless question ``What is a thing?''. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this paper, a key goal is to represent any physical quantity $A$ with an arrow \breve{A}_\phi:\Si_\phi\map\R_\phi where \Si_\phi and $\R_\phi$ are two special objects (the `state-object' and `quantity-value object') in the appropriate topos, $\tau_\phi$. We discuss two different types of language that can be attached to a system, $S$. The first, \PL{S}, is a propositional language; the second, \L{S}, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of \PL{S} we expand and develop some of the earlier work (By CJI and collaborators.) on topos theory and quantum physics. A key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf \Sig--the topos quantum analogue of a classical state space. The topos concerned is \SetH{}: the category of contravariant set-valued functors on the category (partially ordered set) \V{} of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space \Hi.Comment: To appear in ``New Structures in Physics'' ed R. Coeck

Study by MOA of extra-solar planets in gravitational microlensing events of high magnification

A search for extra-solar planets was carried out in three gravitational microlensing events of high magnification, MACHO 98-BLG-35, MACHO 99-LMC-2, and OGLE 00-BUL-12. Photometry was derived from observational images by the MOA and OGLE groups using an image subtraction technique. For MACHO 98-BLG-35, additional photometry derived from the MPS and PLANET groups was included. Planetary modeling of the three events was carried out in a super-cluster computing environment. The estimated probability for explaining the data on MACHO 98-BLG-35 without a planet is <1%. The best planetary model has a planet of mass ~(0.4-1.5) X 10^-5 M_Earth at a projected radius of either ~1.5 or ~2.3 AU. We show how multi-planet models can be applied to the data. We calculated exclusion regions for the three events and found that Jupiter-mass planets can be excluded with projected radii from as wide as about 30 AU to as close as around 0.5 AU for MACHO 98-BLG-35 and OGLE 00-BUL-12. For MACHO 99-LMC-2, the exclusion region extends out to around 10 AU and constitutes the first limit placed on a planetary companion to an extragalactic star. We derive a particularly high peak magnification of ~160 for OGLE 00-BUL-12. We discuss the detectability of planets with masses as low as Mercury in this and similar events.Comment: 14 pages, 16 embedded postscript figures, 3 PNG figures, revised version accepted by MNRA

A logic for abstract data types as existential types

The second-order lambda calculus allows an elegant formalisation of abstract data types (ADT's) using existential types. Plotkin and Abadi's logic for parametricity [PA93] then provides the useful proof principle of simulation for ADT's, which can be used to show equivalence of data representations. However, we show that this logic is not sufficient for reasoning about specifications of ADT's, and we present an extension of the logic that does provide the proof principles for ADT's that we want

Correctness Properties for Workflows with Multiple Starts and/or Ends

5th VERITE : JAIST/TRUST-AIST/CVS joint workshop on VERIfication TEchnologyでの発表資料, 開催：2008年3月3日, 開催場所：北陸先端科学技術大学院大学・情報科学研究科棟 5F コラボレーションルーム７JAIST 21世紀COEシンポジウム 2008「検証進化可能電子社会」と共