Characterisation of Strongly Normalising lambda-mu-Terms

We provide a characterisation of strongly normalising terms of the lambda-mu-calculus by means of a type system that uses intersection and product types. The presence of the latter and a restricted use of the type omega enable us to represent the particular notion of continuation used in the literature for the definition of semantics for the lambda-mu-calculus. This makes it possible to lift the well-known characterisation property for strongly-normalising lambda-terms - that uses intersection types - to the lambda-mu-calculus. From this result an alternative proof of strong normalisation for terms typeable in Parigot's propositional logical system follows, by means of an interpretation of that system into ours.Comment: In Proceedings ITRS 2012, arXiv:1307.784

A journey through resource control lambda calculi and explicit substitution using intersection types (an account)

In this paper we invite the reader to a journey through three lambda calculi with resource control: the lambda calculus, the sequent lambda calculus, and the lambda calculus with explicit substitution. All three calculi enable explicit control of resources due to the presence of weakening and contraction operators. Along this journey, we propose intersection type assignment systems for all three resource control calculi. We recognise the need for three kinds of variables all requiring different kinds of intersection types. Our main contribution is the characterisation of strong normalisation of reductions in all three calculi, using the techniques of reducibility, head subject expansion, a combination of well-orders and suitable embeddings of terms

Infinitary λ\lambda-Calculi from a Linear Perspective (Long Version)

We introduce a linear infinitary $\lambda$-calculus, called $\ell\Lambda_{\infty}$, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applicative $\lambda$-calculus and is universal for computations over infinite strings. What is particularly interesting about $\ell\Lambda_{\infty}$, is that the refinement induced by linear logic allows to restrict both modalities so as to get calculi which are terminating inductively and productive coinductively. We exemplify this idea by analysing a fragment of $\ell\Lambda$ built around the principles of $\mathsf{SLL}$ and $\mathsf{4LL}$. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary $\lambda$-calculi

Resource control and strong normalisation

We introduce the \emph{resource control cube}, a system consisting of eight intuitionistic lambda calculi with either implicit or explicit control of resources and with either natural deduction or sequent calculus. The four calculi of the cube that correspond to natural deduction have been proposed by Kesner and Renaud and the four calculi that correspond to sequent lambda calculi are introduced in this paper. The presentation is parameterized with the set of resources (weakening or contraction), which enables a uniform treatment of the eight calculi of the cube. The simply typed resource control cube, on the one hand, expands the Curry-Howard correspondence to intuitionistic natural deduction and intuitionistic sequent logic with implicit or explicit structural rules and, on the other hand, is related to substructural logics. We propose a general intersection type system for the resource control cube calculi. Our main contribution is a characterisation of strong normalisation of reductions in this cube. First, we prove that typeability implies strong normalisation in the ''natural deduction base" of the cube by adapting the reducibility method. We then prove that typeability implies strong normalisation in the ''sequent base" of the cube by using a combination of well-orders and a suitable embedding in the ''natural deduction base". Finally, we prove that strong normalisation implies typeability in the cube using head subject expansion. All proofs are general and can be made specific to each calculus of the cube by instantiating the set of resources

Mechanising syntax with binders in Coq

Mechanising binders in general-purpose proof assistants such as Coq is cumbersome and difficult. Yet binders, substitutions, and instantiation of terms with substitutions are a critical ingredient of many programming languages. Any practicable mechanisation of the meta-theory of the latter hence requires a lean formalisation of the former. We investigate the topic from three angles: First, we realise formal systems with binders based on both pure and scoped de Bruijn algebras together with basic syntactic rewriting lemmas and automation. We automate this process in a compiler called Autosubst; our final tool supports many-sorted, variadic, and modular syntax. Second, we justify our choice of realisation and mechanise a proof of convergence of the sigma calculus, a calculus of explicit substitutions that is complete for equality of the de Bruijn algebra corresponding to the lambda calculus. Third, to demonstrate the practical usefulness of our approach, we provide concise, transparent, and accessible mechanised proofs for a variety of case studies refined to de Bruijn substitutions.Die Mechanisierung von Bindern in universellen Beweisassistenten wie Coq ist arbeitsaufwändig und schwierig. Binder, Substitutionen und die Instantiierung von Substitutionen sind jedoch kritischer Bestandteil vieler Programmiersprachen. Deshalb setzt eine praktikable Mechanisierung der Metatheorie von Programmiersprachen eine elegante Formalisierung von Bindern voraus. Wir nähern uns dem Thema aus drei Richtungen an: Zuerst realisieren wir formale Systeme mit Bindern mit Hilfe von reinen und indizierten de Bruijn Algebren, zusammen mit grundlegenden syntaktischen Gleichungen und Automatisierung. Wir automatisieren diesen Prozess in einem Kompilierer namens Autosubst. Unser finaler Kompilierer unterstützt Sortenlogik, variadische Syntax und modulare Syntax. Zweitens rechtfertigen wir unsere Repräsentation und mechanisieren einen Beweis der Konvergenz des SP-Kalküls, einem Kalkül expliziter Substitutionen der bezüglich der Gleichheit der puren de Bruijn Algebra des -Kalküls vollständig ist. Drittens entwickeln wir kurze, transparente und leicht zugängliche mechanisierte Beweise für diverse Fallstudien, die wir an de Bruijn Substitutionen angepasst haben. Wir weisen so die praktische Anwendbarkeit unseres Ansatzes nach

Perception-Aware Optimisation Methodologies for Quantum Dot Based Displays and Lighting

Human colour vision acuity is limited. This presents opportunities to leverage these perceptual limits to achieve engineering optimisations for devices and systems that interact with the human vision system. This dissertation presents the results of few investigations we carried out into quantifying these limits and several optimisation methodologies that we devised. The first step in this process is to quantify the acuity of human colour vision. We obtained a large corpus of colour matching data from a mobile video game called Specimen. We examine what questions about human vision this dataset allows us to answer and explore global statistics about colour vision based on this data on 41,000 players from 175 countries. We show that we can use the information in this dataset to infer potential candidate functions for the spectral sensitivities of each person in the dataset. The human eye acts like a many to one function; quantifiably different spectra can look like the same colour. This is referred to as metamerism. From a device perspective, different spectra consume different amounts of energy to generate. We show that we can use these two properties to elicit the same colour sensation using less energy. In the colour samples we evaluated, we show that we can achieve up to 10 times less power consumption while achieving a colour match. Given that one cannot change the emission spectrum of a display after fabrication, we propose the use of a multi-primary colour display to achieve this. We present two indices for quantifying the metameric capacity of such a display and its ability to save energy. The emission spectrum of a quantum dot (QD) based device is very narrow. Previous work in the literature suggested that narrow bandwidth spectra can lead to observer metameric breakdown; different observers disagreeing on the perceived ‘colour’ of a spectrum. We show that this might not be the case, using modern colour science tools, and show how metameric breakdown in a display could be minimised by carefully choosing the primary emission wavelengths. The limited colour acuity of human vision implies that people cannot notice small differences in colour. This fact has been used to create approximate colour transformation algorithms that subtly change colours in images such that they consume less energy when displayed on an emissive pixel display without causing unacceptable visual artefacts. We conducted a user study to gather information about the effect of one such colour transform called Crayon. We present a method for effectively picking the optimal transform parameters for Crayon, based on the user study results. The method presented calculates these parameters based on the properties of the image being transformed such that the power saving can be maximised while minimising the loss of image quality. The user study results show that we can achieve up to 50% power saving with a majority of the study participants reporting a negligible degradation in image quality in the transformed images. We additionally investigate a hypothesis that was presented stating that images with large amounts of highly luminous pixels cause increased power consumption in OLED displays due to localised display heating. We show that this hypothesis is wrong. We also investigate if sub-pixel rendering in Pentile displays can be used to reduce display power consumption by intentionally turning off random sub-pixels. However, we present a negative result showing that even single-pixel artefacts are observable on the test platform and thus, this cannot be used to improve display power efficiency. The narrow-band optical emissions of QD based devices mixed with their ability to be fabricated through solution processing can be used to mix multiple QDs together to build devices that generate arbitrary spectral shapes. We show how to use this property in an numerical optimisation based design framework to create lighting devices with a high colour rendering index (CRI). We evaluate the effects of different cost functions and initialisation strategies, and show that, we are able to design devices with a CRI > 96 using only four different QD primaries. We use a charge-transport based simulator to asses the electric properties of the designed devices. We also showcase initial work done on a modular software interface and a material library we developed for this simulator.EPSRC DTP studentship award RG84040:EP/N509620/