4,474 research outputs found
Dependent Types for Nominal Terms with Atom Substitutions
Nominal terms are an extended first-order language for specifying and verifying properties of syntax with binding. Founded upon the semantics of nominal sets, the success of nominal terms with regard to systems of equational reasoning is already well established. This work first extends the untyped language of nominal terms with a notion of non-capturing atom substitution for object-level names and then proposes a dependent type system for this extended language. Both these contributions are intended to serve as a prelude to a future nominal logical framework based upon nominal equational reasoning and thus an extended example is given to demonstrate that this system is capable of encoding various other formal systems of interest
Nominal Abstraction
Recursive relational specifications are commonly used to describe the
computational structure of formal systems. Recent research in proof theory has
identified two features that facilitate direct, logic-based reasoning about
such descriptions: the interpretation of atomic judgments through recursive
definitions and an encoding of binding constructs via generic judgments.
However, logics encompassing these two features do not currently allow for the
definition of relations that embody dynamic aspects related to binding, a
capability needed in many reasoning tasks. We propose a new relation between
terms called nominal abstraction as a means for overcoming this deficiency. We
incorporate nominal abstraction into a rich logic also including definitions,
generic quantification, induction, and co-induction that we then prove to be
consistent. We present examples to show that this logic can provide elegant
treatments of binding contexts that appear in many proofs, such as those
establishing properties of typing calculi and of arbitrarily cascading
substitutions that play a role in reducibility arguments.Comment: To appear in the Journal of Information and Computatio
Earth Abundant Element Type I Clathrate Phases.
Earth abundant element clathrate phases are of interest for a number of applications ranging from photovoltaics to thermoelectrics. Silicon-containing type I clathrate is a framework structure with the stoichiometry A8-xSi46 (A = guest atom such as alkali metal) that can be tuned by alloying and doping with other elements. The type I clathrate framework can be described as being composed of two types of polyhedral cages made up of tetrahedrally coordinated Si: pentagonal dodecahedra with 20 atoms and tetrakaidecahedra with 24 atoms in the ratio of 2:6. The cation sites, A, are found in the center of each polyhedral cage. This review focuses on the newest discoveries in the group 13-silicon type I clathrate family: A₈E₈Si38 (A = alkali metal; E = Al, Ga) and their properties. Possible approaches to new phases based on earth abundant elements and their potential applications will be discussed
On the Formalisation of the Metatheory of the Lambda Calculus and Languages with Binders
Este trabajo trata sobre el razonamiento formal veri cado por computadora involucrando lenguajes
con operadores de ligadura.
Comenzamos presentando el Cálculo Lambda, para el cual utilizamos la sintaxis histórica, esto es,
sintaxis de primer orden con sólo un tipo de nombres para las variables ligadas y libres. Primeramente
trabajamos con términos concretos, utilizando la operación de sustitución múltiple de nida
por Stoughton como la operación fundamental sobre la cual se de nen las conversiones alfa
y beta. Utilizando esta sintaxis desarrollamos los principales resultados metateóricos del cálculo:
los lemas de sustitución, el teorema de Church-Rosser y el teorema de preservación de tipo (Subject
Reduction) para el sistema de asignación de tipos simples. En una segunda formalización
reproducimos los mismos resultados, esta vez basando la conversion alfa sobre una operación
más sencilla, que es la de permutación de nombres. Utilizando este mecanismo, derivamos principios
de inducción y recursión que permiten trabajar identificando términos alfa equivalentes,
de modo tal de reproducir la llamada convención de variables de Barendregt. De este modo,
podemos imitar las demostraciones al estilo lápiz y papel dentro del riguroso entorno formal
de un asistente de demostración.
Como una generalización de este último enfoque, concluimos utilizando técnicas de programación
genérica para definir una base para razonar sobre estructuras genéricas con operadores de ligadura.
Definimos un universo de tipos de datos regulares con información de variables y operadores
de ligadura, y sobre éstos definimos operadores genéricos de formación, eliminación
e inducción. También introducimos una relación de alfa equivalencia basada en la operación
de permutación y derivamos un principio de iteración/inducción que captura la convención de
variables anteriormente mencionada. A modo de ejemplo, mostramos cómo definir el Cálculo
Lambda y el sistema F en nuestro universo, ilustrando no sólo la reutilización de las pruebas
genéricas, sino también cuán sencillo es el desarrollo de nuevas pruebas en estos casos.
Todas las formalizaciones de esta tesis fueron realizadas en Teoría Constructiva de Tipos y
verificadas utilizando el asistente de pruebas AgdaThis work is about formal, machine-checked reasoning on languages with name binders.
We start by considering the ʎ-calculus using the historical ( rst order) syntax with only one
sort of names for both bound and free variables. We rst work on the concrete terms taking
Stoughton's multiple substitution operation as the fundamental operation upon which the
ά and ß-conversion are de ned. Using this syntax we reach well-known meta-theoretical results,
namely the Substitution lemmas, the Church-Rosser theorem and the Subject Reduction theorem
for the system of assignment of simple types. In a second formalisation we reproduce the same
results, this time using an approach in which -conversion is de ned using the simpler operation
of name permutation. Using this we derive induction and recursion principles that allow us to
work by identifying terms up to -conversion and to reproduce the so-called Barendregt's variable
convention [4]. Thus, we are able to mimic pencil and paper proofs inside the rigorous formal
setting of a proof assistant.
As a generalisation of the latter, we conclude by using generic programming techniques to de ne
a framework for reasoning over generic structures with binders. We de ne a universe of regular
datatypes with variables and binders information, and over these we de ne generic formation,
elimination, and induction operations. We also introduce an ά equivalence relation based on
the swapping operation, and are able to derive an -iteration/induction principle that captures
Barendregt's variable convention. As an example, we show how to de ne the ʎ calculus and
System F in our universe, and thereby we are able to illustrate not only the reuse of the generic
proofs but also how simple the development of new proofs becomes in these instances.
All formalisations in this thesis have been made in Constructive Type Theory and completely
checked using the Agda proof assistan
Mass transport and electrochemical properties of La2Mo2O9 as a fast ionic conductor
La2Mo2O9, as a new fast ionic conductor, has been investigated widely due to its high
ionic conductivity which is comparable to those of the commercialized materials. However,
little work has been reported on the oxygen transport and diffusion in this candidate
electrolyte material. The main purpose of this project was to investigate oxide ion diffusion
in La2Mo2O9 and also the factors which could affect oxygen transport properties.
Oxygen isotope exchange followed by Secondary Ion Mass Spectrometry (SIMS)
measurements were employed to obtain oxygen diffusion profiles. A correlation between
oxygen ion transport and the electrochemical properties such as ionic conductivity was
built upon the Nernst Einstein equation relating the diffusivity to electrical conductivity.
In-situ neutron diffraction and AC impedance measurements were designed and conducted
to investigate the correlation between crystal structure and oxygen transport in the bulk
materials. Other techniques, such as synthesis, microstructure studies, and thermal analysis
were also adopted to study the electrochemical properties of La2Mo2O9.
The results of the study on the effects of microstructure on oxygen diffusion in
La2Mo2O9 revealed that the grain boundary component played a significant role in
electrochemical performance, although the grain size seemed to have little influence on
oxygen transport. The oxygen isotope exchange in 18O2 was successfully carried out by
introducing a silver coating on the sample surface, which solved the main difficulty in
applying oxygen isotope exchange on pure ionic conductors. The ionic conductivity
obtained from the diffusion coefficients was consistent with the result from AC impedance spectroscopy. The number of mobile oxygen ions was estimated to be 5 per unit cell. There
was a difference of oxygen self diffusion coefficient when the isotope exchange was
conducted in 18O2 and H2
18O. The activation energy of oxygen diffusion in humidified
atmosphere was higher than that measured in dry atmosphere. It indicated that the
humidified atmosphere had affected oxygen transport in the material. The studies on
hydroxyl incorporation and transport explained the decreased oxygen diffusion coefficients
in wet atmosphere and also suggested proton conductivity in La2Mo2O9, which leads to
further investigation on applications of La2Mo2O9 as a proton conductor. In-situ neutron
diffraction and AC impedance measurement revealed a close relationship between crystal
structure and ionic conductivity. The successful application of this technique provides a
new method to simultaneously investigate crystal structure and electrical properties in
electro-ceramics in the future
Extensions of nominal terms
This thesis studies two major extensions of nominal terms. In particular, we
study an extension with -abstraction over nominal unknowns and atoms, and an
extension with an arguably better theory of freshness and -equivalence.
Nominal terms possess two levels of variable: atoms a represent variable symbols,
and unknowns X are `real' variables. As a syntax, they are designed to facilitate
metaprogramming; unknowns are used to program on syntax with variable symbols.
Originally, the role of nominal terms was interpreted narrowly. That is, they
were seen solely as a syntax for representing partially-speci ed abstract syntax with
binding.
The main motivation of this thesis is to extend nominal terms so that they can
be used for metaprogramming on proofs, programs, etc. and not just for metaprogramming
on abstract syntax with binding. We therefore extend nominal terms
in two signi cant ways: adding -abstraction over nominal unknowns and atoms|
facilitating functional programing|and improving the theory of -equivalence that
nominal terms possesses.
Neither of the two extensions considered are trivial. The capturing substitution
action of nominal unknowns implies that our notions of scope, intuited from working
with syntax possessing a non-capturing substitution, such as the -calculus, is no
longer applicable. As a result, notions of -abstraction and -equivalence must be
carefully reconsidered.
In particular, the rst research contribution of this thesis is the two-level -
calculus, intuitively an intertwined pair of -calculi. As the name suggests, the
two-level -calculus has two level of variable, modelled by nominal atoms and unknowns,
respectively. Both levels of variable can be -abstracted, and requisite
notions of -reduction are provided. The result is an expressive context-calculus.
The traditional problems of handling -equivalence and the failure of commutation
between instantiation and -reduction in context-calculi are handled through the
use of two distinct levels of variable, swappings, and freshness side-conditions on
unknowns, i.e. `nominal technology'.
The second research contribution of this thesis is permissive nominal terms,
an alternative form of nominal term. They retain the `nominal' rst-order
avour
of nominal terms (in fact, their grammars are almost identical) but forego the use
of explicit freshness contexts. Instead, permissive nominal terms label unknowns
with a permission sort, where permission sorts are in nite and coin nite sets of
atoms. This in nite-coin nite nature means that permissive nominal terms recover
two properties|we call them the `always-fresh' and `always-rename' properties
that nominal terms lack. We argue that these two properties bring the theory of
-equivalence on permissive nominal terms closer to `informal practice'.
The reader may consider -abstraction and -equivalence so familiar as to be
`solved problems'. The work embodied in this thesis stands testament to the fact
that this isn't the case. Considering -abstraction and -equivalence in the context
of two levels of variable poses some new and interesting problems and throws light
on some deep questions related to scope and binding
Alpha-Structural Induction and Recursion for the Lambda Calculus in Constructive Type Theory
We formulate principles of induction and recursion for a variant of lambda calculus in its original syntax (i.e., with only one sort of names) where alpha-conversion is based upon name swapping as in nominal abstract syntax. The principles allow to work modulo alpha-conversion and implement the Barendregt variable convention. We derive them all from the simple structural induction principle on concrete terms and work out applications to some fundamental meta-theoretical results, such as the substitution lemma for alpha-conversion and the lemma on substitution composition. The whole work is implemented in Agda
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