Functions out of Higher Truncations

In homotopy type theory, the truncation operator ||-||n (for a number n > -2) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B which are constant on all (n+1)-st loop spaces. We give one "elementary" proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct "set-based" representations of 1-types, as long as they have "braided" loop spaces. The main result with one of its proofs and the application have been formalised in Agda.Comment: 15 pages; to appear at CSL'1

On Induction, Coinduction and Equality in Martin-L\uf6f and Homotopy Type Theory

Martin L\uf6f Type Theory, having put computation at the center of logicalreasoning, has been shown to be an effective foundation for proof assistants,with applications both in computer science and constructive mathematics. Oneambition though is for MLTT to also double as a practical general purposeprogramming language. Datatypes in type theory come with an induction orcoinduction principle which gives a precise and concise specification of theirinterface. However, such principles can interfere with how we would like toexpress our programs. In this thesis, we investigate more flexible alternativesto direct uses of the (co)induction principles.As a first contribution, we consider the n-truncation of a type in Homo-topy Type Theory. We derive in HoTT an eliminator into (n+1)-truncatedtypes instead of n-truncated ones, assuming extra conditions on the underlyingfunction.As a second contribution, we improve on type-based criteria for terminationand productivity. By augmenting the types with well-foundedness information,such criteria allow function definitions in a style closer to general recursion.We consider two criteria: guarded types, and sized types.Guarded types introduce a modality ”later” to guard the availability ofrecursive calls provided by a general fixed-point combinator. In Guarded Cu-bical Type Theory we equip the fixed-point combinator with a propositionalequality to its one-step unfolding, instead of a definitional equality that wouldbreak normalization. The notion of path from Cubical Type Theory allows usto do so without losing canonicity or decidability of conversion.Sized types, on the other hand, explicitly index datatypes with size boundson the height or depth of their elements. The sizes however can get in theway of the reasoning principles we expect. Our approach is to introduce newquantifiers for ”irrelevant” size quantification. We present a type theory withparametric quantifiers where irrelevance arises as a “free theorem”. We alsodevelop a conversion checking algorithm for a more specific theory where thenew quantifiers are restricted to sizes.Finally, our third contribution is about the operational semantics of typetheory. For the extensions above we would like to devise a practical conversionchecking algorithm suitable for integration into a proof assistant. We formal-ized the correctness of such an algorithm for a small but challenging corecalculus, proving that conversion is decidable. We expect this development toform a good basis to verify more complex theories.The ideas discussed in this thesis are already influencing the developmentof Agda, a proof assistant based on type theory

Decidability of Conversion for Type Theory in Type Theory

Type theory should be able to handle its own meta-theory, both to justify its foundational claims and to obtain a verified implementation. At the core of a type checker for intensional type theory lies an algorithm to check equality of types, or in other words, to check whether two types are convertible. We have formalized in Agda a practical conversion checking algorithm for a dependent type theory with one universe \ue0 la Russell, natural numbers, and η-equality for Π types. We prove the algorithm correct via a Kripke logical relation parameterized by a suitable notion of equivalence of terms. We then instantiate the parameterized fundamental lemma twice: once to obtain canonicity and injectivity of type formers, and once again to prove the completeness of the algorithm. Our proof relies on inductive-recursive definitions, but not on the uniqueness of identity proofs. Thus, it is valid in variants of intensional Martin-L\uf6f Type Theory as long as they support induction-recursion, for instance, Extensional, Observational, or Homotopy Type Theory

Guarded Cubical Type Theory: Path Equality for Guarded Recursion

This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-L\"of type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.Comment: 17 pages, to be published in proceedings of CSL 201

Greatest HITs: Higher Inductive Types in Coinductive Definitions via Induction under Clocks

Guarded recursion is a powerful modal approach to recursion that can be seen as an abstract form of step-indexing. It is currently used extensively in separation logic to model programming languages with advanced features by solving domain equations also with negative occurrences. In its multi-clocked version, guarded recursion can also be used to program with and reason about coinductive types, encoding the productivity condition required for recursive definitions in types. This paper presents the first type theory combining multi-clocked guarded recursion with the features of Cubical Type Theory, as well as a denotational semantics. Using the combination of Higher Inductive Types (HITs) and guarded recursion allows for simple programming and reasoning about coinductive types that are traditionally hard to represent in type theory, such as the type of finitely branching labelled transition systems. For example, our results imply that bisimilarity for these imply path equality, and so proofs can be transported along bisimilarity proofs. Among our technical contributions is a new principle of induction under clocks. This allows universal quantification over clocks to commute with HITs up to equivalence of types, and is crucial for the encoding of coinductive types. Such commutativity requirements have been formulated for inductive types as axioms in previous type theories with multi-clocked guarded recursion, but our present formulation as an induction principle allows for the formulation of general computation rules.Comment: 29 page

Eff ects of Irradiance and Diff erent Nitrogen and Carbon Concentrations on the Minerals Accumulation in Scenedesmus obliquus Biomass

Microalgae are capable of absorbing and concentrating constituent elements that have a wide variety of applications in agriculture, food industry, and medicine. Microalgae chemical composition change according to internal and external factors. In this study, the effect of irradiance, sodium nitrate and sodium acetate concentration on the accumulation of essential minerals in Scenedesmus obliquus biomass were evaluated using 23 factorial screening designs. The simultaneous effect of the three experimental factors was studied using three levels for each parameter (irradiance: 36.71, 69.50, 102.30 μE m-2 s-1, sodium nitrate: 0.27, 44.00, 87.73 g L-1 and sodium acetate: 0.00, 2.50, 5.02 g L-1). The response variables were the minerals concentration of Na, K, Ca, Mg, Fe, Zn and Mn. Results show that each mineral has an optimal operation condition in order to improve its concentration in the microalgae biomass. A signifi cant interaction between the variables was observed, which has direct effects on the minerals accumulation in the microalgae biomass. Under these conditions, the maximum concentration of K (1515.77 [mg (100gdw)-1]), Ca (2744.24 [mg (100gdw)-1]), Mg (9697.65 [mg (100gdw)-1]), Fe (2932.42 [mg (100gdw)-1]), Mn (38.48 [mg (100gdw)-1]), Zn (324.00 [mg (100gdw)-1]) and the minimum concentration of Na (5607.20 [mg (100gdw)- 1 ]) were obtained from the microalga biomass. Thus, Scenedesmus obliquus biomass was characterized as good essential mineral source and confi rmed to be potentially valuable ingredient for utilization in the food industry.Fil: Jimenez Veuthey, Mariana. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Santa Fe. Instituto de Ciencia y Tecnologia de Los Alimentos de Entre Rios. - Universidad Nacional de Entre Rios. Instituto de Ciencia y Tecnologia de Los Alimentos de Entre Rios.; ArgentinaFil: Zapata, Luz Marina. Universidad Nacional de Entre Ríos. Facultad de Ciencias de la Alimentación; ArgentinaFil: Vezzosi Zoto, Gina Fiorella. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Santa Fe. Instituto de Ciencia y Tecnologia de Los Alimentos de Entre Rios. - Universidad Nacional de Entre Rios. Instituto de Ciencia y Tecnologia de Los Alimentos de Entre Rios.; ArgentinaFil: Sacks, Natalia Agustina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Santa Fe. Instituto de Ciencia y Tecnologia de Los Alimentos de Entre Rios. - Universidad Nacional de Entre Rios. Instituto de Ciencia y Tecnologia de Los Alimentos de Entre Rios.; ArgentinaFil: Flores, Agustina. Universidad Nacional de Entre Ríos. Facultad de Ciencias de la Alimentación; ArgentinaFil: Zampedri, Patricia Andrea. Universidad Nacional de Entre Ríos. Facultad de Ciencias de la Alimentación; ArgentinaFil: Zampedri, Carolina Ayelen. Universidad Nacional de Entre Ríos. Facultad de Ciencias de la Alimentación; Argentin

Guarded Recursive Types in Type Theory

In total functional (co)programming valid programs are guaranteed to always produce (part of) their output in a finite number of steps.Enforcing this property while not sacrificing expressivity has beenchallenging. Traditionally systems like Agda and Coq have relied on a syntactic restriction on (co)recursive calls, but this is inherentlyanti-modular.Guarded recursion, first introduced by Nakano, has been recentlyapplied in the coprogramming case to ensure totality through typing instead. The relationship between the consumption and the production of data is captured by a delay modality, which allows to give a safe type to a general fixpoint combinator.This thesis consists of two parts. In the first we formalize, usingthe proof assistant Agda, a result about strong normalization for asmall language extended with guarded recursive types. In the second we extend guarded recursive types to additionally ensure termination of recursive programs: the main result is a model based on relational parametricity for the dependently typed calculus we designed

Guarded Recursive Types in Type Theory

In total functional (co)programming valid programs are guaranteed to always produce (part of) their output in a finite number of steps.Enforcing this property while not sacrificing expressivity has beenchallenging. Traditionally systems like Agda and Coq have relied on a syntactic restriction on (co)recursive calls, but this is inherentlyanti-modular.Guarded recursion, first introduced by Nakano, has been recentlyapplied in the coprogramming case to ensure totality through typing instead. The relationship between the consumption and the production of data is captured by a delay modality, which allows to give a safe type to a general fixpoint combinator.This thesis consists of two parts. In the first we formalize, usingthe proof assistant Agda, a result about strong normalization for asmall language extended with guarded recursive types. In the second we extend guarded recursive types to additionally ensure termination of recursive programs: the main result is a model based on relational parametricity for the dependently typed calculus we designed

Calcolo di stati spin-orbitali in nanofili a semiconduttore core-shell di tipo I, II e broken gap, con un approccio k∙p autoconsistente

I nanofili (NFi) a semiconduttore manifestano proprietà fisiche uniche a causa della bassa dimensionalità e rappresentano una piattaforma promettente per una vasta gamma di applicazioni in nanotecnologia ed elettronica. I NFi possono essere cresciuti in strutture singolo-cristalline, controllando con precisione parametri cruciali come la composizione chimica, le dimensioni e il doping, permettendo così di ingegnerizzare una vasta gamma di dispositivi e tecniche di integrazione basati sui NFi, tra cui FET, giunzioni p-n, dispositivi di cattura della luce e dispositivi termoelettrici. Inoltre, la dimostrazione di NFi con modulazione radiale dei materiali, anche chiamati NFi core-shell (NFiCS), ha aperto la strada a dispositivi di maggiore efficienza e dotati di nuove funzionalità grazie effetti quantistici e tecniche come la wave-function engineering . Tra i metodi teorici utilizzati per descrivere le nanostrutture a semiconduttore, il metodo k∙p autoconsistente si distingue per efficienza computazionale e capacità predittive. Particolarmente rilevante è la capacità di descrivere l'interazione tra l’accoppiamento spin-orbita (ASO) e i parametri strutturali. In questa tesi, abbiamo sviluppato e utilizzato un software originale, allo stato dell’arte e orientato agli oggetti in Python, dove le equazioni di Schrödinger multibanda e di Poisson sono risolte mediante il metodo degli elementi finiti (FEM); L'uso di mesh non strutturate e adattative nel k∙p autoconsistente regola efficacemente il costo numerico e la precisione anche in condizioni di forte confinamento, doping elevato o bassa simmetria. Il modulation-doping è una tecnica chiave per la funzionalizzazione di dispositivi ad alta mobilità, ma relativamente meno sotto controllo nei processi di crescita. Pertanto, studiamo un’eterostruttura radiale AlGaAs/GaAS a diversi regimi di doping. Mostriamo che un doping elevato porta a una forte localizzazione dei portatori verso l'interfaccia fra il core e la shell, così come inversioni di massa e variazioni del carattere spinoriale degli stati di valenza a energia più bassa. Mostriamo che indicazioni sull'evoluzione della struttura a bande con il doping possono essere evidenziate nei pattern di anisotropia degli spettri di assorbimento ottico di luce linearmente polarizzata. I NFi ibridi full-shell - NFi a semiconduttore incorporati in un superconduttore - sono recentemente emersi come candidati nella ricerca dei Majorana zero modes, potenzialmente essenziali per i qubits fault-tolerant grazie alla loro natura topologica e robustezza contro il disordine locale. Fino ad ora, il basso valore di ASO ottenuto in campioni tipici ha ostacolato la possibilità di raggiungere una fase superconduttiva topologica. Qui, proponiamo di sfruttare l’ASO intrinseco delle bande di valenza, esplorando il potenziale di NFiCS di InP/GaSb nelle geometrie full-shell. Calcoli autoconsistenti k∙p prevendono valori dell’ASO intrinseco fino a 20 meV·nm, indipendentemente dal campo elettrico o dallo strain presente all'interfaccia. I NFiCS di InAs/GaSb costituiscono un sistema adatto per applicazioni in elettronica a basso consumo energetico, nonché per studi fondamentali sull'ibridazione elettrone-buca e stati isolanti topologici. Tramite simulazioni k∙p autoconsistenti, identifichiamo un nuovo stato semimetallico rientrante con dispersione di Weyl nel gap di ibridazione di NFi con una lieve inversione di bande, innescato da un campo elettrico trasversale. Utilizzando una Hamiltoniana modello di tipo Bernevig-Hughes-Zhang, mostriamo che la fase semimetallica è dovuta a una compensazione esatta di ASO e interazioni elettrone-buca. Razionalizziamo ulteriormente la chiusura del gap indiretto nei termini dell'apparizione di stati localizzati su entrambi gli estremi del NF.Semiconductor nanowires (NWs) manifest unique physical properties due to their reduced dimensionality and represent a promising platform for a wide range of applications in nanotechnology and electronics. NWs can be reliably manufactured in single crystal structures, with precise control of crucial parameters such as chemical composition, dimensions, and doping, enabling to engineer a vast range of devices and integration techniques, including NW-based FETs, p-n junctions, light-harvesting and thermoelectric devices. Moreover, the demonstration of NWs with radial material modulation, also referred to as core-shell NWs (CSNWs), paved the way to devices with enhanced performance and new functionalities through wave-function engineering and quantum effects. Among the theoretical methods used to describe semiconductor nanostructures, the self-consistent k∙p method, combined with the envelope function approximation, stands out for computational efficiency and predictivity. Of relevance here is the ability to describe the interplay between spin-orbit coupling (SOC) in the underlying compounds, material modulations, and structural parameters. Within this thesis we developed and used an original state-of-the-art, object-oriented Python software where the coupled multiband Schrödinger and Poisson equations are solved by the finite element method (FEM); using unstructured and adaptive meshes in self-consistent k∙p keeps the numerical burden and precision under control also in strong confinement, high-doping, or low-symmetry regimes . The implemented 8-band k∙p Hamiltonian is suitable to describe type-I, type-II, and inverted band heterostructures discussed in this thesis. Modulation doping is a key functionalization technique for high-mobility devices, but relatively less under control in growth processes. Hence, we study a prototypical AlGaAs/GaAS radial heterostructure at different doping regimes. We show that high-doping brings about a strong carrier localization towards the core-shell interface, as well as mass inversions and non-trivial changes in the spinorial character of the low-energy valence states. We show by explicit calculations that indications of the band structure's evolution with doping can be exposed in the anisotropy patterns of linearly polarized optical absorption spectra. Full-shell hybrid NWs -semiconductor NWs embedded in a superconductor- have recently emerged as candidates in the search for Majorana zero modes, possible building-blocks for the implementation of fault-tolerant qubits thanks to their topological nature and ensuing robustness against local disorder. Up to now the primary issue has been the small SOC achieved in typical samples, hindering the possibility to attain a topological superconducting phase. Here, we propose to exploit the inherently strong SOC of the hole valence bands, exploring the potential of InP/GaSb CSNWs in full-shell geometries. Predictive self-consistent k∙p calculations foresee values of the intrinsic SOC as high as 20 meV·nm, regardless of the electric field or strain at the interface. InAs/GaSb CSNWs constitute a suitable system for applications in low-power electronics as well as core studies on electron-hole hybridization and topological insulating states. Using self-consistent k∙p simulations, we identify a new reentrant semimetal state with Weyl dispersion in the hybridization gap of slightly band-inverted NWs, which is triggered by a transverse electric field. Using an effective low-energy Bernevig-Hughes-Zhang model Hamiltonian we show that the semimetallic phase is due to an exact compensation of SOCs and electron-hole interactions. We further rationalize the closure of the indirect gap in terms of the appearance of localized states at both ends of the NW