Algorithmic Randomness

We consider algorithmic randomness in the Cantor space C of the infinite binary sequences. By an algorithmic randomness concept one specifies a set of elements of C, each of which is assigned the property of being random. Miscellaneous notions from computability theory are used in the definitions of randomness concepts that are essentially rooted in the following three intuitive randomness requirements: the initial segments of a random sequence should be effectively incompressible, no random sequence should be an element of an effective measure null set containing sequences with an “exceptional property”, and finally, considering betting games, in which the bits of a sequence are guessed successively, there should be no effective betting strategy that helps a player win an unbounded amount of capital on a random sequence. For various formalizations of these requirements one uses versions of Kolmogorov complexity, of tests, and of martingales, respectively. In case any of these notions is used in the definition of a randomness concept, one may ask in general for fundamental equivalent definitions in terms of the respective other two notions. This was a long-standing open question w.r.t. computable randomness, a central concept that had been introduced by Schnorr via martingales. In this thesis, we introduce bounded tests that we use to give a characterization of computable randomness in terms of tests. Our result was obtained independently of the prior test characterization of computable randomness due to Downey, Griffiths, and LaForte, who defined graded tests for their result. Based on bounded tests, we define bounded machines which give rise to a version of Kolmogorov complexity that we use to prove another characterization of computable randomness. This result, as in analog situations, allows for the introduction of interesting lowness and triviality properties that are, roughly speaking, “anti-randomness” properties. We define and study the notions lowness for bounded machines and bounded triviality. Using a theorem due to Nies, it can be shown that only the computable sequences are low for bounded machines. Further we show some interesting properties of bounded machines, and we demonstrate that every boundedly trivial sequence is K-trivial. Furthermore we define lowness for computable machines, a lowness notion in the setting of Schnorr randomness. We prove that a sequence is low for computable machines if and only if it is computably traceable. Gacs and independently Kucera proved a central theorem which states that every sequence is effectively decodable from a suitable Martin-Löf random sequence. We present a somewhat easier proof of this theorem, where we construct a sequence with the required property by diagonalizing against appropriate martingales. By a variant of that construction we prove that there exists a computably random sequence that is weak truth-table autoreducible. Further, we show that a sequence is computably enumerable self-reducible if and only if its associated real is computably enumerable. Finally we investigate interrelations between the Lebesgue measure and effective measures on C. We prove the following extension of a result due to Book, Lutz, and Wagner: A union of Pi-0-1 classes that is closed under finite variations has Lebesgue measure zero if and only if it contains no Kurtz random real. However we demonstrate that even a Sigma-0-2 class with Lebesgue measure zero need not be a Kurtz null class. Turning to Almost classes, we show among other things that every Almost class with respect to a bounded reducibility has computable packing dimension zero

On Array Noncomputable Degrees, Maximal Pairs and Simplicity Properties

In this thesis, we give contributions to topics which are related to array noncomputable (a.n.c.) Turing degrees, maximal pairs and to simplicity properties. The outline is as follows. In Chapter 2, we introduce a subclass of the a.n.c. Turing degrees, the so called completely array noncomputable (c.a.n.c. for short) Turing degrees. Here, a computably enumerable (c.e.) Turing degree a is c.a.n.c. if any c.e. set A ∈ a is weak truth-table (wtt) equivalent to an a.n.c. set. We show in Section 2.3 that these degrees exist (indeed, there exist infinitely many low c.a.n.c. degrees) and that they cannot be high. Moreover, we apply some of the ideas used to show the existence of c.a.n.c. Turing degrees to show the stronger result that there exists a c.e. Turing degree whose c.e. members are halves of maximal pairs in the c.e. computably Lipschitz (cl) degrees, thereby solving the first part of the first open problem given in the paper by Ambos-Spies, Ding, Fan and Merkle [ASDFM13]. In Chapter 3, we present an approach to extending the notion of array noncomputability to the setting of almost-c.e. sets (these are the sets which correspond to binary representations of left-c.e. reals). This approach is initiated by the Heidelberg Logic Group and it is worked out in detail in an upcoming paper by Ambos-Spies, Losert and Monath [ASLM18], in the thesis of Losert [Los18] and in [ASFL+]. In [ASLM18], the authors introduce the class of sets with the universal similarity property (u.s.p. for short; throughout this thesis, sets with the u.s.p. will shortly be called u.s.p. sets) which is a strong form of array noncomputability in the setting of almost-c.e. sets and they show that sets with this property exist precisely in the c.e. not totally ω-c.e. degrees. Then it is shown that, using u.s.p. sets, one obtains a simplified method for showing the existence of almost-c.e. sets with a property P (for certain properties P) that are contained in c.e. not totally ω-c.e. degrees, namely by showing that u.s.p. sets have property P. This is demonstrated by showing that u.s.p. sets are computably bounded random (CB-random), thereby extending a result from Brodhead, Downey and Ng [BDN12]. Moreover, it is shown that the c.e. not totally ω-c.e. degrees can be characterized as those c.e. degrees which contain an almost-c.e. set which is not cl-reducible to any complex almost-c.e. set. This affirmatively answers a conjecture by Greenberg. For the if-direction of the latter result, we prove a new result on maximal pairs in the almost-c.e. sets by showing the existence of locally almost-c.e. sets which are halves of maximal pairs in the almost-c.e. sets such that the second half can be chosen to be c.e. and arbitrarily sparse. This extends Yun Fan’s result on maximal pairs [Fan09]. By our result, we also get a new proof of one of the main results in Barmpalias, Downey and Greenberg [BDG10], namely that in any c.e. a.n.c. degree there is a left-c.e. real which is not cl-reducible to any ML-random left-c.e. real. In this thesis, we give an overview of some of the results from [ASLM18] and sketch some of the proofs to illustrate this new methodology and, subsequently, we give a detailed proof of the above maximal pair result. In Chapter 4, we look at the interaction between a.n.c. wtt-degrees and the most commonly known simplicity properties by showing that there exists an a.n.c. wtt-degree which contains an r-maximal set. By this result together with the result by Ambos-Spies [AS18] that no a.n.c. wtt-degree contains a dense simple set, we obtain a complete characterization which of the classical simplicity properties may hold for a.n.c. wtt-degrees. The guiding theme for Chapter 5 is a theorem by Barmpalias, Downey and Greenberg [BDG10] in which they characterize the c.e. not totally ω-c.e. degrees as the c.e. degrees which contain a c.e. set which is not wtt-reducible to any hypersimple set. So Ambos-Spies asked what the above characterization would look like if we replaced hypersimple sets by maximal sets in the above theorem. In other words, what are the c.e. Turing degrees that contain c.e. sets which are not wtt-reducible to any maximal set. We completely solve this question on the set level by introducing the new class of eventually uniformly wtt-array computable (e.u.wtt-a.c.) sets and by showing that the c.e. sets with this property are precisely those c.e. sets which are wtt-reducible to maximal sets. Indeed, this characterization can be extended in that we can replace wtt-reducible by ibT-reducible and maximal sets by dense simple sets. By showing that the c.e. e.u.wtt-a.c. sets are closed downwards under wtt-reductions and under the join operation, it follows that the c.e. wtt-degrees containing e.u.wtt-a.c. sets form an ideal in the upper semilattice of the c.e. wtt-degrees and, further, we obtain a characterization of the c.e. wtt-degrees which contain c.e. sets that are not wtt-reducible to any maximal set. Moreover, we give upper and lower bounds (with respect to ⊆) for the class of the c.e. e.u.wtt-a.c. sets. For the upper bound, we show that any c.e. e.u.wtt-a.c. set has array computable wtt-degree. For the lower bound, we introduce the notion of a wtt-superlow set and show that any wtt-superlow c.e. set is e.u.wtt-a.c. Besides, we show that the wtt-superlow c.e. sets can be characterized as the c.e. sets whose bounded jump is ω-computably approximable (ω-c.a. for short); hence, they are precisely the bounded low sets as introduced in the paper by Anderson, Csima and Lange [ACL17]. Furthermore, we prove a hierarchy theorem for the wtt-superlow c.e. sets and we show that there exists a Turing complete set which lies in the intersection of that hierarchy. Finally, it is shown that the above bounds are strict, i.e., there exist c.e. e.u.wtta. c. sets which are not wtt-superlow and that there exist c.e. sets whose wtt-degree is array computable and which are not e.u.wtt-a.c. (where here, we obtain the separation even on the level of Turing degrees). The results from Chapter 5 will be included in a paper which is in preparation by Ambos-Spies, Downey and Monath [ASDM19]

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

In recent years, there have been significant advances in the use of deep learning methods in inverse problems such as denoising, compressive sensing, inpainting, and super-resolution. While this line of works has predominantly been driven by practical algorithms and experiments, it has also given rise to a variety of intriguing theoretical problems. In this paper, we survey some of the prominent theoretical developments in this line of works, focusing in particular on generative priors, untrained neural network priors, and unfolding algorithms. In addition to summarizing existing results in these topics, we highlight several ongoing challenges and open problems

Randomness and Computability

This thesis establishes significant new results in the area of algorithmic randomness. These results elucidate the deep relationship between randomness and computability. A number of results focus on randomness for finite strings. Levin introduced two functions which measure the randomness of finite strings. One function is derived from a universal monotone machine and the other function is derived from an optimal computably enumerable semimeasure. Gacs proved that infinitely often, the gap between these two functions exceeds the inverse Ackermann function (applied to string length). This thesis improves this result to show that infinitely often the difference between these two functions exceeds the double logarithm. Another separation result is proved for two different kinds of process machine. Information about the randomness of finite strings can be used as a computational resource. This information is contained in the overgraph. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not truth-table complete. This question is answered in the negative. Related results are also established. This thesis makes advances in the theory of randomness for infinite binary sequences. A variant of process machines is used to characterise computable randomness, Schnorr randomness and weak randomness. This result is extended to give characterisations of these types of randomness using truthtable reducibility. The computable Lipschitz reducibility measures both the relative randomness and the relative computational power of real numbers. It is proved that the computable Lipschitz degrees of computably enumerable sets are not dense. Infinite binary sequences can be regarded as elements of Cantor space. Most research in randomness for Cantor space has been conducted using the uniform measure. However, the study of non-computable measures has led to interesting results. This thesis shows that the two approaches that have been used to define randomness on Cantor space for non-computable measures: that of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent. Levin established the existence of probability measures for which all infinite sequences are random. These measures are termed neutral measures. It is shown that every PA degree computes a neutral measure. Work of Miller is used to show that the set of atoms of a neutral measure is a countable Scott set and in fact any countable Scott set is the set of atoms of some neutral measure. Neutral measures are used to prove new results in computability theory. For example, it is shown that the low computable enumerable sets are precisely the computably enumerable sets bounded by PA degrees strictly below the halting problem. This thesis applies ideas developed in the study of randomness to computability theory by examining indifferent sets for comeager classes in Cantor space. A number of results are proved. For example, it is shown that there exist 1-generic sets that can compute their own indifferent sets

Probing the top quark couplings within the ATLAS detector and EFT global fits

El treball d'aquesta tesi se centra en les propietats de producció de la partícula elemental (aquelles indivisibles) més massiva del Model Estàndard (ME): el quark top. En particular, s'examinen les interaccions electrofebles del quark top amb altres partícules del ME utilitzant les dades de col·lisions protó--protó principalment lliurades pel Gran Col·lisionador d'Hadrons (LHC) en el laboratori del CERN i enregistrades pel detector ATLAS entre 2015 i 2018. El Model Estàndard de la física de partícules és un model matemàtic que descriu la natura a escala atòmic i subatòmic. És un model altament predictiu, capaç d'explicar la major part dels fenòmens observats, i que ens ha guiat al llarg dels últims 100 anys en la recerca de les partícules que el constitueixen. Tot i això, continuen existint múltiples interrogants que fan del ME una teoria incompleta. Per exemple, no és capaç d'incorporar la força gravitatòria, desentranyar l'origen de la matèria fosca, o explicar l'asimetria de matèria i antimatèria present a l'inici de l'univers. Per això, l'existència de nova física, més enllà del ME, és una de les principals línies d'investigació en la física moderna. La recerca d'aquesta nova física és una de les principals motivacions de l'LHC, l'accelerador de partícules més potent del món. L'LHC està dissenyat per a col·lidir partícules a energies relativistes a un ritme de 40 milions de col·lisions per segon el que permet estudiar la interacció, fins i tot de les partícules més pesants del ME, amb un gran nivell de detall. Un dels assoliments més importants de l'LHC va ser el descobriment del bosó de Higgs l'any 2012: l'última peça del trencaclosques del ME. D'altra banda, les mesures de precisió en el LHC també ens permeten posar a prova el ME i restringir les modificacions dels acoblaments del ME introduïdes per models de nova física. Aquesta tesi consta de tres parts. La primera part explora desviacions de les expectatives del ME en observables de física d'alta energia en el sector electrofeble del quark top utilitzant una extensió de la teoria de camps efectius (EFT, per les seues sigles en anglés). Aquest estudi està fortament motivat per les últimes mesures de precisió de secció eficaç diferencial proporcionades per l'experiment ATLAS, així com per la inclusió de dades del detector CMS i altres experiments com ara Tevatron o LEP/SLC; això permet una caracterització rigorosa de les interaccions electrofebles del quark top. Els resultats obtinguts en els límits dels coeficients de Wilson seleccionats (relacionats amb nova física) oscil·len entre +-0.35 i +-8 TeV^(-2), en acord amb les expectatives del ME. Aquests resultats milloren els estudis anteriors i destaquen la necessitat de noves observables per a limitar els coeficients de l'EFT. La segona part presenta la cerca de l'asimetria de càrrega leptònica (ACL) en la producció de ttW utilitzant les dades de col·lisió protó-protó de 139 fb^(-1) recollides per ATLAS durant el Run 2. A causa de les propietats úniques de la producció de ttW, s'espera que l'asimetria de càrrega siga gran i mostre una sensibilitat significativa a un conjunt reduït d'operadors de l'EFT de quatre fermions, així com a la naturalesa quiral de possibles noves físiques en aquest procés. Es troba que l'asimetria de càrrega reconstruïda és ACL = -0.12 ± 0.14 (estadístic) ± 0.05 (sistemàtic). El resultat s'extrau al nivell de partícules en una regió fiducial que es tria pròxima a la regió de reconstrucció per minimitzar els efectes d'acceptació. Al nivell de partícules, aquesta asimetria és ACL = -0.11 ± 0.17 (estadístic) ± 0.05 (sistemàtic). Ambdós valors són consistents amb l'expectativa del ME. En la part final, s'investiga l'acoblament Yukawa del quark top amb el bosó de Higgs. Aquest acoblament és el més gran del ME i s'espera que siga el més sensible als efectes de nova física. En aquest sentit, pot ser utilitzat per a estudiar una limitació coneguda del ME: la insuficiència de termes de violació de CP per a explicar, per exemple, l'asimetria matèria-antimatèria de l'univers primigeni. Es produeixen simulacions amb termes de violació de CP i s'extreuen límits d'exclusió sobre l'angle de mescla de CP (alpha) usant també les dades recollides per ATLAS durant el Run 2. La hipòtesi de CP imparell (alpha=90) s'exclou amb una significança de 3.9 desviacions estàndard, i s'estableix un límit inferior de |alpha| > 43° amb un nivell de confiança del 95%.The work in this thesis focuses on the properties of the most massive fundamental particle of the Standard Model (SM): the top quark. The electroweak interactions between the top quark and other particles of the SM are studied using the latest proton-proton collision data delivered by the LHC and collected by the ATLAS detector at the CERN laboratory during the Run 2 period (2015-2018). The SM of particle physics is a mathematical model that described nature at the atomic and subatomic levels. It is a highly predictive model, capable of explaining most of the observable phenomena and that has driven the development of particle physics for the last 100 years. However, the SM is not a complete theory. It does not include gravity, nor does it explain the origin of dark matter or the matter-antimatter asymmetry of the universe. For this reason, the existence of new physics beyond the SM is one the main research lines in particle physics. The search for this new physics is one of the main motivations of the LHC, the most powerful particle accelerator in the world. The LHC is designed to collide particles at relativistic energies at a rate of 40 million collisions per second which allows to study the interaction, even of the heaviest particles of the SM, with a high level of detail. One of the most important achievements of the LHC was the discovery of the Higgs boson in 2012: the last piece of the SM puzzle. On the other hand, precision measurements at the LHC also allow us to test the SM and restrict the modifications of the SM couplings introduced by new physics models. This thesis consists of three parts. The first part explores deviations from the Standard Model (ME) expectations in high-energy physics observables in the top quark electroweak sector using an effective field theory (EFT) extension. This study is greatly motivated by the latest differential cross-section precision measurements provided by the ATLAS experiment, as well as including data from the CMS detector and other experiments such as Tevatron or LEP/SLC; which allow for a rigorous characterisation of the electroweak interactions of the top quark. The resulting in the limits on chosen Wilson coefficients (related to new physics) range from ±0.35 to ±8 TeV^(-2), in agreement with the ME expectation. The results improve previous studies and highlight the need for new observables to constrain EFT coefficients. The second part presents the search for the leptonic charge asymmetry (LCA) in ttW production using 139 fb^(-1) proton-proton collision data collected by ATLAS during Run 2. Due to the unique properties of ttW production, the charge asymmetry is expected to be large and to showcase a significant sensitivity to a reduced set of four-fermion EFT operators, and to the chiral nature of possible new physics in this process. The reconstructed charge asymmetry is found to be LCA = -0.12 +- 0.14 (stat.) +- 0.05 (syst.). The result is unfolded to particle level (PL) in a fiducial region which is chosen to be close to the reconstruction level region to minimise acceptance effects. At particle-level, this asymmetry is LCA(PL) = -0.11 +- 0.17 (stat.) +- 0.05 (syst.). Both values are consistent with the ME expectation. In the final part, the thesis investigates the Yukawa coupling of the top quark to the Higgs boson. This coupling is the largest one in the ME and is expected to be the most sensitive to new physics effects. In this regard, it can be used to probe and extract limits on a well-known limitation of the ME: the insufficient CP-violating terms to explain, for example, the matter-antimatter asymmetry of the early universe. MC simulations with a CP-violating term are produced, and exclusion limits on the CP mixing angle (alpha) are extracted using also data collected by ATLAS during Run 2. The CP-odd hypothesis (alpha=90) is excluded at 3.9 standard deviations, and a lower limit of |alpha| > 43° is established at a 95% confidence level

Visual guidance of unmanned aerial manipulators

The ability to fly has greatly expanded the possibilities for robots to perform surveillance, inspection or map generation tasks. Yet it was only in recent years that research in aerial robotics was mature enough to allow active interactions with the environment. The robots responsible for these interactions are called aerial manipulators and usually combine a multirotor platform and one or more robotic arms. The main objective of this thesis is to formalize the concept of aerial manipulator and present guidance methods, using visual information, to provide them with autonomous functionalities. A key competence to control an aerial manipulator is the ability to localize it in the environment. Traditionally, this localization has required external infrastructure of sensors (e.g., GPS or IR cameras), restricting the real applications. Furthermore, localization methods with on-board sensors, exported from other robotics fields such as simultaneous localization and mapping (SLAM), require large computational units becoming a handicap in vehicles where size, load, and power consumption are important restrictions. In this regard, this thesis proposes a method to estimate the state of the vehicle (i.e., position, orientation, velocity and acceleration) by means of on-board, low-cost, light-weight and high-rate sensors. With the physical complexity of these robots, it is required to use advanced control techniques during navigation. Thanks to their redundancy on degrees-of-freedom, they offer the possibility to accomplish not only with mobility requirements but with other tasks simultaneously and hierarchically, prioritizing them depending on their impact to the overall mission success. In this work we present such control laws and define a number of these tasks to drive the vehicle using visual information, guarantee the robot integrity during flight, and improve the platform stability or increase arm operability. The main contributions of this research work are threefold: (1) Present a localization technique to allow autonomous navigation, this method is specifically designed for aerial platforms with size, load and computational burden restrictions. (2) Obtain control commands to drive the vehicle using visual information (visual servo). (3) Integrate the visual servo commands into a hierarchical control law by exploiting the redundancy of the robot to accomplish secondary tasks during flight. These tasks are specific for aerial manipulators and they are also provided. All the techniques presented in this document have been validated throughout extensive experimentation with real robotic platforms.La capacitat de volar ha incrementat molt les possibilitats dels robots per a realitzar tasques de vigilància, inspecció o generació de mapes. Tot i això, no és fins fa pocs anys que la recerca en robòtica aèria ha estat prou madura com per començar a permetre interaccions amb l’entorn d’una manera activa. Els robots per a fer-ho s’anomenen manipuladors aeris i habitualment combinen una plataforma multirotor i un braç robòtic. L’objectiu d’aquesta tesi és formalitzar el concepte de manipulador aeri i presentar mètodes de guiatge, utilitzant informació visual, per dotar d’autonomia aquest tipus de vehicles. Una competència clau per controlar un manipulador aeri és la capacitat de localitzar-se en l’entorn. Tradicionalment aquesta localització ha requerit d’infraestructura sensorial externa (GPS, càmeres IR, etc.), limitant així les aplicacions reals. Pel contrari, sistemes de localització exportats d’altres camps de la robòtica basats en sensors a bord, com per exemple mètodes de localització i mapejat simultànis (SLAM), requereixen de gran capacitat de còmput, característica que penalitza molt en vehicles on la mida, pes i consum elèctric son grans restriccions. En aquest sentit, aquesta tesi proposa un mètode d’estimació d’estat del robot (posició, velocitat, orientació i acceleració) a partir de sensors instal·lats a bord, de baix cost, baix consum computacional i que proporcionen mesures a alta freqüència. Degut a la complexitat física d’aquests robots, és necessari l’ús de tècniques de control avançades. Gràcies a la seva redundància de graus de llibertat, aquests robots ens ofereixen la possibilitat de complir amb els requeriments de mobilitat i, simultàniament, realitzar tasques de manera jeràrquica, ordenant-les segons l’impacte en l’acompliment de la missió. En aquest treball es presenten aquestes lleis de control, juntament amb la descripció de tasques per tal de guiar visualment el vehicle, garantir la integritat del robot durant el vol, millorar de l’estabilitat del vehicle o augmentar la manipulabilitat del braç. Aquesta tesi es centra en tres aspectes fonamentals: (1) Presentar una tècnica de localització per dotar d’autonomia el robot. Aquest mètode està especialment dissenyat per a plataformes amb restriccions de capacitat computacional, mida i pes. (2) Obtenir les comandes de control necessàries per guiar el vehicle a partir d’informació visual. (3) Integrar aquestes accions dins una estructura de control jeràrquica utilitzant la redundància del robot per complir altres tasques durant el vol. Aquestes tasques son específiques per a manipuladors aeris i també es defineixen en aquest document. Totes les tècniques presentades en aquesta tesi han estat avaluades de manera experimental amb plataformes robòtiques real