A Recipe for Symbolic Geometric Computing: Long Geometric Product, BREEFS and Clifford Factorization

In symbolic computing, a major bottleneck is middle expression swell. Symbolic geometric computing based on invariant algebras can alleviate this difficulty. For example, the size of projective geometric computing based on bracket algebra can often be restrained to two terms, using final polynomials, area method, Cayley expansion, etc. This is the "binomial" feature of projective geometric computing in the language of bracket algebra. In this paper we report a stunning discovery in Euclidean geometric computing: the term preservation phenomenon. Input an expression in the language of Null Bracket Algebra (NBA), by the recipe we are to propose in this paper, the computing procedure can often be controlled to within the same number of terms as the input, through to the end. In particular, the conclusions of most Euclidean geometric theorems can be expressed by monomials in NBA, and the expression size in the proving procedure can often be controlled to within one term! Euclidean geometric computing can now be announced as having a "monomial" feature in the language of NBA. The recipe is composed of three parts: use long geometric product to represent and compute multiplicatively, use "BREEFS" to control the expression size locally, and use Clifford factorization for term reduction and transition from algebra to geometry. By the time this paper is being written, the recipe has been tested by 70+ examples from \cite{chou}, among which 30+ have monomial proofs. Among those outside the scope, the famous Miquel's five-circle theorem \cite{chou2}, whose analytic proof is straightforward but very difficult symbolic computing, is discovered to have a 3-termed elegant proof with the recipe

Automated Generation of Geometric Theorems from Images of Diagrams

We propose an approach to generate geometric theorems from electronic images of diagrams automatically. The approach makes use of techniques of Hough transform to recognize geometric objects and their labels and of numeric verification to mine basic geometric relations. Candidate propositions are generated from the retrieved information by using six strategies and geometric theorems are obtained from the candidates via algebraic computation. Experiments with a preliminary implementation illustrate the effectiveness and efficiency of the proposed approach for generating nontrivial theorems from images of diagrams. This work demonstrates the feasibility of automated discovery of profound geometric knowledge from simple image data and has potential applications in geometric knowledge management and education.Comment: 31 pages. Submitted to Annals of Mathematics and Artificial Intelligence (special issue on Geometric Reasoning

Algorithms for detecting dependencies and rigid subsystems for CAD

Geometric constraint systems underly popular Computer Aided Design soft- ware. Automated approaches for detecting dependencies in a design are critical for developing robust solvers and providing informative user feedback, and we provide algorithms for two types of dependencies. First, we give a pebble game algorithm for detecting generic dependencies. Then, we focus on identifying the "special positions" of a design in which generically independent constraints become dependent. We present combinatorial algorithms for identifying subgraphs associated to factors of a particular polynomial, whose vanishing indicates a special position and resulting dependency. Further factoring in the Grassmann- Cayley algebra may allow a geometric interpretation giving conditions (e.g., "these two lines being parallel cause a dependency") determining the special position.Comment: 37 pages, 14 figures (v2 is an expanded version of an AGD'14 abstract based on v1

Restricted Lie (super)algebras, central extensions of non-associative algebras and some tapas

The general framework of this dissertation is the theory of non-associative algebras. We tackle diverse problems regarding restricted Lie algebras and superalgebras, central extensions of different classes of algebras and crossed modules of Lie superalgebras. Namely, we study the relations between the structural properties of a restricted Lie algebra and those of its lattice of restricted subalgebras; we define a non-abelian tensor product for restricted Lie superalgebras and for graded ideal crossed submodules of a crossed module of Lie superalgebras, and explore their properties from structural, categorical and homological points of view; we employ central extensions to classify nilpotent bicommutative algebras; and we compute central extensions of the associative null-filiform algebras and of axial algebras. Also, we include a final chapter devoted to compare the two main methods (Rabinowitsch's trick and saturation) to introduce negative conditions in the standard procedures of the theory of automated proving and discovery

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Efficient and convergent natural gradient based optimization algorithms for machine learning

[eng] Many times machine learning is casted as an optimization problem. This is the case when an objective function assesses the success of an agent in a certain task and hence, learning is accomplished by optimizing that function. Furthermore, gradient descent is an optimization algorithm that has proven to be a powerful tool, becoming the cornerstone to solving most machine learning challenges. Among its strengths, there are the low computational complexity and the convergence guarantee property to the optimum of the function, after certain regularities on the function. Nevertheless, large dimension scenarios show sudden drops in convergence rates which inhibit further improvements in an acceptable amount of time. For this reason, the field has contemplated the natural gradient to tackle this issue. The natural gradient is defined on a Riemannian manifold (M, g). A Riemannian manifold is a manifold M equipped with a metric g. The natural gradient vector of a function f at a point p in (M, g) is a vector in the tangent space at p that points to the direction in which f locally increases its value faster taking into account the metric attached to the manifold. It turns out that the manifold of probability distributions of the same family, usually considered in machine learning, has a natural metric associated, namely the fisher information metric. While natural gradient based algorithms show a better convergence speed in some limited examples, they often fail in providing good estimates or they even diverge. Moreover, they demand more calculations than the ones performed by gradient descent algorithms, increasing the computational complexity order. This thesis explores the natural gradient descent algorithm for the function optimization task. Our research aims at designing a natural gradient based algorithm to solve a function optimization problem, whose computational complexity is comparable to those gradient based and such that it benefits from higher rates of convergence compared to standard gradient based methods. To reach our objectives, the hypothesis formulated in this thesis is that the convergence property guarantee stabilizes natural gradient algorithms and it gives access to fast rates of convergence. Furthermore, the natural gradient can be computed fast for particular manifolds named dually flat manifolds, and hence, fast natural gradient optimization methods become available. The beginning of our research is mainly focused on the convergence property for natural gradient methods. We develop some strategies to define natural gradient methods whose convergence can be proven. The main assumptions require (M, g) to be a Riemannian manifold and f to be a differentiable function on M. Moreover, it turns out that the multinomial logistic regression problem, a widely considered machine learning problem, can be adapted and solved by taking a dually flat manifolds as the model. Hence, this problem is our most promising target in which the objective of the thesis can be completely accomplished.[cat] L’aprenentatge automàtic sovint es relaciona amb un problema d’optimització. Quan l’éxit o l’error d’un agent en una determinada tasca ve donat per una funció, aprendre a realitzar correctament la tasca equival a optimitzar la funció en questió. El descens del gradient és un mètode d’optimització emprat per resoldre la majoria d’aquest tipus de problemes. Aquest algorisme és eficient i, donades certes condicions, convergeix a la solució. No obstant, la convergència pot esdevenir molt lenta en problemes de dimensió alta, on l’algorisme requerix un temps desmesurat. El gradient natural és emprat, sense gaire èxit, per tal d’evitar aquest fet. En una varietat de Riemann (M, g) amb mètrica g, el gradient natural d’una funció "f" en un punt "p" és un vector del espai tangent en "p" que assenyala la direcció on "f" creix localment més intensament, tenint en compte la mètrica del espai. En teoria, el gradient natural té propietats que podrien afavorir la velocitat de convergència, però en problemes pràctics no s’observa cap millora. Alguns algorismes basats en el gradient natural fins i tot divergeixen essent superats pel descens del gradient standard. A més a més, el gradient natural en general té una complexitat computacional més elevada. Aquesta tesis explora els algorismes basats en el gradient natural. En moltes ocasions, l’aprenentatge automàtic es du a terme en families de distribucions de probabilitat, on la mètrica associada a aquest tipus d’espais és la mètrica de Fisher. La nostra hipòtesi és que per obtenir una velocitat de convergència alta és suficient l’assoliment de la propietat de convergència. L’objectiu és definir exemples d’aquest tipus d’algorismes que siguin convergents i amb un cost computacional reduït per tal que pugui ser emprat en problemes actuals de dimensió alta. Per assolir el nostre objectiu, hem trobat indispensable limitar-nos al conjunt de varietats de Riemann anomenades varietats dualment planes. En particular, afrontem el problema de regressió logística multinomial. Aquest espai ens permet definir un algorisme efficient i convergent basat en el gradient natural gràcies a propietats intrínseques de la varietat

CHY Formulae and Soft Theorems in N = 4 Super Yang-Mills Theory

PhDThe study of scattering amplitudes in quantum eld theories (QFTs) is equally important for high energy phenomenology and for theoretical understanding of fundamental physics. Over the last 15 years there has been an explosion of new techniques, inspired by Witten's celebrated twistor string theory [1]. The N = 4 super Yang-Mills theory (SYM) provides a playground for applying and extending these methods, heavily constrained by spacetime, internal and hidden symmetries. Recently, Cachazo, He and Yuan proposed an algebraic construction of scattering amplitudes at tree level in various QFTs, based on the solution of certain scattering equations [2]. This formula was later extended to tree-level form factors of Tr(F2 SD) in four dimensional Yang-Mills theory [3]. In this thesis we show how this result may be naturally supersymmetrised, and derived from a dual connected formulation. Moreover, we relate our results to a geometric construction of form factors via the Grassmannian [4]. Finally, we argue that ambitwistor string theory provides a natural way to lift the result to arbitrary dimensions, paving the way for loop-level results. In complementary work, it was shown that the subleading soft behaviour of tree-level amplitudes in gauge theory and gravity is universal [5{7]. This unexpected property is related to extended symmetries of the theory acting at null in nity. Moreover, the hidden structure provides additional information relevant for resummation of physical observables. In this thesis, we extend the known results to one-loop level in N = 4 SYM, arguing that IR divergences introduce anomaly terms through nite order in the regulator. We constrain these terms using dual superconformal symmetry, and derive explicit formulae in the MHV and NMHV sectors. This thesis contains documentation for two Mathematica packages, illustrating the original calculations we have performed.STFC studentship

Topics in Programming Languages, a Philosophical Analysis through the case of Prolog

[EN]Programming languages seldom find proper anchorage in philosophy of logic, language and science. is more, philosophy of language seems to be restricted to natural languages and linguistics, and even philosophy of logic is rarely framed into programming languages topics. The logic programming paradigm and Prolog are, thus, the most adequate paradigm and programming language to work on this subject, combining natural language processing and linguistics, logic programming and constriction methodology on both algorithms and procedures, on an overall philosophizing declarative status. Not only this, but the dimension of the Fifth Generation Computer system related to strong Al wherein Prolog took a major role. and its historical frame in the very crucial dialectic between procedural and declarative paradigms, structuralist and empiricist biases, serves, in exemplar form, to treat straight ahead philosophy of logic, language and science in the contemporaneous age as well. In recounting Prolog's philosophical, mechanical and algorithmic harbingers, the opportunity is open to various routes. We herein shall exemplify some: - the mechanical-computational background explored by Pascal, Leibniz, Boole, Jacquard, Babbage, Konrad Zuse, until reaching to the ACE (Alan Turing) and EDVAC (von Neumann), offering the backbone in computer architecture, and the work of Turing, Church, Gödel, Kleene, von Neumann, Shannon, and others on computability, in parallel lines, throughly studied in detail, permit us to interpret ahead the evolving realm of programming languages. The proper line from lambda-calculus, to the Algol-family, the declarative and procedural split with the C language and Prolog, and the ensuing branching and programming languages explosion and further delimitation, are thereupon inspected as to relate them with the proper syntax, semantics and philosophical élan of logic programming and Prolog

Mathemathical methods of theoretical physics

Course material for mathematical methods of theoretical physics intended for an undergraduate audience.Comment: 287 pages, revised, some further (relative to previous edition) proofs and sections (on differential operators in orthogonal curvilinear coordinates) adde